research article linear processes in stochastic...
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Research ArticleLinear Processes in Stochastic Population DynamicsTheory and Application to Insect Development
Hernaacuten G Solari1 and Mario A Natiello2
1 Departamento de Fısica FCEN-UBA and IFIBA-CONICET C1428EGA Buenos Aires Argentina2 Centre for Mathematical Sciences Lund University PO Box 118 221 00 Lund Sweden
Correspondence should be addressed to Mario A Natiello marionatiellomathlthse
Received 15 August 2013 Accepted 23 October 2013 Published 17 February 2014
Academic Editors S Casado W Fei and T Nguyen
Copyright copy 2014 H G Solari and M A Natiello This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
We consider stochastic population processes (Markov jump processes) that develop as a consequence of the occurrence of randomevents at random time intervals The population is divided into subpopulations or compartments The events occur at rates thatdepend linearly on the number of individuals in the different described compartments The dynamics is presented in terms ofKolmogorov Forward Equation in the space of events and projected onto the space of populations when needed The generalproperties of the problem are discussed Solutions are obtained using a revised version of the Method of Characteristics Aftera few examples of exact solutions we systematically develop short-time approximations to the problem While the lowest orderapproximation matches the Poisson and multinomial heuristics previously proposed higher-order approximations are completelynew Further we analyse a model for insect development as a sequence of 119864 developmental stages regulated by rates that are linearin the implied subpopulations Transition to the next stage competes with death at all times The process ends at a predeterminedstage for example pupation or adult emergence In its simpler version all the stages are distributed with the same characteristictime
1 Introduction
Stochastic population dynamics deals with the (stochastic)time evolution of groups of similar individuals (called pop-ulations or subpopulations) immersed in a habitat The goalis to describe the evolution of population numbers and thenumber of events associated with changes in the populations
Most of the attention has been paied in the past to thepopulations considering events only by their extrinsic valueas jumps in populations However this perspective must berevised In any study of vector-transmitted diseases suchas in [1 2] the most relevant statistics corresponds to theinteractions between hosts and vectors Such interactionsare associated with events in the life cycle of the vector Inthe case of arthropods transmitting diseases to mammalsblood feeding to complete oogenesis is the opportunity totransmit pathogens In other problems the developmentof insects is experimentally described by the sequences oftransformations in their life cycle (such as moulting) [3ndash5] each transformation must be considered an event To
our knowledge explicit use of the event structure to studypopulation changes in Markov Jump processes has beenintroduced in [6] along with a short-time approximationbased on self-consistent Poisson processes
The present research originates in our efforts to modelvector-transmitted diseases such as Dengue starting from adescription of the life cycle of Aedes aegypti the vector [7ndash9]In modeling insect development in an aquatic phase (suchas the case of Aedes aegypti and Drosophila melanogaster) acrucial point is to represent accurately the statistics of theduration of the aquatic phase (also known as the time ofemergence statistics) as it determines the response of the pop-ulation to climatic events such as rains [10] A deterministicmodel for development was proposed in [11] as a sequentialprocess with associated linear rates The final sections of thispaper describe a stochastic development model
While [6] deals with the stochastic approximation ofgeneral problems (achieving amethodwith error 119900(11990532)) thispaper specifically addresses the problem of formulating andsolving population dynamics problems in event space with
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 873624 15 pageshttpdxdoiorg1011552014873624
2 The Scientific World Journal
linear rates that is rates that depend linearly on the pop-ulations (in the chemical literature they are known as first-order reactions [12]) We provide both exact solutions andgeneral approximation techniques with error 119900(119905
119899) Results
from [6] for the linear case are a modified version of one ofthe approximation schemes in this paper (Poisson approx-imation Section 5) The linear problem is mathematicallysimpler than the general case and a few general results canbe established before resorting to approximations
Linear processes are an important part of most biologicaldynamical descriptions Using an analogy from classicalpopulation dynamics linear (Malthusian) population growth[13] describes the evolution of a population at low densities(eg where the local environment is essentially unaffected bythe population growth and the individuals do not interferewith each other)More generally processes that consider onlyindividuals in a somewhat isolated basis can be satisfactorilydescribed with linear rates However whether to use lineardescriptions or not is ultimately determined by the nature ofthe process (or subprocess) For example finite environmentswill sooner or later force individuals to interact in growingpopulations and nonlinear rates become necessary such as inVerhulstrsquos description of population growth [14]
Perhaps the oldest application of linear processes topopulation dynamics regards birth and death processesHistorically according to Kendall [15] the first to proposebirth and death equations in population dynamics was Furry[16] for a system of physical origin Currently such anapproach is being used in several sciences linear stochasticprocesses have been proposed for the development of tumors[17 18] drug delivery and cell replication are also described inthese terms [19] Concerning the tumors themodel proposedin [17 18] is based on linear individual processes a detailedtreatment of which would require a paper of its own Theaccumulation of individual processes is finally approximatedwith a ldquotunnelling processrdquo corresponding to a maturationcascade that can be easily described with the tools of thispaper (see (71)) Concerning the question of drug deliverywe will come back in detail to [19] in Section 7
A substantial use of population dynamics driven by jumpprocesses has been done also in chemistry particularly afterthe rediscovery by Gillespie [20] of Kendallrsquos simulationmethod [21 22] Poisson [23 24] binomial [25 26] andmultinomial [27] heuristics for the short-time integrals havebeen proposed There is however an important differencebetween chemistry and biology concerning the issues of thispaper On the one hand linear rates are seldom the case inchemical reactions except perhaps in the case of irreversibledecompositions or isomeric recombination Attention to lin-ear rates in chemistry is relatively recent [12] when comparedto over 200 years of Malthusian population growth Onthe other hand detailed event statistics does not appearto be fundamental either highly diluted chemical reactionson a test tube may easily generate 10
14 events of eachtype Consequently attention to event based descriptionsappears to be absent from the chemistry literature Thismight however change in the future since the technicalpossibilities given by for example ldquoin-cell chemistryrdquo andldquonanochemistryrdquo allow dealingwith a lower number of events
where tiny differences in event count could be relevant for thegeneral outcome
In themathematical literature attention to jumpprocessesstarts with Kolmogorovrsquos foundational work [28] and itsfurther elaboration by Feller [29] A substantial effort torelate the stochastic description to deterministic equationswas performed by Kurtz [30ndash34] However attention hasalways focused on the dynamics in population space ratherthan event space
This work focuses on the event statistics associated withlinear processes Even for cases where the detailed eventstatistics is not of central interest such an approach offers analternative to the more traditional formulation in populationspace
Further we discuss in detail the stochastic version ofa linear developmental model Specifically we deal with asubprocess in the life cycle of insects that admits both a lineardescription and an exact solution namely the developmentof immature stages which is a highly individual subprocessFrom egg form to adult form the development of insectsgoes through several transformations at different levels Forexample at the most visible level insects undergoing com-plete metamorphosis evolve in the sequence egg larva pupaand adult Further different substages can be recognised byinspection in the development of larvae (instars) and evenmore stages when observed by other methods [35] Theimmature individual may die along the process or otherwisecomplete it and exit as adult
In Section 2 we formulate the dynamical populationproblem in event space In Section 21 we present the Kol-mogorov Forward Equations (for Markov jump processes)for the most general linear systems of population dynamicsin the form appropriated to probabilities and generatingfunctions The general solution of these equations will bewritten using the Theorem of the Characteristics reformu-lated with respect to the standard geometrical presentation[36] into a dynamical presentation We will analyse thegeneral structure of the equations and in particular theirfirst integrals of motion and the projection onto populationspace (Sections 2 and 3) Section 4 contains two classicalexamples In Section 5 Poisson and multinomial approx-imations (both with error 119900(119905)) are derived and a higher-order approximation with error 119900(119905
119899) (with 119899 as an arbi-
trary positive integer) is elaborated The larval developmentmodel is developed in Sections 6 and 7 treating the twogeneral cases We also revisit in Section 7 the question ofdrug delivery introduced in [19] in terms of the presentapproach Concluding remarks are left for the final Sec-tion
2 Mathematical Formulation
The main tools of stochastic population dynamics are a listof (nonnegative integer) populations 119883
119895 where 119895 = 1 sdot sdot sdot 119873
a list of events 120572 = 1 sdot sdot sdot 119864 an (integer) incidence matrix 120575120572
119895
describing how event 120572 modifies population 119895 and a list oftransition rates 119882
120572(119883
1 119883
119899 119905) describing the probability
rate per unit time of occurrence of each event
The Scientific World Journal 3
An important feature of the present approach is that thedynamical description will be handled in event-space In thissense the ldquostaterdquo of the system is given by the array n(119904 119905) =
(1198991 119899
119864)119905indicating how may events of each class have
occurred from time 119904 up to time 119905 Hence event-space is inthis setup a nonnegative integer lattice of dimension 119864 Theconnection with the actual population values at time 119905 is (119904denotes the initial condition)
119883119895(119905) = 119883
119895(119904) + sum
120572
120575120572
119895119899120572(119904 119905) (1)
Definition 1 (linear independency) One says that the popula-tions are linearly dependent if there is a vector u = 0 such thatfor all 119905 ge 119904 one has sum
119895119906119895119883
119895(119905) = 0 Otherwise one says that
the populations are linearly independent and abbreviate it asLI
In similar form we say that the populations have linearlydependent increments if there is u = 0 so that for all 1199051015840 gt 119905 ge 119904
it holds that sum119895119906119895(119883
119895(1199051015840) minus 119883
119895(119905)) = 0
The populations are said to have linearly independentincrements (abbreviated LII) if they do not have linearlydependent increments
Along this work we will assume the following
Assumption 2 The populations have linearly independentincrements that is all involved populations are necessary toprovide a complete description of the problem none of themcan be expressed as a linear combination of the remainingones for all times
Lemma 3 (a) If the populations have linearly independentincrements then the populations are linearly independent
(b) The matrix 120575 has no nonzero vectors u such thatsum
119895120575120572
119895119906119895= 0 for all 120572 if and only if the populations have linearly
independent increments
Proof (a) The assertion is equivalent to the following if thepopulations are linearly dependent then they have linearlydependent increments which is immediate after the defini-tion of linearly dependent increments
(b) Assume that for all 120572 there exists u = 0 such thatsum
119895119906119895120575120572
119895= 0 then
sum
119895
119906119895(119883
119895(119905
1015840) minus 119883
119895(119905)) = sum
119895
119906119895(sum
120572
120575120572
119895119899120572(119905
1015840 119905)) = 0 (2)
and the populations have linearly dependent incrementsConversely assume that the populations have linearly
dependent increments and select time intervals such that119899120573(1199051015840
120573 119905
120573) = 1 and 119899
120572(1199051015840
120573 119905
120573) = 0 for 120572 = 120573 then we have that
sum
119895
119906119895120575120572
119895= 0 forall120572 (3)
Statement (b) is the negative statement of these facts
Strictly speaking the last step in the above proof takesfor granted that there exist time intervals where only one
event occurs This is assured by the next assumption (thirditem) Also the initial conditions and the problem descriptionshould be such that all defined events are relevant for thedynamics (eg the event ldquocontagionrdquo is irrelevant for anepidemic problem with zero infectives as initial condition)
21 Dynamical EquationmdashBasic Assumptions Populationnumbers evolve in ldquojumpsrdquo given by the occurrence of eachevent (birth death etc) In the sequel we will use Latinindices 119894 119895 119896 and 119898 for populations and Greek indices 120572120573 and 120574 for events
The ultimate goal of stochastic population dynamicsis to calculate the probabilities 119875(119899
1 119899
119864 119904 119905) of having
1198991 119899
119864events in each class up to time 119905 given an initial
condition 119883119895(119904)
The stochastic dynamics we are interested in correspondto a Density Dependent Markov Jump Process [37 38] For asufficiently small time interval ℎ we consider the following
Assumption 4 For each event
(i) event occurrences in disjoint time intervals are inde-pendent
(ii) the Chapman-Kolmogorov equation [28 29] holds
(iii) 119875(119899120572+ 1 119905 + ℎ 119905) = 119882
120572(119883 119905)ℎ + 119900(ℎ)
(iv) 119875(119899120572 119905 + ℎ 119905) = 1 minus 119882
120572(119883 119905)ℎ + 119900(ℎ)
(v) 119875(119899120572(ℎ) + 119896 119905 + ℎ 119905) = 119900(ℎ) 119896 ge 2
Here ℎ is the elapsed time since the time 119905 and 119899120572are the
events of type 120572 accumulated up to 119905
It goes without saying that probabilities 119875(sdot sdot sdot ) areassumed to be differentiable functions of time 119882
120572(119883 119905)
denotes the probability rate for event 120572 which is assumedin the following lemmas to be a smooth function of thepopulations not necessarily linear (yet) Two well-knowngeneral results describe the dynamical evolution under theseassumptions
Lemma 5 (see Theorem 1 in [29]) Under Assumptions 2and 4 the waiting time to the next event is exponentiallydistributed
Theorem6 (see [28] andTheorem 1 in [29]) Under Assump-tions 2 and 4 the dynamics of the process in the space of eventsobeys the Kolmogorov Forward Equation namely
(1198991 119899
119864 119904 119905)
= sum
120572
119875 ( 119899120572minus 1 119904 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119904 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(4)
Proofs of these results are given in Appendix A
4 The Scientific World Journal
22 Linear Rates We will assume that the transition ratesdependmdashin a linear fashionmdashon the populations only Todistinguish from general rates (denoted above by the letter119882) we will use the letter 119877 to denote linear transition ratesHence
119877120572= sum
119896
119903119896
120572119883
119896(119905) (5)
Since rates are nonnegative we have that the proportionalitycoefficient between event 120572 and subpopulation 119896 satisfies119903119896
120572ge 0 These proportionality coefficients convey the
environmental information and as such they may be time-dependent since the environment varieswith timeWheneverpossible we do not write this time dependency explicitly tolighten the notation
Lemma 7 For the case of linear rates population space 119883119896ge
0 is positively invariant under the time evolution given by (1)and the previous assumptions if and only if 120575120572
119894ge minus1 and 120575
120572
119894=
minus1 only if 119877120572= 119903
119894
120572119883
119894(119905) (no sum)
Proof Theprobability of occurrence of event120572 in a small timeinterval ℎ is
ℎ119877120572+ 119900 (ℎ) = ℎsum
119896
119903119896
120572119883
119896(119905) + 119900 (ℎ) (6)
After an occurrence population 119883119896is modified to 119883
119896+ 120575
120572
119896
Population space is positively invariant under time evolutionif and only if the probability of occurrence of event 120572 is zerofor those population values119883
119896such that119883
119896+ 120575
120572
119896lt 0 (we are
just saying that events pushing the populations to negativevalues do not exist) Since this should hold for arbitrary ℎ wehave that 119877
120572= 0
Assume that 120575120572119896
le minus1 Then 119877120572
= 0 for 0 le 119883119896
lt minus120575120572
119896
Letting 119883119896
= 0 we realise that sum119895 = 119896
119903119895
120572119883
119895(119905) = 0 regardless
of the values of 119883119895 and hence it holds that 119903119895
120572= 0 for 119895 = 119896
and therefore 119877120572
= 119903119896
120572119883
119896(no sum) This proves the second
statement If 120575120572119896
lt minus1 we have that 119877120572
= 0 also for 119883119896
= 1Hence 119903
119896
120572= 0 as well 119877
120572equiv 0 and the event 120572 can be ignored
since it does not influence the dynamics
3 Kolmogorov Forward Equation
Let us recast the dynamical equation given by Theorem 6 interms of the generating function [39] in event-space definedas
Ψ (1199111 119911
119864 119904 119905) = sum
119899120572
(prod
120572
119911119899120572
120572)119875 (119899
1 119899
119864 119904 119905) (7)
For the case of linear rates we define
1198770
120572= sum
119896
119903119896
120572119883
119896 (119904)
119865120573
120572= sum
119896
119903119896
120572120575120573
119896
(8)
As a consequence
119877120572= 119877
0
120572+ sum
120573
119865120573
120572119899120573 (9)
Using Theorem 6 to rewrite 120597Ψ120597119905 and using further 119899120573Ψ =
119911120573(120597Ψ120597119911
120573) (a matter of replacing in the definition of Ψ) we
obtain
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
= sum
120572
(119911120572minus 1)(119877
0
120572Ψ + sum
120573
119865120573
120572119911120573120597Ψ120597119911
120573) (10)
Since the generating function reflects a probability distri-bution we have the probability condition Ψ(1 1 119904 119905) =
1 Further the solutions of (10) can be uniquely trans-lated into those given by Theorem 6 for correspondinginitial conditions 119875(119898
1 119898
119864 119904 119904) = 1 corresponding
to Ψ(1199111 119911
119864 119904 119904) = 119911
1198981
11199111198982
2sdot sdot sdot 119911
119898119864
119864 where the integers
1198981 119898
119864denote howmany events of each class had already
occurred at 119905 = 119904 The natural initial condition becomesΨ(119911
1 119911
119864 119904 119904) = 1 corresponding to 119875(0 0 119904 119904) = 1
31 The Method of Characteristics Revisited
Lemma 8 Equation (10) with 1198770
120572= 0 and with the initial
condition Ψ(1199111 119911
119864 119904 119904) = Φ
0(119911
1 119911
119864) admits the
solution
Ψ (1199111 119911
119864 119904 119905) = Φ
0(120601 (119911 119905 119905 minus 119904)) (11)
where 120601(119911 119905 120591) is the flow of the following ODE with initialcondition 119909(0) = 119911 119904(0) = 119905
119889119909120573
119889120591= sum
120572
119865120573
120572(119909
120572minus 1) 119909
120573equiv 119887
120573(119909 119904)
119889119904
119889120591= minus1
(12)
Proof It is convenient to rename 119904 as 119904 = 119909119864+1
introduce119887119864+1
= minus1 and recall that 119887120573 depends only on the extendedvariable array 119909 However when necessary we will writeexplicitly 119909(0) as (119911 119905) We have that
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
119889120601120573(119911 119905 119905 minus 119904)
119889119905
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
times (119887120573(120601 (119911 119905 119905 minus 119904)) +
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
(13)
Further consider the monodromy matrix [40]
119872120573
120574=
120597120601120573(119909 120591)
120597119909120574
(14)
The Scientific World Journal 5
119872 transforms the initial velocity into the final velocity hence
sum
120574
119872120573
120574119887120574(119911 119905) = 119887
120573(120601 (119911 119905 120591)) (15)
Thus we get
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
(
119864+1
sum
120574
119872120573
120574119887120574(119911 119905) +
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
times (
119864
sum
120574
119872120573
120574119887120574(119911 119905) minus 119872
120573
119864+1+
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
=
119864
sum
120574
120597Ψ
120597119911120574
119887120574(119911 119905)
(16)
From the definition of monodromy matrix above it followsthat
minus119872120573
119864+1+
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
= 0 (17)
and by (11) and the chain rule
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)
119872120573
120574= sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)
120597120601120573(119911 119905 119905 minus 119904)
120597119911120574
=120597Ψ
120597119911120574
(18)
Lemma 9 Equation (10) with 1198770
120572= 0 and with the initial
condition Ψ(1199111 119911
119864 119904 119904) = Φ
0(119911
1 119911
119864) admits the
solution
Ψ (1199111 119911
119864 119904 119905) = Φ
0(120601 (119911 119905 119905 minus 119904))Φ
1(119911
1 119911
119864 119904 119905)
(19)
where Φ0(120601(119911 119905 119905 minus 119904)) is a solution of the homogeneous
problem with 1198770
120572= 0 and
Φ1(119911
1 119911
119864 119904 119905)
= expint
119905
119904
(sum
120572
1198770
120572(120591) (120601
120572(119911 119905 119905 minus 120591) minus 1) 119889120591)
(20)
Proof The proof is nothing more than an application of themethod of the variation of the constants The details are asfollows
Introducing the proposed solution Ψ into (10) and usingLemma 8 to eliminateΦ
0we obtain thatΦ
1satisfies (10) with
initial condition Φ1(119911 119904 119904) = 1 Computing directly from
(20) we have that
120597Φ1
120597119905
10038161003816100381610038161003816100381610038161003816119904
= sum
120572
1198770
120572(120601
120572(119911 119905 0) minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573
120597120601120573(119911 119905 119905 minus 120591)
120597119905
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1
+ sum
120573
120597Φ1
120597120601120573(119887
120573(120601 (119911 119905 119905 minus 120591)) +
120597120601120573(119911 119906 119905 minus 120591)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573sum
120574
119872120573
120574119887120574(119911 119905)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573sum
120574
120597120601120573
120597119911120574
119887120574(119911 119905)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120574
120597Φ1
120597119911120574
119887120574(119911 119905)
(21)
after retracing our steps from the previous Lemma
The result presented in Lemma 9 was constructed usingthe method of variation of constants which provides analternative (constructive) proof
32 Basic Properties of the Solution
Definition 10 A nonzero vector v such that sum120572120575120572
119895V120572
= 0for 119895 = 1 119873 is called a structural zero of 119865 Other zeroeigenvalues of 119865 are called nonstructural
Lemma 11 (a)There are structural zeroes if and only if119864 gt 119873(b) The only structurally stable zero eigenvalues of the matrix119865 are the structural zeroes
Proof (a) The ldquoif rdquo part is immediate since dim(ker(120575)) ge
119864minus119873The ldquoonly if rdquo part follows from the assumption of inde-pendent population increments (and therefore independentpopulations) For (b) put 119865 = r120575 If 119865119908 = 0 for 119908 nonzeroand 120575119908 = 119901 = 0 then r119901 = 0 This relation is not structurallystable since by adding 119888 = 0 arbitrarily small to all diagonalelements of r we can assure r1015840119901 = 0 for this modified r1015840 andfor any 119901 = 0
Lemma 12 First integrals of motion for the characteristicequations Let v be a structural zero of 119865 then
prod
120572
(119909120572(119905))
V120572
= 119862 (22)
is a constant of motion for the characteristic equations (12)
6 The Scientific World Journal
Proof Rewrite the first equation (12) as (indices renamed)
119889
119889119905ln119909
120572= sum
120573
119865120572
120573(119909
120573minus 1) (23)
Then
sum
120572
V120572
119889
119889119905ln119909
120572= sum
120573
sum
120572
(V120572119865120572
120573) (119909
120573minus 1) = 0 (24)
Since sum120572(V
120572119865120572
120573) = 0 the expression in the lemma is the
exponential of the integral obtained from
sum
120572
V120572
119889
119889119905ln119909
120572= 0 (25)
Given relationship (1) between initial conditions eventsand populations from the probability generating function inevent space
Ψ (1199111 119911
119864 119904 119905) = sum
119899120572
(prod
120572
119911119899120572
120572)119875 (119899
1sdot sdot sdot 119899
119864 119904 119905) (26)
a corresponding generating function in population space canbe computed namely
Ψ (1199101 119910
119873 119904 119905) = sum
119883119896
(prod
119896
119910119883119896
119896)119875 (119883
1 119883
119873 119904 119905)
(27)
letting 119899120572 119883
119896 be shorthand for (119899
1sdot sdot sdot 119899
119864) (119883
1 119883
119873)
Lemma 13 (projection lemma) Equation 119911120572
= prod119873
119896=1119910120575120572
119896
119896
realises the transformation from event space to populationspace Moreover
Ψ (1199101 119910
119873 119904 119905) = (prod
119896
1199101198830
119896
119896)Ψ(119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896)Ψ(
119873
prod
119896=1
1199101205751
119896
119896
119873
prod
119896=1
119910120575119864
119896
119896 119904 119905)
(28)
Proof Taking logarithms on 119911 and 119910 it is clear that wheneverthere exist structural zeroes the transformation is noninvert-ible In this sense we call it a projection Concerning thestatement we have
(prod
119896
1199101198830
119896
119896)Ψ (119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896) sum
119899120572
(prod
120572
(
119873
prod
119896=1
119910120575120572
119896
119896)
119899120572
)119875 (119899120572 119904 119905)
= sum
119899120572
(
119873
prod
119896=1
1199101198830
119896+sum120572120575120572
119896119899120572
119896)119875 (119899
120572 119904 119905)
(29)
The statement is proved using (1) and rearranging the sumsas
119875 (1198831 119883
119873 119904 119905) = sum
119899120572
1015840
119875 (119899120572 119904 119905) (30)
where the prime denotes sum over 119899120572 such that 119883
0
119895+
sum120572120575120572
119895119899120572= 119883
119895
Corollary 14 The integrals of motion of Lemma 12 projectunder the projection Lemma to 119862 = 1
Proof The above substitution operated on prod119911V120572
120572= 119862 gives
119862 =
119873
prod
119896=1
119910(sum120572V120572120575120572
119896)
119896equiv
119873
prod
119896=1
1199100
119896= 1 (31)
33 Reduced Equation
Assumption 15 There exist integers119898120573120573 = 1 119864 such that
(cf (9))
1198770
120572= sum
120573
119865120573
120572119898
120573 (32)
This assumption amounts to considering that there exists away to mimic the initial condition on the rates with the samematrix 119865 involved in the time evolution
Theorem 16 Under Assumptions 2 4 and 15 equation(10) with natural initial condition Ψ(119911
1 119911
119864 119904 119904 ) =
Φ0(119911
1 119911
119864) = 1 has the solution
Ψ (1199111 119911
119864 119904 119905) = prod
120573
119891119898120573
120573 (33)
where 119911120573119891120573(119911 119904 119905) = 120601
120573(119911 119905 119905 minus 119904) in terms of the solutions
of (12) of Lemma 8 Hence 119891(119911 119904 119905) satisfies (after eliminatingthe auxiliary time 120591)
119889 log (119891120573 (119911 119904 119905))
119889119904
= minussum
120572
119865120573
120572(119904) (119911
120572119891120572(119911 119904 119905) minus 1) 119891
120573(119911 119905 119905) = 1
(34)
Proof The differences between 119891( ) and 120601( ) are (a) factoringout the initial condition 120601
120573(119911 119905 0) = 119911
120573and (b) rearranging
the time arguments since the argument in120601( ) is natural to thead hoc autonomous ODE (12) while the argument in 119891( ) isnatural to the (nonautonomous) PDE (10)
The proof is just a matter of rewriting the previousequations under the present assumptions Because of thenatural initial condition the desired solution to (10) isgiven by (20) in Lemma 9 which can be rewritten usingAssumption 15 as (we use 119911 as shorthand for 119911
1 119911
119864)
Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)
= prod
120573
(expint
119905
119904
sum
120572
119865120573
120572(120591) (120601
120572(119911 119905 119905 minus 120591) minus 1) 119889120591)
119898120573
(35)
The Scientific World Journal 7
Recalling that we defined 119911120572119891120572(119911 119904 119905) = 120601(119911 119905 119905 minus 119904) the
parenthesis can be recognised as the formal solution of (34)integrated from 119905 to 119904
Corollary 17 For v a structural zero of Lemma 12 thegenerating function is unmodified upon the substitution119898
120572rarr
119898120572+ 119896V
120572(119896 isin R)
Remark 18 Having structural zeroes it is amatter of choice toaddress the problem (a) using the variables 119891
120572and imposing
the additional constraints given by structural zeroes or (b)shifting to population coordinates where these constraints areautomatically incorporated Because of Assumption 2 andLemma 12 the transformation mapping one description tothe other corresponds to
119891120572= prod
119895
119908120575120572
119895
119895 (36)
However option (b) proves to bemore efficientwhen it comesto deal with approximate solutions (see Section 5)
4 Examples of Exact Solutions
41 Example I Pure Death Process In the case 119873 = 119864 = 1120575 = minus1 we have 119903 gt 0 119865 = minus119903(119904) and 119898 = minus119872 lt
0 (119872 is the initial population value) Equation (34) reads119889119891119889119904 = 119891119903(119911119891 minus 1) to be integrated in [119905 119904] with 119891(119911 119905 119905) =
1 Hence for constant death rate 119903 we obtain 119891(119911 119904 119905) =
[119911 + (1 minus 119911)119890minus119903(119905minus119904)
]minus1
and
Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)
+ 119911 (1 minus 119890minus119903(119905minus119904)
)]119872
(37)
42 Example II Linear Birth andDeath Process Wehave119873 =
1 119864 = 2 and 1205751
1= minus120575
2
1= 1 while 119903
1denotes the birth rate
and 1199032denotes the death rate 119883
0= 119898
1minus 119898
2is the initial
population There exists one structural zero with vector v =
(1 1)119879 while the corresponding constant of motion results in
the relations 11991111199112= 119862 and119891
11198912= 1The differential equation
(34) for 1198911(to be integrated in [119905 119904]) becomes
1198891198911
119889119904= (119903
111991111198912
1minus (119903
1+ 119903
2) 119891
1+
1198621199032
1199111
) 1198911 (119911 119905 119905) = 1
(38)
with solution (for119862 = 1whichwill be the case after projectionand assuming time-independent birth rates)
1198911(119911 119904 119905) =
1
1199111
(1199032minus 119903
11199111)119882 minus 119903
2(1 minus 119911
1)
(1199032minus 119903
11199111)119882 minus 119903
1(1 minus 119911
1) (39)
where 119882 = exp(1199032
minus 1199031)(119905 minus 119904) Finally Ψ(119911 119904 119905) =
(1198911(119911 119904 119905))
1198981minus1198982 which after projecting in population space
using Lemma 13 and Corollary 14 gives (cf [41])
Ψ (119910 119904 119905) = [(119903
2minus 119903
1119910)119882 minus 119903
2(1 minus 119910)
(1199032minus 119903
1119910)119882 minus 119903
1(1 minus 119910)
]
1198981minus1198982
(40)
Remark 19 A pure death linear process corresponds to abinomial distribution a pure birth process corresponds to anegative binomial and a birth-death process corresponds toa combination of both as independent processes
5 Consistent Approximations
When (34) of Theorem 16 cannot be solved exactly we haveto rely on approximate solutions It is desirable to workwith consistent approximations namely those where basicproperties of the problem are fulfilled exactly rather than ldquoupto an error of size 120598rdquo For example since our solutions shouldbe probabilities we want the coefficients of Ψ(119911
1 119911
119864 119905)
to be always nonnegative and sum up to one Ideally wewant approximations to be constrained to satisfy Lemma 12and to preserve positive invariance in population space (nojumps into negative populations) Whenever consistency isnot automatic it is mandatory to analyse the approximatesolution before jumping into conclusions
51 Poisson Approximation Let us replace 119891120573
equiv 1 in theRHS of (34) (this amounts to propose that 119891
120573is constant and
equal to its initial condition) The solution of the modifiedequation will be a good approximation to the exact solutionfor sufficiently short times such that 119891 is not significantlydifferent from one
For constant rates 119903120572the rhs of (34) does not depend on
time The approximate solution reads 119891120572(119911 119904 119905) = exp((119905 minus
119904)sum120573119865120572
120573(119911
120573minus 1)) and
Ψ (1199111 119911
119864 119904 119905) =
119864
prod
120572
119890119898120572(119905minus119904)sum
120573119865120572
120573(119911120573minus1)
= 119890(119905minus119904)sum
1205731198770
120573(119911120573minus1)
(41)
In other words the generating function is approximatedas a product of individual Poisson generating functions
This approximation has an error of 119900(119905) Simple as it looksthis approach gives a good probability generating functionthat respects structural zeroes but it does not guaranteepositive invariance a Poisson jump could throw the systeminto negative populations However the probability of sucha jump is also 119900(119905) Fortunately there is a natural way toimpose positive invariance and also a natural way to improvethe error up to 119900(119905)
32 in the broader framework of generaltransition rates [6] In this way the approximation can besafely used with error control to address problems that resistexact computation one way or the other
52 Consistent Multinomial Approximations A systematicapproach to obtain a chain of approximations of increasingaccuracy to (34) can be devised via Picard iterations Firstly itis convenient to recast the problem in population coordinates(cf Remark 18) so that the structural zeroes are automaticallytaken into account regardless of other properties of the
8 The Scientific World Journal
adopted approximation From 119891120572
= prod119895119908
120575120572
119895
119895we obtain
119891120572119891
120572= sum
119896120575120572
119896119896119908
119896and (34) can be recast as
sum
119896
120575120572
119896(
119896
119908119896
+ sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1)) = 0
119908119896 (119911 119905 119905) = 1
(42)
Because of the linear independence of population increments(and hence of populations) Lemma 3(b) the counterpart of(34) in population coordinates reads
119896= minus119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896(119911 119905 119905) = 1
(43)
Remark 20 In the case of linearly dependent populationincrements (34) still holds while it is possible to elaboratea more general form in terms of the 119908
119895instead of (43) This
is however outside the scope of this paper
Since sum120572120575120572
119895119898
120572= 119883
119895(119904) we obtain Ψ(119911
1 119911
119864 119904 119905) =
prod119895119908
119883119895(119904)
119895 which is so far an exact result Let us produce
some approximated values for the 119908119896rsquos and correspondingly
for Ψ(1199111 119911
119864 119904 119905)
To avoid further notational complications wewill assumein the sequel that 119903
119896
120573is time independent Hence 119908
119896will
depend only on the time difference 119905 minus 119904 Equation (43) canbe recast as
119896= 119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896 (119911 0) = 1
(44)
now to be integrated along the interval [0 119905 minus 119904] We will evenset 119904 = 0 in what follows
Remark 21 For the case of time-independent proportionalityfactors 119903119896
120573 a correspondingmodification can be introduced in
(34) namely overall change of sign integration in [0 119905 minus 119904]and setting 119904 = 0 to reduce the notational burden
Equations (44) satisfy the Picard-Lindelof theorem sincethe RHS is a Lipschitz function Then the Picard iterationscheme converges to the (unique) solution Taking advantage
of the initial condition the 119899th order truncated version of thePicard iterations reads
119908(0)
119896(ℎ) = 1
119908(1)
119896(ℎ) = 1 + int
ℎ
0
119908(0)
119896(119905)sum
120573
119903119896
120573(119911
120573prod
119895
(119908(0)
119895(119905))
120575120573
119895
minus 1)119889119905
= 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1)
119908(119899)
119896(ℎ) = 1 + int
ℎ
0
119889119905[
[
119908(119899minus1)
119896(119905)
timessum
120573
119903119896
120573(119911
120573prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
minus 1)]
]119899minus1
(45)
where [sdot sdot sdot ]119898denote the Maclaurin polynomial of order119898 in
119905
Lemma 22 For 119905 isin [0 ℎ] ℎ gt 0 being sufficiently smalland for each order of approximation 119899 ge 0 119908(119899)
119896(119905) in (44)
is a polynomial generating function for one individual of thepopulation that is it has the following properties
(a) 119908(119899)
119896(119905) is a polynomial of degree 119899 in 119911
1 119911
119864 where
the coefficient of 11991111989911 119911
119899119864
119864is 119874(119905
119901) with 119901 = 119899
1+
sdot sdot sdot + 119899119864
(b) 119908(119899)
119896(1 1 119905) = 1
(c) Δ(119899)
119896(ℎ) = 119908
(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ) = 119874(ℎ
119899)
(d) the coefficients of 119908(119899)
119896(119905) regarded as a polynomial in
119911120572 are nonnegative functions of time
Proof (a) The statement holds for 119899 = 0 and 119899 = 1Assuming that it holds up to order 119899 minus 1 we have that[119908
(119899minus1)
119896(119905)prod
119895(119908
(119899minus1)
119895(119905))
120575120573
119895 ]119899minus1
is also a polynomial of degree119899 minus 1 in 119911
1 119911
119864with time-dependent coefficients of type
119874(119905119901) since each power 119901 of 119905 carries along a power 119901 in 119911
120572
(as well as lower 119911-powers) A Picard iteration step gives
119908(119899)
119896(119905) = 1 + sum
120573
119903119896
120573119911120573int
ℎ
0
[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
119908(119899minus1)
119896(119905) 119889119905
(46)
The first sum contributes with one time integration and one119911-power which exactly preserves the polynomial structureand the order of the coefficientsThe result is a polynomial ofdegree 119899 in 119911
1 119911
119864 where the coefficients are polynomials
of highest degree at most 119899 in ℎ each one of lowest order119874(ℎ
119901) (as in the statement) The second integral does not
The Scientific World Journal 9
alter these properties since it generates again a polynomial ofdegree 119899minus1 in 119911
1 119911
119864 with time-dependent coefficients of
order119874(ℎ119901+1
) or higher up to ℎ119899 (they are just the coefficients
of 119908(119899minus1)
119896(119905) integrated once in time)
(b) Letting 1199111= 119911
2= sdot sdot sdot = 119911
119864= 1 Picard iteration gives
trivially 119908(119899)
119896(119905) = 1 to all orders again by induction
(c) The statement holds for 119899 = 1 Assuming it holds upto 119899 minus 1 we compute
119908(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ)
= sum
120573
119903119896
120573119911120573
times int
ℎ
0
([(119908(119899minus2)
119896(119905) + Δ
(119899minus1)
119896(119905))
times prod
119895
(119908(119899minus2)
119895(119905) + Δ
(119899minus1)
119895(119905))
120575120573
119895
]
119899minus1
minus[
[
119908(119899minus2)
119896(119905)prod
119895
(119908(119899minus2)
119895(119905))
120575120573
119895]
]119899minus2
)119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
(119908(119899minus1)
119896(119905) minus 119908
(119899minus2)
119896(119905)) 119889119905
(47)
Both expressions in the first integral have the sameMaclaurinpolynomial up to order (119899 minus 2) Hence the difference in thefirst integral contains only terms of order119874(119905
119899minus1) This is also
the case for the second integral After integration we obtainΔ(119899)
119896(ℎ) = 119874(ℎ
119899)
(d) The statement holds for 119899 = 0 and 119899 = 1 and ingeneral for the zero-order coefficient because it is unity for119905 = 0 as a consequence of the initial condition For thehigher-order coefficients we use induction again If all 119911-coefficients in 119908
(119899minus1)
119896(119905) are nonnegative then the same holds
automatically for 119908(119899)
119896(ℎ) up to order 119899 minus 1 since the 119874(ℎ
119899)
contribution adding to each inherited coefficient from theprevious step does not alter its sign for ℎ sufficiently small Itremains to show that the coefficients corresponding to 119901 = 119899
are nonnegative These contributions arise from the (119899 minus 1)thorder of
119903119896
120573119911120573[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
(48)
after time integration If all 120575120573119895are positive then the expres-
sion above is a product of 119911-polynomials with nonnegative
coefficients and the statement follows Suppose that some120575120573
119895= minus1 Then by Lemma 7 119903119896
120573= 0 for 119896 = 119895 Hence
119903119896
120573119911120573119908
(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
=
0 119895 = 119896
119903119896
120573119911120573prod
119895 = 119896
(119908(119899minus1)
119895(119905))
120575120573
119895
119895 = 119896
(49)
However 120575120573119895
= minus 1 for 119895 = 119896 that is they are nonnegative andhence the expression has only nonnegative coefficients
521 Basic Properties of the First- and Second-OrderApproximations Let us analyse the generating functionΨ(119911
1 119911
119864 119905) = prod
119895119908
119883119895(0)
119895 The simplest nontrivial
applications of the approximation scheme are the first- andsecond-order approximations Expanding explicitly up tosecond-order we get
119908119896(ℎ) = 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1) +
ℎ2
2(sum
120573
119903119896
120573(119911
120573minus 1))
times (sum
120574
119903119896
120574(119911
120574minus 1))
+ℎ2
2sum
120573
119903119896
120573119911120573sum
120574
119865120573
120574(119911
120574minus 1)
(50)
We begin by noticing that the first-order approximation isobtained by neglecting all terms in ℎ
2 in the above equationWhenever 119908
119895is linear in 119911 which is always the case for the
first-order approximation the resulting generating functionΨ corresponds to the product of (independent) multinomialdistributions for the events affecting each subpopulation
Let us now consider Ψ(1199111 119911
119864 119905) computed with the
second-order approximated 119908119895rsquos to perform one simulation
step of size ℎCounting powers of 119911
120574on each 119908
119896we have (a) the
probability of no events occurring in time-step ℎ
119875119896(ℎ 0) = 1 minus ℎ(sum
120573
119903119896
120573) +
ℎ2
2(sum
120573
119903119896
120573)
2
(51)
(b) the probability of only one (120573) event occurring in time-step ℎ
119875119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus ℎ
2sum
120572
119903119896
120572minus
ℎ2
2sum
120572
119865120573
120572 (52)
and (c) the probability of another event (120572) occurring afterthe first one
119875119896(ℎ 119868119868 120573 120572) =
ℎ2
2119903119896
120573(119903
119896
120572+ 119865
120573
120572) (53)
10 The Scientific World Journal
522 Simulation Algorithm A simulation step can be oper-ated in the following way Let
sum
120572
(119875119896(ℎ 119868119868 120573 120572)) + 119875
119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus
ℎ2
2sum
120572
119903119896
120572
= 119875119896(ℎ 119868 120573)
(54)
be the probability of one (120573) event occurring alone orfollowed by a second event It follows that
119875119896(ℎ 119868119868 120573 120572) = 119875
119896(ℎ 119868 120573)
ℎ
2
(119903120572
119896+ 119865
120573
120572)
(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ
2)
(55)
The second factor above is the conditional probability119875119896(ℎ 120572120573) for the second event being 120572 given that there was
a first (120573) eventThe algorithm proceeds in two steps First a multino-
mial deviate is computed with probabilities 119875119896(ℎ 119868 120573) (the
probability of no event occurring is still 119875119896(ℎ 0)) This gives
the number 119899120573of occurrences of each event In the second
step for each 119899120573we compute the deviates for the occurrence
of a second event or no more events with the multinomialgiven by the array of probabilities 119875
119896(ℎ 120572120573) defined above
(the probability of no second event occurring is here 1 minus
(ℎ2)(sum120572(119903
120572
119896+ 119865
120573
120572)(1 minus (ℎ2)sum
120572119903119896
120572)))
6 Implementation Developmental Cascadeswithout Mortality
In this and the following Sections we apply the tools devel-oped up to now to the description of a subprocess in insectdevelopment namely the development of immature larvaewhich is in principle a highly individual subprocess Weassume that the evolution of an insect from egg to pupaoccurs via a sequence of 119864 gt 0 maturation stages Biologistsrecognise some substages by inspection in the developmentof larvae (instars) and even more stages when observed byother methods [35] The immature individual may die alongthe process or otherwise complete it and exit as pupa oradult The actual value of apparent maturation stages 119864 willdepend on environmental and experimental conditions andwill ultimately be specified when analying experimental datain Section 71
Empirical evidence based on existing and new experi-mental results as well as biological insight produced by thepresent description will be the subject of an independentwork [42]We discuss however a couple of examples from theliterature in Section 71
The events for this process are maturation events pro-moting individuals from one subpopulation into to the nextand death events for all subpopulations In other wordseach subpopulation represents a given (intermediate) level ofdevelopment and maturation events correspond to progressin the development Development at level 119895 promotes oneindividual to level 119895 + 1 and the event rate is linear in the119895-subpopulation Hence in 119877
120572= sum
119895119903119896
120572119883
119895 the matrix 119903
is square and diagonal for 0 le 120572 le 119864 minus 1 (we haveexactly 119864 + 1 subpopulations counted from 0 to 119864) All ofthe environmental influence is at this point incorporatedthrough 119903
Moreover level 119864 is the matured pupa and we may set119903119864
= 0 since at pupation stage the present mechanism endsand new processes take place Subpopulation 119864 is the exitpoint of the dynamics and we assume it to be determined bythe stochastic evolution of the immature stages 0 le 120572 le 119864minus1
In the same spirit the action of each event on thepopulations modifies just the actual population and the nextone in the chain Hence
120575120572
119895=
minus1 120572 = 119895
+1 120572 = 119895 minus 1(56)
Following Section 3 [1198770
120572]119879
= (1199030119872 0 0) and
119865120573
120572=
minus119903120572 120573 = 120572
+119903120572 120573 = 120572 minus 1
0 otherwise(57)
Finally letting m119879= minus119872(1 1 1) we can write 119877
0
120572=
sum120573119865120573
120572119898
120573 In this framework the index pair (119895 120572) is highly
interconnected and we can achieve a complete descriptionusing just one index We use 0 le 119895 le 119864 minus 1 throughout thissection
The procedure of Section 3 gives Ψ(119911 119905) = prod119895(119891
119895)119898119895 =
prod119895(119891
minus119872
119895) while the dynamical system for 119891
119895reads recalling
the recasting in Remark 21119891119895= 119891
119895(minus119903
119895(119911
119895119891119895minus 1) + 119903
119895+1(119911
119895+1119891119895+1
minus 1))
119895 = 0 119864 minus 2
(58)
and 119891119864minus1
= 119891119864minus1
(minus119903119864minus1
(119911119864minus1
119891119864minus1
minus 1)) The overall initialcondition is 119891
119895(0) = 1 Let 120579
119895= log(119891
119895) and
120595119895=
119864minus1
sum
120573=119895
120579120573 119882
119895= exp (minus120595
119895) (59)
Note that 119882119895119891119895
= 119882119895
sdot exp(120579119895) = 119882
119895+1 Rewriting the
equations120579119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) + 119903
119895+1(119911
119895+1exp (120579
119895+1) minus 1)
119895 lt 119864 minus 1
119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) = 120579
119895+
119895+1 119895 lt 119864 minus 1
120579119864minus1
= 119864minus1
= minus119903119864minus1
(119911119864minus1
exp (120579119864minus1
) minus 1)
(60)
with initial condition 120579119895(0) = 0 for all 119895With this notation the
generating function reads Ψ(119911 119905) = 119882119872
0 and we formulate
hence an ODE for the 119882119895rsquos
119895+ 119903
119895119882
119895= 119903
119895119911119895119882
119895+1 119882
119895 (0) = 1 119895 = 0 119864 minus 2
119864minus1
= 119903119864minus1
(119911119864minus1
minus 119882119864minus1
) 119882119864minus1
(0) = 1
(61)
The Scientific World Journal 11
By defining the auxiliary quantity 119882119864(119905) = 1 the solution
for the 119882119895-differential equations can be written in recursive
form
119882119895(119905) = 119890
minus119903119895119905+ 119911
119895119903119895int
119905
0
119890119903119895(119904minus119905)
119882119895+1
(119904) 119889119904 119895 = 0 119864 minus 1
(62)
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820(119905) = (
119864minus1
prod
119898=0
119911119898)(1 minus
119864minus1
sum
119896=0
exp (minus119903119896119905)
119864minus1
prod
119895=0119895 = 119896
119903119895
119903119895minus 119903
119896
)
+
119864
sum
119896=1
prod119864minus119896
119895=0119911119895
119911119864minus119896
119903119864minus119896
119864minus119896
sum
119898=0
119903119898exp (minus119903
119898119905)
119864minus119896
prod
119897=0119897 =119898
119903119897
119903119897minus 119903
119898
(63)
The solution looksmore compact when all 119903119896are assumed
to take the common value 119903
1198820(119905) =
prod119864minus1
119898=0119911119898
(119864 minus 1)int
119903119905
0
exp (minus119904) 119904119864minus1
119889119904
+
119864
sum
119896=1
prod119864minus119896
119898=0119911119898
119911119864minus119896
(119903119905)119864minus119896 exp (minus119903119905)
(119864 minus 119896)
(64)
7 Developmental Cascades with Mortality
We add to the previous description 119864 subpopulation specificdeath events 119876
120572= 119902
120572119883
120572 where 120572 = 0 119864 minus 1 We
do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883
119864leaves the
present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876
0 119877
0 119876
1 119877
1 119876
119864minus1 and 119877
119864minus1 Then ((119876 119877)
0)119879
=
(1199020119872 119903
0119872 0 0) We have
1205752120572
120572= minus1 (death) 120572 lt 119864
1205752120572+1
120572= minus1
1205752120572+1
120572+1= +1
(development) 120572 lt 119864
119865120573
2120572=
minus119902120572 120573 = 2120572
minus119902120572 120573 = 2120572 + 1
119902120572 120573 = 2120572 minus 1
119865120573
2120572+1=
minus119903120572 120573 = 2120572 + 1
minus119903120572 120573 = 2120572
119903120572 120573 = 2120572 minus 1
120572 lt 119864
(65)
Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891
120572and the 119911
120572in pairs (119892 119891) and
(119911 119910) as (1198920 119891
0 119892
119864minus1 119891
119864minus1) and (119911
0 119910
0 119911
119864minus1 119910
119864minus1)
Also for notational convenience we will assume that thereexist 119902
119864= 119903
119864= 0 when necessary Defining
119867120572(119892
120572 119891
120572) = 119902
120572(119911
120572119892120572minus 1) + 119903
120572(119910
120572119891120572minus 1) (66)
we have (recall again Remark 21 and note that 119867119864equiv 0)
119892120572= minus119867
120572(119892
120572 119891
120572) 119892
120572
119891120572= (minus119867
120572(119892
120572 119891
120572) + 119867
120572+1(119892
120572+1 119891
120572+1)) 119891
120572
120572 = 0 119864 minus 2
119892119864minus1
119892119864minus1
=
119891119864minus1
119891119864minus1
= minus119867119864minus1
(119892119864minus1
119891119864minus1
)
(67)
The overall initial condition is still 119892120572(0) = 1 = 119891
120572(0)
Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881
119879
120572=
(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579
120572= log119892
120572
and 120601120572
= log119891120572 changing 119867
120572and the initial conditions
correspondingly Hence
120601120572= 120579
120572minus 120579
120572+1997904rArr 120601
120572= 120579
120572minus 120579
120572+1 120572 = 0 119864 minus 2
120579120572= minus119867
120572(120579
120572 120579
120572minus 120579
120572+1) 120572 = 0 119864 minus 2
120601119864minus1
= 120579119864minus1
= minus119867119864minus1
(120579119864minus1
120579119864minus1
)
(68)
The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879
= minus119872(1 0 0 0) satisfies 1198770
= 119865119898 FinallyΨ(119911 119910 119905) = 119892
minus119872
0 We now compute 119892
0(119905)
Let119882120572= 119890
minus120579120572 = 119892
minus1
120572 0 le 120572 le 119864minus1 Defining the auxiliary
variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of
119882 read
120572+ (119902
120572+ 119903
120572)119882
120572= 119902
120572119911120572+ 119903
120572119910120572119882
120572+1 119882
120572 (0) = 1
(69)
with solution
119882120572(119905) = 119890
minus(119902120572+119903120572)119905
+ int
119905
0
119890minus(119902120572+119903120572)(119905minus119904)
(119902120572119911120572+ 119903
120572119910120572119882
120572+1(119904)) 119889119904
(70)
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
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Stochastic AnalysisInternational Journal of
2 The Scientific World Journal
linear rates that is rates that depend linearly on the pop-ulations (in the chemical literature they are known as first-order reactions [12]) We provide both exact solutions andgeneral approximation techniques with error 119900(119905
119899) Results
from [6] for the linear case are a modified version of one ofthe approximation schemes in this paper (Poisson approx-imation Section 5) The linear problem is mathematicallysimpler than the general case and a few general results canbe established before resorting to approximations
Linear processes are an important part of most biologicaldynamical descriptions Using an analogy from classicalpopulation dynamics linear (Malthusian) population growth[13] describes the evolution of a population at low densities(eg where the local environment is essentially unaffected bythe population growth and the individuals do not interferewith each other)More generally processes that consider onlyindividuals in a somewhat isolated basis can be satisfactorilydescribed with linear rates However whether to use lineardescriptions or not is ultimately determined by the nature ofthe process (or subprocess) For example finite environmentswill sooner or later force individuals to interact in growingpopulations and nonlinear rates become necessary such as inVerhulstrsquos description of population growth [14]
Perhaps the oldest application of linear processes topopulation dynamics regards birth and death processesHistorically according to Kendall [15] the first to proposebirth and death equations in population dynamics was Furry[16] for a system of physical origin Currently such anapproach is being used in several sciences linear stochasticprocesses have been proposed for the development of tumors[17 18] drug delivery and cell replication are also described inthese terms [19] Concerning the tumors themodel proposedin [17 18] is based on linear individual processes a detailedtreatment of which would require a paper of its own Theaccumulation of individual processes is finally approximatedwith a ldquotunnelling processrdquo corresponding to a maturationcascade that can be easily described with the tools of thispaper (see (71)) Concerning the question of drug deliverywe will come back in detail to [19] in Section 7
A substantial use of population dynamics driven by jumpprocesses has been done also in chemistry particularly afterthe rediscovery by Gillespie [20] of Kendallrsquos simulationmethod [21 22] Poisson [23 24] binomial [25 26] andmultinomial [27] heuristics for the short-time integrals havebeen proposed There is however an important differencebetween chemistry and biology concerning the issues of thispaper On the one hand linear rates are seldom the case inchemical reactions except perhaps in the case of irreversibledecompositions or isomeric recombination Attention to lin-ear rates in chemistry is relatively recent [12] when comparedto over 200 years of Malthusian population growth Onthe other hand detailed event statistics does not appearto be fundamental either highly diluted chemical reactionson a test tube may easily generate 10
14 events of eachtype Consequently attention to event based descriptionsappears to be absent from the chemistry literature Thismight however change in the future since the technicalpossibilities given by for example ldquoin-cell chemistryrdquo andldquonanochemistryrdquo allow dealingwith a lower number of events
where tiny differences in event count could be relevant for thegeneral outcome
In themathematical literature attention to jumpprocessesstarts with Kolmogorovrsquos foundational work [28] and itsfurther elaboration by Feller [29] A substantial effort torelate the stochastic description to deterministic equationswas performed by Kurtz [30ndash34] However attention hasalways focused on the dynamics in population space ratherthan event space
This work focuses on the event statistics associated withlinear processes Even for cases where the detailed eventstatistics is not of central interest such an approach offers analternative to the more traditional formulation in populationspace
Further we discuss in detail the stochastic version ofa linear developmental model Specifically we deal with asubprocess in the life cycle of insects that admits both a lineardescription and an exact solution namely the developmentof immature stages which is a highly individual subprocessFrom egg form to adult form the development of insectsgoes through several transformations at different levels Forexample at the most visible level insects undergoing com-plete metamorphosis evolve in the sequence egg larva pupaand adult Further different substages can be recognised byinspection in the development of larvae (instars) and evenmore stages when observed by other methods [35] Theimmature individual may die along the process or otherwisecomplete it and exit as adult
In Section 2 we formulate the dynamical populationproblem in event space In Section 21 we present the Kol-mogorov Forward Equations (for Markov jump processes)for the most general linear systems of population dynamicsin the form appropriated to probabilities and generatingfunctions The general solution of these equations will bewritten using the Theorem of the Characteristics reformu-lated with respect to the standard geometrical presentation[36] into a dynamical presentation We will analyse thegeneral structure of the equations and in particular theirfirst integrals of motion and the projection onto populationspace (Sections 2 and 3) Section 4 contains two classicalexamples In Section 5 Poisson and multinomial approx-imations (both with error 119900(119905)) are derived and a higher-order approximation with error 119900(119905
119899) (with 119899 as an arbi-
trary positive integer) is elaborated The larval developmentmodel is developed in Sections 6 and 7 treating the twogeneral cases We also revisit in Section 7 the question ofdrug delivery introduced in [19] in terms of the presentapproach Concluding remarks are left for the final Sec-tion
2 Mathematical Formulation
The main tools of stochastic population dynamics are a listof (nonnegative integer) populations 119883
119895 where 119895 = 1 sdot sdot sdot 119873
a list of events 120572 = 1 sdot sdot sdot 119864 an (integer) incidence matrix 120575120572
119895
describing how event 120572 modifies population 119895 and a list oftransition rates 119882
120572(119883
1 119883
119899 119905) describing the probability
rate per unit time of occurrence of each event
The Scientific World Journal 3
An important feature of the present approach is that thedynamical description will be handled in event-space In thissense the ldquostaterdquo of the system is given by the array n(119904 119905) =
(1198991 119899
119864)119905indicating how may events of each class have
occurred from time 119904 up to time 119905 Hence event-space is inthis setup a nonnegative integer lattice of dimension 119864 Theconnection with the actual population values at time 119905 is (119904denotes the initial condition)
119883119895(119905) = 119883
119895(119904) + sum
120572
120575120572
119895119899120572(119904 119905) (1)
Definition 1 (linear independency) One says that the popula-tions are linearly dependent if there is a vector u = 0 such thatfor all 119905 ge 119904 one has sum
119895119906119895119883
119895(119905) = 0 Otherwise one says that
the populations are linearly independent and abbreviate it asLI
In similar form we say that the populations have linearlydependent increments if there is u = 0 so that for all 1199051015840 gt 119905 ge 119904
it holds that sum119895119906119895(119883
119895(1199051015840) minus 119883
119895(119905)) = 0
The populations are said to have linearly independentincrements (abbreviated LII) if they do not have linearlydependent increments
Along this work we will assume the following
Assumption 2 The populations have linearly independentincrements that is all involved populations are necessary toprovide a complete description of the problem none of themcan be expressed as a linear combination of the remainingones for all times
Lemma 3 (a) If the populations have linearly independentincrements then the populations are linearly independent
(b) The matrix 120575 has no nonzero vectors u such thatsum
119895120575120572
119895119906119895= 0 for all 120572 if and only if the populations have linearly
independent increments
Proof (a) The assertion is equivalent to the following if thepopulations are linearly dependent then they have linearlydependent increments which is immediate after the defini-tion of linearly dependent increments
(b) Assume that for all 120572 there exists u = 0 such thatsum
119895119906119895120575120572
119895= 0 then
sum
119895
119906119895(119883
119895(119905
1015840) minus 119883
119895(119905)) = sum
119895
119906119895(sum
120572
120575120572
119895119899120572(119905
1015840 119905)) = 0 (2)
and the populations have linearly dependent incrementsConversely assume that the populations have linearly
dependent increments and select time intervals such that119899120573(1199051015840
120573 119905
120573) = 1 and 119899
120572(1199051015840
120573 119905
120573) = 0 for 120572 = 120573 then we have that
sum
119895
119906119895120575120572
119895= 0 forall120572 (3)
Statement (b) is the negative statement of these facts
Strictly speaking the last step in the above proof takesfor granted that there exist time intervals where only one
event occurs This is assured by the next assumption (thirditem) Also the initial conditions and the problem descriptionshould be such that all defined events are relevant for thedynamics (eg the event ldquocontagionrdquo is irrelevant for anepidemic problem with zero infectives as initial condition)
21 Dynamical EquationmdashBasic Assumptions Populationnumbers evolve in ldquojumpsrdquo given by the occurrence of eachevent (birth death etc) In the sequel we will use Latinindices 119894 119895 119896 and 119898 for populations and Greek indices 120572120573 and 120574 for events
The ultimate goal of stochastic population dynamicsis to calculate the probabilities 119875(119899
1 119899
119864 119904 119905) of having
1198991 119899
119864events in each class up to time 119905 given an initial
condition 119883119895(119904)
The stochastic dynamics we are interested in correspondto a Density Dependent Markov Jump Process [37 38] For asufficiently small time interval ℎ we consider the following
Assumption 4 For each event
(i) event occurrences in disjoint time intervals are inde-pendent
(ii) the Chapman-Kolmogorov equation [28 29] holds
(iii) 119875(119899120572+ 1 119905 + ℎ 119905) = 119882
120572(119883 119905)ℎ + 119900(ℎ)
(iv) 119875(119899120572 119905 + ℎ 119905) = 1 minus 119882
120572(119883 119905)ℎ + 119900(ℎ)
(v) 119875(119899120572(ℎ) + 119896 119905 + ℎ 119905) = 119900(ℎ) 119896 ge 2
Here ℎ is the elapsed time since the time 119905 and 119899120572are the
events of type 120572 accumulated up to 119905
It goes without saying that probabilities 119875(sdot sdot sdot ) areassumed to be differentiable functions of time 119882
120572(119883 119905)
denotes the probability rate for event 120572 which is assumedin the following lemmas to be a smooth function of thepopulations not necessarily linear (yet) Two well-knowngeneral results describe the dynamical evolution under theseassumptions
Lemma 5 (see Theorem 1 in [29]) Under Assumptions 2and 4 the waiting time to the next event is exponentiallydistributed
Theorem6 (see [28] andTheorem 1 in [29]) Under Assump-tions 2 and 4 the dynamics of the process in the space of eventsobeys the Kolmogorov Forward Equation namely
(1198991 119899
119864 119904 119905)
= sum
120572
119875 ( 119899120572minus 1 119904 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119904 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(4)
Proofs of these results are given in Appendix A
4 The Scientific World Journal
22 Linear Rates We will assume that the transition ratesdependmdashin a linear fashionmdashon the populations only Todistinguish from general rates (denoted above by the letter119882) we will use the letter 119877 to denote linear transition ratesHence
119877120572= sum
119896
119903119896
120572119883
119896(119905) (5)
Since rates are nonnegative we have that the proportionalitycoefficient between event 120572 and subpopulation 119896 satisfies119903119896
120572ge 0 These proportionality coefficients convey the
environmental information and as such they may be time-dependent since the environment varieswith timeWheneverpossible we do not write this time dependency explicitly tolighten the notation
Lemma 7 For the case of linear rates population space 119883119896ge
0 is positively invariant under the time evolution given by (1)and the previous assumptions if and only if 120575120572
119894ge minus1 and 120575
120572
119894=
minus1 only if 119877120572= 119903
119894
120572119883
119894(119905) (no sum)
Proof Theprobability of occurrence of event120572 in a small timeinterval ℎ is
ℎ119877120572+ 119900 (ℎ) = ℎsum
119896
119903119896
120572119883
119896(119905) + 119900 (ℎ) (6)
After an occurrence population 119883119896is modified to 119883
119896+ 120575
120572
119896
Population space is positively invariant under time evolutionif and only if the probability of occurrence of event 120572 is zerofor those population values119883
119896such that119883
119896+ 120575
120572
119896lt 0 (we are
just saying that events pushing the populations to negativevalues do not exist) Since this should hold for arbitrary ℎ wehave that 119877
120572= 0
Assume that 120575120572119896
le minus1 Then 119877120572
= 0 for 0 le 119883119896
lt minus120575120572
119896
Letting 119883119896
= 0 we realise that sum119895 = 119896
119903119895
120572119883
119895(119905) = 0 regardless
of the values of 119883119895 and hence it holds that 119903119895
120572= 0 for 119895 = 119896
and therefore 119877120572
= 119903119896
120572119883
119896(no sum) This proves the second
statement If 120575120572119896
lt minus1 we have that 119877120572
= 0 also for 119883119896
= 1Hence 119903
119896
120572= 0 as well 119877
120572equiv 0 and the event 120572 can be ignored
since it does not influence the dynamics
3 Kolmogorov Forward Equation
Let us recast the dynamical equation given by Theorem 6 interms of the generating function [39] in event-space definedas
Ψ (1199111 119911
119864 119904 119905) = sum
119899120572
(prod
120572
119911119899120572
120572)119875 (119899
1 119899
119864 119904 119905) (7)
For the case of linear rates we define
1198770
120572= sum
119896
119903119896
120572119883
119896 (119904)
119865120573
120572= sum
119896
119903119896
120572120575120573
119896
(8)
As a consequence
119877120572= 119877
0
120572+ sum
120573
119865120573
120572119899120573 (9)
Using Theorem 6 to rewrite 120597Ψ120597119905 and using further 119899120573Ψ =
119911120573(120597Ψ120597119911
120573) (a matter of replacing in the definition of Ψ) we
obtain
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
= sum
120572
(119911120572minus 1)(119877
0
120572Ψ + sum
120573
119865120573
120572119911120573120597Ψ120597119911
120573) (10)
Since the generating function reflects a probability distri-bution we have the probability condition Ψ(1 1 119904 119905) =
1 Further the solutions of (10) can be uniquely trans-lated into those given by Theorem 6 for correspondinginitial conditions 119875(119898
1 119898
119864 119904 119904) = 1 corresponding
to Ψ(1199111 119911
119864 119904 119904) = 119911
1198981
11199111198982
2sdot sdot sdot 119911
119898119864
119864 where the integers
1198981 119898
119864denote howmany events of each class had already
occurred at 119905 = 119904 The natural initial condition becomesΨ(119911
1 119911
119864 119904 119904) = 1 corresponding to 119875(0 0 119904 119904) = 1
31 The Method of Characteristics Revisited
Lemma 8 Equation (10) with 1198770
120572= 0 and with the initial
condition Ψ(1199111 119911
119864 119904 119904) = Φ
0(119911
1 119911
119864) admits the
solution
Ψ (1199111 119911
119864 119904 119905) = Φ
0(120601 (119911 119905 119905 minus 119904)) (11)
where 120601(119911 119905 120591) is the flow of the following ODE with initialcondition 119909(0) = 119911 119904(0) = 119905
119889119909120573
119889120591= sum
120572
119865120573
120572(119909
120572minus 1) 119909
120573equiv 119887
120573(119909 119904)
119889119904
119889120591= minus1
(12)
Proof It is convenient to rename 119904 as 119904 = 119909119864+1
introduce119887119864+1
= minus1 and recall that 119887120573 depends only on the extendedvariable array 119909 However when necessary we will writeexplicitly 119909(0) as (119911 119905) We have that
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
119889120601120573(119911 119905 119905 minus 119904)
119889119905
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
times (119887120573(120601 (119911 119905 119905 minus 119904)) +
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
(13)
Further consider the monodromy matrix [40]
119872120573
120574=
120597120601120573(119909 120591)
120597119909120574
(14)
The Scientific World Journal 5
119872 transforms the initial velocity into the final velocity hence
sum
120574
119872120573
120574119887120574(119911 119905) = 119887
120573(120601 (119911 119905 120591)) (15)
Thus we get
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
(
119864+1
sum
120574
119872120573
120574119887120574(119911 119905) +
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
times (
119864
sum
120574
119872120573
120574119887120574(119911 119905) minus 119872
120573
119864+1+
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
=
119864
sum
120574
120597Ψ
120597119911120574
119887120574(119911 119905)
(16)
From the definition of monodromy matrix above it followsthat
minus119872120573
119864+1+
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
= 0 (17)
and by (11) and the chain rule
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)
119872120573
120574= sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)
120597120601120573(119911 119905 119905 minus 119904)
120597119911120574
=120597Ψ
120597119911120574
(18)
Lemma 9 Equation (10) with 1198770
120572= 0 and with the initial
condition Ψ(1199111 119911
119864 119904 119904) = Φ
0(119911
1 119911
119864) admits the
solution
Ψ (1199111 119911
119864 119904 119905) = Φ
0(120601 (119911 119905 119905 minus 119904))Φ
1(119911
1 119911
119864 119904 119905)
(19)
where Φ0(120601(119911 119905 119905 minus 119904)) is a solution of the homogeneous
problem with 1198770
120572= 0 and
Φ1(119911
1 119911
119864 119904 119905)
= expint
119905
119904
(sum
120572
1198770
120572(120591) (120601
120572(119911 119905 119905 minus 120591) minus 1) 119889120591)
(20)
Proof The proof is nothing more than an application of themethod of the variation of the constants The details are asfollows
Introducing the proposed solution Ψ into (10) and usingLemma 8 to eliminateΦ
0we obtain thatΦ
1satisfies (10) with
initial condition Φ1(119911 119904 119904) = 1 Computing directly from
(20) we have that
120597Φ1
120597119905
10038161003816100381610038161003816100381610038161003816119904
= sum
120572
1198770
120572(120601
120572(119911 119905 0) minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573
120597120601120573(119911 119905 119905 minus 120591)
120597119905
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1
+ sum
120573
120597Φ1
120597120601120573(119887
120573(120601 (119911 119905 119905 minus 120591)) +
120597120601120573(119911 119906 119905 minus 120591)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573sum
120574
119872120573
120574119887120574(119911 119905)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573sum
120574
120597120601120573
120597119911120574
119887120574(119911 119905)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120574
120597Φ1
120597119911120574
119887120574(119911 119905)
(21)
after retracing our steps from the previous Lemma
The result presented in Lemma 9 was constructed usingthe method of variation of constants which provides analternative (constructive) proof
32 Basic Properties of the Solution
Definition 10 A nonzero vector v such that sum120572120575120572
119895V120572
= 0for 119895 = 1 119873 is called a structural zero of 119865 Other zeroeigenvalues of 119865 are called nonstructural
Lemma 11 (a)There are structural zeroes if and only if119864 gt 119873(b) The only structurally stable zero eigenvalues of the matrix119865 are the structural zeroes
Proof (a) The ldquoif rdquo part is immediate since dim(ker(120575)) ge
119864minus119873The ldquoonly if rdquo part follows from the assumption of inde-pendent population increments (and therefore independentpopulations) For (b) put 119865 = r120575 If 119865119908 = 0 for 119908 nonzeroand 120575119908 = 119901 = 0 then r119901 = 0 This relation is not structurallystable since by adding 119888 = 0 arbitrarily small to all diagonalelements of r we can assure r1015840119901 = 0 for this modified r1015840 andfor any 119901 = 0
Lemma 12 First integrals of motion for the characteristicequations Let v be a structural zero of 119865 then
prod
120572
(119909120572(119905))
V120572
= 119862 (22)
is a constant of motion for the characteristic equations (12)
6 The Scientific World Journal
Proof Rewrite the first equation (12) as (indices renamed)
119889
119889119905ln119909
120572= sum
120573
119865120572
120573(119909
120573minus 1) (23)
Then
sum
120572
V120572
119889
119889119905ln119909
120572= sum
120573
sum
120572
(V120572119865120572
120573) (119909
120573minus 1) = 0 (24)
Since sum120572(V
120572119865120572
120573) = 0 the expression in the lemma is the
exponential of the integral obtained from
sum
120572
V120572
119889
119889119905ln119909
120572= 0 (25)
Given relationship (1) between initial conditions eventsand populations from the probability generating function inevent space
Ψ (1199111 119911
119864 119904 119905) = sum
119899120572
(prod
120572
119911119899120572
120572)119875 (119899
1sdot sdot sdot 119899
119864 119904 119905) (26)
a corresponding generating function in population space canbe computed namely
Ψ (1199101 119910
119873 119904 119905) = sum
119883119896
(prod
119896
119910119883119896
119896)119875 (119883
1 119883
119873 119904 119905)
(27)
letting 119899120572 119883
119896 be shorthand for (119899
1sdot sdot sdot 119899
119864) (119883
1 119883
119873)
Lemma 13 (projection lemma) Equation 119911120572
= prod119873
119896=1119910120575120572
119896
119896
realises the transformation from event space to populationspace Moreover
Ψ (1199101 119910
119873 119904 119905) = (prod
119896
1199101198830
119896
119896)Ψ(119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896)Ψ(
119873
prod
119896=1
1199101205751
119896
119896
119873
prod
119896=1
119910120575119864
119896
119896 119904 119905)
(28)
Proof Taking logarithms on 119911 and 119910 it is clear that wheneverthere exist structural zeroes the transformation is noninvert-ible In this sense we call it a projection Concerning thestatement we have
(prod
119896
1199101198830
119896
119896)Ψ (119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896) sum
119899120572
(prod
120572
(
119873
prod
119896=1
119910120575120572
119896
119896)
119899120572
)119875 (119899120572 119904 119905)
= sum
119899120572
(
119873
prod
119896=1
1199101198830
119896+sum120572120575120572
119896119899120572
119896)119875 (119899
120572 119904 119905)
(29)
The statement is proved using (1) and rearranging the sumsas
119875 (1198831 119883
119873 119904 119905) = sum
119899120572
1015840
119875 (119899120572 119904 119905) (30)
where the prime denotes sum over 119899120572 such that 119883
0
119895+
sum120572120575120572
119895119899120572= 119883
119895
Corollary 14 The integrals of motion of Lemma 12 projectunder the projection Lemma to 119862 = 1
Proof The above substitution operated on prod119911V120572
120572= 119862 gives
119862 =
119873
prod
119896=1
119910(sum120572V120572120575120572
119896)
119896equiv
119873
prod
119896=1
1199100
119896= 1 (31)
33 Reduced Equation
Assumption 15 There exist integers119898120573120573 = 1 119864 such that
(cf (9))
1198770
120572= sum
120573
119865120573
120572119898
120573 (32)
This assumption amounts to considering that there exists away to mimic the initial condition on the rates with the samematrix 119865 involved in the time evolution
Theorem 16 Under Assumptions 2 4 and 15 equation(10) with natural initial condition Ψ(119911
1 119911
119864 119904 119904 ) =
Φ0(119911
1 119911
119864) = 1 has the solution
Ψ (1199111 119911
119864 119904 119905) = prod
120573
119891119898120573
120573 (33)
where 119911120573119891120573(119911 119904 119905) = 120601
120573(119911 119905 119905 minus 119904) in terms of the solutions
of (12) of Lemma 8 Hence 119891(119911 119904 119905) satisfies (after eliminatingthe auxiliary time 120591)
119889 log (119891120573 (119911 119904 119905))
119889119904
= minussum
120572
119865120573
120572(119904) (119911
120572119891120572(119911 119904 119905) minus 1) 119891
120573(119911 119905 119905) = 1
(34)
Proof The differences between 119891( ) and 120601( ) are (a) factoringout the initial condition 120601
120573(119911 119905 0) = 119911
120573and (b) rearranging
the time arguments since the argument in120601( ) is natural to thead hoc autonomous ODE (12) while the argument in 119891( ) isnatural to the (nonautonomous) PDE (10)
The proof is just a matter of rewriting the previousequations under the present assumptions Because of thenatural initial condition the desired solution to (10) isgiven by (20) in Lemma 9 which can be rewritten usingAssumption 15 as (we use 119911 as shorthand for 119911
1 119911
119864)
Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)
= prod
120573
(expint
119905
119904
sum
120572
119865120573
120572(120591) (120601
120572(119911 119905 119905 minus 120591) minus 1) 119889120591)
119898120573
(35)
The Scientific World Journal 7
Recalling that we defined 119911120572119891120572(119911 119904 119905) = 120601(119911 119905 119905 minus 119904) the
parenthesis can be recognised as the formal solution of (34)integrated from 119905 to 119904
Corollary 17 For v a structural zero of Lemma 12 thegenerating function is unmodified upon the substitution119898
120572rarr
119898120572+ 119896V
120572(119896 isin R)
Remark 18 Having structural zeroes it is amatter of choice toaddress the problem (a) using the variables 119891
120572and imposing
the additional constraints given by structural zeroes or (b)shifting to population coordinates where these constraints areautomatically incorporated Because of Assumption 2 andLemma 12 the transformation mapping one description tothe other corresponds to
119891120572= prod
119895
119908120575120572
119895
119895 (36)
However option (b) proves to bemore efficientwhen it comesto deal with approximate solutions (see Section 5)
4 Examples of Exact Solutions
41 Example I Pure Death Process In the case 119873 = 119864 = 1120575 = minus1 we have 119903 gt 0 119865 = minus119903(119904) and 119898 = minus119872 lt
0 (119872 is the initial population value) Equation (34) reads119889119891119889119904 = 119891119903(119911119891 minus 1) to be integrated in [119905 119904] with 119891(119911 119905 119905) =
1 Hence for constant death rate 119903 we obtain 119891(119911 119904 119905) =
[119911 + (1 minus 119911)119890minus119903(119905minus119904)
]minus1
and
Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)
+ 119911 (1 minus 119890minus119903(119905minus119904)
)]119872
(37)
42 Example II Linear Birth andDeath Process Wehave119873 =
1 119864 = 2 and 1205751
1= minus120575
2
1= 1 while 119903
1denotes the birth rate
and 1199032denotes the death rate 119883
0= 119898
1minus 119898
2is the initial
population There exists one structural zero with vector v =
(1 1)119879 while the corresponding constant of motion results in
the relations 11991111199112= 119862 and119891
11198912= 1The differential equation
(34) for 1198911(to be integrated in [119905 119904]) becomes
1198891198911
119889119904= (119903
111991111198912
1minus (119903
1+ 119903
2) 119891
1+
1198621199032
1199111
) 1198911 (119911 119905 119905) = 1
(38)
with solution (for119862 = 1whichwill be the case after projectionand assuming time-independent birth rates)
1198911(119911 119904 119905) =
1
1199111
(1199032minus 119903
11199111)119882 minus 119903
2(1 minus 119911
1)
(1199032minus 119903
11199111)119882 minus 119903
1(1 minus 119911
1) (39)
where 119882 = exp(1199032
minus 1199031)(119905 minus 119904) Finally Ψ(119911 119904 119905) =
(1198911(119911 119904 119905))
1198981minus1198982 which after projecting in population space
using Lemma 13 and Corollary 14 gives (cf [41])
Ψ (119910 119904 119905) = [(119903
2minus 119903
1119910)119882 minus 119903
2(1 minus 119910)
(1199032minus 119903
1119910)119882 minus 119903
1(1 minus 119910)
]
1198981minus1198982
(40)
Remark 19 A pure death linear process corresponds to abinomial distribution a pure birth process corresponds to anegative binomial and a birth-death process corresponds toa combination of both as independent processes
5 Consistent Approximations
When (34) of Theorem 16 cannot be solved exactly we haveto rely on approximate solutions It is desirable to workwith consistent approximations namely those where basicproperties of the problem are fulfilled exactly rather than ldquoupto an error of size 120598rdquo For example since our solutions shouldbe probabilities we want the coefficients of Ψ(119911
1 119911
119864 119905)
to be always nonnegative and sum up to one Ideally wewant approximations to be constrained to satisfy Lemma 12and to preserve positive invariance in population space (nojumps into negative populations) Whenever consistency isnot automatic it is mandatory to analyse the approximatesolution before jumping into conclusions
51 Poisson Approximation Let us replace 119891120573
equiv 1 in theRHS of (34) (this amounts to propose that 119891
120573is constant and
equal to its initial condition) The solution of the modifiedequation will be a good approximation to the exact solutionfor sufficiently short times such that 119891 is not significantlydifferent from one
For constant rates 119903120572the rhs of (34) does not depend on
time The approximate solution reads 119891120572(119911 119904 119905) = exp((119905 minus
119904)sum120573119865120572
120573(119911
120573minus 1)) and
Ψ (1199111 119911
119864 119904 119905) =
119864
prod
120572
119890119898120572(119905minus119904)sum
120573119865120572
120573(119911120573minus1)
= 119890(119905minus119904)sum
1205731198770
120573(119911120573minus1)
(41)
In other words the generating function is approximatedas a product of individual Poisson generating functions
This approximation has an error of 119900(119905) Simple as it looksthis approach gives a good probability generating functionthat respects structural zeroes but it does not guaranteepositive invariance a Poisson jump could throw the systeminto negative populations However the probability of sucha jump is also 119900(119905) Fortunately there is a natural way toimpose positive invariance and also a natural way to improvethe error up to 119900(119905)
32 in the broader framework of generaltransition rates [6] In this way the approximation can besafely used with error control to address problems that resistexact computation one way or the other
52 Consistent Multinomial Approximations A systematicapproach to obtain a chain of approximations of increasingaccuracy to (34) can be devised via Picard iterations Firstly itis convenient to recast the problem in population coordinates(cf Remark 18) so that the structural zeroes are automaticallytaken into account regardless of other properties of the
8 The Scientific World Journal
adopted approximation From 119891120572
= prod119895119908
120575120572
119895
119895we obtain
119891120572119891
120572= sum
119896120575120572
119896119896119908
119896and (34) can be recast as
sum
119896
120575120572
119896(
119896
119908119896
+ sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1)) = 0
119908119896 (119911 119905 119905) = 1
(42)
Because of the linear independence of population increments(and hence of populations) Lemma 3(b) the counterpart of(34) in population coordinates reads
119896= minus119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896(119911 119905 119905) = 1
(43)
Remark 20 In the case of linearly dependent populationincrements (34) still holds while it is possible to elaboratea more general form in terms of the 119908
119895instead of (43) This
is however outside the scope of this paper
Since sum120572120575120572
119895119898
120572= 119883
119895(119904) we obtain Ψ(119911
1 119911
119864 119904 119905) =
prod119895119908
119883119895(119904)
119895 which is so far an exact result Let us produce
some approximated values for the 119908119896rsquos and correspondingly
for Ψ(1199111 119911
119864 119904 119905)
To avoid further notational complications wewill assumein the sequel that 119903
119896
120573is time independent Hence 119908
119896will
depend only on the time difference 119905 minus 119904 Equation (43) canbe recast as
119896= 119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896 (119911 0) = 1
(44)
now to be integrated along the interval [0 119905 minus 119904] We will evenset 119904 = 0 in what follows
Remark 21 For the case of time-independent proportionalityfactors 119903119896
120573 a correspondingmodification can be introduced in
(34) namely overall change of sign integration in [0 119905 minus 119904]and setting 119904 = 0 to reduce the notational burden
Equations (44) satisfy the Picard-Lindelof theorem sincethe RHS is a Lipschitz function Then the Picard iterationscheme converges to the (unique) solution Taking advantage
of the initial condition the 119899th order truncated version of thePicard iterations reads
119908(0)
119896(ℎ) = 1
119908(1)
119896(ℎ) = 1 + int
ℎ
0
119908(0)
119896(119905)sum
120573
119903119896
120573(119911
120573prod
119895
(119908(0)
119895(119905))
120575120573
119895
minus 1)119889119905
= 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1)
119908(119899)
119896(ℎ) = 1 + int
ℎ
0
119889119905[
[
119908(119899minus1)
119896(119905)
timessum
120573
119903119896
120573(119911
120573prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
minus 1)]
]119899minus1
(45)
where [sdot sdot sdot ]119898denote the Maclaurin polynomial of order119898 in
119905
Lemma 22 For 119905 isin [0 ℎ] ℎ gt 0 being sufficiently smalland for each order of approximation 119899 ge 0 119908(119899)
119896(119905) in (44)
is a polynomial generating function for one individual of thepopulation that is it has the following properties
(a) 119908(119899)
119896(119905) is a polynomial of degree 119899 in 119911
1 119911
119864 where
the coefficient of 11991111989911 119911
119899119864
119864is 119874(119905
119901) with 119901 = 119899
1+
sdot sdot sdot + 119899119864
(b) 119908(119899)
119896(1 1 119905) = 1
(c) Δ(119899)
119896(ℎ) = 119908
(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ) = 119874(ℎ
119899)
(d) the coefficients of 119908(119899)
119896(119905) regarded as a polynomial in
119911120572 are nonnegative functions of time
Proof (a) The statement holds for 119899 = 0 and 119899 = 1Assuming that it holds up to order 119899 minus 1 we have that[119908
(119899minus1)
119896(119905)prod
119895(119908
(119899minus1)
119895(119905))
120575120573
119895 ]119899minus1
is also a polynomial of degree119899 minus 1 in 119911
1 119911
119864with time-dependent coefficients of type
119874(119905119901) since each power 119901 of 119905 carries along a power 119901 in 119911
120572
(as well as lower 119911-powers) A Picard iteration step gives
119908(119899)
119896(119905) = 1 + sum
120573
119903119896
120573119911120573int
ℎ
0
[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
119908(119899minus1)
119896(119905) 119889119905
(46)
The first sum contributes with one time integration and one119911-power which exactly preserves the polynomial structureand the order of the coefficientsThe result is a polynomial ofdegree 119899 in 119911
1 119911
119864 where the coefficients are polynomials
of highest degree at most 119899 in ℎ each one of lowest order119874(ℎ
119901) (as in the statement) The second integral does not
The Scientific World Journal 9
alter these properties since it generates again a polynomial ofdegree 119899minus1 in 119911
1 119911
119864 with time-dependent coefficients of
order119874(ℎ119901+1
) or higher up to ℎ119899 (they are just the coefficients
of 119908(119899minus1)
119896(119905) integrated once in time)
(b) Letting 1199111= 119911
2= sdot sdot sdot = 119911
119864= 1 Picard iteration gives
trivially 119908(119899)
119896(119905) = 1 to all orders again by induction
(c) The statement holds for 119899 = 1 Assuming it holds upto 119899 minus 1 we compute
119908(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ)
= sum
120573
119903119896
120573119911120573
times int
ℎ
0
([(119908(119899minus2)
119896(119905) + Δ
(119899minus1)
119896(119905))
times prod
119895
(119908(119899minus2)
119895(119905) + Δ
(119899minus1)
119895(119905))
120575120573
119895
]
119899minus1
minus[
[
119908(119899minus2)
119896(119905)prod
119895
(119908(119899minus2)
119895(119905))
120575120573
119895]
]119899minus2
)119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
(119908(119899minus1)
119896(119905) minus 119908
(119899minus2)
119896(119905)) 119889119905
(47)
Both expressions in the first integral have the sameMaclaurinpolynomial up to order (119899 minus 2) Hence the difference in thefirst integral contains only terms of order119874(119905
119899minus1) This is also
the case for the second integral After integration we obtainΔ(119899)
119896(ℎ) = 119874(ℎ
119899)
(d) The statement holds for 119899 = 0 and 119899 = 1 and ingeneral for the zero-order coefficient because it is unity for119905 = 0 as a consequence of the initial condition For thehigher-order coefficients we use induction again If all 119911-coefficients in 119908
(119899minus1)
119896(119905) are nonnegative then the same holds
automatically for 119908(119899)
119896(ℎ) up to order 119899 minus 1 since the 119874(ℎ
119899)
contribution adding to each inherited coefficient from theprevious step does not alter its sign for ℎ sufficiently small Itremains to show that the coefficients corresponding to 119901 = 119899
are nonnegative These contributions arise from the (119899 minus 1)thorder of
119903119896
120573119911120573[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
(48)
after time integration If all 120575120573119895are positive then the expres-
sion above is a product of 119911-polynomials with nonnegative
coefficients and the statement follows Suppose that some120575120573
119895= minus1 Then by Lemma 7 119903119896
120573= 0 for 119896 = 119895 Hence
119903119896
120573119911120573119908
(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
=
0 119895 = 119896
119903119896
120573119911120573prod
119895 = 119896
(119908(119899minus1)
119895(119905))
120575120573
119895
119895 = 119896
(49)
However 120575120573119895
= minus 1 for 119895 = 119896 that is they are nonnegative andhence the expression has only nonnegative coefficients
521 Basic Properties of the First- and Second-OrderApproximations Let us analyse the generating functionΨ(119911
1 119911
119864 119905) = prod
119895119908
119883119895(0)
119895 The simplest nontrivial
applications of the approximation scheme are the first- andsecond-order approximations Expanding explicitly up tosecond-order we get
119908119896(ℎ) = 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1) +
ℎ2
2(sum
120573
119903119896
120573(119911
120573minus 1))
times (sum
120574
119903119896
120574(119911
120574minus 1))
+ℎ2
2sum
120573
119903119896
120573119911120573sum
120574
119865120573
120574(119911
120574minus 1)
(50)
We begin by noticing that the first-order approximation isobtained by neglecting all terms in ℎ
2 in the above equationWhenever 119908
119895is linear in 119911 which is always the case for the
first-order approximation the resulting generating functionΨ corresponds to the product of (independent) multinomialdistributions for the events affecting each subpopulation
Let us now consider Ψ(1199111 119911
119864 119905) computed with the
second-order approximated 119908119895rsquos to perform one simulation
step of size ℎCounting powers of 119911
120574on each 119908
119896we have (a) the
probability of no events occurring in time-step ℎ
119875119896(ℎ 0) = 1 minus ℎ(sum
120573
119903119896
120573) +
ℎ2
2(sum
120573
119903119896
120573)
2
(51)
(b) the probability of only one (120573) event occurring in time-step ℎ
119875119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus ℎ
2sum
120572
119903119896
120572minus
ℎ2
2sum
120572
119865120573
120572 (52)
and (c) the probability of another event (120572) occurring afterthe first one
119875119896(ℎ 119868119868 120573 120572) =
ℎ2
2119903119896
120573(119903
119896
120572+ 119865
120573
120572) (53)
10 The Scientific World Journal
522 Simulation Algorithm A simulation step can be oper-ated in the following way Let
sum
120572
(119875119896(ℎ 119868119868 120573 120572)) + 119875
119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus
ℎ2
2sum
120572
119903119896
120572
= 119875119896(ℎ 119868 120573)
(54)
be the probability of one (120573) event occurring alone orfollowed by a second event It follows that
119875119896(ℎ 119868119868 120573 120572) = 119875
119896(ℎ 119868 120573)
ℎ
2
(119903120572
119896+ 119865
120573
120572)
(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ
2)
(55)
The second factor above is the conditional probability119875119896(ℎ 120572120573) for the second event being 120572 given that there was
a first (120573) eventThe algorithm proceeds in two steps First a multino-
mial deviate is computed with probabilities 119875119896(ℎ 119868 120573) (the
probability of no event occurring is still 119875119896(ℎ 0)) This gives
the number 119899120573of occurrences of each event In the second
step for each 119899120573we compute the deviates for the occurrence
of a second event or no more events with the multinomialgiven by the array of probabilities 119875
119896(ℎ 120572120573) defined above
(the probability of no second event occurring is here 1 minus
(ℎ2)(sum120572(119903
120572
119896+ 119865
120573
120572)(1 minus (ℎ2)sum
120572119903119896
120572)))
6 Implementation Developmental Cascadeswithout Mortality
In this and the following Sections we apply the tools devel-oped up to now to the description of a subprocess in insectdevelopment namely the development of immature larvaewhich is in principle a highly individual subprocess Weassume that the evolution of an insect from egg to pupaoccurs via a sequence of 119864 gt 0 maturation stages Biologistsrecognise some substages by inspection in the developmentof larvae (instars) and even more stages when observed byother methods [35] The immature individual may die alongthe process or otherwise complete it and exit as pupa oradult The actual value of apparent maturation stages 119864 willdepend on environmental and experimental conditions andwill ultimately be specified when analying experimental datain Section 71
Empirical evidence based on existing and new experi-mental results as well as biological insight produced by thepresent description will be the subject of an independentwork [42]We discuss however a couple of examples from theliterature in Section 71
The events for this process are maturation events pro-moting individuals from one subpopulation into to the nextand death events for all subpopulations In other wordseach subpopulation represents a given (intermediate) level ofdevelopment and maturation events correspond to progressin the development Development at level 119895 promotes oneindividual to level 119895 + 1 and the event rate is linear in the119895-subpopulation Hence in 119877
120572= sum
119895119903119896
120572119883
119895 the matrix 119903
is square and diagonal for 0 le 120572 le 119864 minus 1 (we haveexactly 119864 + 1 subpopulations counted from 0 to 119864) All ofthe environmental influence is at this point incorporatedthrough 119903
Moreover level 119864 is the matured pupa and we may set119903119864
= 0 since at pupation stage the present mechanism endsand new processes take place Subpopulation 119864 is the exitpoint of the dynamics and we assume it to be determined bythe stochastic evolution of the immature stages 0 le 120572 le 119864minus1
In the same spirit the action of each event on thepopulations modifies just the actual population and the nextone in the chain Hence
120575120572
119895=
minus1 120572 = 119895
+1 120572 = 119895 minus 1(56)
Following Section 3 [1198770
120572]119879
= (1199030119872 0 0) and
119865120573
120572=
minus119903120572 120573 = 120572
+119903120572 120573 = 120572 minus 1
0 otherwise(57)
Finally letting m119879= minus119872(1 1 1) we can write 119877
0
120572=
sum120573119865120573
120572119898
120573 In this framework the index pair (119895 120572) is highly
interconnected and we can achieve a complete descriptionusing just one index We use 0 le 119895 le 119864 minus 1 throughout thissection
The procedure of Section 3 gives Ψ(119911 119905) = prod119895(119891
119895)119898119895 =
prod119895(119891
minus119872
119895) while the dynamical system for 119891
119895reads recalling
the recasting in Remark 21119891119895= 119891
119895(minus119903
119895(119911
119895119891119895minus 1) + 119903
119895+1(119911
119895+1119891119895+1
minus 1))
119895 = 0 119864 minus 2
(58)
and 119891119864minus1
= 119891119864minus1
(minus119903119864minus1
(119911119864minus1
119891119864minus1
minus 1)) The overall initialcondition is 119891
119895(0) = 1 Let 120579
119895= log(119891
119895) and
120595119895=
119864minus1
sum
120573=119895
120579120573 119882
119895= exp (minus120595
119895) (59)
Note that 119882119895119891119895
= 119882119895
sdot exp(120579119895) = 119882
119895+1 Rewriting the
equations120579119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) + 119903
119895+1(119911
119895+1exp (120579
119895+1) minus 1)
119895 lt 119864 minus 1
119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) = 120579
119895+
119895+1 119895 lt 119864 minus 1
120579119864minus1
= 119864minus1
= minus119903119864minus1
(119911119864minus1
exp (120579119864minus1
) minus 1)
(60)
with initial condition 120579119895(0) = 0 for all 119895With this notation the
generating function reads Ψ(119911 119905) = 119882119872
0 and we formulate
hence an ODE for the 119882119895rsquos
119895+ 119903
119895119882
119895= 119903
119895119911119895119882
119895+1 119882
119895 (0) = 1 119895 = 0 119864 minus 2
119864minus1
= 119903119864minus1
(119911119864minus1
minus 119882119864minus1
) 119882119864minus1
(0) = 1
(61)
The Scientific World Journal 11
By defining the auxiliary quantity 119882119864(119905) = 1 the solution
for the 119882119895-differential equations can be written in recursive
form
119882119895(119905) = 119890
minus119903119895119905+ 119911
119895119903119895int
119905
0
119890119903119895(119904minus119905)
119882119895+1
(119904) 119889119904 119895 = 0 119864 minus 1
(62)
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820(119905) = (
119864minus1
prod
119898=0
119911119898)(1 minus
119864minus1
sum
119896=0
exp (minus119903119896119905)
119864minus1
prod
119895=0119895 = 119896
119903119895
119903119895minus 119903
119896
)
+
119864
sum
119896=1
prod119864minus119896
119895=0119911119895
119911119864minus119896
119903119864minus119896
119864minus119896
sum
119898=0
119903119898exp (minus119903
119898119905)
119864minus119896
prod
119897=0119897 =119898
119903119897
119903119897minus 119903
119898
(63)
The solution looksmore compact when all 119903119896are assumed
to take the common value 119903
1198820(119905) =
prod119864minus1
119898=0119911119898
(119864 minus 1)int
119903119905
0
exp (minus119904) 119904119864minus1
119889119904
+
119864
sum
119896=1
prod119864minus119896
119898=0119911119898
119911119864minus119896
(119903119905)119864minus119896 exp (minus119903119905)
(119864 minus 119896)
(64)
7 Developmental Cascades with Mortality
We add to the previous description 119864 subpopulation specificdeath events 119876
120572= 119902
120572119883
120572 where 120572 = 0 119864 minus 1 We
do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883
119864leaves the
present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876
0 119877
0 119876
1 119877
1 119876
119864minus1 and 119877
119864minus1 Then ((119876 119877)
0)119879
=
(1199020119872 119903
0119872 0 0) We have
1205752120572
120572= minus1 (death) 120572 lt 119864
1205752120572+1
120572= minus1
1205752120572+1
120572+1= +1
(development) 120572 lt 119864
119865120573
2120572=
minus119902120572 120573 = 2120572
minus119902120572 120573 = 2120572 + 1
119902120572 120573 = 2120572 minus 1
119865120573
2120572+1=
minus119903120572 120573 = 2120572 + 1
minus119903120572 120573 = 2120572
119903120572 120573 = 2120572 minus 1
120572 lt 119864
(65)
Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891
120572and the 119911
120572in pairs (119892 119891) and
(119911 119910) as (1198920 119891
0 119892
119864minus1 119891
119864minus1) and (119911
0 119910
0 119911
119864minus1 119910
119864minus1)
Also for notational convenience we will assume that thereexist 119902
119864= 119903
119864= 0 when necessary Defining
119867120572(119892
120572 119891
120572) = 119902
120572(119911
120572119892120572minus 1) + 119903
120572(119910
120572119891120572minus 1) (66)
we have (recall again Remark 21 and note that 119867119864equiv 0)
119892120572= minus119867
120572(119892
120572 119891
120572) 119892
120572
119891120572= (minus119867
120572(119892
120572 119891
120572) + 119867
120572+1(119892
120572+1 119891
120572+1)) 119891
120572
120572 = 0 119864 minus 2
119892119864minus1
119892119864minus1
=
119891119864minus1
119891119864minus1
= minus119867119864minus1
(119892119864minus1
119891119864minus1
)
(67)
The overall initial condition is still 119892120572(0) = 1 = 119891
120572(0)
Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881
119879
120572=
(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579
120572= log119892
120572
and 120601120572
= log119891120572 changing 119867
120572and the initial conditions
correspondingly Hence
120601120572= 120579
120572minus 120579
120572+1997904rArr 120601
120572= 120579
120572minus 120579
120572+1 120572 = 0 119864 minus 2
120579120572= minus119867
120572(120579
120572 120579
120572minus 120579
120572+1) 120572 = 0 119864 minus 2
120601119864minus1
= 120579119864minus1
= minus119867119864minus1
(120579119864minus1
120579119864minus1
)
(68)
The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879
= minus119872(1 0 0 0) satisfies 1198770
= 119865119898 FinallyΨ(119911 119910 119905) = 119892
minus119872
0 We now compute 119892
0(119905)
Let119882120572= 119890
minus120579120572 = 119892
minus1
120572 0 le 120572 le 119864minus1 Defining the auxiliary
variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of
119882 read
120572+ (119902
120572+ 119903
120572)119882
120572= 119902
120572119911120572+ 119903
120572119910120572119882
120572+1 119882
120572 (0) = 1
(69)
with solution
119882120572(119905) = 119890
minus(119902120572+119903120572)119905
+ int
119905
0
119890minus(119902120572+119903120572)(119905minus119904)
(119902120572119911120572+ 119903
120572119910120572119882
120572+1(119904)) 119889119904
(70)
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
The Scientific World Journal 3
An important feature of the present approach is that thedynamical description will be handled in event-space In thissense the ldquostaterdquo of the system is given by the array n(119904 119905) =
(1198991 119899
119864)119905indicating how may events of each class have
occurred from time 119904 up to time 119905 Hence event-space is inthis setup a nonnegative integer lattice of dimension 119864 Theconnection with the actual population values at time 119905 is (119904denotes the initial condition)
119883119895(119905) = 119883
119895(119904) + sum
120572
120575120572
119895119899120572(119904 119905) (1)
Definition 1 (linear independency) One says that the popula-tions are linearly dependent if there is a vector u = 0 such thatfor all 119905 ge 119904 one has sum
119895119906119895119883
119895(119905) = 0 Otherwise one says that
the populations are linearly independent and abbreviate it asLI
In similar form we say that the populations have linearlydependent increments if there is u = 0 so that for all 1199051015840 gt 119905 ge 119904
it holds that sum119895119906119895(119883
119895(1199051015840) minus 119883
119895(119905)) = 0
The populations are said to have linearly independentincrements (abbreviated LII) if they do not have linearlydependent increments
Along this work we will assume the following
Assumption 2 The populations have linearly independentincrements that is all involved populations are necessary toprovide a complete description of the problem none of themcan be expressed as a linear combination of the remainingones for all times
Lemma 3 (a) If the populations have linearly independentincrements then the populations are linearly independent
(b) The matrix 120575 has no nonzero vectors u such thatsum
119895120575120572
119895119906119895= 0 for all 120572 if and only if the populations have linearly
independent increments
Proof (a) The assertion is equivalent to the following if thepopulations are linearly dependent then they have linearlydependent increments which is immediate after the defini-tion of linearly dependent increments
(b) Assume that for all 120572 there exists u = 0 such thatsum
119895119906119895120575120572
119895= 0 then
sum
119895
119906119895(119883
119895(119905
1015840) minus 119883
119895(119905)) = sum
119895
119906119895(sum
120572
120575120572
119895119899120572(119905
1015840 119905)) = 0 (2)
and the populations have linearly dependent incrementsConversely assume that the populations have linearly
dependent increments and select time intervals such that119899120573(1199051015840
120573 119905
120573) = 1 and 119899
120572(1199051015840
120573 119905
120573) = 0 for 120572 = 120573 then we have that
sum
119895
119906119895120575120572
119895= 0 forall120572 (3)
Statement (b) is the negative statement of these facts
Strictly speaking the last step in the above proof takesfor granted that there exist time intervals where only one
event occurs This is assured by the next assumption (thirditem) Also the initial conditions and the problem descriptionshould be such that all defined events are relevant for thedynamics (eg the event ldquocontagionrdquo is irrelevant for anepidemic problem with zero infectives as initial condition)
21 Dynamical EquationmdashBasic Assumptions Populationnumbers evolve in ldquojumpsrdquo given by the occurrence of eachevent (birth death etc) In the sequel we will use Latinindices 119894 119895 119896 and 119898 for populations and Greek indices 120572120573 and 120574 for events
The ultimate goal of stochastic population dynamicsis to calculate the probabilities 119875(119899
1 119899
119864 119904 119905) of having
1198991 119899
119864events in each class up to time 119905 given an initial
condition 119883119895(119904)
The stochastic dynamics we are interested in correspondto a Density Dependent Markov Jump Process [37 38] For asufficiently small time interval ℎ we consider the following
Assumption 4 For each event
(i) event occurrences in disjoint time intervals are inde-pendent
(ii) the Chapman-Kolmogorov equation [28 29] holds
(iii) 119875(119899120572+ 1 119905 + ℎ 119905) = 119882
120572(119883 119905)ℎ + 119900(ℎ)
(iv) 119875(119899120572 119905 + ℎ 119905) = 1 minus 119882
120572(119883 119905)ℎ + 119900(ℎ)
(v) 119875(119899120572(ℎ) + 119896 119905 + ℎ 119905) = 119900(ℎ) 119896 ge 2
Here ℎ is the elapsed time since the time 119905 and 119899120572are the
events of type 120572 accumulated up to 119905
It goes without saying that probabilities 119875(sdot sdot sdot ) areassumed to be differentiable functions of time 119882
120572(119883 119905)
denotes the probability rate for event 120572 which is assumedin the following lemmas to be a smooth function of thepopulations not necessarily linear (yet) Two well-knowngeneral results describe the dynamical evolution under theseassumptions
Lemma 5 (see Theorem 1 in [29]) Under Assumptions 2and 4 the waiting time to the next event is exponentiallydistributed
Theorem6 (see [28] andTheorem 1 in [29]) Under Assump-tions 2 and 4 the dynamics of the process in the space of eventsobeys the Kolmogorov Forward Equation namely
(1198991 119899
119864 119904 119905)
= sum
120572
119875 ( 119899120572minus 1 119904 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119904 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(4)
Proofs of these results are given in Appendix A
4 The Scientific World Journal
22 Linear Rates We will assume that the transition ratesdependmdashin a linear fashionmdashon the populations only Todistinguish from general rates (denoted above by the letter119882) we will use the letter 119877 to denote linear transition ratesHence
119877120572= sum
119896
119903119896
120572119883
119896(119905) (5)
Since rates are nonnegative we have that the proportionalitycoefficient between event 120572 and subpopulation 119896 satisfies119903119896
120572ge 0 These proportionality coefficients convey the
environmental information and as such they may be time-dependent since the environment varieswith timeWheneverpossible we do not write this time dependency explicitly tolighten the notation
Lemma 7 For the case of linear rates population space 119883119896ge
0 is positively invariant under the time evolution given by (1)and the previous assumptions if and only if 120575120572
119894ge minus1 and 120575
120572
119894=
minus1 only if 119877120572= 119903
119894
120572119883
119894(119905) (no sum)
Proof Theprobability of occurrence of event120572 in a small timeinterval ℎ is
ℎ119877120572+ 119900 (ℎ) = ℎsum
119896
119903119896
120572119883
119896(119905) + 119900 (ℎ) (6)
After an occurrence population 119883119896is modified to 119883
119896+ 120575
120572
119896
Population space is positively invariant under time evolutionif and only if the probability of occurrence of event 120572 is zerofor those population values119883
119896such that119883
119896+ 120575
120572
119896lt 0 (we are
just saying that events pushing the populations to negativevalues do not exist) Since this should hold for arbitrary ℎ wehave that 119877
120572= 0
Assume that 120575120572119896
le minus1 Then 119877120572
= 0 for 0 le 119883119896
lt minus120575120572
119896
Letting 119883119896
= 0 we realise that sum119895 = 119896
119903119895
120572119883
119895(119905) = 0 regardless
of the values of 119883119895 and hence it holds that 119903119895
120572= 0 for 119895 = 119896
and therefore 119877120572
= 119903119896
120572119883
119896(no sum) This proves the second
statement If 120575120572119896
lt minus1 we have that 119877120572
= 0 also for 119883119896
= 1Hence 119903
119896
120572= 0 as well 119877
120572equiv 0 and the event 120572 can be ignored
since it does not influence the dynamics
3 Kolmogorov Forward Equation
Let us recast the dynamical equation given by Theorem 6 interms of the generating function [39] in event-space definedas
Ψ (1199111 119911
119864 119904 119905) = sum
119899120572
(prod
120572
119911119899120572
120572)119875 (119899
1 119899
119864 119904 119905) (7)
For the case of linear rates we define
1198770
120572= sum
119896
119903119896
120572119883
119896 (119904)
119865120573
120572= sum
119896
119903119896
120572120575120573
119896
(8)
As a consequence
119877120572= 119877
0
120572+ sum
120573
119865120573
120572119899120573 (9)
Using Theorem 6 to rewrite 120597Ψ120597119905 and using further 119899120573Ψ =
119911120573(120597Ψ120597119911
120573) (a matter of replacing in the definition of Ψ) we
obtain
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
= sum
120572
(119911120572minus 1)(119877
0
120572Ψ + sum
120573
119865120573
120572119911120573120597Ψ120597119911
120573) (10)
Since the generating function reflects a probability distri-bution we have the probability condition Ψ(1 1 119904 119905) =
1 Further the solutions of (10) can be uniquely trans-lated into those given by Theorem 6 for correspondinginitial conditions 119875(119898
1 119898
119864 119904 119904) = 1 corresponding
to Ψ(1199111 119911
119864 119904 119904) = 119911
1198981
11199111198982
2sdot sdot sdot 119911
119898119864
119864 where the integers
1198981 119898
119864denote howmany events of each class had already
occurred at 119905 = 119904 The natural initial condition becomesΨ(119911
1 119911
119864 119904 119904) = 1 corresponding to 119875(0 0 119904 119904) = 1
31 The Method of Characteristics Revisited
Lemma 8 Equation (10) with 1198770
120572= 0 and with the initial
condition Ψ(1199111 119911
119864 119904 119904) = Φ
0(119911
1 119911
119864) admits the
solution
Ψ (1199111 119911
119864 119904 119905) = Φ
0(120601 (119911 119905 119905 minus 119904)) (11)
where 120601(119911 119905 120591) is the flow of the following ODE with initialcondition 119909(0) = 119911 119904(0) = 119905
119889119909120573
119889120591= sum
120572
119865120573
120572(119909
120572minus 1) 119909
120573equiv 119887
120573(119909 119904)
119889119904
119889120591= minus1
(12)
Proof It is convenient to rename 119904 as 119904 = 119909119864+1
introduce119887119864+1
= minus1 and recall that 119887120573 depends only on the extendedvariable array 119909 However when necessary we will writeexplicitly 119909(0) as (119911 119905) We have that
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
119889120601120573(119911 119905 119905 minus 119904)
119889119905
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
times (119887120573(120601 (119911 119905 119905 minus 119904)) +
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
(13)
Further consider the monodromy matrix [40]
119872120573
120574=
120597120601120573(119909 120591)
120597119909120574
(14)
The Scientific World Journal 5
119872 transforms the initial velocity into the final velocity hence
sum
120574
119872120573
120574119887120574(119911 119905) = 119887
120573(120601 (119911 119905 120591)) (15)
Thus we get
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
(
119864+1
sum
120574
119872120573
120574119887120574(119911 119905) +
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
times (
119864
sum
120574
119872120573
120574119887120574(119911 119905) minus 119872
120573
119864+1+
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
=
119864
sum
120574
120597Ψ
120597119911120574
119887120574(119911 119905)
(16)
From the definition of monodromy matrix above it followsthat
minus119872120573
119864+1+
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
= 0 (17)
and by (11) and the chain rule
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)
119872120573
120574= sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)
120597120601120573(119911 119905 119905 minus 119904)
120597119911120574
=120597Ψ
120597119911120574
(18)
Lemma 9 Equation (10) with 1198770
120572= 0 and with the initial
condition Ψ(1199111 119911
119864 119904 119904) = Φ
0(119911
1 119911
119864) admits the
solution
Ψ (1199111 119911
119864 119904 119905) = Φ
0(120601 (119911 119905 119905 minus 119904))Φ
1(119911
1 119911
119864 119904 119905)
(19)
where Φ0(120601(119911 119905 119905 minus 119904)) is a solution of the homogeneous
problem with 1198770
120572= 0 and
Φ1(119911
1 119911
119864 119904 119905)
= expint
119905
119904
(sum
120572
1198770
120572(120591) (120601
120572(119911 119905 119905 minus 120591) minus 1) 119889120591)
(20)
Proof The proof is nothing more than an application of themethod of the variation of the constants The details are asfollows
Introducing the proposed solution Ψ into (10) and usingLemma 8 to eliminateΦ
0we obtain thatΦ
1satisfies (10) with
initial condition Φ1(119911 119904 119904) = 1 Computing directly from
(20) we have that
120597Φ1
120597119905
10038161003816100381610038161003816100381610038161003816119904
= sum
120572
1198770
120572(120601
120572(119911 119905 0) minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573
120597120601120573(119911 119905 119905 minus 120591)
120597119905
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1
+ sum
120573
120597Φ1
120597120601120573(119887
120573(120601 (119911 119905 119905 minus 120591)) +
120597120601120573(119911 119906 119905 minus 120591)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573sum
120574
119872120573
120574119887120574(119911 119905)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573sum
120574
120597120601120573
120597119911120574
119887120574(119911 119905)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120574
120597Φ1
120597119911120574
119887120574(119911 119905)
(21)
after retracing our steps from the previous Lemma
The result presented in Lemma 9 was constructed usingthe method of variation of constants which provides analternative (constructive) proof
32 Basic Properties of the Solution
Definition 10 A nonzero vector v such that sum120572120575120572
119895V120572
= 0for 119895 = 1 119873 is called a structural zero of 119865 Other zeroeigenvalues of 119865 are called nonstructural
Lemma 11 (a)There are structural zeroes if and only if119864 gt 119873(b) The only structurally stable zero eigenvalues of the matrix119865 are the structural zeroes
Proof (a) The ldquoif rdquo part is immediate since dim(ker(120575)) ge
119864minus119873The ldquoonly if rdquo part follows from the assumption of inde-pendent population increments (and therefore independentpopulations) For (b) put 119865 = r120575 If 119865119908 = 0 for 119908 nonzeroand 120575119908 = 119901 = 0 then r119901 = 0 This relation is not structurallystable since by adding 119888 = 0 arbitrarily small to all diagonalelements of r we can assure r1015840119901 = 0 for this modified r1015840 andfor any 119901 = 0
Lemma 12 First integrals of motion for the characteristicequations Let v be a structural zero of 119865 then
prod
120572
(119909120572(119905))
V120572
= 119862 (22)
is a constant of motion for the characteristic equations (12)
6 The Scientific World Journal
Proof Rewrite the first equation (12) as (indices renamed)
119889
119889119905ln119909
120572= sum
120573
119865120572
120573(119909
120573minus 1) (23)
Then
sum
120572
V120572
119889
119889119905ln119909
120572= sum
120573
sum
120572
(V120572119865120572
120573) (119909
120573minus 1) = 0 (24)
Since sum120572(V
120572119865120572
120573) = 0 the expression in the lemma is the
exponential of the integral obtained from
sum
120572
V120572
119889
119889119905ln119909
120572= 0 (25)
Given relationship (1) between initial conditions eventsand populations from the probability generating function inevent space
Ψ (1199111 119911
119864 119904 119905) = sum
119899120572
(prod
120572
119911119899120572
120572)119875 (119899
1sdot sdot sdot 119899
119864 119904 119905) (26)
a corresponding generating function in population space canbe computed namely
Ψ (1199101 119910
119873 119904 119905) = sum
119883119896
(prod
119896
119910119883119896
119896)119875 (119883
1 119883
119873 119904 119905)
(27)
letting 119899120572 119883
119896 be shorthand for (119899
1sdot sdot sdot 119899
119864) (119883
1 119883
119873)
Lemma 13 (projection lemma) Equation 119911120572
= prod119873
119896=1119910120575120572
119896
119896
realises the transformation from event space to populationspace Moreover
Ψ (1199101 119910
119873 119904 119905) = (prod
119896
1199101198830
119896
119896)Ψ(119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896)Ψ(
119873
prod
119896=1
1199101205751
119896
119896
119873
prod
119896=1
119910120575119864
119896
119896 119904 119905)
(28)
Proof Taking logarithms on 119911 and 119910 it is clear that wheneverthere exist structural zeroes the transformation is noninvert-ible In this sense we call it a projection Concerning thestatement we have
(prod
119896
1199101198830
119896
119896)Ψ (119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896) sum
119899120572
(prod
120572
(
119873
prod
119896=1
119910120575120572
119896
119896)
119899120572
)119875 (119899120572 119904 119905)
= sum
119899120572
(
119873
prod
119896=1
1199101198830
119896+sum120572120575120572
119896119899120572
119896)119875 (119899
120572 119904 119905)
(29)
The statement is proved using (1) and rearranging the sumsas
119875 (1198831 119883
119873 119904 119905) = sum
119899120572
1015840
119875 (119899120572 119904 119905) (30)
where the prime denotes sum over 119899120572 such that 119883
0
119895+
sum120572120575120572
119895119899120572= 119883
119895
Corollary 14 The integrals of motion of Lemma 12 projectunder the projection Lemma to 119862 = 1
Proof The above substitution operated on prod119911V120572
120572= 119862 gives
119862 =
119873
prod
119896=1
119910(sum120572V120572120575120572
119896)
119896equiv
119873
prod
119896=1
1199100
119896= 1 (31)
33 Reduced Equation
Assumption 15 There exist integers119898120573120573 = 1 119864 such that
(cf (9))
1198770
120572= sum
120573
119865120573
120572119898
120573 (32)
This assumption amounts to considering that there exists away to mimic the initial condition on the rates with the samematrix 119865 involved in the time evolution
Theorem 16 Under Assumptions 2 4 and 15 equation(10) with natural initial condition Ψ(119911
1 119911
119864 119904 119904 ) =
Φ0(119911
1 119911
119864) = 1 has the solution
Ψ (1199111 119911
119864 119904 119905) = prod
120573
119891119898120573
120573 (33)
where 119911120573119891120573(119911 119904 119905) = 120601
120573(119911 119905 119905 minus 119904) in terms of the solutions
of (12) of Lemma 8 Hence 119891(119911 119904 119905) satisfies (after eliminatingthe auxiliary time 120591)
119889 log (119891120573 (119911 119904 119905))
119889119904
= minussum
120572
119865120573
120572(119904) (119911
120572119891120572(119911 119904 119905) minus 1) 119891
120573(119911 119905 119905) = 1
(34)
Proof The differences between 119891( ) and 120601( ) are (a) factoringout the initial condition 120601
120573(119911 119905 0) = 119911
120573and (b) rearranging
the time arguments since the argument in120601( ) is natural to thead hoc autonomous ODE (12) while the argument in 119891( ) isnatural to the (nonautonomous) PDE (10)
The proof is just a matter of rewriting the previousequations under the present assumptions Because of thenatural initial condition the desired solution to (10) isgiven by (20) in Lemma 9 which can be rewritten usingAssumption 15 as (we use 119911 as shorthand for 119911
1 119911
119864)
Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)
= prod
120573
(expint
119905
119904
sum
120572
119865120573
120572(120591) (120601
120572(119911 119905 119905 minus 120591) minus 1) 119889120591)
119898120573
(35)
The Scientific World Journal 7
Recalling that we defined 119911120572119891120572(119911 119904 119905) = 120601(119911 119905 119905 minus 119904) the
parenthesis can be recognised as the formal solution of (34)integrated from 119905 to 119904
Corollary 17 For v a structural zero of Lemma 12 thegenerating function is unmodified upon the substitution119898
120572rarr
119898120572+ 119896V
120572(119896 isin R)
Remark 18 Having structural zeroes it is amatter of choice toaddress the problem (a) using the variables 119891
120572and imposing
the additional constraints given by structural zeroes or (b)shifting to population coordinates where these constraints areautomatically incorporated Because of Assumption 2 andLemma 12 the transformation mapping one description tothe other corresponds to
119891120572= prod
119895
119908120575120572
119895
119895 (36)
However option (b) proves to bemore efficientwhen it comesto deal with approximate solutions (see Section 5)
4 Examples of Exact Solutions
41 Example I Pure Death Process In the case 119873 = 119864 = 1120575 = minus1 we have 119903 gt 0 119865 = minus119903(119904) and 119898 = minus119872 lt
0 (119872 is the initial population value) Equation (34) reads119889119891119889119904 = 119891119903(119911119891 minus 1) to be integrated in [119905 119904] with 119891(119911 119905 119905) =
1 Hence for constant death rate 119903 we obtain 119891(119911 119904 119905) =
[119911 + (1 minus 119911)119890minus119903(119905minus119904)
]minus1
and
Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)
+ 119911 (1 minus 119890minus119903(119905minus119904)
)]119872
(37)
42 Example II Linear Birth andDeath Process Wehave119873 =
1 119864 = 2 and 1205751
1= minus120575
2
1= 1 while 119903
1denotes the birth rate
and 1199032denotes the death rate 119883
0= 119898
1minus 119898
2is the initial
population There exists one structural zero with vector v =
(1 1)119879 while the corresponding constant of motion results in
the relations 11991111199112= 119862 and119891
11198912= 1The differential equation
(34) for 1198911(to be integrated in [119905 119904]) becomes
1198891198911
119889119904= (119903
111991111198912
1minus (119903
1+ 119903
2) 119891
1+
1198621199032
1199111
) 1198911 (119911 119905 119905) = 1
(38)
with solution (for119862 = 1whichwill be the case after projectionand assuming time-independent birth rates)
1198911(119911 119904 119905) =
1
1199111
(1199032minus 119903
11199111)119882 minus 119903
2(1 minus 119911
1)
(1199032minus 119903
11199111)119882 minus 119903
1(1 minus 119911
1) (39)
where 119882 = exp(1199032
minus 1199031)(119905 minus 119904) Finally Ψ(119911 119904 119905) =
(1198911(119911 119904 119905))
1198981minus1198982 which after projecting in population space
using Lemma 13 and Corollary 14 gives (cf [41])
Ψ (119910 119904 119905) = [(119903
2minus 119903
1119910)119882 minus 119903
2(1 minus 119910)
(1199032minus 119903
1119910)119882 minus 119903
1(1 minus 119910)
]
1198981minus1198982
(40)
Remark 19 A pure death linear process corresponds to abinomial distribution a pure birth process corresponds to anegative binomial and a birth-death process corresponds toa combination of both as independent processes
5 Consistent Approximations
When (34) of Theorem 16 cannot be solved exactly we haveto rely on approximate solutions It is desirable to workwith consistent approximations namely those where basicproperties of the problem are fulfilled exactly rather than ldquoupto an error of size 120598rdquo For example since our solutions shouldbe probabilities we want the coefficients of Ψ(119911
1 119911
119864 119905)
to be always nonnegative and sum up to one Ideally wewant approximations to be constrained to satisfy Lemma 12and to preserve positive invariance in population space (nojumps into negative populations) Whenever consistency isnot automatic it is mandatory to analyse the approximatesolution before jumping into conclusions
51 Poisson Approximation Let us replace 119891120573
equiv 1 in theRHS of (34) (this amounts to propose that 119891
120573is constant and
equal to its initial condition) The solution of the modifiedequation will be a good approximation to the exact solutionfor sufficiently short times such that 119891 is not significantlydifferent from one
For constant rates 119903120572the rhs of (34) does not depend on
time The approximate solution reads 119891120572(119911 119904 119905) = exp((119905 minus
119904)sum120573119865120572
120573(119911
120573minus 1)) and
Ψ (1199111 119911
119864 119904 119905) =
119864
prod
120572
119890119898120572(119905minus119904)sum
120573119865120572
120573(119911120573minus1)
= 119890(119905minus119904)sum
1205731198770
120573(119911120573minus1)
(41)
In other words the generating function is approximatedas a product of individual Poisson generating functions
This approximation has an error of 119900(119905) Simple as it looksthis approach gives a good probability generating functionthat respects structural zeroes but it does not guaranteepositive invariance a Poisson jump could throw the systeminto negative populations However the probability of sucha jump is also 119900(119905) Fortunately there is a natural way toimpose positive invariance and also a natural way to improvethe error up to 119900(119905)
32 in the broader framework of generaltransition rates [6] In this way the approximation can besafely used with error control to address problems that resistexact computation one way or the other
52 Consistent Multinomial Approximations A systematicapproach to obtain a chain of approximations of increasingaccuracy to (34) can be devised via Picard iterations Firstly itis convenient to recast the problem in population coordinates(cf Remark 18) so that the structural zeroes are automaticallytaken into account regardless of other properties of the
8 The Scientific World Journal
adopted approximation From 119891120572
= prod119895119908
120575120572
119895
119895we obtain
119891120572119891
120572= sum
119896120575120572
119896119896119908
119896and (34) can be recast as
sum
119896
120575120572
119896(
119896
119908119896
+ sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1)) = 0
119908119896 (119911 119905 119905) = 1
(42)
Because of the linear independence of population increments(and hence of populations) Lemma 3(b) the counterpart of(34) in population coordinates reads
119896= minus119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896(119911 119905 119905) = 1
(43)
Remark 20 In the case of linearly dependent populationincrements (34) still holds while it is possible to elaboratea more general form in terms of the 119908
119895instead of (43) This
is however outside the scope of this paper
Since sum120572120575120572
119895119898
120572= 119883
119895(119904) we obtain Ψ(119911
1 119911
119864 119904 119905) =
prod119895119908
119883119895(119904)
119895 which is so far an exact result Let us produce
some approximated values for the 119908119896rsquos and correspondingly
for Ψ(1199111 119911
119864 119904 119905)
To avoid further notational complications wewill assumein the sequel that 119903
119896
120573is time independent Hence 119908
119896will
depend only on the time difference 119905 minus 119904 Equation (43) canbe recast as
119896= 119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896 (119911 0) = 1
(44)
now to be integrated along the interval [0 119905 minus 119904] We will evenset 119904 = 0 in what follows
Remark 21 For the case of time-independent proportionalityfactors 119903119896
120573 a correspondingmodification can be introduced in
(34) namely overall change of sign integration in [0 119905 minus 119904]and setting 119904 = 0 to reduce the notational burden
Equations (44) satisfy the Picard-Lindelof theorem sincethe RHS is a Lipschitz function Then the Picard iterationscheme converges to the (unique) solution Taking advantage
of the initial condition the 119899th order truncated version of thePicard iterations reads
119908(0)
119896(ℎ) = 1
119908(1)
119896(ℎ) = 1 + int
ℎ
0
119908(0)
119896(119905)sum
120573
119903119896
120573(119911
120573prod
119895
(119908(0)
119895(119905))
120575120573
119895
minus 1)119889119905
= 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1)
119908(119899)
119896(ℎ) = 1 + int
ℎ
0
119889119905[
[
119908(119899minus1)
119896(119905)
timessum
120573
119903119896
120573(119911
120573prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
minus 1)]
]119899minus1
(45)
where [sdot sdot sdot ]119898denote the Maclaurin polynomial of order119898 in
119905
Lemma 22 For 119905 isin [0 ℎ] ℎ gt 0 being sufficiently smalland for each order of approximation 119899 ge 0 119908(119899)
119896(119905) in (44)
is a polynomial generating function for one individual of thepopulation that is it has the following properties
(a) 119908(119899)
119896(119905) is a polynomial of degree 119899 in 119911
1 119911
119864 where
the coefficient of 11991111989911 119911
119899119864
119864is 119874(119905
119901) with 119901 = 119899
1+
sdot sdot sdot + 119899119864
(b) 119908(119899)
119896(1 1 119905) = 1
(c) Δ(119899)
119896(ℎ) = 119908
(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ) = 119874(ℎ
119899)
(d) the coefficients of 119908(119899)
119896(119905) regarded as a polynomial in
119911120572 are nonnegative functions of time
Proof (a) The statement holds for 119899 = 0 and 119899 = 1Assuming that it holds up to order 119899 minus 1 we have that[119908
(119899minus1)
119896(119905)prod
119895(119908
(119899minus1)
119895(119905))
120575120573
119895 ]119899minus1
is also a polynomial of degree119899 minus 1 in 119911
1 119911
119864with time-dependent coefficients of type
119874(119905119901) since each power 119901 of 119905 carries along a power 119901 in 119911
120572
(as well as lower 119911-powers) A Picard iteration step gives
119908(119899)
119896(119905) = 1 + sum
120573
119903119896
120573119911120573int
ℎ
0
[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
119908(119899minus1)
119896(119905) 119889119905
(46)
The first sum contributes with one time integration and one119911-power which exactly preserves the polynomial structureand the order of the coefficientsThe result is a polynomial ofdegree 119899 in 119911
1 119911
119864 where the coefficients are polynomials
of highest degree at most 119899 in ℎ each one of lowest order119874(ℎ
119901) (as in the statement) The second integral does not
The Scientific World Journal 9
alter these properties since it generates again a polynomial ofdegree 119899minus1 in 119911
1 119911
119864 with time-dependent coefficients of
order119874(ℎ119901+1
) or higher up to ℎ119899 (they are just the coefficients
of 119908(119899minus1)
119896(119905) integrated once in time)
(b) Letting 1199111= 119911
2= sdot sdot sdot = 119911
119864= 1 Picard iteration gives
trivially 119908(119899)
119896(119905) = 1 to all orders again by induction
(c) The statement holds for 119899 = 1 Assuming it holds upto 119899 minus 1 we compute
119908(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ)
= sum
120573
119903119896
120573119911120573
times int
ℎ
0
([(119908(119899minus2)
119896(119905) + Δ
(119899minus1)
119896(119905))
times prod
119895
(119908(119899minus2)
119895(119905) + Δ
(119899minus1)
119895(119905))
120575120573
119895
]
119899minus1
minus[
[
119908(119899minus2)
119896(119905)prod
119895
(119908(119899minus2)
119895(119905))
120575120573
119895]
]119899minus2
)119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
(119908(119899minus1)
119896(119905) minus 119908
(119899minus2)
119896(119905)) 119889119905
(47)
Both expressions in the first integral have the sameMaclaurinpolynomial up to order (119899 minus 2) Hence the difference in thefirst integral contains only terms of order119874(119905
119899minus1) This is also
the case for the second integral After integration we obtainΔ(119899)
119896(ℎ) = 119874(ℎ
119899)
(d) The statement holds for 119899 = 0 and 119899 = 1 and ingeneral for the zero-order coefficient because it is unity for119905 = 0 as a consequence of the initial condition For thehigher-order coefficients we use induction again If all 119911-coefficients in 119908
(119899minus1)
119896(119905) are nonnegative then the same holds
automatically for 119908(119899)
119896(ℎ) up to order 119899 minus 1 since the 119874(ℎ
119899)
contribution adding to each inherited coefficient from theprevious step does not alter its sign for ℎ sufficiently small Itremains to show that the coefficients corresponding to 119901 = 119899
are nonnegative These contributions arise from the (119899 minus 1)thorder of
119903119896
120573119911120573[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
(48)
after time integration If all 120575120573119895are positive then the expres-
sion above is a product of 119911-polynomials with nonnegative
coefficients and the statement follows Suppose that some120575120573
119895= minus1 Then by Lemma 7 119903119896
120573= 0 for 119896 = 119895 Hence
119903119896
120573119911120573119908
(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
=
0 119895 = 119896
119903119896
120573119911120573prod
119895 = 119896
(119908(119899minus1)
119895(119905))
120575120573
119895
119895 = 119896
(49)
However 120575120573119895
= minus 1 for 119895 = 119896 that is they are nonnegative andhence the expression has only nonnegative coefficients
521 Basic Properties of the First- and Second-OrderApproximations Let us analyse the generating functionΨ(119911
1 119911
119864 119905) = prod
119895119908
119883119895(0)
119895 The simplest nontrivial
applications of the approximation scheme are the first- andsecond-order approximations Expanding explicitly up tosecond-order we get
119908119896(ℎ) = 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1) +
ℎ2
2(sum
120573
119903119896
120573(119911
120573minus 1))
times (sum
120574
119903119896
120574(119911
120574minus 1))
+ℎ2
2sum
120573
119903119896
120573119911120573sum
120574
119865120573
120574(119911
120574minus 1)
(50)
We begin by noticing that the first-order approximation isobtained by neglecting all terms in ℎ
2 in the above equationWhenever 119908
119895is linear in 119911 which is always the case for the
first-order approximation the resulting generating functionΨ corresponds to the product of (independent) multinomialdistributions for the events affecting each subpopulation
Let us now consider Ψ(1199111 119911
119864 119905) computed with the
second-order approximated 119908119895rsquos to perform one simulation
step of size ℎCounting powers of 119911
120574on each 119908
119896we have (a) the
probability of no events occurring in time-step ℎ
119875119896(ℎ 0) = 1 minus ℎ(sum
120573
119903119896
120573) +
ℎ2
2(sum
120573
119903119896
120573)
2
(51)
(b) the probability of only one (120573) event occurring in time-step ℎ
119875119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus ℎ
2sum
120572
119903119896
120572minus
ℎ2
2sum
120572
119865120573
120572 (52)
and (c) the probability of another event (120572) occurring afterthe first one
119875119896(ℎ 119868119868 120573 120572) =
ℎ2
2119903119896
120573(119903
119896
120572+ 119865
120573
120572) (53)
10 The Scientific World Journal
522 Simulation Algorithm A simulation step can be oper-ated in the following way Let
sum
120572
(119875119896(ℎ 119868119868 120573 120572)) + 119875
119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus
ℎ2
2sum
120572
119903119896
120572
= 119875119896(ℎ 119868 120573)
(54)
be the probability of one (120573) event occurring alone orfollowed by a second event It follows that
119875119896(ℎ 119868119868 120573 120572) = 119875
119896(ℎ 119868 120573)
ℎ
2
(119903120572
119896+ 119865
120573
120572)
(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ
2)
(55)
The second factor above is the conditional probability119875119896(ℎ 120572120573) for the second event being 120572 given that there was
a first (120573) eventThe algorithm proceeds in two steps First a multino-
mial deviate is computed with probabilities 119875119896(ℎ 119868 120573) (the
probability of no event occurring is still 119875119896(ℎ 0)) This gives
the number 119899120573of occurrences of each event In the second
step for each 119899120573we compute the deviates for the occurrence
of a second event or no more events with the multinomialgiven by the array of probabilities 119875
119896(ℎ 120572120573) defined above
(the probability of no second event occurring is here 1 minus
(ℎ2)(sum120572(119903
120572
119896+ 119865
120573
120572)(1 minus (ℎ2)sum
120572119903119896
120572)))
6 Implementation Developmental Cascadeswithout Mortality
In this and the following Sections we apply the tools devel-oped up to now to the description of a subprocess in insectdevelopment namely the development of immature larvaewhich is in principle a highly individual subprocess Weassume that the evolution of an insect from egg to pupaoccurs via a sequence of 119864 gt 0 maturation stages Biologistsrecognise some substages by inspection in the developmentof larvae (instars) and even more stages when observed byother methods [35] The immature individual may die alongthe process or otherwise complete it and exit as pupa oradult The actual value of apparent maturation stages 119864 willdepend on environmental and experimental conditions andwill ultimately be specified when analying experimental datain Section 71
Empirical evidence based on existing and new experi-mental results as well as biological insight produced by thepresent description will be the subject of an independentwork [42]We discuss however a couple of examples from theliterature in Section 71
The events for this process are maturation events pro-moting individuals from one subpopulation into to the nextand death events for all subpopulations In other wordseach subpopulation represents a given (intermediate) level ofdevelopment and maturation events correspond to progressin the development Development at level 119895 promotes oneindividual to level 119895 + 1 and the event rate is linear in the119895-subpopulation Hence in 119877
120572= sum
119895119903119896
120572119883
119895 the matrix 119903
is square and diagonal for 0 le 120572 le 119864 minus 1 (we haveexactly 119864 + 1 subpopulations counted from 0 to 119864) All ofthe environmental influence is at this point incorporatedthrough 119903
Moreover level 119864 is the matured pupa and we may set119903119864
= 0 since at pupation stage the present mechanism endsand new processes take place Subpopulation 119864 is the exitpoint of the dynamics and we assume it to be determined bythe stochastic evolution of the immature stages 0 le 120572 le 119864minus1
In the same spirit the action of each event on thepopulations modifies just the actual population and the nextone in the chain Hence
120575120572
119895=
minus1 120572 = 119895
+1 120572 = 119895 minus 1(56)
Following Section 3 [1198770
120572]119879
= (1199030119872 0 0) and
119865120573
120572=
minus119903120572 120573 = 120572
+119903120572 120573 = 120572 minus 1
0 otherwise(57)
Finally letting m119879= minus119872(1 1 1) we can write 119877
0
120572=
sum120573119865120573
120572119898
120573 In this framework the index pair (119895 120572) is highly
interconnected and we can achieve a complete descriptionusing just one index We use 0 le 119895 le 119864 minus 1 throughout thissection
The procedure of Section 3 gives Ψ(119911 119905) = prod119895(119891
119895)119898119895 =
prod119895(119891
minus119872
119895) while the dynamical system for 119891
119895reads recalling
the recasting in Remark 21119891119895= 119891
119895(minus119903
119895(119911
119895119891119895minus 1) + 119903
119895+1(119911
119895+1119891119895+1
minus 1))
119895 = 0 119864 minus 2
(58)
and 119891119864minus1
= 119891119864minus1
(minus119903119864minus1
(119911119864minus1
119891119864minus1
minus 1)) The overall initialcondition is 119891
119895(0) = 1 Let 120579
119895= log(119891
119895) and
120595119895=
119864minus1
sum
120573=119895
120579120573 119882
119895= exp (minus120595
119895) (59)
Note that 119882119895119891119895
= 119882119895
sdot exp(120579119895) = 119882
119895+1 Rewriting the
equations120579119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) + 119903
119895+1(119911
119895+1exp (120579
119895+1) minus 1)
119895 lt 119864 minus 1
119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) = 120579
119895+
119895+1 119895 lt 119864 minus 1
120579119864minus1
= 119864minus1
= minus119903119864minus1
(119911119864minus1
exp (120579119864minus1
) minus 1)
(60)
with initial condition 120579119895(0) = 0 for all 119895With this notation the
generating function reads Ψ(119911 119905) = 119882119872
0 and we formulate
hence an ODE for the 119882119895rsquos
119895+ 119903
119895119882
119895= 119903
119895119911119895119882
119895+1 119882
119895 (0) = 1 119895 = 0 119864 minus 2
119864minus1
= 119903119864minus1
(119911119864minus1
minus 119882119864minus1
) 119882119864minus1
(0) = 1
(61)
The Scientific World Journal 11
By defining the auxiliary quantity 119882119864(119905) = 1 the solution
for the 119882119895-differential equations can be written in recursive
form
119882119895(119905) = 119890
minus119903119895119905+ 119911
119895119903119895int
119905
0
119890119903119895(119904minus119905)
119882119895+1
(119904) 119889119904 119895 = 0 119864 minus 1
(62)
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820(119905) = (
119864minus1
prod
119898=0
119911119898)(1 minus
119864minus1
sum
119896=0
exp (minus119903119896119905)
119864minus1
prod
119895=0119895 = 119896
119903119895
119903119895minus 119903
119896
)
+
119864
sum
119896=1
prod119864minus119896
119895=0119911119895
119911119864minus119896
119903119864minus119896
119864minus119896
sum
119898=0
119903119898exp (minus119903
119898119905)
119864minus119896
prod
119897=0119897 =119898
119903119897
119903119897minus 119903
119898
(63)
The solution looksmore compact when all 119903119896are assumed
to take the common value 119903
1198820(119905) =
prod119864minus1
119898=0119911119898
(119864 minus 1)int
119903119905
0
exp (minus119904) 119904119864minus1
119889119904
+
119864
sum
119896=1
prod119864minus119896
119898=0119911119898
119911119864minus119896
(119903119905)119864minus119896 exp (minus119903119905)
(119864 minus 119896)
(64)
7 Developmental Cascades with Mortality
We add to the previous description 119864 subpopulation specificdeath events 119876
120572= 119902
120572119883
120572 where 120572 = 0 119864 minus 1 We
do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883
119864leaves the
present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876
0 119877
0 119876
1 119877
1 119876
119864minus1 and 119877
119864minus1 Then ((119876 119877)
0)119879
=
(1199020119872 119903
0119872 0 0) We have
1205752120572
120572= minus1 (death) 120572 lt 119864
1205752120572+1
120572= minus1
1205752120572+1
120572+1= +1
(development) 120572 lt 119864
119865120573
2120572=
minus119902120572 120573 = 2120572
minus119902120572 120573 = 2120572 + 1
119902120572 120573 = 2120572 minus 1
119865120573
2120572+1=
minus119903120572 120573 = 2120572 + 1
minus119903120572 120573 = 2120572
119903120572 120573 = 2120572 minus 1
120572 lt 119864
(65)
Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891
120572and the 119911
120572in pairs (119892 119891) and
(119911 119910) as (1198920 119891
0 119892
119864minus1 119891
119864minus1) and (119911
0 119910
0 119911
119864minus1 119910
119864minus1)
Also for notational convenience we will assume that thereexist 119902
119864= 119903
119864= 0 when necessary Defining
119867120572(119892
120572 119891
120572) = 119902
120572(119911
120572119892120572minus 1) + 119903
120572(119910
120572119891120572minus 1) (66)
we have (recall again Remark 21 and note that 119867119864equiv 0)
119892120572= minus119867
120572(119892
120572 119891
120572) 119892
120572
119891120572= (minus119867
120572(119892
120572 119891
120572) + 119867
120572+1(119892
120572+1 119891
120572+1)) 119891
120572
120572 = 0 119864 minus 2
119892119864minus1
119892119864minus1
=
119891119864minus1
119891119864minus1
= minus119867119864minus1
(119892119864minus1
119891119864minus1
)
(67)
The overall initial condition is still 119892120572(0) = 1 = 119891
120572(0)
Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881
119879
120572=
(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579
120572= log119892
120572
and 120601120572
= log119891120572 changing 119867
120572and the initial conditions
correspondingly Hence
120601120572= 120579
120572minus 120579
120572+1997904rArr 120601
120572= 120579
120572minus 120579
120572+1 120572 = 0 119864 minus 2
120579120572= minus119867
120572(120579
120572 120579
120572minus 120579
120572+1) 120572 = 0 119864 minus 2
120601119864minus1
= 120579119864minus1
= minus119867119864minus1
(120579119864minus1
120579119864minus1
)
(68)
The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879
= minus119872(1 0 0 0) satisfies 1198770
= 119865119898 FinallyΨ(119911 119910 119905) = 119892
minus119872
0 We now compute 119892
0(119905)
Let119882120572= 119890
minus120579120572 = 119892
minus1
120572 0 le 120572 le 119864minus1 Defining the auxiliary
variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of
119882 read
120572+ (119902
120572+ 119903
120572)119882
120572= 119902
120572119911120572+ 119903
120572119910120572119882
120572+1 119882
120572 (0) = 1
(69)
with solution
119882120572(119905) = 119890
minus(119902120572+119903120572)119905
+ int
119905
0
119890minus(119902120572+119903120572)(119905minus119904)
(119902120572119911120572+ 119903
120572119910120572119882
120572+1(119904)) 119889119904
(70)
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 The Scientific World Journal
22 Linear Rates We will assume that the transition ratesdependmdashin a linear fashionmdashon the populations only Todistinguish from general rates (denoted above by the letter119882) we will use the letter 119877 to denote linear transition ratesHence
119877120572= sum
119896
119903119896
120572119883
119896(119905) (5)
Since rates are nonnegative we have that the proportionalitycoefficient between event 120572 and subpopulation 119896 satisfies119903119896
120572ge 0 These proportionality coefficients convey the
environmental information and as such they may be time-dependent since the environment varieswith timeWheneverpossible we do not write this time dependency explicitly tolighten the notation
Lemma 7 For the case of linear rates population space 119883119896ge
0 is positively invariant under the time evolution given by (1)and the previous assumptions if and only if 120575120572
119894ge minus1 and 120575
120572
119894=
minus1 only if 119877120572= 119903
119894
120572119883
119894(119905) (no sum)
Proof Theprobability of occurrence of event120572 in a small timeinterval ℎ is
ℎ119877120572+ 119900 (ℎ) = ℎsum
119896
119903119896
120572119883
119896(119905) + 119900 (ℎ) (6)
After an occurrence population 119883119896is modified to 119883
119896+ 120575
120572
119896
Population space is positively invariant under time evolutionif and only if the probability of occurrence of event 120572 is zerofor those population values119883
119896such that119883
119896+ 120575
120572
119896lt 0 (we are
just saying that events pushing the populations to negativevalues do not exist) Since this should hold for arbitrary ℎ wehave that 119877
120572= 0
Assume that 120575120572119896
le minus1 Then 119877120572
= 0 for 0 le 119883119896
lt minus120575120572
119896
Letting 119883119896
= 0 we realise that sum119895 = 119896
119903119895
120572119883
119895(119905) = 0 regardless
of the values of 119883119895 and hence it holds that 119903119895
120572= 0 for 119895 = 119896
and therefore 119877120572
= 119903119896
120572119883
119896(no sum) This proves the second
statement If 120575120572119896
lt minus1 we have that 119877120572
= 0 also for 119883119896
= 1Hence 119903
119896
120572= 0 as well 119877
120572equiv 0 and the event 120572 can be ignored
since it does not influence the dynamics
3 Kolmogorov Forward Equation
Let us recast the dynamical equation given by Theorem 6 interms of the generating function [39] in event-space definedas
Ψ (1199111 119911
119864 119904 119905) = sum
119899120572
(prod
120572
119911119899120572
120572)119875 (119899
1 119899
119864 119904 119905) (7)
For the case of linear rates we define
1198770
120572= sum
119896
119903119896
120572119883
119896 (119904)
119865120573
120572= sum
119896
119903119896
120572120575120573
119896
(8)
As a consequence
119877120572= 119877
0
120572+ sum
120573
119865120573
120572119899120573 (9)
Using Theorem 6 to rewrite 120597Ψ120597119905 and using further 119899120573Ψ =
119911120573(120597Ψ120597119911
120573) (a matter of replacing in the definition of Ψ) we
obtain
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
= sum
120572
(119911120572minus 1)(119877
0
120572Ψ + sum
120573
119865120573
120572119911120573120597Ψ120597119911
120573) (10)
Since the generating function reflects a probability distri-bution we have the probability condition Ψ(1 1 119904 119905) =
1 Further the solutions of (10) can be uniquely trans-lated into those given by Theorem 6 for correspondinginitial conditions 119875(119898
1 119898
119864 119904 119904) = 1 corresponding
to Ψ(1199111 119911
119864 119904 119904) = 119911
1198981
11199111198982
2sdot sdot sdot 119911
119898119864
119864 where the integers
1198981 119898
119864denote howmany events of each class had already
occurred at 119905 = 119904 The natural initial condition becomesΨ(119911
1 119911
119864 119904 119904) = 1 corresponding to 119875(0 0 119904 119904) = 1
31 The Method of Characteristics Revisited
Lemma 8 Equation (10) with 1198770
120572= 0 and with the initial
condition Ψ(1199111 119911
119864 119904 119904) = Φ
0(119911
1 119911
119864) admits the
solution
Ψ (1199111 119911
119864 119904 119905) = Φ
0(120601 (119911 119905 119905 minus 119904)) (11)
where 120601(119911 119905 120591) is the flow of the following ODE with initialcondition 119909(0) = 119911 119904(0) = 119905
119889119909120573
119889120591= sum
120572
119865120573
120572(119909
120572minus 1) 119909
120573equiv 119887
120573(119909 119904)
119889119904
119889120591= minus1
(12)
Proof It is convenient to rename 119904 as 119904 = 119909119864+1
introduce119887119864+1
= minus1 and recall that 119887120573 depends only on the extendedvariable array 119909 However when necessary we will writeexplicitly 119909(0) as (119911 119905) We have that
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
119889120601120573(119911 119905 119905 minus 119904)
119889119905
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
times (119887120573(120601 (119911 119905 119905 minus 119904)) +
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
(13)
Further consider the monodromy matrix [40]
119872120573
120574=
120597120601120573(119909 120591)
120597119909120574
(14)
The Scientific World Journal 5
119872 transforms the initial velocity into the final velocity hence
sum
120574
119872120573
120574119887120574(119911 119905) = 119887
120573(120601 (119911 119905 120591)) (15)
Thus we get
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
(
119864+1
sum
120574
119872120573
120574119887120574(119911 119905) +
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
times (
119864
sum
120574
119872120573
120574119887120574(119911 119905) minus 119872
120573
119864+1+
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
=
119864
sum
120574
120597Ψ
120597119911120574
119887120574(119911 119905)
(16)
From the definition of monodromy matrix above it followsthat
minus119872120573
119864+1+
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
= 0 (17)
and by (11) and the chain rule
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)
119872120573
120574= sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)
120597120601120573(119911 119905 119905 minus 119904)
120597119911120574
=120597Ψ
120597119911120574
(18)
Lemma 9 Equation (10) with 1198770
120572= 0 and with the initial
condition Ψ(1199111 119911
119864 119904 119904) = Φ
0(119911
1 119911
119864) admits the
solution
Ψ (1199111 119911
119864 119904 119905) = Φ
0(120601 (119911 119905 119905 minus 119904))Φ
1(119911
1 119911
119864 119904 119905)
(19)
where Φ0(120601(119911 119905 119905 minus 119904)) is a solution of the homogeneous
problem with 1198770
120572= 0 and
Φ1(119911
1 119911
119864 119904 119905)
= expint
119905
119904
(sum
120572
1198770
120572(120591) (120601
120572(119911 119905 119905 minus 120591) minus 1) 119889120591)
(20)
Proof The proof is nothing more than an application of themethod of the variation of the constants The details are asfollows
Introducing the proposed solution Ψ into (10) and usingLemma 8 to eliminateΦ
0we obtain thatΦ
1satisfies (10) with
initial condition Φ1(119911 119904 119904) = 1 Computing directly from
(20) we have that
120597Φ1
120597119905
10038161003816100381610038161003816100381610038161003816119904
= sum
120572
1198770
120572(120601
120572(119911 119905 0) minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573
120597120601120573(119911 119905 119905 minus 120591)
120597119905
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1
+ sum
120573
120597Φ1
120597120601120573(119887
120573(120601 (119911 119905 119905 minus 120591)) +
120597120601120573(119911 119906 119905 minus 120591)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573sum
120574
119872120573
120574119887120574(119911 119905)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573sum
120574
120597120601120573
120597119911120574
119887120574(119911 119905)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120574
120597Φ1
120597119911120574
119887120574(119911 119905)
(21)
after retracing our steps from the previous Lemma
The result presented in Lemma 9 was constructed usingthe method of variation of constants which provides analternative (constructive) proof
32 Basic Properties of the Solution
Definition 10 A nonzero vector v such that sum120572120575120572
119895V120572
= 0for 119895 = 1 119873 is called a structural zero of 119865 Other zeroeigenvalues of 119865 are called nonstructural
Lemma 11 (a)There are structural zeroes if and only if119864 gt 119873(b) The only structurally stable zero eigenvalues of the matrix119865 are the structural zeroes
Proof (a) The ldquoif rdquo part is immediate since dim(ker(120575)) ge
119864minus119873The ldquoonly if rdquo part follows from the assumption of inde-pendent population increments (and therefore independentpopulations) For (b) put 119865 = r120575 If 119865119908 = 0 for 119908 nonzeroand 120575119908 = 119901 = 0 then r119901 = 0 This relation is not structurallystable since by adding 119888 = 0 arbitrarily small to all diagonalelements of r we can assure r1015840119901 = 0 for this modified r1015840 andfor any 119901 = 0
Lemma 12 First integrals of motion for the characteristicequations Let v be a structural zero of 119865 then
prod
120572
(119909120572(119905))
V120572
= 119862 (22)
is a constant of motion for the characteristic equations (12)
6 The Scientific World Journal
Proof Rewrite the first equation (12) as (indices renamed)
119889
119889119905ln119909
120572= sum
120573
119865120572
120573(119909
120573minus 1) (23)
Then
sum
120572
V120572
119889
119889119905ln119909
120572= sum
120573
sum
120572
(V120572119865120572
120573) (119909
120573minus 1) = 0 (24)
Since sum120572(V
120572119865120572
120573) = 0 the expression in the lemma is the
exponential of the integral obtained from
sum
120572
V120572
119889
119889119905ln119909
120572= 0 (25)
Given relationship (1) between initial conditions eventsand populations from the probability generating function inevent space
Ψ (1199111 119911
119864 119904 119905) = sum
119899120572
(prod
120572
119911119899120572
120572)119875 (119899
1sdot sdot sdot 119899
119864 119904 119905) (26)
a corresponding generating function in population space canbe computed namely
Ψ (1199101 119910
119873 119904 119905) = sum
119883119896
(prod
119896
119910119883119896
119896)119875 (119883
1 119883
119873 119904 119905)
(27)
letting 119899120572 119883
119896 be shorthand for (119899
1sdot sdot sdot 119899
119864) (119883
1 119883
119873)
Lemma 13 (projection lemma) Equation 119911120572
= prod119873
119896=1119910120575120572
119896
119896
realises the transformation from event space to populationspace Moreover
Ψ (1199101 119910
119873 119904 119905) = (prod
119896
1199101198830
119896
119896)Ψ(119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896)Ψ(
119873
prod
119896=1
1199101205751
119896
119896
119873
prod
119896=1
119910120575119864
119896
119896 119904 119905)
(28)
Proof Taking logarithms on 119911 and 119910 it is clear that wheneverthere exist structural zeroes the transformation is noninvert-ible In this sense we call it a projection Concerning thestatement we have
(prod
119896
1199101198830
119896
119896)Ψ (119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896) sum
119899120572
(prod
120572
(
119873
prod
119896=1
119910120575120572
119896
119896)
119899120572
)119875 (119899120572 119904 119905)
= sum
119899120572
(
119873
prod
119896=1
1199101198830
119896+sum120572120575120572
119896119899120572
119896)119875 (119899
120572 119904 119905)
(29)
The statement is proved using (1) and rearranging the sumsas
119875 (1198831 119883
119873 119904 119905) = sum
119899120572
1015840
119875 (119899120572 119904 119905) (30)
where the prime denotes sum over 119899120572 such that 119883
0
119895+
sum120572120575120572
119895119899120572= 119883
119895
Corollary 14 The integrals of motion of Lemma 12 projectunder the projection Lemma to 119862 = 1
Proof The above substitution operated on prod119911V120572
120572= 119862 gives
119862 =
119873
prod
119896=1
119910(sum120572V120572120575120572
119896)
119896equiv
119873
prod
119896=1
1199100
119896= 1 (31)
33 Reduced Equation
Assumption 15 There exist integers119898120573120573 = 1 119864 such that
(cf (9))
1198770
120572= sum
120573
119865120573
120572119898
120573 (32)
This assumption amounts to considering that there exists away to mimic the initial condition on the rates with the samematrix 119865 involved in the time evolution
Theorem 16 Under Assumptions 2 4 and 15 equation(10) with natural initial condition Ψ(119911
1 119911
119864 119904 119904 ) =
Φ0(119911
1 119911
119864) = 1 has the solution
Ψ (1199111 119911
119864 119904 119905) = prod
120573
119891119898120573
120573 (33)
where 119911120573119891120573(119911 119904 119905) = 120601
120573(119911 119905 119905 minus 119904) in terms of the solutions
of (12) of Lemma 8 Hence 119891(119911 119904 119905) satisfies (after eliminatingthe auxiliary time 120591)
119889 log (119891120573 (119911 119904 119905))
119889119904
= minussum
120572
119865120573
120572(119904) (119911
120572119891120572(119911 119904 119905) minus 1) 119891
120573(119911 119905 119905) = 1
(34)
Proof The differences between 119891( ) and 120601( ) are (a) factoringout the initial condition 120601
120573(119911 119905 0) = 119911
120573and (b) rearranging
the time arguments since the argument in120601( ) is natural to thead hoc autonomous ODE (12) while the argument in 119891( ) isnatural to the (nonautonomous) PDE (10)
The proof is just a matter of rewriting the previousequations under the present assumptions Because of thenatural initial condition the desired solution to (10) isgiven by (20) in Lemma 9 which can be rewritten usingAssumption 15 as (we use 119911 as shorthand for 119911
1 119911
119864)
Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)
= prod
120573
(expint
119905
119904
sum
120572
119865120573
120572(120591) (120601
120572(119911 119905 119905 minus 120591) minus 1) 119889120591)
119898120573
(35)
The Scientific World Journal 7
Recalling that we defined 119911120572119891120572(119911 119904 119905) = 120601(119911 119905 119905 minus 119904) the
parenthesis can be recognised as the formal solution of (34)integrated from 119905 to 119904
Corollary 17 For v a structural zero of Lemma 12 thegenerating function is unmodified upon the substitution119898
120572rarr
119898120572+ 119896V
120572(119896 isin R)
Remark 18 Having structural zeroes it is amatter of choice toaddress the problem (a) using the variables 119891
120572and imposing
the additional constraints given by structural zeroes or (b)shifting to population coordinates where these constraints areautomatically incorporated Because of Assumption 2 andLemma 12 the transformation mapping one description tothe other corresponds to
119891120572= prod
119895
119908120575120572
119895
119895 (36)
However option (b) proves to bemore efficientwhen it comesto deal with approximate solutions (see Section 5)
4 Examples of Exact Solutions
41 Example I Pure Death Process In the case 119873 = 119864 = 1120575 = minus1 we have 119903 gt 0 119865 = minus119903(119904) and 119898 = minus119872 lt
0 (119872 is the initial population value) Equation (34) reads119889119891119889119904 = 119891119903(119911119891 minus 1) to be integrated in [119905 119904] with 119891(119911 119905 119905) =
1 Hence for constant death rate 119903 we obtain 119891(119911 119904 119905) =
[119911 + (1 minus 119911)119890minus119903(119905minus119904)
]minus1
and
Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)
+ 119911 (1 minus 119890minus119903(119905minus119904)
)]119872
(37)
42 Example II Linear Birth andDeath Process Wehave119873 =
1 119864 = 2 and 1205751
1= minus120575
2
1= 1 while 119903
1denotes the birth rate
and 1199032denotes the death rate 119883
0= 119898
1minus 119898
2is the initial
population There exists one structural zero with vector v =
(1 1)119879 while the corresponding constant of motion results in
the relations 11991111199112= 119862 and119891
11198912= 1The differential equation
(34) for 1198911(to be integrated in [119905 119904]) becomes
1198891198911
119889119904= (119903
111991111198912
1minus (119903
1+ 119903
2) 119891
1+
1198621199032
1199111
) 1198911 (119911 119905 119905) = 1
(38)
with solution (for119862 = 1whichwill be the case after projectionand assuming time-independent birth rates)
1198911(119911 119904 119905) =
1
1199111
(1199032minus 119903
11199111)119882 minus 119903
2(1 minus 119911
1)
(1199032minus 119903
11199111)119882 minus 119903
1(1 minus 119911
1) (39)
where 119882 = exp(1199032
minus 1199031)(119905 minus 119904) Finally Ψ(119911 119904 119905) =
(1198911(119911 119904 119905))
1198981minus1198982 which after projecting in population space
using Lemma 13 and Corollary 14 gives (cf [41])
Ψ (119910 119904 119905) = [(119903
2minus 119903
1119910)119882 minus 119903
2(1 minus 119910)
(1199032minus 119903
1119910)119882 minus 119903
1(1 minus 119910)
]
1198981minus1198982
(40)
Remark 19 A pure death linear process corresponds to abinomial distribution a pure birth process corresponds to anegative binomial and a birth-death process corresponds toa combination of both as independent processes
5 Consistent Approximations
When (34) of Theorem 16 cannot be solved exactly we haveto rely on approximate solutions It is desirable to workwith consistent approximations namely those where basicproperties of the problem are fulfilled exactly rather than ldquoupto an error of size 120598rdquo For example since our solutions shouldbe probabilities we want the coefficients of Ψ(119911
1 119911
119864 119905)
to be always nonnegative and sum up to one Ideally wewant approximations to be constrained to satisfy Lemma 12and to preserve positive invariance in population space (nojumps into negative populations) Whenever consistency isnot automatic it is mandatory to analyse the approximatesolution before jumping into conclusions
51 Poisson Approximation Let us replace 119891120573
equiv 1 in theRHS of (34) (this amounts to propose that 119891
120573is constant and
equal to its initial condition) The solution of the modifiedequation will be a good approximation to the exact solutionfor sufficiently short times such that 119891 is not significantlydifferent from one
For constant rates 119903120572the rhs of (34) does not depend on
time The approximate solution reads 119891120572(119911 119904 119905) = exp((119905 minus
119904)sum120573119865120572
120573(119911
120573minus 1)) and
Ψ (1199111 119911
119864 119904 119905) =
119864
prod
120572
119890119898120572(119905minus119904)sum
120573119865120572
120573(119911120573minus1)
= 119890(119905minus119904)sum
1205731198770
120573(119911120573minus1)
(41)
In other words the generating function is approximatedas a product of individual Poisson generating functions
This approximation has an error of 119900(119905) Simple as it looksthis approach gives a good probability generating functionthat respects structural zeroes but it does not guaranteepositive invariance a Poisson jump could throw the systeminto negative populations However the probability of sucha jump is also 119900(119905) Fortunately there is a natural way toimpose positive invariance and also a natural way to improvethe error up to 119900(119905)
32 in the broader framework of generaltransition rates [6] In this way the approximation can besafely used with error control to address problems that resistexact computation one way or the other
52 Consistent Multinomial Approximations A systematicapproach to obtain a chain of approximations of increasingaccuracy to (34) can be devised via Picard iterations Firstly itis convenient to recast the problem in population coordinates(cf Remark 18) so that the structural zeroes are automaticallytaken into account regardless of other properties of the
8 The Scientific World Journal
adopted approximation From 119891120572
= prod119895119908
120575120572
119895
119895we obtain
119891120572119891
120572= sum
119896120575120572
119896119896119908
119896and (34) can be recast as
sum
119896
120575120572
119896(
119896
119908119896
+ sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1)) = 0
119908119896 (119911 119905 119905) = 1
(42)
Because of the linear independence of population increments(and hence of populations) Lemma 3(b) the counterpart of(34) in population coordinates reads
119896= minus119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896(119911 119905 119905) = 1
(43)
Remark 20 In the case of linearly dependent populationincrements (34) still holds while it is possible to elaboratea more general form in terms of the 119908
119895instead of (43) This
is however outside the scope of this paper
Since sum120572120575120572
119895119898
120572= 119883
119895(119904) we obtain Ψ(119911
1 119911
119864 119904 119905) =
prod119895119908
119883119895(119904)
119895 which is so far an exact result Let us produce
some approximated values for the 119908119896rsquos and correspondingly
for Ψ(1199111 119911
119864 119904 119905)
To avoid further notational complications wewill assumein the sequel that 119903
119896
120573is time independent Hence 119908
119896will
depend only on the time difference 119905 minus 119904 Equation (43) canbe recast as
119896= 119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896 (119911 0) = 1
(44)
now to be integrated along the interval [0 119905 minus 119904] We will evenset 119904 = 0 in what follows
Remark 21 For the case of time-independent proportionalityfactors 119903119896
120573 a correspondingmodification can be introduced in
(34) namely overall change of sign integration in [0 119905 minus 119904]and setting 119904 = 0 to reduce the notational burden
Equations (44) satisfy the Picard-Lindelof theorem sincethe RHS is a Lipschitz function Then the Picard iterationscheme converges to the (unique) solution Taking advantage
of the initial condition the 119899th order truncated version of thePicard iterations reads
119908(0)
119896(ℎ) = 1
119908(1)
119896(ℎ) = 1 + int
ℎ
0
119908(0)
119896(119905)sum
120573
119903119896
120573(119911
120573prod
119895
(119908(0)
119895(119905))
120575120573
119895
minus 1)119889119905
= 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1)
119908(119899)
119896(ℎ) = 1 + int
ℎ
0
119889119905[
[
119908(119899minus1)
119896(119905)
timessum
120573
119903119896
120573(119911
120573prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
minus 1)]
]119899minus1
(45)
where [sdot sdot sdot ]119898denote the Maclaurin polynomial of order119898 in
119905
Lemma 22 For 119905 isin [0 ℎ] ℎ gt 0 being sufficiently smalland for each order of approximation 119899 ge 0 119908(119899)
119896(119905) in (44)
is a polynomial generating function for one individual of thepopulation that is it has the following properties
(a) 119908(119899)
119896(119905) is a polynomial of degree 119899 in 119911
1 119911
119864 where
the coefficient of 11991111989911 119911
119899119864
119864is 119874(119905
119901) with 119901 = 119899
1+
sdot sdot sdot + 119899119864
(b) 119908(119899)
119896(1 1 119905) = 1
(c) Δ(119899)
119896(ℎ) = 119908
(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ) = 119874(ℎ
119899)
(d) the coefficients of 119908(119899)
119896(119905) regarded as a polynomial in
119911120572 are nonnegative functions of time
Proof (a) The statement holds for 119899 = 0 and 119899 = 1Assuming that it holds up to order 119899 minus 1 we have that[119908
(119899minus1)
119896(119905)prod
119895(119908
(119899minus1)
119895(119905))
120575120573
119895 ]119899minus1
is also a polynomial of degree119899 minus 1 in 119911
1 119911
119864with time-dependent coefficients of type
119874(119905119901) since each power 119901 of 119905 carries along a power 119901 in 119911
120572
(as well as lower 119911-powers) A Picard iteration step gives
119908(119899)
119896(119905) = 1 + sum
120573
119903119896
120573119911120573int
ℎ
0
[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
119908(119899minus1)
119896(119905) 119889119905
(46)
The first sum contributes with one time integration and one119911-power which exactly preserves the polynomial structureand the order of the coefficientsThe result is a polynomial ofdegree 119899 in 119911
1 119911
119864 where the coefficients are polynomials
of highest degree at most 119899 in ℎ each one of lowest order119874(ℎ
119901) (as in the statement) The second integral does not
The Scientific World Journal 9
alter these properties since it generates again a polynomial ofdegree 119899minus1 in 119911
1 119911
119864 with time-dependent coefficients of
order119874(ℎ119901+1
) or higher up to ℎ119899 (they are just the coefficients
of 119908(119899minus1)
119896(119905) integrated once in time)
(b) Letting 1199111= 119911
2= sdot sdot sdot = 119911
119864= 1 Picard iteration gives
trivially 119908(119899)
119896(119905) = 1 to all orders again by induction
(c) The statement holds for 119899 = 1 Assuming it holds upto 119899 minus 1 we compute
119908(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ)
= sum
120573
119903119896
120573119911120573
times int
ℎ
0
([(119908(119899minus2)
119896(119905) + Δ
(119899minus1)
119896(119905))
times prod
119895
(119908(119899minus2)
119895(119905) + Δ
(119899minus1)
119895(119905))
120575120573
119895
]
119899minus1
minus[
[
119908(119899minus2)
119896(119905)prod
119895
(119908(119899minus2)
119895(119905))
120575120573
119895]
]119899minus2
)119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
(119908(119899minus1)
119896(119905) minus 119908
(119899minus2)
119896(119905)) 119889119905
(47)
Both expressions in the first integral have the sameMaclaurinpolynomial up to order (119899 minus 2) Hence the difference in thefirst integral contains only terms of order119874(119905
119899minus1) This is also
the case for the second integral After integration we obtainΔ(119899)
119896(ℎ) = 119874(ℎ
119899)
(d) The statement holds for 119899 = 0 and 119899 = 1 and ingeneral for the zero-order coefficient because it is unity for119905 = 0 as a consequence of the initial condition For thehigher-order coefficients we use induction again If all 119911-coefficients in 119908
(119899minus1)
119896(119905) are nonnegative then the same holds
automatically for 119908(119899)
119896(ℎ) up to order 119899 minus 1 since the 119874(ℎ
119899)
contribution adding to each inherited coefficient from theprevious step does not alter its sign for ℎ sufficiently small Itremains to show that the coefficients corresponding to 119901 = 119899
are nonnegative These contributions arise from the (119899 minus 1)thorder of
119903119896
120573119911120573[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
(48)
after time integration If all 120575120573119895are positive then the expres-
sion above is a product of 119911-polynomials with nonnegative
coefficients and the statement follows Suppose that some120575120573
119895= minus1 Then by Lemma 7 119903119896
120573= 0 for 119896 = 119895 Hence
119903119896
120573119911120573119908
(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
=
0 119895 = 119896
119903119896
120573119911120573prod
119895 = 119896
(119908(119899minus1)
119895(119905))
120575120573
119895
119895 = 119896
(49)
However 120575120573119895
= minus 1 for 119895 = 119896 that is they are nonnegative andhence the expression has only nonnegative coefficients
521 Basic Properties of the First- and Second-OrderApproximations Let us analyse the generating functionΨ(119911
1 119911
119864 119905) = prod
119895119908
119883119895(0)
119895 The simplest nontrivial
applications of the approximation scheme are the first- andsecond-order approximations Expanding explicitly up tosecond-order we get
119908119896(ℎ) = 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1) +
ℎ2
2(sum
120573
119903119896
120573(119911
120573minus 1))
times (sum
120574
119903119896
120574(119911
120574minus 1))
+ℎ2
2sum
120573
119903119896
120573119911120573sum
120574
119865120573
120574(119911
120574minus 1)
(50)
We begin by noticing that the first-order approximation isobtained by neglecting all terms in ℎ
2 in the above equationWhenever 119908
119895is linear in 119911 which is always the case for the
first-order approximation the resulting generating functionΨ corresponds to the product of (independent) multinomialdistributions for the events affecting each subpopulation
Let us now consider Ψ(1199111 119911
119864 119905) computed with the
second-order approximated 119908119895rsquos to perform one simulation
step of size ℎCounting powers of 119911
120574on each 119908
119896we have (a) the
probability of no events occurring in time-step ℎ
119875119896(ℎ 0) = 1 minus ℎ(sum
120573
119903119896
120573) +
ℎ2
2(sum
120573
119903119896
120573)
2
(51)
(b) the probability of only one (120573) event occurring in time-step ℎ
119875119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus ℎ
2sum
120572
119903119896
120572minus
ℎ2
2sum
120572
119865120573
120572 (52)
and (c) the probability of another event (120572) occurring afterthe first one
119875119896(ℎ 119868119868 120573 120572) =
ℎ2
2119903119896
120573(119903
119896
120572+ 119865
120573
120572) (53)
10 The Scientific World Journal
522 Simulation Algorithm A simulation step can be oper-ated in the following way Let
sum
120572
(119875119896(ℎ 119868119868 120573 120572)) + 119875
119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus
ℎ2
2sum
120572
119903119896
120572
= 119875119896(ℎ 119868 120573)
(54)
be the probability of one (120573) event occurring alone orfollowed by a second event It follows that
119875119896(ℎ 119868119868 120573 120572) = 119875
119896(ℎ 119868 120573)
ℎ
2
(119903120572
119896+ 119865
120573
120572)
(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ
2)
(55)
The second factor above is the conditional probability119875119896(ℎ 120572120573) for the second event being 120572 given that there was
a first (120573) eventThe algorithm proceeds in two steps First a multino-
mial deviate is computed with probabilities 119875119896(ℎ 119868 120573) (the
probability of no event occurring is still 119875119896(ℎ 0)) This gives
the number 119899120573of occurrences of each event In the second
step for each 119899120573we compute the deviates for the occurrence
of a second event or no more events with the multinomialgiven by the array of probabilities 119875
119896(ℎ 120572120573) defined above
(the probability of no second event occurring is here 1 minus
(ℎ2)(sum120572(119903
120572
119896+ 119865
120573
120572)(1 minus (ℎ2)sum
120572119903119896
120572)))
6 Implementation Developmental Cascadeswithout Mortality
In this and the following Sections we apply the tools devel-oped up to now to the description of a subprocess in insectdevelopment namely the development of immature larvaewhich is in principle a highly individual subprocess Weassume that the evolution of an insect from egg to pupaoccurs via a sequence of 119864 gt 0 maturation stages Biologistsrecognise some substages by inspection in the developmentof larvae (instars) and even more stages when observed byother methods [35] The immature individual may die alongthe process or otherwise complete it and exit as pupa oradult The actual value of apparent maturation stages 119864 willdepend on environmental and experimental conditions andwill ultimately be specified when analying experimental datain Section 71
Empirical evidence based on existing and new experi-mental results as well as biological insight produced by thepresent description will be the subject of an independentwork [42]We discuss however a couple of examples from theliterature in Section 71
The events for this process are maturation events pro-moting individuals from one subpopulation into to the nextand death events for all subpopulations In other wordseach subpopulation represents a given (intermediate) level ofdevelopment and maturation events correspond to progressin the development Development at level 119895 promotes oneindividual to level 119895 + 1 and the event rate is linear in the119895-subpopulation Hence in 119877
120572= sum
119895119903119896
120572119883
119895 the matrix 119903
is square and diagonal for 0 le 120572 le 119864 minus 1 (we haveexactly 119864 + 1 subpopulations counted from 0 to 119864) All ofthe environmental influence is at this point incorporatedthrough 119903
Moreover level 119864 is the matured pupa and we may set119903119864
= 0 since at pupation stage the present mechanism endsand new processes take place Subpopulation 119864 is the exitpoint of the dynamics and we assume it to be determined bythe stochastic evolution of the immature stages 0 le 120572 le 119864minus1
In the same spirit the action of each event on thepopulations modifies just the actual population and the nextone in the chain Hence
120575120572
119895=
minus1 120572 = 119895
+1 120572 = 119895 minus 1(56)
Following Section 3 [1198770
120572]119879
= (1199030119872 0 0) and
119865120573
120572=
minus119903120572 120573 = 120572
+119903120572 120573 = 120572 minus 1
0 otherwise(57)
Finally letting m119879= minus119872(1 1 1) we can write 119877
0
120572=
sum120573119865120573
120572119898
120573 In this framework the index pair (119895 120572) is highly
interconnected and we can achieve a complete descriptionusing just one index We use 0 le 119895 le 119864 minus 1 throughout thissection
The procedure of Section 3 gives Ψ(119911 119905) = prod119895(119891
119895)119898119895 =
prod119895(119891
minus119872
119895) while the dynamical system for 119891
119895reads recalling
the recasting in Remark 21119891119895= 119891
119895(minus119903
119895(119911
119895119891119895minus 1) + 119903
119895+1(119911
119895+1119891119895+1
minus 1))
119895 = 0 119864 minus 2
(58)
and 119891119864minus1
= 119891119864minus1
(minus119903119864minus1
(119911119864minus1
119891119864minus1
minus 1)) The overall initialcondition is 119891
119895(0) = 1 Let 120579
119895= log(119891
119895) and
120595119895=
119864minus1
sum
120573=119895
120579120573 119882
119895= exp (minus120595
119895) (59)
Note that 119882119895119891119895
= 119882119895
sdot exp(120579119895) = 119882
119895+1 Rewriting the
equations120579119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) + 119903
119895+1(119911
119895+1exp (120579
119895+1) minus 1)
119895 lt 119864 minus 1
119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) = 120579
119895+
119895+1 119895 lt 119864 minus 1
120579119864minus1
= 119864minus1
= minus119903119864minus1
(119911119864minus1
exp (120579119864minus1
) minus 1)
(60)
with initial condition 120579119895(0) = 0 for all 119895With this notation the
generating function reads Ψ(119911 119905) = 119882119872
0 and we formulate
hence an ODE for the 119882119895rsquos
119895+ 119903
119895119882
119895= 119903
119895119911119895119882
119895+1 119882
119895 (0) = 1 119895 = 0 119864 minus 2
119864minus1
= 119903119864minus1
(119911119864minus1
minus 119882119864minus1
) 119882119864minus1
(0) = 1
(61)
The Scientific World Journal 11
By defining the auxiliary quantity 119882119864(119905) = 1 the solution
for the 119882119895-differential equations can be written in recursive
form
119882119895(119905) = 119890
minus119903119895119905+ 119911
119895119903119895int
119905
0
119890119903119895(119904minus119905)
119882119895+1
(119904) 119889119904 119895 = 0 119864 minus 1
(62)
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820(119905) = (
119864minus1
prod
119898=0
119911119898)(1 minus
119864minus1
sum
119896=0
exp (minus119903119896119905)
119864minus1
prod
119895=0119895 = 119896
119903119895
119903119895minus 119903
119896
)
+
119864
sum
119896=1
prod119864minus119896
119895=0119911119895
119911119864minus119896
119903119864minus119896
119864minus119896
sum
119898=0
119903119898exp (minus119903
119898119905)
119864minus119896
prod
119897=0119897 =119898
119903119897
119903119897minus 119903
119898
(63)
The solution looksmore compact when all 119903119896are assumed
to take the common value 119903
1198820(119905) =
prod119864minus1
119898=0119911119898
(119864 minus 1)int
119903119905
0
exp (minus119904) 119904119864minus1
119889119904
+
119864
sum
119896=1
prod119864minus119896
119898=0119911119898
119911119864minus119896
(119903119905)119864minus119896 exp (minus119903119905)
(119864 minus 119896)
(64)
7 Developmental Cascades with Mortality
We add to the previous description 119864 subpopulation specificdeath events 119876
120572= 119902
120572119883
120572 where 120572 = 0 119864 minus 1 We
do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883
119864leaves the
present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876
0 119877
0 119876
1 119877
1 119876
119864minus1 and 119877
119864minus1 Then ((119876 119877)
0)119879
=
(1199020119872 119903
0119872 0 0) We have
1205752120572
120572= minus1 (death) 120572 lt 119864
1205752120572+1
120572= minus1
1205752120572+1
120572+1= +1
(development) 120572 lt 119864
119865120573
2120572=
minus119902120572 120573 = 2120572
minus119902120572 120573 = 2120572 + 1
119902120572 120573 = 2120572 minus 1
119865120573
2120572+1=
minus119903120572 120573 = 2120572 + 1
minus119903120572 120573 = 2120572
119903120572 120573 = 2120572 minus 1
120572 lt 119864
(65)
Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891
120572and the 119911
120572in pairs (119892 119891) and
(119911 119910) as (1198920 119891
0 119892
119864minus1 119891
119864minus1) and (119911
0 119910
0 119911
119864minus1 119910
119864minus1)
Also for notational convenience we will assume that thereexist 119902
119864= 119903
119864= 0 when necessary Defining
119867120572(119892
120572 119891
120572) = 119902
120572(119911
120572119892120572minus 1) + 119903
120572(119910
120572119891120572minus 1) (66)
we have (recall again Remark 21 and note that 119867119864equiv 0)
119892120572= minus119867
120572(119892
120572 119891
120572) 119892
120572
119891120572= (minus119867
120572(119892
120572 119891
120572) + 119867
120572+1(119892
120572+1 119891
120572+1)) 119891
120572
120572 = 0 119864 minus 2
119892119864minus1
119892119864minus1
=
119891119864minus1
119891119864minus1
= minus119867119864minus1
(119892119864minus1
119891119864minus1
)
(67)
The overall initial condition is still 119892120572(0) = 1 = 119891
120572(0)
Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881
119879
120572=
(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579
120572= log119892
120572
and 120601120572
= log119891120572 changing 119867
120572and the initial conditions
correspondingly Hence
120601120572= 120579
120572minus 120579
120572+1997904rArr 120601
120572= 120579
120572minus 120579
120572+1 120572 = 0 119864 minus 2
120579120572= minus119867
120572(120579
120572 120579
120572minus 120579
120572+1) 120572 = 0 119864 minus 2
120601119864minus1
= 120579119864minus1
= minus119867119864minus1
(120579119864minus1
120579119864minus1
)
(68)
The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879
= minus119872(1 0 0 0) satisfies 1198770
= 119865119898 FinallyΨ(119911 119910 119905) = 119892
minus119872
0 We now compute 119892
0(119905)
Let119882120572= 119890
minus120579120572 = 119892
minus1
120572 0 le 120572 le 119864minus1 Defining the auxiliary
variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of
119882 read
120572+ (119902
120572+ 119903
120572)119882
120572= 119902
120572119911120572+ 119903
120572119910120572119882
120572+1 119882
120572 (0) = 1
(69)
with solution
119882120572(119905) = 119890
minus(119902120572+119903120572)119905
+ int
119905
0
119890minus(119902120572+119903120572)(119905minus119904)
(119902120572119911120572+ 119903
120572119910120572119882
120572+1(119904)) 119889119904
(70)
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 5
119872 transforms the initial velocity into the final velocity hence
sum
120574
119872120573
120574119887120574(119911 119905) = 119887
120573(120601 (119911 119905 120591)) (15)
Thus we get
120597Ψ
120597119905
10038161003816100381610038161003816100381610038161003816119904
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
(
119864+1
sum
120574
119872120573
120574119887120574(119911 119905) +
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
=
119864+1
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)
times (
119864
sum
120574
119872120573
120574119887120574(119911 119905) minus 119872
120573
119864+1+
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
=
119864
sum
120574
120597Ψ
120597119911120574
119887120574(119911 119905)
(16)
From the definition of monodromy matrix above it followsthat
minus119872120573
119864+1+
120597120601120573(119911 119906 119905 minus 119904)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
= 0 (17)
and by (11) and the chain rule
sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)
119872120573
120574= sum
120573
120597Φ0
120597119909120573
1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)
120597120601120573(119911 119905 119905 minus 119904)
120597119911120574
=120597Ψ
120597119911120574
(18)
Lemma 9 Equation (10) with 1198770
120572= 0 and with the initial
condition Ψ(1199111 119911
119864 119904 119904) = Φ
0(119911
1 119911
119864) admits the
solution
Ψ (1199111 119911
119864 119904 119905) = Φ
0(120601 (119911 119905 119905 minus 119904))Φ
1(119911
1 119911
119864 119904 119905)
(19)
where Φ0(120601(119911 119905 119905 minus 119904)) is a solution of the homogeneous
problem with 1198770
120572= 0 and
Φ1(119911
1 119911
119864 119904 119905)
= expint
119905
119904
(sum
120572
1198770
120572(120591) (120601
120572(119911 119905 119905 minus 120591) minus 1) 119889120591)
(20)
Proof The proof is nothing more than an application of themethod of the variation of the constants The details are asfollows
Introducing the proposed solution Ψ into (10) and usingLemma 8 to eliminateΦ
0we obtain thatΦ
1satisfies (10) with
initial condition Φ1(119911 119904 119904) = 1 Computing directly from
(20) we have that
120597Φ1
120597119905
10038161003816100381610038161003816100381610038161003816119904
= sum
120572
1198770
120572(120601
120572(119911 119905 0) minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573
120597120601120573(119911 119905 119905 minus 120591)
120597119905
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1
+ sum
120573
120597Φ1
120597120601120573(119887
120573(120601 (119911 119905 119905 minus 120591)) +
120597120601120573(119911 119906 119905 minus 120591)
120597119906
100381610038161003816100381610038161003816100381610038161003816119906=119905
)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573sum
120574
119872120573
120574119887120574(119911 119905)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120573
120597Φ1
120597120601120573sum
120574
120597120601120573
120597119911120574
119887120574(119911 119905)
= sum
120572
1198770
120572(119911
120572minus 1)Φ
1+ sum
120574
120597Φ1
120597119911120574
119887120574(119911 119905)
(21)
after retracing our steps from the previous Lemma
The result presented in Lemma 9 was constructed usingthe method of variation of constants which provides analternative (constructive) proof
32 Basic Properties of the Solution
Definition 10 A nonzero vector v such that sum120572120575120572
119895V120572
= 0for 119895 = 1 119873 is called a structural zero of 119865 Other zeroeigenvalues of 119865 are called nonstructural
Lemma 11 (a)There are structural zeroes if and only if119864 gt 119873(b) The only structurally stable zero eigenvalues of the matrix119865 are the structural zeroes
Proof (a) The ldquoif rdquo part is immediate since dim(ker(120575)) ge
119864minus119873The ldquoonly if rdquo part follows from the assumption of inde-pendent population increments (and therefore independentpopulations) For (b) put 119865 = r120575 If 119865119908 = 0 for 119908 nonzeroand 120575119908 = 119901 = 0 then r119901 = 0 This relation is not structurallystable since by adding 119888 = 0 arbitrarily small to all diagonalelements of r we can assure r1015840119901 = 0 for this modified r1015840 andfor any 119901 = 0
Lemma 12 First integrals of motion for the characteristicequations Let v be a structural zero of 119865 then
prod
120572
(119909120572(119905))
V120572
= 119862 (22)
is a constant of motion for the characteristic equations (12)
6 The Scientific World Journal
Proof Rewrite the first equation (12) as (indices renamed)
119889
119889119905ln119909
120572= sum
120573
119865120572
120573(119909
120573minus 1) (23)
Then
sum
120572
V120572
119889
119889119905ln119909
120572= sum
120573
sum
120572
(V120572119865120572
120573) (119909
120573minus 1) = 0 (24)
Since sum120572(V
120572119865120572
120573) = 0 the expression in the lemma is the
exponential of the integral obtained from
sum
120572
V120572
119889
119889119905ln119909
120572= 0 (25)
Given relationship (1) between initial conditions eventsand populations from the probability generating function inevent space
Ψ (1199111 119911
119864 119904 119905) = sum
119899120572
(prod
120572
119911119899120572
120572)119875 (119899
1sdot sdot sdot 119899
119864 119904 119905) (26)
a corresponding generating function in population space canbe computed namely
Ψ (1199101 119910
119873 119904 119905) = sum
119883119896
(prod
119896
119910119883119896
119896)119875 (119883
1 119883
119873 119904 119905)
(27)
letting 119899120572 119883
119896 be shorthand for (119899
1sdot sdot sdot 119899
119864) (119883
1 119883
119873)
Lemma 13 (projection lemma) Equation 119911120572
= prod119873
119896=1119910120575120572
119896
119896
realises the transformation from event space to populationspace Moreover
Ψ (1199101 119910
119873 119904 119905) = (prod
119896
1199101198830
119896
119896)Ψ(119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896)Ψ(
119873
prod
119896=1
1199101205751
119896
119896
119873
prod
119896=1
119910120575119864
119896
119896 119904 119905)
(28)
Proof Taking logarithms on 119911 and 119910 it is clear that wheneverthere exist structural zeroes the transformation is noninvert-ible In this sense we call it a projection Concerning thestatement we have
(prod
119896
1199101198830
119896
119896)Ψ (119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896) sum
119899120572
(prod
120572
(
119873
prod
119896=1
119910120575120572
119896
119896)
119899120572
)119875 (119899120572 119904 119905)
= sum
119899120572
(
119873
prod
119896=1
1199101198830
119896+sum120572120575120572
119896119899120572
119896)119875 (119899
120572 119904 119905)
(29)
The statement is proved using (1) and rearranging the sumsas
119875 (1198831 119883
119873 119904 119905) = sum
119899120572
1015840
119875 (119899120572 119904 119905) (30)
where the prime denotes sum over 119899120572 such that 119883
0
119895+
sum120572120575120572
119895119899120572= 119883
119895
Corollary 14 The integrals of motion of Lemma 12 projectunder the projection Lemma to 119862 = 1
Proof The above substitution operated on prod119911V120572
120572= 119862 gives
119862 =
119873
prod
119896=1
119910(sum120572V120572120575120572
119896)
119896equiv
119873
prod
119896=1
1199100
119896= 1 (31)
33 Reduced Equation
Assumption 15 There exist integers119898120573120573 = 1 119864 such that
(cf (9))
1198770
120572= sum
120573
119865120573
120572119898
120573 (32)
This assumption amounts to considering that there exists away to mimic the initial condition on the rates with the samematrix 119865 involved in the time evolution
Theorem 16 Under Assumptions 2 4 and 15 equation(10) with natural initial condition Ψ(119911
1 119911
119864 119904 119904 ) =
Φ0(119911
1 119911
119864) = 1 has the solution
Ψ (1199111 119911
119864 119904 119905) = prod
120573
119891119898120573
120573 (33)
where 119911120573119891120573(119911 119904 119905) = 120601
120573(119911 119905 119905 minus 119904) in terms of the solutions
of (12) of Lemma 8 Hence 119891(119911 119904 119905) satisfies (after eliminatingthe auxiliary time 120591)
119889 log (119891120573 (119911 119904 119905))
119889119904
= minussum
120572
119865120573
120572(119904) (119911
120572119891120572(119911 119904 119905) minus 1) 119891
120573(119911 119905 119905) = 1
(34)
Proof The differences between 119891( ) and 120601( ) are (a) factoringout the initial condition 120601
120573(119911 119905 0) = 119911
120573and (b) rearranging
the time arguments since the argument in120601( ) is natural to thead hoc autonomous ODE (12) while the argument in 119891( ) isnatural to the (nonautonomous) PDE (10)
The proof is just a matter of rewriting the previousequations under the present assumptions Because of thenatural initial condition the desired solution to (10) isgiven by (20) in Lemma 9 which can be rewritten usingAssumption 15 as (we use 119911 as shorthand for 119911
1 119911
119864)
Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)
= prod
120573
(expint
119905
119904
sum
120572
119865120573
120572(120591) (120601
120572(119911 119905 119905 minus 120591) minus 1) 119889120591)
119898120573
(35)
The Scientific World Journal 7
Recalling that we defined 119911120572119891120572(119911 119904 119905) = 120601(119911 119905 119905 minus 119904) the
parenthesis can be recognised as the formal solution of (34)integrated from 119905 to 119904
Corollary 17 For v a structural zero of Lemma 12 thegenerating function is unmodified upon the substitution119898
120572rarr
119898120572+ 119896V
120572(119896 isin R)
Remark 18 Having structural zeroes it is amatter of choice toaddress the problem (a) using the variables 119891
120572and imposing
the additional constraints given by structural zeroes or (b)shifting to population coordinates where these constraints areautomatically incorporated Because of Assumption 2 andLemma 12 the transformation mapping one description tothe other corresponds to
119891120572= prod
119895
119908120575120572
119895
119895 (36)
However option (b) proves to bemore efficientwhen it comesto deal with approximate solutions (see Section 5)
4 Examples of Exact Solutions
41 Example I Pure Death Process In the case 119873 = 119864 = 1120575 = minus1 we have 119903 gt 0 119865 = minus119903(119904) and 119898 = minus119872 lt
0 (119872 is the initial population value) Equation (34) reads119889119891119889119904 = 119891119903(119911119891 minus 1) to be integrated in [119905 119904] with 119891(119911 119905 119905) =
1 Hence for constant death rate 119903 we obtain 119891(119911 119904 119905) =
[119911 + (1 minus 119911)119890minus119903(119905minus119904)
]minus1
and
Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)
+ 119911 (1 minus 119890minus119903(119905minus119904)
)]119872
(37)
42 Example II Linear Birth andDeath Process Wehave119873 =
1 119864 = 2 and 1205751
1= minus120575
2
1= 1 while 119903
1denotes the birth rate
and 1199032denotes the death rate 119883
0= 119898
1minus 119898
2is the initial
population There exists one structural zero with vector v =
(1 1)119879 while the corresponding constant of motion results in
the relations 11991111199112= 119862 and119891
11198912= 1The differential equation
(34) for 1198911(to be integrated in [119905 119904]) becomes
1198891198911
119889119904= (119903
111991111198912
1minus (119903
1+ 119903
2) 119891
1+
1198621199032
1199111
) 1198911 (119911 119905 119905) = 1
(38)
with solution (for119862 = 1whichwill be the case after projectionand assuming time-independent birth rates)
1198911(119911 119904 119905) =
1
1199111
(1199032minus 119903
11199111)119882 minus 119903
2(1 minus 119911
1)
(1199032minus 119903
11199111)119882 minus 119903
1(1 minus 119911
1) (39)
where 119882 = exp(1199032
minus 1199031)(119905 minus 119904) Finally Ψ(119911 119904 119905) =
(1198911(119911 119904 119905))
1198981minus1198982 which after projecting in population space
using Lemma 13 and Corollary 14 gives (cf [41])
Ψ (119910 119904 119905) = [(119903
2minus 119903
1119910)119882 minus 119903
2(1 minus 119910)
(1199032minus 119903
1119910)119882 minus 119903
1(1 minus 119910)
]
1198981minus1198982
(40)
Remark 19 A pure death linear process corresponds to abinomial distribution a pure birth process corresponds to anegative binomial and a birth-death process corresponds toa combination of both as independent processes
5 Consistent Approximations
When (34) of Theorem 16 cannot be solved exactly we haveto rely on approximate solutions It is desirable to workwith consistent approximations namely those where basicproperties of the problem are fulfilled exactly rather than ldquoupto an error of size 120598rdquo For example since our solutions shouldbe probabilities we want the coefficients of Ψ(119911
1 119911
119864 119905)
to be always nonnegative and sum up to one Ideally wewant approximations to be constrained to satisfy Lemma 12and to preserve positive invariance in population space (nojumps into negative populations) Whenever consistency isnot automatic it is mandatory to analyse the approximatesolution before jumping into conclusions
51 Poisson Approximation Let us replace 119891120573
equiv 1 in theRHS of (34) (this amounts to propose that 119891
120573is constant and
equal to its initial condition) The solution of the modifiedequation will be a good approximation to the exact solutionfor sufficiently short times such that 119891 is not significantlydifferent from one
For constant rates 119903120572the rhs of (34) does not depend on
time The approximate solution reads 119891120572(119911 119904 119905) = exp((119905 minus
119904)sum120573119865120572
120573(119911
120573minus 1)) and
Ψ (1199111 119911
119864 119904 119905) =
119864
prod
120572
119890119898120572(119905minus119904)sum
120573119865120572
120573(119911120573minus1)
= 119890(119905minus119904)sum
1205731198770
120573(119911120573minus1)
(41)
In other words the generating function is approximatedas a product of individual Poisson generating functions
This approximation has an error of 119900(119905) Simple as it looksthis approach gives a good probability generating functionthat respects structural zeroes but it does not guaranteepositive invariance a Poisson jump could throw the systeminto negative populations However the probability of sucha jump is also 119900(119905) Fortunately there is a natural way toimpose positive invariance and also a natural way to improvethe error up to 119900(119905)
32 in the broader framework of generaltransition rates [6] In this way the approximation can besafely used with error control to address problems that resistexact computation one way or the other
52 Consistent Multinomial Approximations A systematicapproach to obtain a chain of approximations of increasingaccuracy to (34) can be devised via Picard iterations Firstly itis convenient to recast the problem in population coordinates(cf Remark 18) so that the structural zeroes are automaticallytaken into account regardless of other properties of the
8 The Scientific World Journal
adopted approximation From 119891120572
= prod119895119908
120575120572
119895
119895we obtain
119891120572119891
120572= sum
119896120575120572
119896119896119908
119896and (34) can be recast as
sum
119896
120575120572
119896(
119896
119908119896
+ sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1)) = 0
119908119896 (119911 119905 119905) = 1
(42)
Because of the linear independence of population increments(and hence of populations) Lemma 3(b) the counterpart of(34) in population coordinates reads
119896= minus119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896(119911 119905 119905) = 1
(43)
Remark 20 In the case of linearly dependent populationincrements (34) still holds while it is possible to elaboratea more general form in terms of the 119908
119895instead of (43) This
is however outside the scope of this paper
Since sum120572120575120572
119895119898
120572= 119883
119895(119904) we obtain Ψ(119911
1 119911
119864 119904 119905) =
prod119895119908
119883119895(119904)
119895 which is so far an exact result Let us produce
some approximated values for the 119908119896rsquos and correspondingly
for Ψ(1199111 119911
119864 119904 119905)
To avoid further notational complications wewill assumein the sequel that 119903
119896
120573is time independent Hence 119908
119896will
depend only on the time difference 119905 minus 119904 Equation (43) canbe recast as
119896= 119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896 (119911 0) = 1
(44)
now to be integrated along the interval [0 119905 minus 119904] We will evenset 119904 = 0 in what follows
Remark 21 For the case of time-independent proportionalityfactors 119903119896
120573 a correspondingmodification can be introduced in
(34) namely overall change of sign integration in [0 119905 minus 119904]and setting 119904 = 0 to reduce the notational burden
Equations (44) satisfy the Picard-Lindelof theorem sincethe RHS is a Lipschitz function Then the Picard iterationscheme converges to the (unique) solution Taking advantage
of the initial condition the 119899th order truncated version of thePicard iterations reads
119908(0)
119896(ℎ) = 1
119908(1)
119896(ℎ) = 1 + int
ℎ
0
119908(0)
119896(119905)sum
120573
119903119896
120573(119911
120573prod
119895
(119908(0)
119895(119905))
120575120573
119895
minus 1)119889119905
= 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1)
119908(119899)
119896(ℎ) = 1 + int
ℎ
0
119889119905[
[
119908(119899minus1)
119896(119905)
timessum
120573
119903119896
120573(119911
120573prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
minus 1)]
]119899minus1
(45)
where [sdot sdot sdot ]119898denote the Maclaurin polynomial of order119898 in
119905
Lemma 22 For 119905 isin [0 ℎ] ℎ gt 0 being sufficiently smalland for each order of approximation 119899 ge 0 119908(119899)
119896(119905) in (44)
is a polynomial generating function for one individual of thepopulation that is it has the following properties
(a) 119908(119899)
119896(119905) is a polynomial of degree 119899 in 119911
1 119911
119864 where
the coefficient of 11991111989911 119911
119899119864
119864is 119874(119905
119901) with 119901 = 119899
1+
sdot sdot sdot + 119899119864
(b) 119908(119899)
119896(1 1 119905) = 1
(c) Δ(119899)
119896(ℎ) = 119908
(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ) = 119874(ℎ
119899)
(d) the coefficients of 119908(119899)
119896(119905) regarded as a polynomial in
119911120572 are nonnegative functions of time
Proof (a) The statement holds for 119899 = 0 and 119899 = 1Assuming that it holds up to order 119899 minus 1 we have that[119908
(119899minus1)
119896(119905)prod
119895(119908
(119899minus1)
119895(119905))
120575120573
119895 ]119899minus1
is also a polynomial of degree119899 minus 1 in 119911
1 119911
119864with time-dependent coefficients of type
119874(119905119901) since each power 119901 of 119905 carries along a power 119901 in 119911
120572
(as well as lower 119911-powers) A Picard iteration step gives
119908(119899)
119896(119905) = 1 + sum
120573
119903119896
120573119911120573int
ℎ
0
[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
119908(119899minus1)
119896(119905) 119889119905
(46)
The first sum contributes with one time integration and one119911-power which exactly preserves the polynomial structureand the order of the coefficientsThe result is a polynomial ofdegree 119899 in 119911
1 119911
119864 where the coefficients are polynomials
of highest degree at most 119899 in ℎ each one of lowest order119874(ℎ
119901) (as in the statement) The second integral does not
The Scientific World Journal 9
alter these properties since it generates again a polynomial ofdegree 119899minus1 in 119911
1 119911
119864 with time-dependent coefficients of
order119874(ℎ119901+1
) or higher up to ℎ119899 (they are just the coefficients
of 119908(119899minus1)
119896(119905) integrated once in time)
(b) Letting 1199111= 119911
2= sdot sdot sdot = 119911
119864= 1 Picard iteration gives
trivially 119908(119899)
119896(119905) = 1 to all orders again by induction
(c) The statement holds for 119899 = 1 Assuming it holds upto 119899 minus 1 we compute
119908(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ)
= sum
120573
119903119896
120573119911120573
times int
ℎ
0
([(119908(119899minus2)
119896(119905) + Δ
(119899minus1)
119896(119905))
times prod
119895
(119908(119899minus2)
119895(119905) + Δ
(119899minus1)
119895(119905))
120575120573
119895
]
119899minus1
minus[
[
119908(119899minus2)
119896(119905)prod
119895
(119908(119899minus2)
119895(119905))
120575120573
119895]
]119899minus2
)119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
(119908(119899minus1)
119896(119905) minus 119908
(119899minus2)
119896(119905)) 119889119905
(47)
Both expressions in the first integral have the sameMaclaurinpolynomial up to order (119899 minus 2) Hence the difference in thefirst integral contains only terms of order119874(119905
119899minus1) This is also
the case for the second integral After integration we obtainΔ(119899)
119896(ℎ) = 119874(ℎ
119899)
(d) The statement holds for 119899 = 0 and 119899 = 1 and ingeneral for the zero-order coefficient because it is unity for119905 = 0 as a consequence of the initial condition For thehigher-order coefficients we use induction again If all 119911-coefficients in 119908
(119899minus1)
119896(119905) are nonnegative then the same holds
automatically for 119908(119899)
119896(ℎ) up to order 119899 minus 1 since the 119874(ℎ
119899)
contribution adding to each inherited coefficient from theprevious step does not alter its sign for ℎ sufficiently small Itremains to show that the coefficients corresponding to 119901 = 119899
are nonnegative These contributions arise from the (119899 minus 1)thorder of
119903119896
120573119911120573[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
(48)
after time integration If all 120575120573119895are positive then the expres-
sion above is a product of 119911-polynomials with nonnegative
coefficients and the statement follows Suppose that some120575120573
119895= minus1 Then by Lemma 7 119903119896
120573= 0 for 119896 = 119895 Hence
119903119896
120573119911120573119908
(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
=
0 119895 = 119896
119903119896
120573119911120573prod
119895 = 119896
(119908(119899minus1)
119895(119905))
120575120573
119895
119895 = 119896
(49)
However 120575120573119895
= minus 1 for 119895 = 119896 that is they are nonnegative andhence the expression has only nonnegative coefficients
521 Basic Properties of the First- and Second-OrderApproximations Let us analyse the generating functionΨ(119911
1 119911
119864 119905) = prod
119895119908
119883119895(0)
119895 The simplest nontrivial
applications of the approximation scheme are the first- andsecond-order approximations Expanding explicitly up tosecond-order we get
119908119896(ℎ) = 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1) +
ℎ2
2(sum
120573
119903119896
120573(119911
120573minus 1))
times (sum
120574
119903119896
120574(119911
120574minus 1))
+ℎ2
2sum
120573
119903119896
120573119911120573sum
120574
119865120573
120574(119911
120574minus 1)
(50)
We begin by noticing that the first-order approximation isobtained by neglecting all terms in ℎ
2 in the above equationWhenever 119908
119895is linear in 119911 which is always the case for the
first-order approximation the resulting generating functionΨ corresponds to the product of (independent) multinomialdistributions for the events affecting each subpopulation
Let us now consider Ψ(1199111 119911
119864 119905) computed with the
second-order approximated 119908119895rsquos to perform one simulation
step of size ℎCounting powers of 119911
120574on each 119908
119896we have (a) the
probability of no events occurring in time-step ℎ
119875119896(ℎ 0) = 1 minus ℎ(sum
120573
119903119896
120573) +
ℎ2
2(sum
120573
119903119896
120573)
2
(51)
(b) the probability of only one (120573) event occurring in time-step ℎ
119875119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus ℎ
2sum
120572
119903119896
120572minus
ℎ2
2sum
120572
119865120573
120572 (52)
and (c) the probability of another event (120572) occurring afterthe first one
119875119896(ℎ 119868119868 120573 120572) =
ℎ2
2119903119896
120573(119903
119896
120572+ 119865
120573
120572) (53)
10 The Scientific World Journal
522 Simulation Algorithm A simulation step can be oper-ated in the following way Let
sum
120572
(119875119896(ℎ 119868119868 120573 120572)) + 119875
119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus
ℎ2
2sum
120572
119903119896
120572
= 119875119896(ℎ 119868 120573)
(54)
be the probability of one (120573) event occurring alone orfollowed by a second event It follows that
119875119896(ℎ 119868119868 120573 120572) = 119875
119896(ℎ 119868 120573)
ℎ
2
(119903120572
119896+ 119865
120573
120572)
(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ
2)
(55)
The second factor above is the conditional probability119875119896(ℎ 120572120573) for the second event being 120572 given that there was
a first (120573) eventThe algorithm proceeds in two steps First a multino-
mial deviate is computed with probabilities 119875119896(ℎ 119868 120573) (the
probability of no event occurring is still 119875119896(ℎ 0)) This gives
the number 119899120573of occurrences of each event In the second
step for each 119899120573we compute the deviates for the occurrence
of a second event or no more events with the multinomialgiven by the array of probabilities 119875
119896(ℎ 120572120573) defined above
(the probability of no second event occurring is here 1 minus
(ℎ2)(sum120572(119903
120572
119896+ 119865
120573
120572)(1 minus (ℎ2)sum
120572119903119896
120572)))
6 Implementation Developmental Cascadeswithout Mortality
In this and the following Sections we apply the tools devel-oped up to now to the description of a subprocess in insectdevelopment namely the development of immature larvaewhich is in principle a highly individual subprocess Weassume that the evolution of an insect from egg to pupaoccurs via a sequence of 119864 gt 0 maturation stages Biologistsrecognise some substages by inspection in the developmentof larvae (instars) and even more stages when observed byother methods [35] The immature individual may die alongthe process or otherwise complete it and exit as pupa oradult The actual value of apparent maturation stages 119864 willdepend on environmental and experimental conditions andwill ultimately be specified when analying experimental datain Section 71
Empirical evidence based on existing and new experi-mental results as well as biological insight produced by thepresent description will be the subject of an independentwork [42]We discuss however a couple of examples from theliterature in Section 71
The events for this process are maturation events pro-moting individuals from one subpopulation into to the nextand death events for all subpopulations In other wordseach subpopulation represents a given (intermediate) level ofdevelopment and maturation events correspond to progressin the development Development at level 119895 promotes oneindividual to level 119895 + 1 and the event rate is linear in the119895-subpopulation Hence in 119877
120572= sum
119895119903119896
120572119883
119895 the matrix 119903
is square and diagonal for 0 le 120572 le 119864 minus 1 (we haveexactly 119864 + 1 subpopulations counted from 0 to 119864) All ofthe environmental influence is at this point incorporatedthrough 119903
Moreover level 119864 is the matured pupa and we may set119903119864
= 0 since at pupation stage the present mechanism endsand new processes take place Subpopulation 119864 is the exitpoint of the dynamics and we assume it to be determined bythe stochastic evolution of the immature stages 0 le 120572 le 119864minus1
In the same spirit the action of each event on thepopulations modifies just the actual population and the nextone in the chain Hence
120575120572
119895=
minus1 120572 = 119895
+1 120572 = 119895 minus 1(56)
Following Section 3 [1198770
120572]119879
= (1199030119872 0 0) and
119865120573
120572=
minus119903120572 120573 = 120572
+119903120572 120573 = 120572 minus 1
0 otherwise(57)
Finally letting m119879= minus119872(1 1 1) we can write 119877
0
120572=
sum120573119865120573
120572119898
120573 In this framework the index pair (119895 120572) is highly
interconnected and we can achieve a complete descriptionusing just one index We use 0 le 119895 le 119864 minus 1 throughout thissection
The procedure of Section 3 gives Ψ(119911 119905) = prod119895(119891
119895)119898119895 =
prod119895(119891
minus119872
119895) while the dynamical system for 119891
119895reads recalling
the recasting in Remark 21119891119895= 119891
119895(minus119903
119895(119911
119895119891119895minus 1) + 119903
119895+1(119911
119895+1119891119895+1
minus 1))
119895 = 0 119864 minus 2
(58)
and 119891119864minus1
= 119891119864minus1
(minus119903119864minus1
(119911119864minus1
119891119864minus1
minus 1)) The overall initialcondition is 119891
119895(0) = 1 Let 120579
119895= log(119891
119895) and
120595119895=
119864minus1
sum
120573=119895
120579120573 119882
119895= exp (minus120595
119895) (59)
Note that 119882119895119891119895
= 119882119895
sdot exp(120579119895) = 119882
119895+1 Rewriting the
equations120579119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) + 119903
119895+1(119911
119895+1exp (120579
119895+1) minus 1)
119895 lt 119864 minus 1
119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) = 120579
119895+
119895+1 119895 lt 119864 minus 1
120579119864minus1
= 119864minus1
= minus119903119864minus1
(119911119864minus1
exp (120579119864minus1
) minus 1)
(60)
with initial condition 120579119895(0) = 0 for all 119895With this notation the
generating function reads Ψ(119911 119905) = 119882119872
0 and we formulate
hence an ODE for the 119882119895rsquos
119895+ 119903
119895119882
119895= 119903
119895119911119895119882
119895+1 119882
119895 (0) = 1 119895 = 0 119864 minus 2
119864minus1
= 119903119864minus1
(119911119864minus1
minus 119882119864minus1
) 119882119864minus1
(0) = 1
(61)
The Scientific World Journal 11
By defining the auxiliary quantity 119882119864(119905) = 1 the solution
for the 119882119895-differential equations can be written in recursive
form
119882119895(119905) = 119890
minus119903119895119905+ 119911
119895119903119895int
119905
0
119890119903119895(119904minus119905)
119882119895+1
(119904) 119889119904 119895 = 0 119864 minus 1
(62)
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820(119905) = (
119864minus1
prod
119898=0
119911119898)(1 minus
119864minus1
sum
119896=0
exp (minus119903119896119905)
119864minus1
prod
119895=0119895 = 119896
119903119895
119903119895minus 119903
119896
)
+
119864
sum
119896=1
prod119864minus119896
119895=0119911119895
119911119864minus119896
119903119864minus119896
119864minus119896
sum
119898=0
119903119898exp (minus119903
119898119905)
119864minus119896
prod
119897=0119897 =119898
119903119897
119903119897minus 119903
119898
(63)
The solution looksmore compact when all 119903119896are assumed
to take the common value 119903
1198820(119905) =
prod119864minus1
119898=0119911119898
(119864 minus 1)int
119903119905
0
exp (minus119904) 119904119864minus1
119889119904
+
119864
sum
119896=1
prod119864minus119896
119898=0119911119898
119911119864minus119896
(119903119905)119864minus119896 exp (minus119903119905)
(119864 minus 119896)
(64)
7 Developmental Cascades with Mortality
We add to the previous description 119864 subpopulation specificdeath events 119876
120572= 119902
120572119883
120572 where 120572 = 0 119864 minus 1 We
do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883
119864leaves the
present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876
0 119877
0 119876
1 119877
1 119876
119864minus1 and 119877
119864minus1 Then ((119876 119877)
0)119879
=
(1199020119872 119903
0119872 0 0) We have
1205752120572
120572= minus1 (death) 120572 lt 119864
1205752120572+1
120572= minus1
1205752120572+1
120572+1= +1
(development) 120572 lt 119864
119865120573
2120572=
minus119902120572 120573 = 2120572
minus119902120572 120573 = 2120572 + 1
119902120572 120573 = 2120572 minus 1
119865120573
2120572+1=
minus119903120572 120573 = 2120572 + 1
minus119903120572 120573 = 2120572
119903120572 120573 = 2120572 minus 1
120572 lt 119864
(65)
Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891
120572and the 119911
120572in pairs (119892 119891) and
(119911 119910) as (1198920 119891
0 119892
119864minus1 119891
119864minus1) and (119911
0 119910
0 119911
119864minus1 119910
119864minus1)
Also for notational convenience we will assume that thereexist 119902
119864= 119903
119864= 0 when necessary Defining
119867120572(119892
120572 119891
120572) = 119902
120572(119911
120572119892120572minus 1) + 119903
120572(119910
120572119891120572minus 1) (66)
we have (recall again Remark 21 and note that 119867119864equiv 0)
119892120572= minus119867
120572(119892
120572 119891
120572) 119892
120572
119891120572= (minus119867
120572(119892
120572 119891
120572) + 119867
120572+1(119892
120572+1 119891
120572+1)) 119891
120572
120572 = 0 119864 minus 2
119892119864minus1
119892119864minus1
=
119891119864minus1
119891119864minus1
= minus119867119864minus1
(119892119864minus1
119891119864minus1
)
(67)
The overall initial condition is still 119892120572(0) = 1 = 119891
120572(0)
Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881
119879
120572=
(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579
120572= log119892
120572
and 120601120572
= log119891120572 changing 119867
120572and the initial conditions
correspondingly Hence
120601120572= 120579
120572minus 120579
120572+1997904rArr 120601
120572= 120579
120572minus 120579
120572+1 120572 = 0 119864 minus 2
120579120572= minus119867
120572(120579
120572 120579
120572minus 120579
120572+1) 120572 = 0 119864 minus 2
120601119864minus1
= 120579119864minus1
= minus119867119864minus1
(120579119864minus1
120579119864minus1
)
(68)
The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879
= minus119872(1 0 0 0) satisfies 1198770
= 119865119898 FinallyΨ(119911 119910 119905) = 119892
minus119872
0 We now compute 119892
0(119905)
Let119882120572= 119890
minus120579120572 = 119892
minus1
120572 0 le 120572 le 119864minus1 Defining the auxiliary
variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of
119882 read
120572+ (119902
120572+ 119903
120572)119882
120572= 119902
120572119911120572+ 119903
120572119910120572119882
120572+1 119882
120572 (0) = 1
(69)
with solution
119882120572(119905) = 119890
minus(119902120572+119903120572)119905
+ int
119905
0
119890minus(119902120572+119903120572)(119905minus119904)
(119902120572119911120572+ 119903
120572119910120572119882
120572+1(119904)) 119889119904
(70)
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 The Scientific World Journal
Proof Rewrite the first equation (12) as (indices renamed)
119889
119889119905ln119909
120572= sum
120573
119865120572
120573(119909
120573minus 1) (23)
Then
sum
120572
V120572
119889
119889119905ln119909
120572= sum
120573
sum
120572
(V120572119865120572
120573) (119909
120573minus 1) = 0 (24)
Since sum120572(V
120572119865120572
120573) = 0 the expression in the lemma is the
exponential of the integral obtained from
sum
120572
V120572
119889
119889119905ln119909
120572= 0 (25)
Given relationship (1) between initial conditions eventsand populations from the probability generating function inevent space
Ψ (1199111 119911
119864 119904 119905) = sum
119899120572
(prod
120572
119911119899120572
120572)119875 (119899
1sdot sdot sdot 119899
119864 119904 119905) (26)
a corresponding generating function in population space canbe computed namely
Ψ (1199101 119910
119873 119904 119905) = sum
119883119896
(prod
119896
119910119883119896
119896)119875 (119883
1 119883
119873 119904 119905)
(27)
letting 119899120572 119883
119896 be shorthand for (119899
1sdot sdot sdot 119899
119864) (119883
1 119883
119873)
Lemma 13 (projection lemma) Equation 119911120572
= prod119873
119896=1119910120575120572
119896
119896
realises the transformation from event space to populationspace Moreover
Ψ (1199101 119910
119873 119904 119905) = (prod
119896
1199101198830
119896
119896)Ψ(119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896)Ψ(
119873
prod
119896=1
1199101205751
119896
119896
119873
prod
119896=1
119910120575119864
119896
119896 119904 119905)
(28)
Proof Taking logarithms on 119911 and 119910 it is clear that wheneverthere exist structural zeroes the transformation is noninvert-ible In this sense we call it a projection Concerning thestatement we have
(prod
119896
1199101198830
119896
119896)Ψ (119911
1 119911
119864 119904 119905)
= (prod
119896
1199101198830
119896
119896) sum
119899120572
(prod
120572
(
119873
prod
119896=1
119910120575120572
119896
119896)
119899120572
)119875 (119899120572 119904 119905)
= sum
119899120572
(
119873
prod
119896=1
1199101198830
119896+sum120572120575120572
119896119899120572
119896)119875 (119899
120572 119904 119905)
(29)
The statement is proved using (1) and rearranging the sumsas
119875 (1198831 119883
119873 119904 119905) = sum
119899120572
1015840
119875 (119899120572 119904 119905) (30)
where the prime denotes sum over 119899120572 such that 119883
0
119895+
sum120572120575120572
119895119899120572= 119883
119895
Corollary 14 The integrals of motion of Lemma 12 projectunder the projection Lemma to 119862 = 1
Proof The above substitution operated on prod119911V120572
120572= 119862 gives
119862 =
119873
prod
119896=1
119910(sum120572V120572120575120572
119896)
119896equiv
119873
prod
119896=1
1199100
119896= 1 (31)
33 Reduced Equation
Assumption 15 There exist integers119898120573120573 = 1 119864 such that
(cf (9))
1198770
120572= sum
120573
119865120573
120572119898
120573 (32)
This assumption amounts to considering that there exists away to mimic the initial condition on the rates with the samematrix 119865 involved in the time evolution
Theorem 16 Under Assumptions 2 4 and 15 equation(10) with natural initial condition Ψ(119911
1 119911
119864 119904 119904 ) =
Φ0(119911
1 119911
119864) = 1 has the solution
Ψ (1199111 119911
119864 119904 119905) = prod
120573
119891119898120573
120573 (33)
where 119911120573119891120573(119911 119904 119905) = 120601
120573(119911 119905 119905 minus 119904) in terms of the solutions
of (12) of Lemma 8 Hence 119891(119911 119904 119905) satisfies (after eliminatingthe auxiliary time 120591)
119889 log (119891120573 (119911 119904 119905))
119889119904
= minussum
120572
119865120573
120572(119904) (119911
120572119891120572(119911 119904 119905) minus 1) 119891
120573(119911 119905 119905) = 1
(34)
Proof The differences between 119891( ) and 120601( ) are (a) factoringout the initial condition 120601
120573(119911 119905 0) = 119911
120573and (b) rearranging
the time arguments since the argument in120601( ) is natural to thead hoc autonomous ODE (12) while the argument in 119891( ) isnatural to the (nonautonomous) PDE (10)
The proof is just a matter of rewriting the previousequations under the present assumptions Because of thenatural initial condition the desired solution to (10) isgiven by (20) in Lemma 9 which can be rewritten usingAssumption 15 as (we use 119911 as shorthand for 119911
1 119911
119864)
Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)
= prod
120573
(expint
119905
119904
sum
120572
119865120573
120572(120591) (120601
120572(119911 119905 119905 minus 120591) minus 1) 119889120591)
119898120573
(35)
The Scientific World Journal 7
Recalling that we defined 119911120572119891120572(119911 119904 119905) = 120601(119911 119905 119905 minus 119904) the
parenthesis can be recognised as the formal solution of (34)integrated from 119905 to 119904
Corollary 17 For v a structural zero of Lemma 12 thegenerating function is unmodified upon the substitution119898
120572rarr
119898120572+ 119896V
120572(119896 isin R)
Remark 18 Having structural zeroes it is amatter of choice toaddress the problem (a) using the variables 119891
120572and imposing
the additional constraints given by structural zeroes or (b)shifting to population coordinates where these constraints areautomatically incorporated Because of Assumption 2 andLemma 12 the transformation mapping one description tothe other corresponds to
119891120572= prod
119895
119908120575120572
119895
119895 (36)
However option (b) proves to bemore efficientwhen it comesto deal with approximate solutions (see Section 5)
4 Examples of Exact Solutions
41 Example I Pure Death Process In the case 119873 = 119864 = 1120575 = minus1 we have 119903 gt 0 119865 = minus119903(119904) and 119898 = minus119872 lt
0 (119872 is the initial population value) Equation (34) reads119889119891119889119904 = 119891119903(119911119891 minus 1) to be integrated in [119905 119904] with 119891(119911 119905 119905) =
1 Hence for constant death rate 119903 we obtain 119891(119911 119904 119905) =
[119911 + (1 minus 119911)119890minus119903(119905minus119904)
]minus1
and
Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)
+ 119911 (1 minus 119890minus119903(119905minus119904)
)]119872
(37)
42 Example II Linear Birth andDeath Process Wehave119873 =
1 119864 = 2 and 1205751
1= minus120575
2
1= 1 while 119903
1denotes the birth rate
and 1199032denotes the death rate 119883
0= 119898
1minus 119898
2is the initial
population There exists one structural zero with vector v =
(1 1)119879 while the corresponding constant of motion results in
the relations 11991111199112= 119862 and119891
11198912= 1The differential equation
(34) for 1198911(to be integrated in [119905 119904]) becomes
1198891198911
119889119904= (119903
111991111198912
1minus (119903
1+ 119903
2) 119891
1+
1198621199032
1199111
) 1198911 (119911 119905 119905) = 1
(38)
with solution (for119862 = 1whichwill be the case after projectionand assuming time-independent birth rates)
1198911(119911 119904 119905) =
1
1199111
(1199032minus 119903
11199111)119882 minus 119903
2(1 minus 119911
1)
(1199032minus 119903
11199111)119882 minus 119903
1(1 minus 119911
1) (39)
where 119882 = exp(1199032
minus 1199031)(119905 minus 119904) Finally Ψ(119911 119904 119905) =
(1198911(119911 119904 119905))
1198981minus1198982 which after projecting in population space
using Lemma 13 and Corollary 14 gives (cf [41])
Ψ (119910 119904 119905) = [(119903
2minus 119903
1119910)119882 minus 119903
2(1 minus 119910)
(1199032minus 119903
1119910)119882 minus 119903
1(1 minus 119910)
]
1198981minus1198982
(40)
Remark 19 A pure death linear process corresponds to abinomial distribution a pure birth process corresponds to anegative binomial and a birth-death process corresponds toa combination of both as independent processes
5 Consistent Approximations
When (34) of Theorem 16 cannot be solved exactly we haveto rely on approximate solutions It is desirable to workwith consistent approximations namely those where basicproperties of the problem are fulfilled exactly rather than ldquoupto an error of size 120598rdquo For example since our solutions shouldbe probabilities we want the coefficients of Ψ(119911
1 119911
119864 119905)
to be always nonnegative and sum up to one Ideally wewant approximations to be constrained to satisfy Lemma 12and to preserve positive invariance in population space (nojumps into negative populations) Whenever consistency isnot automatic it is mandatory to analyse the approximatesolution before jumping into conclusions
51 Poisson Approximation Let us replace 119891120573
equiv 1 in theRHS of (34) (this amounts to propose that 119891
120573is constant and
equal to its initial condition) The solution of the modifiedequation will be a good approximation to the exact solutionfor sufficiently short times such that 119891 is not significantlydifferent from one
For constant rates 119903120572the rhs of (34) does not depend on
time The approximate solution reads 119891120572(119911 119904 119905) = exp((119905 minus
119904)sum120573119865120572
120573(119911
120573minus 1)) and
Ψ (1199111 119911
119864 119904 119905) =
119864
prod
120572
119890119898120572(119905minus119904)sum
120573119865120572
120573(119911120573minus1)
= 119890(119905minus119904)sum
1205731198770
120573(119911120573minus1)
(41)
In other words the generating function is approximatedas a product of individual Poisson generating functions
This approximation has an error of 119900(119905) Simple as it looksthis approach gives a good probability generating functionthat respects structural zeroes but it does not guaranteepositive invariance a Poisson jump could throw the systeminto negative populations However the probability of sucha jump is also 119900(119905) Fortunately there is a natural way toimpose positive invariance and also a natural way to improvethe error up to 119900(119905)
32 in the broader framework of generaltransition rates [6] In this way the approximation can besafely used with error control to address problems that resistexact computation one way or the other
52 Consistent Multinomial Approximations A systematicapproach to obtain a chain of approximations of increasingaccuracy to (34) can be devised via Picard iterations Firstly itis convenient to recast the problem in population coordinates(cf Remark 18) so that the structural zeroes are automaticallytaken into account regardless of other properties of the
8 The Scientific World Journal
adopted approximation From 119891120572
= prod119895119908
120575120572
119895
119895we obtain
119891120572119891
120572= sum
119896120575120572
119896119896119908
119896and (34) can be recast as
sum
119896
120575120572
119896(
119896
119908119896
+ sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1)) = 0
119908119896 (119911 119905 119905) = 1
(42)
Because of the linear independence of population increments(and hence of populations) Lemma 3(b) the counterpart of(34) in population coordinates reads
119896= minus119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896(119911 119905 119905) = 1
(43)
Remark 20 In the case of linearly dependent populationincrements (34) still holds while it is possible to elaboratea more general form in terms of the 119908
119895instead of (43) This
is however outside the scope of this paper
Since sum120572120575120572
119895119898
120572= 119883
119895(119904) we obtain Ψ(119911
1 119911
119864 119904 119905) =
prod119895119908
119883119895(119904)
119895 which is so far an exact result Let us produce
some approximated values for the 119908119896rsquos and correspondingly
for Ψ(1199111 119911
119864 119904 119905)
To avoid further notational complications wewill assumein the sequel that 119903
119896
120573is time independent Hence 119908
119896will
depend only on the time difference 119905 minus 119904 Equation (43) canbe recast as
119896= 119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896 (119911 0) = 1
(44)
now to be integrated along the interval [0 119905 minus 119904] We will evenset 119904 = 0 in what follows
Remark 21 For the case of time-independent proportionalityfactors 119903119896
120573 a correspondingmodification can be introduced in
(34) namely overall change of sign integration in [0 119905 minus 119904]and setting 119904 = 0 to reduce the notational burden
Equations (44) satisfy the Picard-Lindelof theorem sincethe RHS is a Lipschitz function Then the Picard iterationscheme converges to the (unique) solution Taking advantage
of the initial condition the 119899th order truncated version of thePicard iterations reads
119908(0)
119896(ℎ) = 1
119908(1)
119896(ℎ) = 1 + int
ℎ
0
119908(0)
119896(119905)sum
120573
119903119896
120573(119911
120573prod
119895
(119908(0)
119895(119905))
120575120573
119895
minus 1)119889119905
= 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1)
119908(119899)
119896(ℎ) = 1 + int
ℎ
0
119889119905[
[
119908(119899minus1)
119896(119905)
timessum
120573
119903119896
120573(119911
120573prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
minus 1)]
]119899minus1
(45)
where [sdot sdot sdot ]119898denote the Maclaurin polynomial of order119898 in
119905
Lemma 22 For 119905 isin [0 ℎ] ℎ gt 0 being sufficiently smalland for each order of approximation 119899 ge 0 119908(119899)
119896(119905) in (44)
is a polynomial generating function for one individual of thepopulation that is it has the following properties
(a) 119908(119899)
119896(119905) is a polynomial of degree 119899 in 119911
1 119911
119864 where
the coefficient of 11991111989911 119911
119899119864
119864is 119874(119905
119901) with 119901 = 119899
1+
sdot sdot sdot + 119899119864
(b) 119908(119899)
119896(1 1 119905) = 1
(c) Δ(119899)
119896(ℎ) = 119908
(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ) = 119874(ℎ
119899)
(d) the coefficients of 119908(119899)
119896(119905) regarded as a polynomial in
119911120572 are nonnegative functions of time
Proof (a) The statement holds for 119899 = 0 and 119899 = 1Assuming that it holds up to order 119899 minus 1 we have that[119908
(119899minus1)
119896(119905)prod
119895(119908
(119899minus1)
119895(119905))
120575120573
119895 ]119899minus1
is also a polynomial of degree119899 minus 1 in 119911
1 119911
119864with time-dependent coefficients of type
119874(119905119901) since each power 119901 of 119905 carries along a power 119901 in 119911
120572
(as well as lower 119911-powers) A Picard iteration step gives
119908(119899)
119896(119905) = 1 + sum
120573
119903119896
120573119911120573int
ℎ
0
[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
119908(119899minus1)
119896(119905) 119889119905
(46)
The first sum contributes with one time integration and one119911-power which exactly preserves the polynomial structureand the order of the coefficientsThe result is a polynomial ofdegree 119899 in 119911
1 119911
119864 where the coefficients are polynomials
of highest degree at most 119899 in ℎ each one of lowest order119874(ℎ
119901) (as in the statement) The second integral does not
The Scientific World Journal 9
alter these properties since it generates again a polynomial ofdegree 119899minus1 in 119911
1 119911
119864 with time-dependent coefficients of
order119874(ℎ119901+1
) or higher up to ℎ119899 (they are just the coefficients
of 119908(119899minus1)
119896(119905) integrated once in time)
(b) Letting 1199111= 119911
2= sdot sdot sdot = 119911
119864= 1 Picard iteration gives
trivially 119908(119899)
119896(119905) = 1 to all orders again by induction
(c) The statement holds for 119899 = 1 Assuming it holds upto 119899 minus 1 we compute
119908(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ)
= sum
120573
119903119896
120573119911120573
times int
ℎ
0
([(119908(119899minus2)
119896(119905) + Δ
(119899minus1)
119896(119905))
times prod
119895
(119908(119899minus2)
119895(119905) + Δ
(119899minus1)
119895(119905))
120575120573
119895
]
119899minus1
minus[
[
119908(119899minus2)
119896(119905)prod
119895
(119908(119899minus2)
119895(119905))
120575120573
119895]
]119899minus2
)119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
(119908(119899minus1)
119896(119905) minus 119908
(119899minus2)
119896(119905)) 119889119905
(47)
Both expressions in the first integral have the sameMaclaurinpolynomial up to order (119899 minus 2) Hence the difference in thefirst integral contains only terms of order119874(119905
119899minus1) This is also
the case for the second integral After integration we obtainΔ(119899)
119896(ℎ) = 119874(ℎ
119899)
(d) The statement holds for 119899 = 0 and 119899 = 1 and ingeneral for the zero-order coefficient because it is unity for119905 = 0 as a consequence of the initial condition For thehigher-order coefficients we use induction again If all 119911-coefficients in 119908
(119899minus1)
119896(119905) are nonnegative then the same holds
automatically for 119908(119899)
119896(ℎ) up to order 119899 minus 1 since the 119874(ℎ
119899)
contribution adding to each inherited coefficient from theprevious step does not alter its sign for ℎ sufficiently small Itremains to show that the coefficients corresponding to 119901 = 119899
are nonnegative These contributions arise from the (119899 minus 1)thorder of
119903119896
120573119911120573[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
(48)
after time integration If all 120575120573119895are positive then the expres-
sion above is a product of 119911-polynomials with nonnegative
coefficients and the statement follows Suppose that some120575120573
119895= minus1 Then by Lemma 7 119903119896
120573= 0 for 119896 = 119895 Hence
119903119896
120573119911120573119908
(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
=
0 119895 = 119896
119903119896
120573119911120573prod
119895 = 119896
(119908(119899minus1)
119895(119905))
120575120573
119895
119895 = 119896
(49)
However 120575120573119895
= minus 1 for 119895 = 119896 that is they are nonnegative andhence the expression has only nonnegative coefficients
521 Basic Properties of the First- and Second-OrderApproximations Let us analyse the generating functionΨ(119911
1 119911
119864 119905) = prod
119895119908
119883119895(0)
119895 The simplest nontrivial
applications of the approximation scheme are the first- andsecond-order approximations Expanding explicitly up tosecond-order we get
119908119896(ℎ) = 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1) +
ℎ2
2(sum
120573
119903119896
120573(119911
120573minus 1))
times (sum
120574
119903119896
120574(119911
120574minus 1))
+ℎ2
2sum
120573
119903119896
120573119911120573sum
120574
119865120573
120574(119911
120574minus 1)
(50)
We begin by noticing that the first-order approximation isobtained by neglecting all terms in ℎ
2 in the above equationWhenever 119908
119895is linear in 119911 which is always the case for the
first-order approximation the resulting generating functionΨ corresponds to the product of (independent) multinomialdistributions for the events affecting each subpopulation
Let us now consider Ψ(1199111 119911
119864 119905) computed with the
second-order approximated 119908119895rsquos to perform one simulation
step of size ℎCounting powers of 119911
120574on each 119908
119896we have (a) the
probability of no events occurring in time-step ℎ
119875119896(ℎ 0) = 1 minus ℎ(sum
120573
119903119896
120573) +
ℎ2
2(sum
120573
119903119896
120573)
2
(51)
(b) the probability of only one (120573) event occurring in time-step ℎ
119875119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus ℎ
2sum
120572
119903119896
120572minus
ℎ2
2sum
120572
119865120573
120572 (52)
and (c) the probability of another event (120572) occurring afterthe first one
119875119896(ℎ 119868119868 120573 120572) =
ℎ2
2119903119896
120573(119903
119896
120572+ 119865
120573
120572) (53)
10 The Scientific World Journal
522 Simulation Algorithm A simulation step can be oper-ated in the following way Let
sum
120572
(119875119896(ℎ 119868119868 120573 120572)) + 119875
119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus
ℎ2
2sum
120572
119903119896
120572
= 119875119896(ℎ 119868 120573)
(54)
be the probability of one (120573) event occurring alone orfollowed by a second event It follows that
119875119896(ℎ 119868119868 120573 120572) = 119875
119896(ℎ 119868 120573)
ℎ
2
(119903120572
119896+ 119865
120573
120572)
(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ
2)
(55)
The second factor above is the conditional probability119875119896(ℎ 120572120573) for the second event being 120572 given that there was
a first (120573) eventThe algorithm proceeds in two steps First a multino-
mial deviate is computed with probabilities 119875119896(ℎ 119868 120573) (the
probability of no event occurring is still 119875119896(ℎ 0)) This gives
the number 119899120573of occurrences of each event In the second
step for each 119899120573we compute the deviates for the occurrence
of a second event or no more events with the multinomialgiven by the array of probabilities 119875
119896(ℎ 120572120573) defined above
(the probability of no second event occurring is here 1 minus
(ℎ2)(sum120572(119903
120572
119896+ 119865
120573
120572)(1 minus (ℎ2)sum
120572119903119896
120572)))
6 Implementation Developmental Cascadeswithout Mortality
In this and the following Sections we apply the tools devel-oped up to now to the description of a subprocess in insectdevelopment namely the development of immature larvaewhich is in principle a highly individual subprocess Weassume that the evolution of an insect from egg to pupaoccurs via a sequence of 119864 gt 0 maturation stages Biologistsrecognise some substages by inspection in the developmentof larvae (instars) and even more stages when observed byother methods [35] The immature individual may die alongthe process or otherwise complete it and exit as pupa oradult The actual value of apparent maturation stages 119864 willdepend on environmental and experimental conditions andwill ultimately be specified when analying experimental datain Section 71
Empirical evidence based on existing and new experi-mental results as well as biological insight produced by thepresent description will be the subject of an independentwork [42]We discuss however a couple of examples from theliterature in Section 71
The events for this process are maturation events pro-moting individuals from one subpopulation into to the nextand death events for all subpopulations In other wordseach subpopulation represents a given (intermediate) level ofdevelopment and maturation events correspond to progressin the development Development at level 119895 promotes oneindividual to level 119895 + 1 and the event rate is linear in the119895-subpopulation Hence in 119877
120572= sum
119895119903119896
120572119883
119895 the matrix 119903
is square and diagonal for 0 le 120572 le 119864 minus 1 (we haveexactly 119864 + 1 subpopulations counted from 0 to 119864) All ofthe environmental influence is at this point incorporatedthrough 119903
Moreover level 119864 is the matured pupa and we may set119903119864
= 0 since at pupation stage the present mechanism endsand new processes take place Subpopulation 119864 is the exitpoint of the dynamics and we assume it to be determined bythe stochastic evolution of the immature stages 0 le 120572 le 119864minus1
In the same spirit the action of each event on thepopulations modifies just the actual population and the nextone in the chain Hence
120575120572
119895=
minus1 120572 = 119895
+1 120572 = 119895 minus 1(56)
Following Section 3 [1198770
120572]119879
= (1199030119872 0 0) and
119865120573
120572=
minus119903120572 120573 = 120572
+119903120572 120573 = 120572 minus 1
0 otherwise(57)
Finally letting m119879= minus119872(1 1 1) we can write 119877
0
120572=
sum120573119865120573
120572119898
120573 In this framework the index pair (119895 120572) is highly
interconnected and we can achieve a complete descriptionusing just one index We use 0 le 119895 le 119864 minus 1 throughout thissection
The procedure of Section 3 gives Ψ(119911 119905) = prod119895(119891
119895)119898119895 =
prod119895(119891
minus119872
119895) while the dynamical system for 119891
119895reads recalling
the recasting in Remark 21119891119895= 119891
119895(minus119903
119895(119911
119895119891119895minus 1) + 119903
119895+1(119911
119895+1119891119895+1
minus 1))
119895 = 0 119864 minus 2
(58)
and 119891119864minus1
= 119891119864minus1
(minus119903119864minus1
(119911119864minus1
119891119864minus1
minus 1)) The overall initialcondition is 119891
119895(0) = 1 Let 120579
119895= log(119891
119895) and
120595119895=
119864minus1
sum
120573=119895
120579120573 119882
119895= exp (minus120595
119895) (59)
Note that 119882119895119891119895
= 119882119895
sdot exp(120579119895) = 119882
119895+1 Rewriting the
equations120579119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) + 119903
119895+1(119911
119895+1exp (120579
119895+1) minus 1)
119895 lt 119864 minus 1
119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) = 120579
119895+
119895+1 119895 lt 119864 minus 1
120579119864minus1
= 119864minus1
= minus119903119864minus1
(119911119864minus1
exp (120579119864minus1
) minus 1)
(60)
with initial condition 120579119895(0) = 0 for all 119895With this notation the
generating function reads Ψ(119911 119905) = 119882119872
0 and we formulate
hence an ODE for the 119882119895rsquos
119895+ 119903
119895119882
119895= 119903
119895119911119895119882
119895+1 119882
119895 (0) = 1 119895 = 0 119864 minus 2
119864minus1
= 119903119864minus1
(119911119864minus1
minus 119882119864minus1
) 119882119864minus1
(0) = 1
(61)
The Scientific World Journal 11
By defining the auxiliary quantity 119882119864(119905) = 1 the solution
for the 119882119895-differential equations can be written in recursive
form
119882119895(119905) = 119890
minus119903119895119905+ 119911
119895119903119895int
119905
0
119890119903119895(119904minus119905)
119882119895+1
(119904) 119889119904 119895 = 0 119864 minus 1
(62)
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820(119905) = (
119864minus1
prod
119898=0
119911119898)(1 minus
119864minus1
sum
119896=0
exp (minus119903119896119905)
119864minus1
prod
119895=0119895 = 119896
119903119895
119903119895minus 119903
119896
)
+
119864
sum
119896=1
prod119864minus119896
119895=0119911119895
119911119864minus119896
119903119864minus119896
119864minus119896
sum
119898=0
119903119898exp (minus119903
119898119905)
119864minus119896
prod
119897=0119897 =119898
119903119897
119903119897minus 119903
119898
(63)
The solution looksmore compact when all 119903119896are assumed
to take the common value 119903
1198820(119905) =
prod119864minus1
119898=0119911119898
(119864 minus 1)int
119903119905
0
exp (minus119904) 119904119864minus1
119889119904
+
119864
sum
119896=1
prod119864minus119896
119898=0119911119898
119911119864minus119896
(119903119905)119864minus119896 exp (minus119903119905)
(119864 minus 119896)
(64)
7 Developmental Cascades with Mortality
We add to the previous description 119864 subpopulation specificdeath events 119876
120572= 119902
120572119883
120572 where 120572 = 0 119864 minus 1 We
do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883
119864leaves the
present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876
0 119877
0 119876
1 119877
1 119876
119864minus1 and 119877
119864minus1 Then ((119876 119877)
0)119879
=
(1199020119872 119903
0119872 0 0) We have
1205752120572
120572= minus1 (death) 120572 lt 119864
1205752120572+1
120572= minus1
1205752120572+1
120572+1= +1
(development) 120572 lt 119864
119865120573
2120572=
minus119902120572 120573 = 2120572
minus119902120572 120573 = 2120572 + 1
119902120572 120573 = 2120572 minus 1
119865120573
2120572+1=
minus119903120572 120573 = 2120572 + 1
minus119903120572 120573 = 2120572
119903120572 120573 = 2120572 minus 1
120572 lt 119864
(65)
Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891
120572and the 119911
120572in pairs (119892 119891) and
(119911 119910) as (1198920 119891
0 119892
119864minus1 119891
119864minus1) and (119911
0 119910
0 119911
119864minus1 119910
119864minus1)
Also for notational convenience we will assume that thereexist 119902
119864= 119903
119864= 0 when necessary Defining
119867120572(119892
120572 119891
120572) = 119902
120572(119911
120572119892120572minus 1) + 119903
120572(119910
120572119891120572minus 1) (66)
we have (recall again Remark 21 and note that 119867119864equiv 0)
119892120572= minus119867
120572(119892
120572 119891
120572) 119892
120572
119891120572= (minus119867
120572(119892
120572 119891
120572) + 119867
120572+1(119892
120572+1 119891
120572+1)) 119891
120572
120572 = 0 119864 minus 2
119892119864minus1
119892119864minus1
=
119891119864minus1
119891119864minus1
= minus119867119864minus1
(119892119864minus1
119891119864minus1
)
(67)
The overall initial condition is still 119892120572(0) = 1 = 119891
120572(0)
Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881
119879
120572=
(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579
120572= log119892
120572
and 120601120572
= log119891120572 changing 119867
120572and the initial conditions
correspondingly Hence
120601120572= 120579
120572minus 120579
120572+1997904rArr 120601
120572= 120579
120572minus 120579
120572+1 120572 = 0 119864 minus 2
120579120572= minus119867
120572(120579
120572 120579
120572minus 120579
120572+1) 120572 = 0 119864 minus 2
120601119864minus1
= 120579119864minus1
= minus119867119864minus1
(120579119864minus1
120579119864minus1
)
(68)
The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879
= minus119872(1 0 0 0) satisfies 1198770
= 119865119898 FinallyΨ(119911 119910 119905) = 119892
minus119872
0 We now compute 119892
0(119905)
Let119882120572= 119890
minus120579120572 = 119892
minus1
120572 0 le 120572 le 119864minus1 Defining the auxiliary
variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of
119882 read
120572+ (119902
120572+ 119903
120572)119882
120572= 119902
120572119911120572+ 119903
120572119910120572119882
120572+1 119882
120572 (0) = 1
(69)
with solution
119882120572(119905) = 119890
minus(119902120572+119903120572)119905
+ int
119905
0
119890minus(119902120572+119903120572)(119905minus119904)
(119902120572119911120572+ 119903
120572119910120572119882
120572+1(119904)) 119889119904
(70)
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 7
Recalling that we defined 119911120572119891120572(119911 119904 119905) = 120601(119911 119905 119905 minus 119904) the
parenthesis can be recognised as the formal solution of (34)integrated from 119905 to 119904
Corollary 17 For v a structural zero of Lemma 12 thegenerating function is unmodified upon the substitution119898
120572rarr
119898120572+ 119896V
120572(119896 isin R)
Remark 18 Having structural zeroes it is amatter of choice toaddress the problem (a) using the variables 119891
120572and imposing
the additional constraints given by structural zeroes or (b)shifting to population coordinates where these constraints areautomatically incorporated Because of Assumption 2 andLemma 12 the transformation mapping one description tothe other corresponds to
119891120572= prod
119895
119908120575120572
119895
119895 (36)
However option (b) proves to bemore efficientwhen it comesto deal with approximate solutions (see Section 5)
4 Examples of Exact Solutions
41 Example I Pure Death Process In the case 119873 = 119864 = 1120575 = minus1 we have 119903 gt 0 119865 = minus119903(119904) and 119898 = minus119872 lt
0 (119872 is the initial population value) Equation (34) reads119889119891119889119904 = 119891119903(119911119891 minus 1) to be integrated in [119905 119904] with 119891(119911 119905 119905) =
1 Hence for constant death rate 119903 we obtain 119891(119911 119904 119905) =
[119911 + (1 minus 119911)119890minus119903(119905minus119904)
]minus1
and
Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)
+ 119911 (1 minus 119890minus119903(119905minus119904)
)]119872
(37)
42 Example II Linear Birth andDeath Process Wehave119873 =
1 119864 = 2 and 1205751
1= minus120575
2
1= 1 while 119903
1denotes the birth rate
and 1199032denotes the death rate 119883
0= 119898
1minus 119898
2is the initial
population There exists one structural zero with vector v =
(1 1)119879 while the corresponding constant of motion results in
the relations 11991111199112= 119862 and119891
11198912= 1The differential equation
(34) for 1198911(to be integrated in [119905 119904]) becomes
1198891198911
119889119904= (119903
111991111198912
1minus (119903
1+ 119903
2) 119891
1+
1198621199032
1199111
) 1198911 (119911 119905 119905) = 1
(38)
with solution (for119862 = 1whichwill be the case after projectionand assuming time-independent birth rates)
1198911(119911 119904 119905) =
1
1199111
(1199032minus 119903
11199111)119882 minus 119903
2(1 minus 119911
1)
(1199032minus 119903
11199111)119882 minus 119903
1(1 minus 119911
1) (39)
where 119882 = exp(1199032
minus 1199031)(119905 minus 119904) Finally Ψ(119911 119904 119905) =
(1198911(119911 119904 119905))
1198981minus1198982 which after projecting in population space
using Lemma 13 and Corollary 14 gives (cf [41])
Ψ (119910 119904 119905) = [(119903
2minus 119903
1119910)119882 minus 119903
2(1 minus 119910)
(1199032minus 119903
1119910)119882 minus 119903
1(1 minus 119910)
]
1198981minus1198982
(40)
Remark 19 A pure death linear process corresponds to abinomial distribution a pure birth process corresponds to anegative binomial and a birth-death process corresponds toa combination of both as independent processes
5 Consistent Approximations
When (34) of Theorem 16 cannot be solved exactly we haveto rely on approximate solutions It is desirable to workwith consistent approximations namely those where basicproperties of the problem are fulfilled exactly rather than ldquoupto an error of size 120598rdquo For example since our solutions shouldbe probabilities we want the coefficients of Ψ(119911
1 119911
119864 119905)
to be always nonnegative and sum up to one Ideally wewant approximations to be constrained to satisfy Lemma 12and to preserve positive invariance in population space (nojumps into negative populations) Whenever consistency isnot automatic it is mandatory to analyse the approximatesolution before jumping into conclusions
51 Poisson Approximation Let us replace 119891120573
equiv 1 in theRHS of (34) (this amounts to propose that 119891
120573is constant and
equal to its initial condition) The solution of the modifiedequation will be a good approximation to the exact solutionfor sufficiently short times such that 119891 is not significantlydifferent from one
For constant rates 119903120572the rhs of (34) does not depend on
time The approximate solution reads 119891120572(119911 119904 119905) = exp((119905 minus
119904)sum120573119865120572
120573(119911
120573minus 1)) and
Ψ (1199111 119911
119864 119904 119905) =
119864
prod
120572
119890119898120572(119905minus119904)sum
120573119865120572
120573(119911120573minus1)
= 119890(119905minus119904)sum
1205731198770
120573(119911120573minus1)
(41)
In other words the generating function is approximatedas a product of individual Poisson generating functions
This approximation has an error of 119900(119905) Simple as it looksthis approach gives a good probability generating functionthat respects structural zeroes but it does not guaranteepositive invariance a Poisson jump could throw the systeminto negative populations However the probability of sucha jump is also 119900(119905) Fortunately there is a natural way toimpose positive invariance and also a natural way to improvethe error up to 119900(119905)
32 in the broader framework of generaltransition rates [6] In this way the approximation can besafely used with error control to address problems that resistexact computation one way or the other
52 Consistent Multinomial Approximations A systematicapproach to obtain a chain of approximations of increasingaccuracy to (34) can be devised via Picard iterations Firstly itis convenient to recast the problem in population coordinates(cf Remark 18) so that the structural zeroes are automaticallytaken into account regardless of other properties of the
8 The Scientific World Journal
adopted approximation From 119891120572
= prod119895119908
120575120572
119895
119895we obtain
119891120572119891
120572= sum
119896120575120572
119896119896119908
119896and (34) can be recast as
sum
119896
120575120572
119896(
119896
119908119896
+ sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1)) = 0
119908119896 (119911 119905 119905) = 1
(42)
Because of the linear independence of population increments(and hence of populations) Lemma 3(b) the counterpart of(34) in population coordinates reads
119896= minus119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896(119911 119905 119905) = 1
(43)
Remark 20 In the case of linearly dependent populationincrements (34) still holds while it is possible to elaboratea more general form in terms of the 119908
119895instead of (43) This
is however outside the scope of this paper
Since sum120572120575120572
119895119898
120572= 119883
119895(119904) we obtain Ψ(119911
1 119911
119864 119904 119905) =
prod119895119908
119883119895(119904)
119895 which is so far an exact result Let us produce
some approximated values for the 119908119896rsquos and correspondingly
for Ψ(1199111 119911
119864 119904 119905)
To avoid further notational complications wewill assumein the sequel that 119903
119896
120573is time independent Hence 119908
119896will
depend only on the time difference 119905 minus 119904 Equation (43) canbe recast as
119896= 119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896 (119911 0) = 1
(44)
now to be integrated along the interval [0 119905 minus 119904] We will evenset 119904 = 0 in what follows
Remark 21 For the case of time-independent proportionalityfactors 119903119896
120573 a correspondingmodification can be introduced in
(34) namely overall change of sign integration in [0 119905 minus 119904]and setting 119904 = 0 to reduce the notational burden
Equations (44) satisfy the Picard-Lindelof theorem sincethe RHS is a Lipschitz function Then the Picard iterationscheme converges to the (unique) solution Taking advantage
of the initial condition the 119899th order truncated version of thePicard iterations reads
119908(0)
119896(ℎ) = 1
119908(1)
119896(ℎ) = 1 + int
ℎ
0
119908(0)
119896(119905)sum
120573
119903119896
120573(119911
120573prod
119895
(119908(0)
119895(119905))
120575120573
119895
minus 1)119889119905
= 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1)
119908(119899)
119896(ℎ) = 1 + int
ℎ
0
119889119905[
[
119908(119899minus1)
119896(119905)
timessum
120573
119903119896
120573(119911
120573prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
minus 1)]
]119899minus1
(45)
where [sdot sdot sdot ]119898denote the Maclaurin polynomial of order119898 in
119905
Lemma 22 For 119905 isin [0 ℎ] ℎ gt 0 being sufficiently smalland for each order of approximation 119899 ge 0 119908(119899)
119896(119905) in (44)
is a polynomial generating function for one individual of thepopulation that is it has the following properties
(a) 119908(119899)
119896(119905) is a polynomial of degree 119899 in 119911
1 119911
119864 where
the coefficient of 11991111989911 119911
119899119864
119864is 119874(119905
119901) with 119901 = 119899
1+
sdot sdot sdot + 119899119864
(b) 119908(119899)
119896(1 1 119905) = 1
(c) Δ(119899)
119896(ℎ) = 119908
(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ) = 119874(ℎ
119899)
(d) the coefficients of 119908(119899)
119896(119905) regarded as a polynomial in
119911120572 are nonnegative functions of time
Proof (a) The statement holds for 119899 = 0 and 119899 = 1Assuming that it holds up to order 119899 minus 1 we have that[119908
(119899minus1)
119896(119905)prod
119895(119908
(119899minus1)
119895(119905))
120575120573
119895 ]119899minus1
is also a polynomial of degree119899 minus 1 in 119911
1 119911
119864with time-dependent coefficients of type
119874(119905119901) since each power 119901 of 119905 carries along a power 119901 in 119911
120572
(as well as lower 119911-powers) A Picard iteration step gives
119908(119899)
119896(119905) = 1 + sum
120573
119903119896
120573119911120573int
ℎ
0
[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
119908(119899minus1)
119896(119905) 119889119905
(46)
The first sum contributes with one time integration and one119911-power which exactly preserves the polynomial structureand the order of the coefficientsThe result is a polynomial ofdegree 119899 in 119911
1 119911
119864 where the coefficients are polynomials
of highest degree at most 119899 in ℎ each one of lowest order119874(ℎ
119901) (as in the statement) The second integral does not
The Scientific World Journal 9
alter these properties since it generates again a polynomial ofdegree 119899minus1 in 119911
1 119911
119864 with time-dependent coefficients of
order119874(ℎ119901+1
) or higher up to ℎ119899 (they are just the coefficients
of 119908(119899minus1)
119896(119905) integrated once in time)
(b) Letting 1199111= 119911
2= sdot sdot sdot = 119911
119864= 1 Picard iteration gives
trivially 119908(119899)
119896(119905) = 1 to all orders again by induction
(c) The statement holds for 119899 = 1 Assuming it holds upto 119899 minus 1 we compute
119908(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ)
= sum
120573
119903119896
120573119911120573
times int
ℎ
0
([(119908(119899minus2)
119896(119905) + Δ
(119899minus1)
119896(119905))
times prod
119895
(119908(119899minus2)
119895(119905) + Δ
(119899minus1)
119895(119905))
120575120573
119895
]
119899minus1
minus[
[
119908(119899minus2)
119896(119905)prod
119895
(119908(119899minus2)
119895(119905))
120575120573
119895]
]119899minus2
)119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
(119908(119899minus1)
119896(119905) minus 119908
(119899minus2)
119896(119905)) 119889119905
(47)
Both expressions in the first integral have the sameMaclaurinpolynomial up to order (119899 minus 2) Hence the difference in thefirst integral contains only terms of order119874(119905
119899minus1) This is also
the case for the second integral After integration we obtainΔ(119899)
119896(ℎ) = 119874(ℎ
119899)
(d) The statement holds for 119899 = 0 and 119899 = 1 and ingeneral for the zero-order coefficient because it is unity for119905 = 0 as a consequence of the initial condition For thehigher-order coefficients we use induction again If all 119911-coefficients in 119908
(119899minus1)
119896(119905) are nonnegative then the same holds
automatically for 119908(119899)
119896(ℎ) up to order 119899 minus 1 since the 119874(ℎ
119899)
contribution adding to each inherited coefficient from theprevious step does not alter its sign for ℎ sufficiently small Itremains to show that the coefficients corresponding to 119901 = 119899
are nonnegative These contributions arise from the (119899 minus 1)thorder of
119903119896
120573119911120573[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
(48)
after time integration If all 120575120573119895are positive then the expres-
sion above is a product of 119911-polynomials with nonnegative
coefficients and the statement follows Suppose that some120575120573
119895= minus1 Then by Lemma 7 119903119896
120573= 0 for 119896 = 119895 Hence
119903119896
120573119911120573119908
(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
=
0 119895 = 119896
119903119896
120573119911120573prod
119895 = 119896
(119908(119899minus1)
119895(119905))
120575120573
119895
119895 = 119896
(49)
However 120575120573119895
= minus 1 for 119895 = 119896 that is they are nonnegative andhence the expression has only nonnegative coefficients
521 Basic Properties of the First- and Second-OrderApproximations Let us analyse the generating functionΨ(119911
1 119911
119864 119905) = prod
119895119908
119883119895(0)
119895 The simplest nontrivial
applications of the approximation scheme are the first- andsecond-order approximations Expanding explicitly up tosecond-order we get
119908119896(ℎ) = 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1) +
ℎ2
2(sum
120573
119903119896
120573(119911
120573minus 1))
times (sum
120574
119903119896
120574(119911
120574minus 1))
+ℎ2
2sum
120573
119903119896
120573119911120573sum
120574
119865120573
120574(119911
120574minus 1)
(50)
We begin by noticing that the first-order approximation isobtained by neglecting all terms in ℎ
2 in the above equationWhenever 119908
119895is linear in 119911 which is always the case for the
first-order approximation the resulting generating functionΨ corresponds to the product of (independent) multinomialdistributions for the events affecting each subpopulation
Let us now consider Ψ(1199111 119911
119864 119905) computed with the
second-order approximated 119908119895rsquos to perform one simulation
step of size ℎCounting powers of 119911
120574on each 119908
119896we have (a) the
probability of no events occurring in time-step ℎ
119875119896(ℎ 0) = 1 minus ℎ(sum
120573
119903119896
120573) +
ℎ2
2(sum
120573
119903119896
120573)
2
(51)
(b) the probability of only one (120573) event occurring in time-step ℎ
119875119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus ℎ
2sum
120572
119903119896
120572minus
ℎ2
2sum
120572
119865120573
120572 (52)
and (c) the probability of another event (120572) occurring afterthe first one
119875119896(ℎ 119868119868 120573 120572) =
ℎ2
2119903119896
120573(119903
119896
120572+ 119865
120573
120572) (53)
10 The Scientific World Journal
522 Simulation Algorithm A simulation step can be oper-ated in the following way Let
sum
120572
(119875119896(ℎ 119868119868 120573 120572)) + 119875
119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus
ℎ2
2sum
120572
119903119896
120572
= 119875119896(ℎ 119868 120573)
(54)
be the probability of one (120573) event occurring alone orfollowed by a second event It follows that
119875119896(ℎ 119868119868 120573 120572) = 119875
119896(ℎ 119868 120573)
ℎ
2
(119903120572
119896+ 119865
120573
120572)
(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ
2)
(55)
The second factor above is the conditional probability119875119896(ℎ 120572120573) for the second event being 120572 given that there was
a first (120573) eventThe algorithm proceeds in two steps First a multino-
mial deviate is computed with probabilities 119875119896(ℎ 119868 120573) (the
probability of no event occurring is still 119875119896(ℎ 0)) This gives
the number 119899120573of occurrences of each event In the second
step for each 119899120573we compute the deviates for the occurrence
of a second event or no more events with the multinomialgiven by the array of probabilities 119875
119896(ℎ 120572120573) defined above
(the probability of no second event occurring is here 1 minus
(ℎ2)(sum120572(119903
120572
119896+ 119865
120573
120572)(1 minus (ℎ2)sum
120572119903119896
120572)))
6 Implementation Developmental Cascadeswithout Mortality
In this and the following Sections we apply the tools devel-oped up to now to the description of a subprocess in insectdevelopment namely the development of immature larvaewhich is in principle a highly individual subprocess Weassume that the evolution of an insect from egg to pupaoccurs via a sequence of 119864 gt 0 maturation stages Biologistsrecognise some substages by inspection in the developmentof larvae (instars) and even more stages when observed byother methods [35] The immature individual may die alongthe process or otherwise complete it and exit as pupa oradult The actual value of apparent maturation stages 119864 willdepend on environmental and experimental conditions andwill ultimately be specified when analying experimental datain Section 71
Empirical evidence based on existing and new experi-mental results as well as biological insight produced by thepresent description will be the subject of an independentwork [42]We discuss however a couple of examples from theliterature in Section 71
The events for this process are maturation events pro-moting individuals from one subpopulation into to the nextand death events for all subpopulations In other wordseach subpopulation represents a given (intermediate) level ofdevelopment and maturation events correspond to progressin the development Development at level 119895 promotes oneindividual to level 119895 + 1 and the event rate is linear in the119895-subpopulation Hence in 119877
120572= sum
119895119903119896
120572119883
119895 the matrix 119903
is square and diagonal for 0 le 120572 le 119864 minus 1 (we haveexactly 119864 + 1 subpopulations counted from 0 to 119864) All ofthe environmental influence is at this point incorporatedthrough 119903
Moreover level 119864 is the matured pupa and we may set119903119864
= 0 since at pupation stage the present mechanism endsand new processes take place Subpopulation 119864 is the exitpoint of the dynamics and we assume it to be determined bythe stochastic evolution of the immature stages 0 le 120572 le 119864minus1
In the same spirit the action of each event on thepopulations modifies just the actual population and the nextone in the chain Hence
120575120572
119895=
minus1 120572 = 119895
+1 120572 = 119895 minus 1(56)
Following Section 3 [1198770
120572]119879
= (1199030119872 0 0) and
119865120573
120572=
minus119903120572 120573 = 120572
+119903120572 120573 = 120572 minus 1
0 otherwise(57)
Finally letting m119879= minus119872(1 1 1) we can write 119877
0
120572=
sum120573119865120573
120572119898
120573 In this framework the index pair (119895 120572) is highly
interconnected and we can achieve a complete descriptionusing just one index We use 0 le 119895 le 119864 minus 1 throughout thissection
The procedure of Section 3 gives Ψ(119911 119905) = prod119895(119891
119895)119898119895 =
prod119895(119891
minus119872
119895) while the dynamical system for 119891
119895reads recalling
the recasting in Remark 21119891119895= 119891
119895(minus119903
119895(119911
119895119891119895minus 1) + 119903
119895+1(119911
119895+1119891119895+1
minus 1))
119895 = 0 119864 minus 2
(58)
and 119891119864minus1
= 119891119864minus1
(minus119903119864minus1
(119911119864minus1
119891119864minus1
minus 1)) The overall initialcondition is 119891
119895(0) = 1 Let 120579
119895= log(119891
119895) and
120595119895=
119864minus1
sum
120573=119895
120579120573 119882
119895= exp (minus120595
119895) (59)
Note that 119882119895119891119895
= 119882119895
sdot exp(120579119895) = 119882
119895+1 Rewriting the
equations120579119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) + 119903
119895+1(119911
119895+1exp (120579
119895+1) minus 1)
119895 lt 119864 minus 1
119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) = 120579
119895+
119895+1 119895 lt 119864 minus 1
120579119864minus1
= 119864minus1
= minus119903119864minus1
(119911119864minus1
exp (120579119864minus1
) minus 1)
(60)
with initial condition 120579119895(0) = 0 for all 119895With this notation the
generating function reads Ψ(119911 119905) = 119882119872
0 and we formulate
hence an ODE for the 119882119895rsquos
119895+ 119903
119895119882
119895= 119903
119895119911119895119882
119895+1 119882
119895 (0) = 1 119895 = 0 119864 minus 2
119864minus1
= 119903119864minus1
(119911119864minus1
minus 119882119864minus1
) 119882119864minus1
(0) = 1
(61)
The Scientific World Journal 11
By defining the auxiliary quantity 119882119864(119905) = 1 the solution
for the 119882119895-differential equations can be written in recursive
form
119882119895(119905) = 119890
minus119903119895119905+ 119911
119895119903119895int
119905
0
119890119903119895(119904minus119905)
119882119895+1
(119904) 119889119904 119895 = 0 119864 minus 1
(62)
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820(119905) = (
119864minus1
prod
119898=0
119911119898)(1 minus
119864minus1
sum
119896=0
exp (minus119903119896119905)
119864minus1
prod
119895=0119895 = 119896
119903119895
119903119895minus 119903
119896
)
+
119864
sum
119896=1
prod119864minus119896
119895=0119911119895
119911119864minus119896
119903119864minus119896
119864minus119896
sum
119898=0
119903119898exp (minus119903
119898119905)
119864minus119896
prod
119897=0119897 =119898
119903119897
119903119897minus 119903
119898
(63)
The solution looksmore compact when all 119903119896are assumed
to take the common value 119903
1198820(119905) =
prod119864minus1
119898=0119911119898
(119864 minus 1)int
119903119905
0
exp (minus119904) 119904119864minus1
119889119904
+
119864
sum
119896=1
prod119864minus119896
119898=0119911119898
119911119864minus119896
(119903119905)119864minus119896 exp (minus119903119905)
(119864 minus 119896)
(64)
7 Developmental Cascades with Mortality
We add to the previous description 119864 subpopulation specificdeath events 119876
120572= 119902
120572119883
120572 where 120572 = 0 119864 minus 1 We
do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883
119864leaves the
present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876
0 119877
0 119876
1 119877
1 119876
119864minus1 and 119877
119864minus1 Then ((119876 119877)
0)119879
=
(1199020119872 119903
0119872 0 0) We have
1205752120572
120572= minus1 (death) 120572 lt 119864
1205752120572+1
120572= minus1
1205752120572+1
120572+1= +1
(development) 120572 lt 119864
119865120573
2120572=
minus119902120572 120573 = 2120572
minus119902120572 120573 = 2120572 + 1
119902120572 120573 = 2120572 minus 1
119865120573
2120572+1=
minus119903120572 120573 = 2120572 + 1
minus119903120572 120573 = 2120572
119903120572 120573 = 2120572 minus 1
120572 lt 119864
(65)
Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891
120572and the 119911
120572in pairs (119892 119891) and
(119911 119910) as (1198920 119891
0 119892
119864minus1 119891
119864minus1) and (119911
0 119910
0 119911
119864minus1 119910
119864minus1)
Also for notational convenience we will assume that thereexist 119902
119864= 119903
119864= 0 when necessary Defining
119867120572(119892
120572 119891
120572) = 119902
120572(119911
120572119892120572minus 1) + 119903
120572(119910
120572119891120572minus 1) (66)
we have (recall again Remark 21 and note that 119867119864equiv 0)
119892120572= minus119867
120572(119892
120572 119891
120572) 119892
120572
119891120572= (minus119867
120572(119892
120572 119891
120572) + 119867
120572+1(119892
120572+1 119891
120572+1)) 119891
120572
120572 = 0 119864 minus 2
119892119864minus1
119892119864minus1
=
119891119864minus1
119891119864minus1
= minus119867119864minus1
(119892119864minus1
119891119864minus1
)
(67)
The overall initial condition is still 119892120572(0) = 1 = 119891
120572(0)
Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881
119879
120572=
(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579
120572= log119892
120572
and 120601120572
= log119891120572 changing 119867
120572and the initial conditions
correspondingly Hence
120601120572= 120579
120572minus 120579
120572+1997904rArr 120601
120572= 120579
120572minus 120579
120572+1 120572 = 0 119864 minus 2
120579120572= minus119867
120572(120579
120572 120579
120572minus 120579
120572+1) 120572 = 0 119864 minus 2
120601119864minus1
= 120579119864minus1
= minus119867119864minus1
(120579119864minus1
120579119864minus1
)
(68)
The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879
= minus119872(1 0 0 0) satisfies 1198770
= 119865119898 FinallyΨ(119911 119910 119905) = 119892
minus119872
0 We now compute 119892
0(119905)
Let119882120572= 119890
minus120579120572 = 119892
minus1
120572 0 le 120572 le 119864minus1 Defining the auxiliary
variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of
119882 read
120572+ (119902
120572+ 119903
120572)119882
120572= 119902
120572119911120572+ 119903
120572119910120572119882
120572+1 119882
120572 (0) = 1
(69)
with solution
119882120572(119905) = 119890
minus(119902120572+119903120572)119905
+ int
119905
0
119890minus(119902120572+119903120572)(119905minus119904)
(119902120572119911120572+ 119903
120572119910120572119882
120572+1(119904)) 119889119904
(70)
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 The Scientific World Journal
adopted approximation From 119891120572
= prod119895119908
120575120572
119895
119895we obtain
119891120572119891
120572= sum
119896120575120572
119896119896119908
119896and (34) can be recast as
sum
119896
120575120572
119896(
119896
119908119896
+ sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1)) = 0
119908119896 (119911 119905 119905) = 1
(42)
Because of the linear independence of population increments(and hence of populations) Lemma 3(b) the counterpart of(34) in population coordinates reads
119896= minus119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896(119911 119905 119905) = 1
(43)
Remark 20 In the case of linearly dependent populationincrements (34) still holds while it is possible to elaboratea more general form in terms of the 119908
119895instead of (43) This
is however outside the scope of this paper
Since sum120572120575120572
119895119898
120572= 119883
119895(119904) we obtain Ψ(119911
1 119911
119864 119904 119905) =
prod119895119908
119883119895(119904)
119895 which is so far an exact result Let us produce
some approximated values for the 119908119896rsquos and correspondingly
for Ψ(1199111 119911
119864 119904 119905)
To avoid further notational complications wewill assumein the sequel that 119903
119896
120573is time independent Hence 119908
119896will
depend only on the time difference 119905 minus 119904 Equation (43) canbe recast as
119896= 119908
119896sum
120573
119903119896
120573(119911
120573(prod
119895
119908120575120573
119895
119895) minus 1) 119908
119896 (119911 0) = 1
(44)
now to be integrated along the interval [0 119905 minus 119904] We will evenset 119904 = 0 in what follows
Remark 21 For the case of time-independent proportionalityfactors 119903119896
120573 a correspondingmodification can be introduced in
(34) namely overall change of sign integration in [0 119905 minus 119904]and setting 119904 = 0 to reduce the notational burden
Equations (44) satisfy the Picard-Lindelof theorem sincethe RHS is a Lipschitz function Then the Picard iterationscheme converges to the (unique) solution Taking advantage
of the initial condition the 119899th order truncated version of thePicard iterations reads
119908(0)
119896(ℎ) = 1
119908(1)
119896(ℎ) = 1 + int
ℎ
0
119908(0)
119896(119905)sum
120573
119903119896
120573(119911
120573prod
119895
(119908(0)
119895(119905))
120575120573
119895
minus 1)119889119905
= 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1)
119908(119899)
119896(ℎ) = 1 + int
ℎ
0
119889119905[
[
119908(119899minus1)
119896(119905)
timessum
120573
119903119896
120573(119911
120573prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
minus 1)]
]119899minus1
(45)
where [sdot sdot sdot ]119898denote the Maclaurin polynomial of order119898 in
119905
Lemma 22 For 119905 isin [0 ℎ] ℎ gt 0 being sufficiently smalland for each order of approximation 119899 ge 0 119908(119899)
119896(119905) in (44)
is a polynomial generating function for one individual of thepopulation that is it has the following properties
(a) 119908(119899)
119896(119905) is a polynomial of degree 119899 in 119911
1 119911
119864 where
the coefficient of 11991111989911 119911
119899119864
119864is 119874(119905
119901) with 119901 = 119899
1+
sdot sdot sdot + 119899119864
(b) 119908(119899)
119896(1 1 119905) = 1
(c) Δ(119899)
119896(ℎ) = 119908
(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ) = 119874(ℎ
119899)
(d) the coefficients of 119908(119899)
119896(119905) regarded as a polynomial in
119911120572 are nonnegative functions of time
Proof (a) The statement holds for 119899 = 0 and 119899 = 1Assuming that it holds up to order 119899 minus 1 we have that[119908
(119899minus1)
119896(119905)prod
119895(119908
(119899minus1)
119895(119905))
120575120573
119895 ]119899minus1
is also a polynomial of degree119899 minus 1 in 119911
1 119911
119864with time-dependent coefficients of type
119874(119905119901) since each power 119901 of 119905 carries along a power 119901 in 119911
120572
(as well as lower 119911-powers) A Picard iteration step gives
119908(119899)
119896(119905) = 1 + sum
120573
119903119896
120573119911120573int
ℎ
0
[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
119908(119899minus1)
119896(119905) 119889119905
(46)
The first sum contributes with one time integration and one119911-power which exactly preserves the polynomial structureand the order of the coefficientsThe result is a polynomial ofdegree 119899 in 119911
1 119911
119864 where the coefficients are polynomials
of highest degree at most 119899 in ℎ each one of lowest order119874(ℎ
119901) (as in the statement) The second integral does not
The Scientific World Journal 9
alter these properties since it generates again a polynomial ofdegree 119899minus1 in 119911
1 119911
119864 with time-dependent coefficients of
order119874(ℎ119901+1
) or higher up to ℎ119899 (they are just the coefficients
of 119908(119899minus1)
119896(119905) integrated once in time)
(b) Letting 1199111= 119911
2= sdot sdot sdot = 119911
119864= 1 Picard iteration gives
trivially 119908(119899)
119896(119905) = 1 to all orders again by induction
(c) The statement holds for 119899 = 1 Assuming it holds upto 119899 minus 1 we compute
119908(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ)
= sum
120573
119903119896
120573119911120573
times int
ℎ
0
([(119908(119899minus2)
119896(119905) + Δ
(119899minus1)
119896(119905))
times prod
119895
(119908(119899minus2)
119895(119905) + Δ
(119899minus1)
119895(119905))
120575120573
119895
]
119899minus1
minus[
[
119908(119899minus2)
119896(119905)prod
119895
(119908(119899minus2)
119895(119905))
120575120573
119895]
]119899minus2
)119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
(119908(119899minus1)
119896(119905) minus 119908
(119899minus2)
119896(119905)) 119889119905
(47)
Both expressions in the first integral have the sameMaclaurinpolynomial up to order (119899 minus 2) Hence the difference in thefirst integral contains only terms of order119874(119905
119899minus1) This is also
the case for the second integral After integration we obtainΔ(119899)
119896(ℎ) = 119874(ℎ
119899)
(d) The statement holds for 119899 = 0 and 119899 = 1 and ingeneral for the zero-order coefficient because it is unity for119905 = 0 as a consequence of the initial condition For thehigher-order coefficients we use induction again If all 119911-coefficients in 119908
(119899minus1)
119896(119905) are nonnegative then the same holds
automatically for 119908(119899)
119896(ℎ) up to order 119899 minus 1 since the 119874(ℎ
119899)
contribution adding to each inherited coefficient from theprevious step does not alter its sign for ℎ sufficiently small Itremains to show that the coefficients corresponding to 119901 = 119899
are nonnegative These contributions arise from the (119899 minus 1)thorder of
119903119896
120573119911120573[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
(48)
after time integration If all 120575120573119895are positive then the expres-
sion above is a product of 119911-polynomials with nonnegative
coefficients and the statement follows Suppose that some120575120573
119895= minus1 Then by Lemma 7 119903119896
120573= 0 for 119896 = 119895 Hence
119903119896
120573119911120573119908
(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
=
0 119895 = 119896
119903119896
120573119911120573prod
119895 = 119896
(119908(119899minus1)
119895(119905))
120575120573
119895
119895 = 119896
(49)
However 120575120573119895
= minus 1 for 119895 = 119896 that is they are nonnegative andhence the expression has only nonnegative coefficients
521 Basic Properties of the First- and Second-OrderApproximations Let us analyse the generating functionΨ(119911
1 119911
119864 119905) = prod
119895119908
119883119895(0)
119895 The simplest nontrivial
applications of the approximation scheme are the first- andsecond-order approximations Expanding explicitly up tosecond-order we get
119908119896(ℎ) = 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1) +
ℎ2
2(sum
120573
119903119896
120573(119911
120573minus 1))
times (sum
120574
119903119896
120574(119911
120574minus 1))
+ℎ2
2sum
120573
119903119896
120573119911120573sum
120574
119865120573
120574(119911
120574minus 1)
(50)
We begin by noticing that the first-order approximation isobtained by neglecting all terms in ℎ
2 in the above equationWhenever 119908
119895is linear in 119911 which is always the case for the
first-order approximation the resulting generating functionΨ corresponds to the product of (independent) multinomialdistributions for the events affecting each subpopulation
Let us now consider Ψ(1199111 119911
119864 119905) computed with the
second-order approximated 119908119895rsquos to perform one simulation
step of size ℎCounting powers of 119911
120574on each 119908
119896we have (a) the
probability of no events occurring in time-step ℎ
119875119896(ℎ 0) = 1 minus ℎ(sum
120573
119903119896
120573) +
ℎ2
2(sum
120573
119903119896
120573)
2
(51)
(b) the probability of only one (120573) event occurring in time-step ℎ
119875119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus ℎ
2sum
120572
119903119896
120572minus
ℎ2
2sum
120572
119865120573
120572 (52)
and (c) the probability of another event (120572) occurring afterthe first one
119875119896(ℎ 119868119868 120573 120572) =
ℎ2
2119903119896
120573(119903
119896
120572+ 119865
120573
120572) (53)
10 The Scientific World Journal
522 Simulation Algorithm A simulation step can be oper-ated in the following way Let
sum
120572
(119875119896(ℎ 119868119868 120573 120572)) + 119875
119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus
ℎ2
2sum
120572
119903119896
120572
= 119875119896(ℎ 119868 120573)
(54)
be the probability of one (120573) event occurring alone orfollowed by a second event It follows that
119875119896(ℎ 119868119868 120573 120572) = 119875
119896(ℎ 119868 120573)
ℎ
2
(119903120572
119896+ 119865
120573
120572)
(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ
2)
(55)
The second factor above is the conditional probability119875119896(ℎ 120572120573) for the second event being 120572 given that there was
a first (120573) eventThe algorithm proceeds in two steps First a multino-
mial deviate is computed with probabilities 119875119896(ℎ 119868 120573) (the
probability of no event occurring is still 119875119896(ℎ 0)) This gives
the number 119899120573of occurrences of each event In the second
step for each 119899120573we compute the deviates for the occurrence
of a second event or no more events with the multinomialgiven by the array of probabilities 119875
119896(ℎ 120572120573) defined above
(the probability of no second event occurring is here 1 minus
(ℎ2)(sum120572(119903
120572
119896+ 119865
120573
120572)(1 minus (ℎ2)sum
120572119903119896
120572)))
6 Implementation Developmental Cascadeswithout Mortality
In this and the following Sections we apply the tools devel-oped up to now to the description of a subprocess in insectdevelopment namely the development of immature larvaewhich is in principle a highly individual subprocess Weassume that the evolution of an insect from egg to pupaoccurs via a sequence of 119864 gt 0 maturation stages Biologistsrecognise some substages by inspection in the developmentof larvae (instars) and even more stages when observed byother methods [35] The immature individual may die alongthe process or otherwise complete it and exit as pupa oradult The actual value of apparent maturation stages 119864 willdepend on environmental and experimental conditions andwill ultimately be specified when analying experimental datain Section 71
Empirical evidence based on existing and new experi-mental results as well as biological insight produced by thepresent description will be the subject of an independentwork [42]We discuss however a couple of examples from theliterature in Section 71
The events for this process are maturation events pro-moting individuals from one subpopulation into to the nextand death events for all subpopulations In other wordseach subpopulation represents a given (intermediate) level ofdevelopment and maturation events correspond to progressin the development Development at level 119895 promotes oneindividual to level 119895 + 1 and the event rate is linear in the119895-subpopulation Hence in 119877
120572= sum
119895119903119896
120572119883
119895 the matrix 119903
is square and diagonal for 0 le 120572 le 119864 minus 1 (we haveexactly 119864 + 1 subpopulations counted from 0 to 119864) All ofthe environmental influence is at this point incorporatedthrough 119903
Moreover level 119864 is the matured pupa and we may set119903119864
= 0 since at pupation stage the present mechanism endsand new processes take place Subpopulation 119864 is the exitpoint of the dynamics and we assume it to be determined bythe stochastic evolution of the immature stages 0 le 120572 le 119864minus1
In the same spirit the action of each event on thepopulations modifies just the actual population and the nextone in the chain Hence
120575120572
119895=
minus1 120572 = 119895
+1 120572 = 119895 minus 1(56)
Following Section 3 [1198770
120572]119879
= (1199030119872 0 0) and
119865120573
120572=
minus119903120572 120573 = 120572
+119903120572 120573 = 120572 minus 1
0 otherwise(57)
Finally letting m119879= minus119872(1 1 1) we can write 119877
0
120572=
sum120573119865120573
120572119898
120573 In this framework the index pair (119895 120572) is highly
interconnected and we can achieve a complete descriptionusing just one index We use 0 le 119895 le 119864 minus 1 throughout thissection
The procedure of Section 3 gives Ψ(119911 119905) = prod119895(119891
119895)119898119895 =
prod119895(119891
minus119872
119895) while the dynamical system for 119891
119895reads recalling
the recasting in Remark 21119891119895= 119891
119895(minus119903
119895(119911
119895119891119895minus 1) + 119903
119895+1(119911
119895+1119891119895+1
minus 1))
119895 = 0 119864 minus 2
(58)
and 119891119864minus1
= 119891119864minus1
(minus119903119864minus1
(119911119864minus1
119891119864minus1
minus 1)) The overall initialcondition is 119891
119895(0) = 1 Let 120579
119895= log(119891
119895) and
120595119895=
119864minus1
sum
120573=119895
120579120573 119882
119895= exp (minus120595
119895) (59)
Note that 119882119895119891119895
= 119882119895
sdot exp(120579119895) = 119882
119895+1 Rewriting the
equations120579119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) + 119903
119895+1(119911
119895+1exp (120579
119895+1) minus 1)
119895 lt 119864 minus 1
119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) = 120579
119895+
119895+1 119895 lt 119864 minus 1
120579119864minus1
= 119864minus1
= minus119903119864minus1
(119911119864minus1
exp (120579119864minus1
) minus 1)
(60)
with initial condition 120579119895(0) = 0 for all 119895With this notation the
generating function reads Ψ(119911 119905) = 119882119872
0 and we formulate
hence an ODE for the 119882119895rsquos
119895+ 119903
119895119882
119895= 119903
119895119911119895119882
119895+1 119882
119895 (0) = 1 119895 = 0 119864 minus 2
119864minus1
= 119903119864minus1
(119911119864minus1
minus 119882119864minus1
) 119882119864minus1
(0) = 1
(61)
The Scientific World Journal 11
By defining the auxiliary quantity 119882119864(119905) = 1 the solution
for the 119882119895-differential equations can be written in recursive
form
119882119895(119905) = 119890
minus119903119895119905+ 119911
119895119903119895int
119905
0
119890119903119895(119904minus119905)
119882119895+1
(119904) 119889119904 119895 = 0 119864 minus 1
(62)
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820(119905) = (
119864minus1
prod
119898=0
119911119898)(1 minus
119864minus1
sum
119896=0
exp (minus119903119896119905)
119864minus1
prod
119895=0119895 = 119896
119903119895
119903119895minus 119903
119896
)
+
119864
sum
119896=1
prod119864minus119896
119895=0119911119895
119911119864minus119896
119903119864minus119896
119864minus119896
sum
119898=0
119903119898exp (minus119903
119898119905)
119864minus119896
prod
119897=0119897 =119898
119903119897
119903119897minus 119903
119898
(63)
The solution looksmore compact when all 119903119896are assumed
to take the common value 119903
1198820(119905) =
prod119864minus1
119898=0119911119898
(119864 minus 1)int
119903119905
0
exp (minus119904) 119904119864minus1
119889119904
+
119864
sum
119896=1
prod119864minus119896
119898=0119911119898
119911119864minus119896
(119903119905)119864minus119896 exp (minus119903119905)
(119864 minus 119896)
(64)
7 Developmental Cascades with Mortality
We add to the previous description 119864 subpopulation specificdeath events 119876
120572= 119902
120572119883
120572 where 120572 = 0 119864 minus 1 We
do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883
119864leaves the
present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876
0 119877
0 119876
1 119877
1 119876
119864minus1 and 119877
119864minus1 Then ((119876 119877)
0)119879
=
(1199020119872 119903
0119872 0 0) We have
1205752120572
120572= minus1 (death) 120572 lt 119864
1205752120572+1
120572= minus1
1205752120572+1
120572+1= +1
(development) 120572 lt 119864
119865120573
2120572=
minus119902120572 120573 = 2120572
minus119902120572 120573 = 2120572 + 1
119902120572 120573 = 2120572 minus 1
119865120573
2120572+1=
minus119903120572 120573 = 2120572 + 1
minus119903120572 120573 = 2120572
119903120572 120573 = 2120572 minus 1
120572 lt 119864
(65)
Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891
120572and the 119911
120572in pairs (119892 119891) and
(119911 119910) as (1198920 119891
0 119892
119864minus1 119891
119864minus1) and (119911
0 119910
0 119911
119864minus1 119910
119864minus1)
Also for notational convenience we will assume that thereexist 119902
119864= 119903
119864= 0 when necessary Defining
119867120572(119892
120572 119891
120572) = 119902
120572(119911
120572119892120572minus 1) + 119903
120572(119910
120572119891120572minus 1) (66)
we have (recall again Remark 21 and note that 119867119864equiv 0)
119892120572= minus119867
120572(119892
120572 119891
120572) 119892
120572
119891120572= (minus119867
120572(119892
120572 119891
120572) + 119867
120572+1(119892
120572+1 119891
120572+1)) 119891
120572
120572 = 0 119864 minus 2
119892119864minus1
119892119864minus1
=
119891119864minus1
119891119864minus1
= minus119867119864minus1
(119892119864minus1
119891119864minus1
)
(67)
The overall initial condition is still 119892120572(0) = 1 = 119891
120572(0)
Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881
119879
120572=
(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579
120572= log119892
120572
and 120601120572
= log119891120572 changing 119867
120572and the initial conditions
correspondingly Hence
120601120572= 120579
120572minus 120579
120572+1997904rArr 120601
120572= 120579
120572minus 120579
120572+1 120572 = 0 119864 minus 2
120579120572= minus119867
120572(120579
120572 120579
120572minus 120579
120572+1) 120572 = 0 119864 minus 2
120601119864minus1
= 120579119864minus1
= minus119867119864minus1
(120579119864minus1
120579119864minus1
)
(68)
The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879
= minus119872(1 0 0 0) satisfies 1198770
= 119865119898 FinallyΨ(119911 119910 119905) = 119892
minus119872
0 We now compute 119892
0(119905)
Let119882120572= 119890
minus120579120572 = 119892
minus1
120572 0 le 120572 le 119864minus1 Defining the auxiliary
variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of
119882 read
120572+ (119902
120572+ 119903
120572)119882
120572= 119902
120572119911120572+ 119903
120572119910120572119882
120572+1 119882
120572 (0) = 1
(69)
with solution
119882120572(119905) = 119890
minus(119902120572+119903120572)119905
+ int
119905
0
119890minus(119902120572+119903120572)(119905minus119904)
(119902120572119911120572+ 119903
120572119910120572119882
120572+1(119904)) 119889119904
(70)
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 9
alter these properties since it generates again a polynomial ofdegree 119899minus1 in 119911
1 119911
119864 with time-dependent coefficients of
order119874(ℎ119901+1
) or higher up to ℎ119899 (they are just the coefficients
of 119908(119899minus1)
119896(119905) integrated once in time)
(b) Letting 1199111= 119911
2= sdot sdot sdot = 119911
119864= 1 Picard iteration gives
trivially 119908(119899)
119896(119905) = 1 to all orders again by induction
(c) The statement holds for 119899 = 1 Assuming it holds upto 119899 minus 1 we compute
119908(119899)
119896(ℎ) minus 119908
(119899minus1)
119896(ℎ)
= sum
120573
119903119896
120573119911120573
times int
ℎ
0
([(119908(119899minus2)
119896(119905) + Δ
(119899minus1)
119896(119905))
times prod
119895
(119908(119899minus2)
119895(119905) + Δ
(119899minus1)
119895(119905))
120575120573
119895
]
119899minus1
minus[
[
119908(119899minus2)
119896(119905)prod
119895
(119908(119899minus2)
119895(119905))
120575120573
119895]
]119899minus2
)119889119905
minus (sum
120573
119903119896
120573)int
ℎ
0
(119908(119899minus1)
119896(119905) minus 119908
(119899minus2)
119896(119905)) 119889119905
(47)
Both expressions in the first integral have the sameMaclaurinpolynomial up to order (119899 minus 2) Hence the difference in thefirst integral contains only terms of order119874(119905
119899minus1) This is also
the case for the second integral After integration we obtainΔ(119899)
119896(ℎ) = 119874(ℎ
119899)
(d) The statement holds for 119899 = 0 and 119899 = 1 and ingeneral for the zero-order coefficient because it is unity for119905 = 0 as a consequence of the initial condition For thehigher-order coefficients we use induction again If all 119911-coefficients in 119908
(119899minus1)
119896(119905) are nonnegative then the same holds
automatically for 119908(119899)
119896(ℎ) up to order 119899 minus 1 since the 119874(ℎ
119899)
contribution adding to each inherited coefficient from theprevious step does not alter its sign for ℎ sufficiently small Itremains to show that the coefficients corresponding to 119901 = 119899
are nonnegative These contributions arise from the (119899 minus 1)thorder of
119903119896
120573119911120573[
[
119908(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895]
]119899minus1
(48)
after time integration If all 120575120573119895are positive then the expres-
sion above is a product of 119911-polynomials with nonnegative
coefficients and the statement follows Suppose that some120575120573
119895= minus1 Then by Lemma 7 119903119896
120573= 0 for 119896 = 119895 Hence
119903119896
120573119911120573119908
(119899minus1)
119896(119905)prod
119895
(119908(119899minus1)
119895(119905))
120575120573
119895
=
0 119895 = 119896
119903119896
120573119911120573prod
119895 = 119896
(119908(119899minus1)
119895(119905))
120575120573
119895
119895 = 119896
(49)
However 120575120573119895
= minus 1 for 119895 = 119896 that is they are nonnegative andhence the expression has only nonnegative coefficients
521 Basic Properties of the First- and Second-OrderApproximations Let us analyse the generating functionΨ(119911
1 119911
119864 119905) = prod
119895119908
119883119895(0)
119895 The simplest nontrivial
applications of the approximation scheme are the first- andsecond-order approximations Expanding explicitly up tosecond-order we get
119908119896(ℎ) = 1 + ℎsum
120573
119903119896
120573(119911
120573minus 1) +
ℎ2
2(sum
120573
119903119896
120573(119911
120573minus 1))
times (sum
120574
119903119896
120574(119911
120574minus 1))
+ℎ2
2sum
120573
119903119896
120573119911120573sum
120574
119865120573
120574(119911
120574minus 1)
(50)
We begin by noticing that the first-order approximation isobtained by neglecting all terms in ℎ
2 in the above equationWhenever 119908
119895is linear in 119911 which is always the case for the
first-order approximation the resulting generating functionΨ corresponds to the product of (independent) multinomialdistributions for the events affecting each subpopulation
Let us now consider Ψ(1199111 119911
119864 119905) computed with the
second-order approximated 119908119895rsquos to perform one simulation
step of size ℎCounting powers of 119911
120574on each 119908
119896we have (a) the
probability of no events occurring in time-step ℎ
119875119896(ℎ 0) = 1 minus ℎ(sum
120573
119903119896
120573) +
ℎ2
2(sum
120573
119903119896
120573)
2
(51)
(b) the probability of only one (120573) event occurring in time-step ℎ
119875119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus ℎ
2sum
120572
119903119896
120572minus
ℎ2
2sum
120572
119865120573
120572 (52)
and (c) the probability of another event (120572) occurring afterthe first one
119875119896(ℎ 119868119868 120573 120572) =
ℎ2
2119903119896
120573(119903
119896
120572+ 119865
120573
120572) (53)
10 The Scientific World Journal
522 Simulation Algorithm A simulation step can be oper-ated in the following way Let
sum
120572
(119875119896(ℎ 119868119868 120573 120572)) + 119875
119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus
ℎ2
2sum
120572
119903119896
120572
= 119875119896(ℎ 119868 120573)
(54)
be the probability of one (120573) event occurring alone orfollowed by a second event It follows that
119875119896(ℎ 119868119868 120573 120572) = 119875
119896(ℎ 119868 120573)
ℎ
2
(119903120572
119896+ 119865
120573
120572)
(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ
2)
(55)
The second factor above is the conditional probability119875119896(ℎ 120572120573) for the second event being 120572 given that there was
a first (120573) eventThe algorithm proceeds in two steps First a multino-
mial deviate is computed with probabilities 119875119896(ℎ 119868 120573) (the
probability of no event occurring is still 119875119896(ℎ 0)) This gives
the number 119899120573of occurrences of each event In the second
step for each 119899120573we compute the deviates for the occurrence
of a second event or no more events with the multinomialgiven by the array of probabilities 119875
119896(ℎ 120572120573) defined above
(the probability of no second event occurring is here 1 minus
(ℎ2)(sum120572(119903
120572
119896+ 119865
120573
120572)(1 minus (ℎ2)sum
120572119903119896
120572)))
6 Implementation Developmental Cascadeswithout Mortality
In this and the following Sections we apply the tools devel-oped up to now to the description of a subprocess in insectdevelopment namely the development of immature larvaewhich is in principle a highly individual subprocess Weassume that the evolution of an insect from egg to pupaoccurs via a sequence of 119864 gt 0 maturation stages Biologistsrecognise some substages by inspection in the developmentof larvae (instars) and even more stages when observed byother methods [35] The immature individual may die alongthe process or otherwise complete it and exit as pupa oradult The actual value of apparent maturation stages 119864 willdepend on environmental and experimental conditions andwill ultimately be specified when analying experimental datain Section 71
Empirical evidence based on existing and new experi-mental results as well as biological insight produced by thepresent description will be the subject of an independentwork [42]We discuss however a couple of examples from theliterature in Section 71
The events for this process are maturation events pro-moting individuals from one subpopulation into to the nextand death events for all subpopulations In other wordseach subpopulation represents a given (intermediate) level ofdevelopment and maturation events correspond to progressin the development Development at level 119895 promotes oneindividual to level 119895 + 1 and the event rate is linear in the119895-subpopulation Hence in 119877
120572= sum
119895119903119896
120572119883
119895 the matrix 119903
is square and diagonal for 0 le 120572 le 119864 minus 1 (we haveexactly 119864 + 1 subpopulations counted from 0 to 119864) All ofthe environmental influence is at this point incorporatedthrough 119903
Moreover level 119864 is the matured pupa and we may set119903119864
= 0 since at pupation stage the present mechanism endsand new processes take place Subpopulation 119864 is the exitpoint of the dynamics and we assume it to be determined bythe stochastic evolution of the immature stages 0 le 120572 le 119864minus1
In the same spirit the action of each event on thepopulations modifies just the actual population and the nextone in the chain Hence
120575120572
119895=
minus1 120572 = 119895
+1 120572 = 119895 minus 1(56)
Following Section 3 [1198770
120572]119879
= (1199030119872 0 0) and
119865120573
120572=
minus119903120572 120573 = 120572
+119903120572 120573 = 120572 minus 1
0 otherwise(57)
Finally letting m119879= minus119872(1 1 1) we can write 119877
0
120572=
sum120573119865120573
120572119898
120573 In this framework the index pair (119895 120572) is highly
interconnected and we can achieve a complete descriptionusing just one index We use 0 le 119895 le 119864 minus 1 throughout thissection
The procedure of Section 3 gives Ψ(119911 119905) = prod119895(119891
119895)119898119895 =
prod119895(119891
minus119872
119895) while the dynamical system for 119891
119895reads recalling
the recasting in Remark 21119891119895= 119891
119895(minus119903
119895(119911
119895119891119895minus 1) + 119903
119895+1(119911
119895+1119891119895+1
minus 1))
119895 = 0 119864 minus 2
(58)
and 119891119864minus1
= 119891119864minus1
(minus119903119864minus1
(119911119864minus1
119891119864minus1
minus 1)) The overall initialcondition is 119891
119895(0) = 1 Let 120579
119895= log(119891
119895) and
120595119895=
119864minus1
sum
120573=119895
120579120573 119882
119895= exp (minus120595
119895) (59)
Note that 119882119895119891119895
= 119882119895
sdot exp(120579119895) = 119882
119895+1 Rewriting the
equations120579119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) + 119903
119895+1(119911
119895+1exp (120579
119895+1) minus 1)
119895 lt 119864 minus 1
119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) = 120579
119895+
119895+1 119895 lt 119864 minus 1
120579119864minus1
= 119864minus1
= minus119903119864minus1
(119911119864minus1
exp (120579119864minus1
) minus 1)
(60)
with initial condition 120579119895(0) = 0 for all 119895With this notation the
generating function reads Ψ(119911 119905) = 119882119872
0 and we formulate
hence an ODE for the 119882119895rsquos
119895+ 119903
119895119882
119895= 119903
119895119911119895119882
119895+1 119882
119895 (0) = 1 119895 = 0 119864 minus 2
119864minus1
= 119903119864minus1
(119911119864minus1
minus 119882119864minus1
) 119882119864minus1
(0) = 1
(61)
The Scientific World Journal 11
By defining the auxiliary quantity 119882119864(119905) = 1 the solution
for the 119882119895-differential equations can be written in recursive
form
119882119895(119905) = 119890
minus119903119895119905+ 119911
119895119903119895int
119905
0
119890119903119895(119904minus119905)
119882119895+1
(119904) 119889119904 119895 = 0 119864 minus 1
(62)
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820(119905) = (
119864minus1
prod
119898=0
119911119898)(1 minus
119864minus1
sum
119896=0
exp (minus119903119896119905)
119864minus1
prod
119895=0119895 = 119896
119903119895
119903119895minus 119903
119896
)
+
119864
sum
119896=1
prod119864minus119896
119895=0119911119895
119911119864minus119896
119903119864minus119896
119864minus119896
sum
119898=0
119903119898exp (minus119903
119898119905)
119864minus119896
prod
119897=0119897 =119898
119903119897
119903119897minus 119903
119898
(63)
The solution looksmore compact when all 119903119896are assumed
to take the common value 119903
1198820(119905) =
prod119864minus1
119898=0119911119898
(119864 minus 1)int
119903119905
0
exp (minus119904) 119904119864minus1
119889119904
+
119864
sum
119896=1
prod119864minus119896
119898=0119911119898
119911119864minus119896
(119903119905)119864minus119896 exp (minus119903119905)
(119864 minus 119896)
(64)
7 Developmental Cascades with Mortality
We add to the previous description 119864 subpopulation specificdeath events 119876
120572= 119902
120572119883
120572 where 120572 = 0 119864 minus 1 We
do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883
119864leaves the
present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876
0 119877
0 119876
1 119877
1 119876
119864minus1 and 119877
119864minus1 Then ((119876 119877)
0)119879
=
(1199020119872 119903
0119872 0 0) We have
1205752120572
120572= minus1 (death) 120572 lt 119864
1205752120572+1
120572= minus1
1205752120572+1
120572+1= +1
(development) 120572 lt 119864
119865120573
2120572=
minus119902120572 120573 = 2120572
minus119902120572 120573 = 2120572 + 1
119902120572 120573 = 2120572 minus 1
119865120573
2120572+1=
minus119903120572 120573 = 2120572 + 1
minus119903120572 120573 = 2120572
119903120572 120573 = 2120572 minus 1
120572 lt 119864
(65)
Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891
120572and the 119911
120572in pairs (119892 119891) and
(119911 119910) as (1198920 119891
0 119892
119864minus1 119891
119864minus1) and (119911
0 119910
0 119911
119864minus1 119910
119864minus1)
Also for notational convenience we will assume that thereexist 119902
119864= 119903
119864= 0 when necessary Defining
119867120572(119892
120572 119891
120572) = 119902
120572(119911
120572119892120572minus 1) + 119903
120572(119910
120572119891120572minus 1) (66)
we have (recall again Remark 21 and note that 119867119864equiv 0)
119892120572= minus119867
120572(119892
120572 119891
120572) 119892
120572
119891120572= (minus119867
120572(119892
120572 119891
120572) + 119867
120572+1(119892
120572+1 119891
120572+1)) 119891
120572
120572 = 0 119864 minus 2
119892119864minus1
119892119864minus1
=
119891119864minus1
119891119864minus1
= minus119867119864minus1
(119892119864minus1
119891119864minus1
)
(67)
The overall initial condition is still 119892120572(0) = 1 = 119891
120572(0)
Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881
119879
120572=
(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579
120572= log119892
120572
and 120601120572
= log119891120572 changing 119867
120572and the initial conditions
correspondingly Hence
120601120572= 120579
120572minus 120579
120572+1997904rArr 120601
120572= 120579
120572minus 120579
120572+1 120572 = 0 119864 minus 2
120579120572= minus119867
120572(120579
120572 120579
120572minus 120579
120572+1) 120572 = 0 119864 minus 2
120601119864minus1
= 120579119864minus1
= minus119867119864minus1
(120579119864minus1
120579119864minus1
)
(68)
The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879
= minus119872(1 0 0 0) satisfies 1198770
= 119865119898 FinallyΨ(119911 119910 119905) = 119892
minus119872
0 We now compute 119892
0(119905)
Let119882120572= 119890
minus120579120572 = 119892
minus1
120572 0 le 120572 le 119864minus1 Defining the auxiliary
variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of
119882 read
120572+ (119902
120572+ 119903
120572)119882
120572= 119902
120572119911120572+ 119903
120572119910120572119882
120572+1 119882
120572 (0) = 1
(69)
with solution
119882120572(119905) = 119890
minus(119902120572+119903120572)119905
+ int
119905
0
119890minus(119902120572+119903120572)(119905minus119904)
(119902120572119911120572+ 119903
120572119910120572119882
120572+1(119904)) 119889119904
(70)
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 The Scientific World Journal
522 Simulation Algorithm A simulation step can be oper-ated in the following way Let
sum
120572
(119875119896(ℎ 119868119868 120573 120572)) + 119875
119896(ℎ 119868 120573) = 119903
119896
120573ℎ minus
ℎ2
2sum
120572
119903119896
120572
= 119875119896(ℎ 119868 120573)
(54)
be the probability of one (120573) event occurring alone orfollowed by a second event It follows that
119875119896(ℎ 119868119868 120573 120572) = 119875
119896(ℎ 119868 120573)
ℎ
2
(119903120572
119896+ 119865
120573
120572)
(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ
2)
(55)
The second factor above is the conditional probability119875119896(ℎ 120572120573) for the second event being 120572 given that there was
a first (120573) eventThe algorithm proceeds in two steps First a multino-
mial deviate is computed with probabilities 119875119896(ℎ 119868 120573) (the
probability of no event occurring is still 119875119896(ℎ 0)) This gives
the number 119899120573of occurrences of each event In the second
step for each 119899120573we compute the deviates for the occurrence
of a second event or no more events with the multinomialgiven by the array of probabilities 119875
119896(ℎ 120572120573) defined above
(the probability of no second event occurring is here 1 minus
(ℎ2)(sum120572(119903
120572
119896+ 119865
120573
120572)(1 minus (ℎ2)sum
120572119903119896
120572)))
6 Implementation Developmental Cascadeswithout Mortality
In this and the following Sections we apply the tools devel-oped up to now to the description of a subprocess in insectdevelopment namely the development of immature larvaewhich is in principle a highly individual subprocess Weassume that the evolution of an insect from egg to pupaoccurs via a sequence of 119864 gt 0 maturation stages Biologistsrecognise some substages by inspection in the developmentof larvae (instars) and even more stages when observed byother methods [35] The immature individual may die alongthe process or otherwise complete it and exit as pupa oradult The actual value of apparent maturation stages 119864 willdepend on environmental and experimental conditions andwill ultimately be specified when analying experimental datain Section 71
Empirical evidence based on existing and new experi-mental results as well as biological insight produced by thepresent description will be the subject of an independentwork [42]We discuss however a couple of examples from theliterature in Section 71
The events for this process are maturation events pro-moting individuals from one subpopulation into to the nextand death events for all subpopulations In other wordseach subpopulation represents a given (intermediate) level ofdevelopment and maturation events correspond to progressin the development Development at level 119895 promotes oneindividual to level 119895 + 1 and the event rate is linear in the119895-subpopulation Hence in 119877
120572= sum
119895119903119896
120572119883
119895 the matrix 119903
is square and diagonal for 0 le 120572 le 119864 minus 1 (we haveexactly 119864 + 1 subpopulations counted from 0 to 119864) All ofthe environmental influence is at this point incorporatedthrough 119903
Moreover level 119864 is the matured pupa and we may set119903119864
= 0 since at pupation stage the present mechanism endsand new processes take place Subpopulation 119864 is the exitpoint of the dynamics and we assume it to be determined bythe stochastic evolution of the immature stages 0 le 120572 le 119864minus1
In the same spirit the action of each event on thepopulations modifies just the actual population and the nextone in the chain Hence
120575120572
119895=
minus1 120572 = 119895
+1 120572 = 119895 minus 1(56)
Following Section 3 [1198770
120572]119879
= (1199030119872 0 0) and
119865120573
120572=
minus119903120572 120573 = 120572
+119903120572 120573 = 120572 minus 1
0 otherwise(57)
Finally letting m119879= minus119872(1 1 1) we can write 119877
0
120572=
sum120573119865120573
120572119898
120573 In this framework the index pair (119895 120572) is highly
interconnected and we can achieve a complete descriptionusing just one index We use 0 le 119895 le 119864 minus 1 throughout thissection
The procedure of Section 3 gives Ψ(119911 119905) = prod119895(119891
119895)119898119895 =
prod119895(119891
minus119872
119895) while the dynamical system for 119891
119895reads recalling
the recasting in Remark 21119891119895= 119891
119895(minus119903
119895(119911
119895119891119895minus 1) + 119903
119895+1(119911
119895+1119891119895+1
minus 1))
119895 = 0 119864 minus 2
(58)
and 119891119864minus1
= 119891119864minus1
(minus119903119864minus1
(119911119864minus1
119891119864minus1
minus 1)) The overall initialcondition is 119891
119895(0) = 1 Let 120579
119895= log(119891
119895) and
120595119895=
119864minus1
sum
120573=119895
120579120573 119882
119895= exp (minus120595
119895) (59)
Note that 119882119895119891119895
= 119882119895
sdot exp(120579119895) = 119882
119895+1 Rewriting the
equations120579119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) + 119903
119895+1(119911
119895+1exp (120579
119895+1) minus 1)
119895 lt 119864 minus 1
119895= minus119903
119895(119911
119895exp (120579
119895) minus 1) = 120579
119895+
119895+1 119895 lt 119864 minus 1
120579119864minus1
= 119864minus1
= minus119903119864minus1
(119911119864minus1
exp (120579119864minus1
) minus 1)
(60)
with initial condition 120579119895(0) = 0 for all 119895With this notation the
generating function reads Ψ(119911 119905) = 119882119872
0 and we formulate
hence an ODE for the 119882119895rsquos
119895+ 119903
119895119882
119895= 119903
119895119911119895119882
119895+1 119882
119895 (0) = 1 119895 = 0 119864 minus 2
119864minus1
= 119903119864minus1
(119911119864minus1
minus 119882119864minus1
) 119882119864minus1
(0) = 1
(61)
The Scientific World Journal 11
By defining the auxiliary quantity 119882119864(119905) = 1 the solution
for the 119882119895-differential equations can be written in recursive
form
119882119895(119905) = 119890
minus119903119895119905+ 119911
119895119903119895int
119905
0
119890119903119895(119904minus119905)
119882119895+1
(119904) 119889119904 119895 = 0 119864 minus 1
(62)
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820(119905) = (
119864minus1
prod
119898=0
119911119898)(1 minus
119864minus1
sum
119896=0
exp (minus119903119896119905)
119864minus1
prod
119895=0119895 = 119896
119903119895
119903119895minus 119903
119896
)
+
119864
sum
119896=1
prod119864minus119896
119895=0119911119895
119911119864minus119896
119903119864minus119896
119864minus119896
sum
119898=0
119903119898exp (minus119903
119898119905)
119864minus119896
prod
119897=0119897 =119898
119903119897
119903119897minus 119903
119898
(63)
The solution looksmore compact when all 119903119896are assumed
to take the common value 119903
1198820(119905) =
prod119864minus1
119898=0119911119898
(119864 minus 1)int
119903119905
0
exp (minus119904) 119904119864minus1
119889119904
+
119864
sum
119896=1
prod119864minus119896
119898=0119911119898
119911119864minus119896
(119903119905)119864minus119896 exp (minus119903119905)
(119864 minus 119896)
(64)
7 Developmental Cascades with Mortality
We add to the previous description 119864 subpopulation specificdeath events 119876
120572= 119902
120572119883
120572 where 120572 = 0 119864 minus 1 We
do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883
119864leaves the
present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876
0 119877
0 119876
1 119877
1 119876
119864minus1 and 119877
119864minus1 Then ((119876 119877)
0)119879
=
(1199020119872 119903
0119872 0 0) We have
1205752120572
120572= minus1 (death) 120572 lt 119864
1205752120572+1
120572= minus1
1205752120572+1
120572+1= +1
(development) 120572 lt 119864
119865120573
2120572=
minus119902120572 120573 = 2120572
minus119902120572 120573 = 2120572 + 1
119902120572 120573 = 2120572 minus 1
119865120573
2120572+1=
minus119903120572 120573 = 2120572 + 1
minus119903120572 120573 = 2120572
119903120572 120573 = 2120572 minus 1
120572 lt 119864
(65)
Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891
120572and the 119911
120572in pairs (119892 119891) and
(119911 119910) as (1198920 119891
0 119892
119864minus1 119891
119864minus1) and (119911
0 119910
0 119911
119864minus1 119910
119864minus1)
Also for notational convenience we will assume that thereexist 119902
119864= 119903
119864= 0 when necessary Defining
119867120572(119892
120572 119891
120572) = 119902
120572(119911
120572119892120572minus 1) + 119903
120572(119910
120572119891120572minus 1) (66)
we have (recall again Remark 21 and note that 119867119864equiv 0)
119892120572= minus119867
120572(119892
120572 119891
120572) 119892
120572
119891120572= (minus119867
120572(119892
120572 119891
120572) + 119867
120572+1(119892
120572+1 119891
120572+1)) 119891
120572
120572 = 0 119864 minus 2
119892119864minus1
119892119864minus1
=
119891119864minus1
119891119864minus1
= minus119867119864minus1
(119892119864minus1
119891119864minus1
)
(67)
The overall initial condition is still 119892120572(0) = 1 = 119891
120572(0)
Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881
119879
120572=
(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579
120572= log119892
120572
and 120601120572
= log119891120572 changing 119867
120572and the initial conditions
correspondingly Hence
120601120572= 120579
120572minus 120579
120572+1997904rArr 120601
120572= 120579
120572minus 120579
120572+1 120572 = 0 119864 minus 2
120579120572= minus119867
120572(120579
120572 120579
120572minus 120579
120572+1) 120572 = 0 119864 minus 2
120601119864minus1
= 120579119864minus1
= minus119867119864minus1
(120579119864minus1
120579119864minus1
)
(68)
The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879
= minus119872(1 0 0 0) satisfies 1198770
= 119865119898 FinallyΨ(119911 119910 119905) = 119892
minus119872
0 We now compute 119892
0(119905)
Let119882120572= 119890
minus120579120572 = 119892
minus1
120572 0 le 120572 le 119864minus1 Defining the auxiliary
variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of
119882 read
120572+ (119902
120572+ 119903
120572)119882
120572= 119902
120572119911120572+ 119903
120572119910120572119882
120572+1 119882
120572 (0) = 1
(69)
with solution
119882120572(119905) = 119890
minus(119902120572+119903120572)119905
+ int
119905
0
119890minus(119902120572+119903120572)(119905minus119904)
(119902120572119911120572+ 119903
120572119910120572119882
120572+1(119904)) 119889119904
(70)
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 11
By defining the auxiliary quantity 119882119864(119905) = 1 the solution
for the 119882119895-differential equations can be written in recursive
form
119882119895(119905) = 119890
minus119903119895119905+ 119911
119895119903119895int
119905
0
119890119903119895(119904minus119905)
119882119895+1
(119904) 119889119904 119895 = 0 119864 minus 1
(62)
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820(119905) = (
119864minus1
prod
119898=0
119911119898)(1 minus
119864minus1
sum
119896=0
exp (minus119903119896119905)
119864minus1
prod
119895=0119895 = 119896
119903119895
119903119895minus 119903
119896
)
+
119864
sum
119896=1
prod119864minus119896
119895=0119911119895
119911119864minus119896
119903119864minus119896
119864minus119896
sum
119898=0
119903119898exp (minus119903
119898119905)
119864minus119896
prod
119897=0119897 =119898
119903119897
119903119897minus 119903
119898
(63)
The solution looksmore compact when all 119903119896are assumed
to take the common value 119903
1198820(119905) =
prod119864minus1
119898=0119911119898
(119864 minus 1)int
119903119905
0
exp (minus119904) 119904119864minus1
119889119904
+
119864
sum
119896=1
prod119864minus119896
119898=0119911119898
119911119864minus119896
(119903119905)119864minus119896 exp (minus119903119905)
(119864 minus 119896)
(64)
7 Developmental Cascades with Mortality
We add to the previous description 119864 subpopulation specificdeath events 119876
120572= 119902
120572119883
120572 where 120572 = 0 119864 minus 1 We
do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883
119864leaves the
present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876
0 119877
0 119876
1 119877
1 119876
119864minus1 and 119877
119864minus1 Then ((119876 119877)
0)119879
=
(1199020119872 119903
0119872 0 0) We have
1205752120572
120572= minus1 (death) 120572 lt 119864
1205752120572+1
120572= minus1
1205752120572+1
120572+1= +1
(development) 120572 lt 119864
119865120573
2120572=
minus119902120572 120573 = 2120572
minus119902120572 120573 = 2120572 + 1
119902120572 120573 = 2120572 minus 1
119865120573
2120572+1=
minus119903120572 120573 = 2120572 + 1
minus119903120572 120573 = 2120572
119903120572 120573 = 2120572 minus 1
120572 lt 119864
(65)
Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891
120572and the 119911
120572in pairs (119892 119891) and
(119911 119910) as (1198920 119891
0 119892
119864minus1 119891
119864minus1) and (119911
0 119910
0 119911
119864minus1 119910
119864minus1)
Also for notational convenience we will assume that thereexist 119902
119864= 119903
119864= 0 when necessary Defining
119867120572(119892
120572 119891
120572) = 119902
120572(119911
120572119892120572minus 1) + 119903
120572(119910
120572119891120572minus 1) (66)
we have (recall again Remark 21 and note that 119867119864equiv 0)
119892120572= minus119867
120572(119892
120572 119891
120572) 119892
120572
119891120572= (minus119867
120572(119892
120572 119891
120572) + 119867
120572+1(119892
120572+1 119891
120572+1)) 119891
120572
120572 = 0 119864 minus 2
119892119864minus1
119892119864minus1
=
119891119864minus1
119891119864minus1
= minus119867119864minus1
(119892119864minus1
119891119864minus1
)
(67)
The overall initial condition is still 119892120572(0) = 1 = 119891
120572(0)
Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881
119879
120572=
(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579
120572= log119892
120572
and 120601120572
= log119891120572 changing 119867
120572and the initial conditions
correspondingly Hence
120601120572= 120579
120572minus 120579
120572+1997904rArr 120601
120572= 120579
120572minus 120579
120572+1 120572 = 0 119864 minus 2
120579120572= minus119867
120572(120579
120572 120579
120572minus 120579
120572+1) 120572 = 0 119864 minus 2
120601119864minus1
= 120579119864minus1
= minus119867119864minus1
(120579119864minus1
120579119864minus1
)
(68)
The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879
= minus119872(1 0 0 0) satisfies 1198770
= 119865119898 FinallyΨ(119911 119910 119905) = 119892
minus119872
0 We now compute 119892
0(119905)
Let119882120572= 119890
minus120579120572 = 119892
minus1
120572 0 le 120572 le 119864minus1 Defining the auxiliary
variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of
119882 read
120572+ (119902
120572+ 119903
120572)119882
120572= 119902
120572119911120572+ 119903
120572119910120572119882
120572+1 119882
120572 (0) = 1
(69)
with solution
119882120572(119905) = 119890
minus(119902120572+119903120572)119905
+ int
119905
0
119890minus(119902120572+119903120572)(119905minus119904)
(119902120572119911120572+ 119903
120572119910120572119882
120572+1(119904)) 119889119904
(70)
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 The Scientific World Journal
In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain
1198820 (119905) =
119864minus1
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
minus
119864minus1
sum
119898=0
119903119898119910119898119890minus(119902119898+119903119898)119905
119902119898
+ 119903119898
times
119864minus1
prod
119896=0119896 =119898
119903119896119910119896
119902119896minus 119902
119898+ 119903
119896minus 119903
119898
+
119864
sum
119896=1
1
[119903119910]119864minus119896
times [
[
119864minus119896
prod
119898=0
119903119898119910119898
119902119898
+ 119903119898
[119902119911]119864minus119896
+
119864minus119896
sum
119898=0
(1 minus[119902119911]
119864minus119896
119902119898
+ 119903119898
)
times 119903119898119910119898119890minus(119902119898+119903119898)119905
times
119864minus119896
prod
119895=0119895 =119898
119903119895119910119895
119902119895minus 119902
119898+ 119903
119895minus 119903
119898
]
]
(71)
Again we have that Ψ(119911 119910 119905) = 119882119872
0 Specialising for the
case where all 119903 and all 119902 coefficients are equal the solutionreads
1198820(119905) =
prod119864minus1
119898=0119903119910
119898
(119864 minus 1)int
119905
0
119890minus(119902+119903)119909
119909119864minus1
119889119909
+
119864minus1
sum
119896=0
prod119896
119898=0119903119910
119898
119903119910119896119896
[119905119896119890minus(119902+119903)119905
+ 119902119911119896int
119905
0
119890minus(119902+119903)119909
119909119896119889119909]
(72)
71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882
0for different powers of
119911119896and 119910
119896 As an example we compute the probabilities 119901
119894(119905)
of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod
119904
119898=0120582119898 119904 =
0 1 2 119876(119904119898) = prod119904
119896=0119896 =119898(120582
119896minus 120582
119898+ 120583
119896minus 120583
119898) 119904 = 1 2 3
for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583
119904
119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877
0) while for 119894 = 1 2 3 we have 119901
119894(119905) =
119875(119894 minus 1)sum119894
119898=0(exp(minus119905119877
119898)119876(119894 119898))
8 Concluding Remarks
The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one
When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population
The probability generating function for the number ofevents 119899
120573 conditioned to an initial state described by
population values 119883119894(0) has been obtained as a function
of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods
We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented
The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment
Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered
Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations
Appendices
A Proofs of Traditional Results
The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 13
A1 Proof of Lemma 5
Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905
1minus 119905 ge 0 be
sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591
For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that
119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum
120572
119882120572(119883 119905) ℎ) + 119900 (ℎ)
(A1)
Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ= minus119875 (119899 = 0 119905)
times (sum
120572
119882120572 (119883 119905)) +
119900 (ℎ)
ℎ
(A2)
and finally
limℎrarr0
119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)
ℎ
= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum
120572
119882120572(119883 119905))
(A3)
Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is
119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)
(A4)
A2 Proof of Theorem 6
Proof Again the Chapman-Kolmogorov equation gives
119875 (1198991 119899
119864 119905 + ℎ)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905)) ℎ
+ 119875 (1198991 119899
119864 119905) (1 minus sum
120572
119882120572(119883 (119899
1 119899
119864 119905)) ℎ)
+ 119900 (ℎ)
(A5)
Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation
(1198991 119899
119864 119905)
= sum
120572
119875 ( 119899120572minus 1 119905)119882
120572(119883 ( 119899
120572minus 1 119905))
minus 119875 (1198991 119899
119864 119905) (sum
120572
119882120572(119883 (119899
1 119899
119864 119905)))
(A6)
B Solution via Laplace Transform
After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866
119896(119904) =
L(119867(119905)119882119896(119905)) the equation can be rewritten as
119866119896(119904)=
1
119904 + 119903119896
(1 + 119911119896119903119896119866119896+1
(119904)) 119866119864(119904)=
1
119904 0 le 119896 lt 119864
(B1)
since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction
1198660(119904) =
1
119904
119864minus1
prod
119898=0
(119911119898119903119898
119904 + 119903119898
) +
119864
sum
119896=1
(1
119911119864minus119896
119903119864minus119896
119864minus119896
prod
119898=0
(119911119898119903119898
119904 + 119903119898
))
(B2)
Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903
119896are taken to be equal the corresponding119866
0(119904)
still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))
The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)
119866119896(119904) =
1
119904 + 119902119896+ 119903
119896
(1 +119902119896119911119896
119904+ 119903
119896119910119896119866119896+1
(119904))
119866119864(119904) =
1
119904
(B3)
with solution
1198660(119904) =
1
119904
119864minus1
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
+
119864
sum
119896=1
[119904 + 119902119911
119904119903119910]
119864minus119896
119864minus119896
prod
119898=0
119903119898119910119898
119904 + 119902119898
+ 119903119898
(B4)
Again inverse Laplace transform leads to (71)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 The Scientific World Journal
References
[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993
[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010
[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001
[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972
[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990
[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003
[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006
[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008
[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011
[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013
[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976
[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005
[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg
[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008
[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949
[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937
[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002
[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004
[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003
[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977
[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950
[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956
[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001
[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003
[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004
[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007
[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007
[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931
[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940
[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970
[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971
[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976
[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978
[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981
[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003
[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953
[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986
[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001
[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 15
[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996
[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964
[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of