a theoretical model for the energy dynamics of the

A Theoretical Model for the Energy Dynamicsof the Parasite Fasciola hepatica

Anthony AylwardCCS Mathematics

University of California, Santa Barbara

AcknowledgementsBjorn Birnir

Department of MathematicsUniversity of California, Santa Barbara

Kevin LaffertyEcological Parasitology


Armand KurisEcological Parasitology


June 1, 2012

1

Abstract

We apply a Dynamic Energy Budget model to the common liverfluke (Fasciola hepatica), as well as its intermediate and final hosts.First, an introduction to the DEB theory is given. We divide thefluke’s seven-stage life cycle into three phases and construct an inde-pendent model for each. We link DEB variables to measurable quanti-ties like weight, glycogen content, and oxygen consumption, then findmodel parameters that match predictions to data from existing stud-ies. We compare the present model to existing models. We discussthe effects of environmental temperature and intra-host crowding onthe parasite’s life cycle as a whole.

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Contents

1 Introduction 41.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The Life Cycle of F. hepatica . . . . . . . . . . . . . . . . . . 41.3 The DEB Theory. . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 The Adult Phase 92.1 An existing DEB model for parasitism . . . . . . . . . . . . . 102.2 The present model . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Parametrization of the host model . . . . . . . . . . . . . . . . 122.4 Adding the parasite . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Multiple infections . . . . . . . . . . . . . . . . . . . . . . . . 15

3 The Miracidial Phase 193.1 The egg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 The miracidium . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 The Intramolluscan Phase 254.1 The host model . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Host modification . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 The snail-trematode system: a bifurcation in the model . . . . 264.4 Intra-host interaction between parasites . . . . . . . . . . . . . 27

5 Temperature Dependence 305.1 Arrhenius temperature . . . . . . . . . . . . . . . . . . . . . . 30

6 Discussion 316.1 Application of the model and further study . . . . . . . . . . . 31

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1 Introduction

1.1 Background

In [10], S.A.L.M Kooijman develops the Dynamic Energy Budget (DEB),a theoretical model for the uptake and use of energy by living organisms.He ambitiously touts the applicability of the model to all living things, andindeed it has been applied to free-living organisms with considerable success[5] [17]. However, the application of the model to parasites has been onlylittle explored [6]. After reading [5] we became curious about the constructionand behavior of a DEB model for a parasitic system. Hall et al.’s DEB modelfor parasitic castrators [6] is promising but somewhat narrow in its scope:their focus was on a parasitoid system, in which the ’parasite’ grows largerand larger until it kills the host. In contrast, this paper focuses on a systemin which the parasite and host have a long-term coexisting relationship.

We wanted not only to study the theoretical behavior of a host-parasiterelationship, but also to make that study concrete by applying it to an organ-ism actually found in nature. After some background reading on parasitologywe settled on the common liver fluke Fasciola hepatica as a subject. It be-longs to the trematodes, a class of parasitic worms that have complex lifecycles. F. hepatica is large compared to other trematodes and has economicsignificance as a cause of disease in cows and sheep. Occasionally it is alsofound in humans. Due to its size and medical and economic importance,Fasciola hepatica has been known of for hundreds of years and has been thesubject of much scientific study [12]. In the present paper, we combine var-ious data on the growth and energy metabolism of this parasite with theframework provided by DEB to model its entire life cycle. For the reader’sconvenience, a brief overview of the life cycle follows.

1.2 The Life Cycle of F. hepatica

F. hepatica begins its life as a microscopic egg, which develops and hatchesif immersed in fresh water. When it hatches, a tiny larva called a miracid-ium emerges. The miracidium is covered with cillia, which it uses to swimconstantly in search of a snail, the parasite’s intermediate host. If it findsa snail, the miracidium penetrates it and immediately metamorphoses intoits next form, called the sporocyst. The sporocyst is little more than anelongated sac, about twice the size of the miracidium. It containins embryos

4

Figure 1: Life cycle of F. hepatica, courtesy of pathologyoutlines.com.

which grow into the next stage, the redia, similar to the sporocyst but witha mouth and digestive tract. The rediae leave the sporocyst, feed on thesnail, and produce more rediae. After a few weeks, they begin to producecercariae, the next life stage. A cercaria leaves the snail and attaches itselfto a plant, forming a cyst called a metacercaria. It waits to be eaten bya mammalian host like a cow or a sheep. If eaten, the fluke metamorphosesagain into its adult form. The adult fluke grows to a length of about 20 mm,much larger than any of its larval stages. It travels to the host’s liver, whereit feeds and produces eggs for the rest of its lifetime, which may be a fewmonths or several years. The eggs are excreted by the host, and if they fallinto fresh water they restart the cycle [12].

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1.3 The DEB Theory.

The Dynamic Energy Budget model is a general theory for the energy dy-namics of living organisms. It treats an individual organism as a nonlineardynamical system, and traces its physiological state over time. It also hasapplications at the cellular and population levels.

A DEB model describes an individual organism of a given species by it’svolume (denoted V ) and energy reserve (denoted E ). As the individual ages,V and E are functions of time that depend on species-specific parameters andenvironment-specific ones like temperature and food density.

V and E are known as the state variables of the individual. For modelingpurposes, we are interested in d

dtV and d

dtE. These two functions are derived

in the following way.The function E represents the organism’s stored energy (in the form of

sugars, fats, and proteins) which is ready for use. An organism takes inenergy at a certain (not fixed) rate A, for anabolic process. It uses up theenergy at rate C, for catabolic process. This means that at any point intime,

d

dtE = A− C

where E, A, and C are all functions of time. Now, we wish to learn moreabout A and C. By Kooijman’s derivation

A = afV 2/3

where a is a proportionality constant and f represents the availability offood. The term V 2/3 arises from the fact that the organism absorbs foodacross a membrane, the size of which is proportional to the organism’s totalsurface area. The result is that if the food supply f is constant, then theorganism takes in energy at a rate proportional to its surface area.

Kooijman’s derivation of C is more daunting. Remember now that Crepresents the sum of all the ways our organism uses energy. In the standardDEB model, there are four:

C = M +G+R + J

Where M is the maintenance rate, the rate at which the organism’s cellsuse energy for upkeep. G is the rate at which energy is used for growth,

6

R is the energy-usage rate for reproduction, and J is the maturity mainte-nance rate, which is required for the organism to maintain a certain level ofcomplexity.

Kooijman derives the following expressions for these three rates. Alllowercase letters are proportionality constants.

M = mV

G = gd

dtV

R + J = r(M +G)

We should note that Kooijman uses a slightly different notation for thereproduction rate, 1−κ

κ(M +G). The constant κ is an important part of the

model and will be referred to later. ddtE is now given by

d

dtE = A−M −G−R− J

d

dtE = afV 2/3 − (1 + r)(mV + g

d

dtV ) (1)

Energy Density We can learn more about this system by considering theorganism’s energy density, that is, the quantity [E] = E/V . After mechanis-tic arguments, Kooijman derives the following formula:

d

dt[E] = (af − c[E])V −1/3) (2)

Now, note that

d

dtE = [E]

d

dtV + V

d

dt[E]

By our previous discussion, this means

[E]d

dtV + V

d

dt[E] = afV 2/3 − (1 + r)(mV + g

d

dtV )

Substituting the expression for ddt

[E] gives

[E]d

dtV + V (af − c[E])V −1/3) = afV 2/3 − (1 + r)(mV + g

d

dtV )

7

We can now solve for ddtV :

d

dtV =

c[E]V 2/3 −m(1 + r)V

g(1 + r) + [E](3)

The nonlinear system formed by ddt

[E] and ddtV is the core of the DEB

theory.By Kooijman’s derivation, [E] and V have maximum values at steady

state [E]max = ac

and Vmax = ( am(1+r)

)3. To non-dimensionalize the system,

we write e = [E][E]max

and v = VVmax

, so that:

d

dte = µ(f − e)v−1/3 (4)

d

dtv =

µ(ev2/3 − v)

γ + e(5)

Where µ = cam(1 + r), γ = c

ag(1 + r).

Note that e and v are functions of time, f is a function of food density,and all other letters are constants.

The scaled functional response In the above system, the function f isgiven by :

f =X

Xhalf +X(6)

where X is the density of food in an organism’s environment, and Xhalf isthe value of X for which f = 1

2.f is called the scaled functional response.

It determines how the availability of food influences the organism’s ingestionrate.

The Von Bertalanffy curve Kooijman found that when food density isconstant, energy density approaches a constant e = f . When energy densityis constant, we must have e = f and the value of V then follows a verypredictable curve, known as a Von Bertalanffy curve, given by

V 1/3 = L(t) = L∞ − (L∞ − Lb)exp{−rBt} (7)

which non-dimensionalizes to

8

Food Waste

Energy Reserve

Volume

Structural Maintenance Maturity Maintenance

Reproduction

A

C

κ 1−κG R

M J

Figure 2: Conceptual diagram of the energy flow through a DEB model.

l = f − (f − lb)exp{−rBt} (8)

where l = ( VVmax

)1/3. Lb stands for the volumetric length at birth, L∞ =fLmax ,and rB is a parameter. This means that when energy density isconstant we have an explicit solution for the system, which will be useful forinterpreting data.

Applying the model to F. hepatica F. Hepatica has a total of seven lifestages, some free-living and some parasitic. Here we divide the life cycle intothree phases: the miracidial phase, which covers the egg and miracidium, theintramolluscan phase, which covers the sporocyst and rediae and the adultphase. In terms of energy and growth, very little happens during the cercarialand metacercarial stages, so they are skipped. The following sections detailthree distinct model formulations for the three different phases.

2 The Adult Phase

Adult F. hepatica live in mammalian hosts. The growth of adult flukes hasbeen studied in rat hosts (Rattus norvegicus), so for our model we will assume

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the host is a rat.If an organism is infected with a parasite (like a trematode), it suffers

from a drain on its energy reserve. The parasite treats the host’s energy asits own food supply. This energy stolen from the host is then stored by theparasite in its own reserve and used for its own catabolic processes.

2.1 An existing DEB model for parasitism

Hall, Becker, and Caceres adapted the DEB model to include parasitism,including parasitic castrators. Their model simplifies the dynamics of theparasite, but has some drawbacks. It includes parasite biomass, but notenergy. It is designed for parasites of the water flea Daphnia magna, whichgrow until they kill their host after a relatively short time. If allowed tocoexist with their host, parasites under the model formulated by Hall et almay achieve unrealistically large volumes [6].

In the present model, we couple a DEB for the rat host with one foran individual parasite, including its volume and energy density. This makessense since F. hepatica is a macroparasite, which means that being infectedwith one is not the same as being infected with many. Applying a DEBmodel to the parasite also gives it a built-in maximum volume and allowslong-term coexistence of the parasite and host.

2.2 The present model

Let’s consider equations for the state variables of the adult parasite (Ep, Vp)and its host (Eh, Vh) . We have:

d

dtEh = Ah − Ch − Ap

d

dtEp = Ap − Cp

From our previous discussion we know the composition of Ah and Ap.

Ah = ahfhV2/3h

Ap = apfpV2/3p

For simplicity, we can let fh be a constant. (i.e., the host’s food will bereplenished as fast as the host eats it.). Note, however, that the parasite’s

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Food Waste

Host Energy Reserve

Host Volume

Host Structural Maintenance Host Maturity Maintenance

Host Reproduction

A p

Ch

κh 1−κhG h Rh

M h J h

Parasite Energy Reserve

Parasite Volume

Parasite Structural Maintenance

Parasite Maturity Maintenance

Parasite Reproduction

C p

1−κpκ pG p

M p

R p

J p

Ah

Figure 3: DEB flowchart with parasite included. The arrows colored red collapseif parasitic castration occurs, see sec. 4.2.

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environment is the host itelf, and its food is the host’s energy reserve. Hence,the density of the parasite’s food in its environment is the quantity [E]. Thismeans that fp is given by

fp =[Eh]

[Eh]half + [Eh]=

ehehhalf + eh

(9)

We now have that Ap depends on Eh and Vh, so the state of the parasiteis tied to that of the host. Of course, the parasite also influences the rat bythe effect of Ap on Eh. We can now write:

d

dt[E] = (af − c[E])V −1/3)− apfp

V 2/3p

Vhso

d

dteh = µh(fh − eh)v−1/3h − αfp

v2/3p

vh(10)

where α = apchah

V2/3pmax

Vhmax, while d

dtvh,

ddtep, and d

dtvp have the standard forms

of equations eq:nondime and eq:nondimv. This completes our formulation ofa DEB model for the host-parasite system.

2.3 Parametrization of the host model

[10] includes parameters for the Von Bertalanffy growth curve of Rattusnorvegicus :

L∞ = 7.523cm

rB = 9.286yr−1

From these we can deduce µh and γh as follows.Animal tissue has a density approximately the same as water, about 1

gram per cubic centimeter [17]. Hence, we can find the volume of a rat fromits weight in grams.

The maximum weight of fully grown R. norvegicus is about 900 g. Hencewe will set Vhmax = 900cm3, which means Lhmax = V

1/3hmax = 9.655cm. We

can infer

fh =Lh∞Lhmax

= 0.779

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0.4

0.5

0.6

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scal

ed v

alue

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Student Version of MATLAB

Figure 4: Evolution of the state variables of R. norvegicus from birth to age 1year. The dashed light blue line is the energy density eh while the solid blue lineis the volume vh. Both are shown in non-dimensionalized form as a percentage oftheir maximum values. Note that eh is constant, so vh is a Von Bertalanffy curve.Parameter values: µh = 49.559, γh = 1, fh = 0.779.

We now have the scaled Von Bertalanffy curve

lh = fh − fhexp{−rBt}

We have let lhb = 0. We can safely do this since changing lhb does notchange the shape of the curve. Our model will match this curve providedthe relation rB = µh

γh+fh)holds. We arbitrarily set γh = 1 and deduce muh =

49.559. This completes our (simple) DEB model for R. norvegicus. See figure4.

2.4 Adding the parasite

The maximum volume and energy density used for scaling here are

[Epmax] = 0.13gglucosegbiomass

Vpmax = 0.571cm3

To complete the model of the host-parasite system, we need to determinethe values of µp, γp, α and ehhalf . Once again, we assume for simplicity that

13

γp = 1. Then we fit the system to data on growth of F. hepatica in rats [15][14] and reduction of weight in infected rats [7] to find

γp = 1

µp = 30.354

α = 1.060

ehhalf = 0.169

This, combined with the parameters from the previous section, producesa model for the rat and the parasite infecting it.

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2.5 Multiple infections

If we assume all individuals of F. hepatica are identical, we can model asimultaneous infection with multiple parasites by replacing α with nα wheren is the number of parasites. This allows us to extract the true value ofα from data on rats infected with 2 parasites, and extrapolate to modelpathogenicity as a function of parasite burden. This model also includes theeffect of crowding, wherein the size of individual parasites decreases when thenumber of parasites per host increases. The rat-adult parasite model withseveral values of n is shown in fig. 5. Relative error is defined as

‖ε‖‖x‖

Where x is a vector containing the y-values of the data points and ε avector containing the difference of each data point from the model prediction.The norm is the standard 2-norm.

15

Figure 5: Plots of state variables in the rat-adult parasite system from time ofinfection to one year later, at various levels of parasite burden (n). Curves on theleft in blue refer to the host, those on the right in red refer to the parasite. Dashedlines represent energy density and solid lines volume. Data points for parasitegrowth are from [14] and [15]. Data is consistent for the first four months ofgrowth but begins to scatter after, so error estimates are given both for the wholeyear and for the first four months. Since the mean volume of parasites seems todecline after 5-6 months, it is possible that the data are showing the effects offactors that are not included in the model (i.e. parasite mortality.) Parametervalues are as in fig. 4 along with: µp = 30.354, γp = 1, α = 1.060, ehhalf = 0.169Relative errors including all data points are, from top to bottom, 0.426, 0.430,0.654. Restricted to the first four months of growth, they are 0.149, 0.106, 0.107

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Figure 6: Effect of parasite burden on the steady-state values of state variablesin the rat-adult parasite system. The blue plot on the left represents the host,and the red plot on the right represents the parasite. Note the substantial effectof increased parasite burden on host energy density. The data points are themaximum sizes reached by parasites at differing levels of crowding, from [15]. Thedata point at n = 8 lies significantly below the model’s prediction, which could bedue to the small sample size of data at that crowding level (only 6 data points).Relative error is 0.224 against all three data points, 0.017 if the point at n = 8 isexcluded. Parameter values are as in fig. 5.

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3 The Miracidial Phase

3.1 The egg

Adult F. hepatica lay microscopic eggs which leave the host and hatch inaquatic environments. The energy flow within the egg is much simpler thanthat of the adult:

EC−−−→ V (11)

In short, the energy stored in the egg is used for the growth and mainte-nance of the embryo. The egg does not eat, so we can say f = 0. Hence, oursystem for d

dte and d

dtv reduces to:

d

dte = −µev−1/3 (12)

d

dtv =

µ(ev2/3 − v)

γ + e(13)

Data for the growth of aF. hepatica embryo over time are available [16].We fit the model with the values:

µ = 2.28 ∗ 10−3

γ = 2.23

e0 = 68.132

Where e0 is the non-dimensionalized energy density of the embryo whenits volume is 3.5 micrometers, the first data point in [16]. In theory v(0) = 0,which implies e(0) =∞. e0 was inferred using data from [9] scaled to [E]pmaxfound for the adult model.

3.2 The miracidium

The first of F. hepatica’s larval stages, the miracidium is a microorganismabout 100 microns in length. It swims freely in water and searches for asusceptible snail host. If it does not find one within a few hours of hatching,it dies.

19

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−7

cubi

c cm

days


Figure 7: Model prediction for growth of the embryo of F. hepatica, with datapoints from [16]. Relative error: 0.144. Parameter values: µ = 2.28 ∗ 10−3, γ =2.23, e0 = 68.132.

The miracidium under the standard DEB model The energy dynam-ics of the miracidium under a standard DEB model are only slightly differentfrom those for the egg:

EM+J−−−→ V (14)

Like the egg, the miracidum does not feed. We lose the terms G and Rfrom C since no growth or reproduction occurs during the miracidial stage.It has fixed volume Vl, where the l stands for larva. The terms M and Jremain because the miracidium must still spend energy on structural andmaturity maintenance. Both are now constants and J = rM .

When we apply the standard DEB model to the miracidium, we run intosome difficulties. Firstly, there are two possible derivations for d

dte. We

denote them by e1 (Standard DEB model) and e2 (Modified DEB model).Since f = 0 we can write (as in the egg model):

d

dte1 = −µe1v−1/3

since v−1/3 is now constant, this implies

e1(t) = k1exp{−µv−1/3t}

20

Where k1 is the constant resulting from integration.The second equation is derived as follows. We have:

d

dtE2 = −M − J

d

dtE2 = −m(1 + r)Vl

d

dt[E2] = −m(1 + r)

d

dte2 = −µ

e2(t) = −µt+ k2

An inconsistency has arisen from forcing f = 0 and ddtV = 0, the con-

ditions which define the miracidium. Kooijman recognized this discrepancyand found that some organisms follow e1(t) when starving, while others fol-low e2(t) (i.e. the LD vs MD snails in [17]). However, neither e1(t) nor e2(t)results in a catabolic rate that matches data on the miracidium’s oxygenconsumption [4].

Comparison with data The miracidium converts its energy reserve (glyco-gen) to maintainence and movement through aerobic respiration (at least inpart.) In respiration, energy is released from glucose by combining it withoxygen, and the amount of glucose used is proportional to the amount ofoxygen consumed. This means that the oxygen consumption rate of a F.hepatica miracidium should be proportional to its catabolic rate .

data from [4] show that the miracidium’s oxygen consumption starts outlow, but ‘spools up’ quickly at the beginning of its lifetime, then reaches apeak and decreases exponentially thereafter. Neither e1(t) nor e2(t) fit thisdata, since both are monotonically decreasing.

Actually, e1(t) models the energy consumption of the miracidium fairlywell after about the 6-hour mark. The rate of energy consumption decreaseswith time, implying that swimming speed decreases as the miracidium ap-proaches the end of its life. This matches biological observations [12]. Theproblem is found in the first six hours after hatching. e1(t) Implies thatenergy consumption during this time should be high but decreasing, whilethe data show that it is low but increasing. This also creates a rather large

21

discontinuity in the model, since the rate of energy consumption by the em-bryo at hatching is approximately µ, which is close to zero, while the rate ofenergy consumption by the miracidium at hatching is much higher accordingto e1(t).

e2(t) is even less successful than e1(t), since under e2(t) energy consump-tion (therefore swimming speed) should be constant, which is not supportedby observations.

Something is missing from the standard model that prevents it from accu-rately describing the miracidium. We claim that this is because the standardmodel does not account for energy used for swimming. Kooijman himselfwrites that ‘sustained powered movement’ requires special treatment [10].To resolve this difficulty, we modify the model as follows.

Present DEB model for the miracidium To improve the model, wepick up a new term , the rate at which energy is used for swimming. We canexpress the miracidium’s energy use as :

d

dtE = −m(1 + r)Vl − S

d

dt[E] = −m(1 + r)− S

Vl

d

dte = −µ− σS

where σ = caVl

. Since S is the only non-constant term of ddte, we can learn

about its shape from data on oxygen consumption by F. hepatica miracidia[4]. Since oxygen is required for respiration, the rate of its consuption is pro-portional to the rate at which energy reserves are consumed by the miracid-ium’s (aerobic) metabolism.

According to the data (at 25 degrees C), the miracidium’s catabolic rateincreases rapidly after hatching, peaks after about six hours, and then de-creases until the miracidium dies at the 24 hour mark. It makes sense toattribute this pattern to swimming costs, since miracidia swim quickly atthe beginning of their lifetime but slow down as they run out of energy re-serves.

We can capture the observed pattern by writing:

d

dtS = e− τS

22

Which is to say that when the miracidium has a lot of energy relative toits swimming speed, it speeds up, but as its energy density decreases, it slowsdown to extend its life. Constructed in this way, the coupled system formedby d

dte and d

dtS fits the data better than either of the formulas resulting from

the standard model. It fits with parameters:

µ = 2.28 ∗ 10−3

σ = 50

τ = 20

The value of µ and the initial energy density of the miracidium wereinferred from the egg model. The model is compared to data from [4], afterconverting respiratory oxygen consumption to glucose consumption. Thiscompletes our formulation of the model for the miracidial phase.

23

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

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valu

e

days


Figure 8: Plots of energy density and energy consumption for miracidia of F.hepatica under the standard DEB model (top left) modified DEB model (top right)and present model (bottom). The purple dashed lines represent energy densitywhile the pink dot-dashed lines are the rate of energy use. They are compared todata on oxygen consumption from [4]. Relative errors are: 0.454 (standard) 0.519(modified) and 0.440 (present). Parameter values: µ = 2.28∗10−3, σ = 50, τ = 20.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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Figure 9: Growth of L. stagnalis from hatching to age 1 year at constant fooddensity. Parameter values: µs = 31.284, γs = 1, fs = 0.7.

4 The Intramolluscan Phase

The sporocyst is a sack-like form that lives within the snail host, absorbingnutrients and producing rediae. The energy flow for this system is the sameas that of the rat-adult system. We will denote the processes of the snailby a subscript ‘s’, and those of the parasite by ‘r’ for redia. In the followingdiscussion, we treat the entire population of parasites within the snail as asingle biomass. This makes sense since all of the rediae are really just asexu-ally produced clones of the ‘mother sporocyst’, and in growth and behaviorthey act together very much like a single individual.

4.1 The host model

[17] found Von Bertalanffy parameters for Lymnaea stagnalis, a snail that canact as a host for F. hepatica larvae. We match our model to those parametersby the same method used for R. norvegicus, and set:

µs = 31.284

γs = 1

25

4.2 Host modification

F. hepatica larvae influence their snail hosts by a process called parasiticcastration. By a somewhat mysterious means, the parasite causes the snailto stop reproducing and instead become larger. This is probably related tothe fact that F. hepatica rediae occupy and consume the reproductive tissueof their snail host [13]. According to [11], the transfer of the snail’s energyfrom reproduction to growth may be the result of a combination of (a) theparasite physically attacking the snail’s reproductive organs, (b) the para-site producing hormones that alter the snail’s physiology, and (c) the snailmodifying its own energy allocation as a survival strategy (according to [17],uninfected snails may modify their energy allocation to increase longevityduring starvation). In DEB terms, all three of the above mechanisms havethe effect that that once the snail is infected, rs becomes smaller, eventuallyreaching zero. (Kooijman refers to this parameter by κs = 1

1+rs). Increasing

κs causes an increase in ddtVs and results in gigantism. This may be beneficial

to both the parasite and the snail [6] [11] .In model terms, this results in the reduction of µs and γs to κsµs and

κsγs . It also changes Vsmax to 1κ3sVsmax: recall

Vsmax = (as

ms(1 + rs))3 = (

asms

κs)3

This is what allows the snail to grow larger. As an approximation, we canassume that the parasite alters the host model immediately upon infection.[17] found κs = 0.13 for the snail, so these changes are quite substantial. Thesmall value of κs is conducive to castration in this system: since the snail’senergy allocation to reproduction is large, the parasite has a lot to gain bytaking it.

4.3 The snail-trematode system: a bifurcation in themodel

When we modify the snail model and add the parasite, we find a rough butnot unrealistic fit with the parameters:

µr = 110

γr = 1

26

α = 0.09

eshalf = 0.02

The biomass of the rediae vr grows close to its maximum size very quickly,which explains the large value of µr and the small eshalf .

Here α is much smaller than in the rat-adult system, which is explained by

the greatly increased value of Vsmax (recall α = arcsas

V2/3rmax

Vsmax). However, in this

system small changes in α have drastic consequences for the model, whichwas not the case in the rat-adult system. When α = 0.090, the trematodegrows quickly and drives the snail’s energy density to a very low level, butstill allows the snail to grow to over twice its normal size. If α = 0.089,however, the snail seems to break free from the constraints of the parasiteand grow unrealistically huge. If α = 0.180, both the snail and the parasitebecome stunted and the snail’s energy density drops to a level that seemsunrealistically low.

4.4 Intra-host interaction between parasites

According to [8] and [11], larval trematodes of distinct species rarely sharea snail host. Instead, one trematode drives the other to a very small size orto death according to their places in a dominance heirarchy [11]. F. hepaticais likely to be relatively dominant compared with other trematodes, sinceits sporocysts produce rediae as offspring [12]. The substantial effects ofincreased parasite burden on the snail-trematode system, shown in fig. 12,suggests that larval trematodes should completely exclude competitors tomaintain their fitness. This matches their observed behavior [11]. Comparewith fig. 6, in which a single rat host can support several parasites withrelative ease.

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Figure 10: Plots of the state variables of the snail-host system from time ofinfection to one year after, when the parasite burden is 1 (top) or 2 (bottom).The green plots on the left indicate the state variables of the snail host, the redones on the right refer to the parasite. Dashed lines are energy densities and solidlines are volumes. Note the substantial impact of adding a second parasite tothe system. Data points on parasite growth are from [2] and [3]. Relative error:0.157. Paramater values are as in fig. 9 together with: µr = 110, γp = 1, α =0.09, eshalf = 0.02.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

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Figure 11: Snail-trematode system with α = 0.089. Other parameter values areas in fig. 10. This illustrates the sensitivity of the model to small changes in α.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

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Figure 12: Effect of increased parasite burden on the steady-state values of statevariables in the snail-trematode system, from 1 parasite to 3. Note the severeimpact of adding extra parasites to the snail. Parameter values are as in fig. 10.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

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5 Temperature Dependence

Adult F. hepatica are insulated from temperature variation by their endother-mic host, but the other life stages are not. As temperature goes up, the de-velopment and survival times of eggs and miracidia decrease while the growthand reproduction rates of the sporocyst and rediae increase [1]. The DEBmodel describes the effect of temperature on physiological rates via Arrheniustemperatures [10].

5.1 Arrhenius temperature

We can include temperature dependence in the above models by making theparameters which define rates (µ, σ) functions of temperature (T ). Accordingto [10], the dependence of physiological rates on temperature (within a certainrange) is usually described well by:

µ(T ) = µ1exp{TAT1− TA

T}

Where T1 is an arbitrary reference temperature, µ1 = µ(T1), and TA is theArrhenius temperature. In the case of F. hepatica, the temperature range forwhich this applies is 283-303 K. The experiments in [16] had a temperatureof 298 K, so we will use that as the reference temperature. If we assumethat hatching occurs when the volume of the embryo reaches 10−7cm3 (thevolume of the miracidium), our model for egg development fits data from [1]with an Arrhenius temperature of 12990 K.

Data from [1] on lifespan of miracidia do not match those from [4]. Thisis likely because the experiments of [4] were on miracidia concentrated at5000 individuals per milliliter of water, so their data represents a maximumlifespan rather than a mean. Accordingly, we modify σ in our model ofthe miracidium to better reflect [1], setting σ1 = 160.782 at T1 = 298K.We use the same value of µ1 as in the egg model. Different physiologicalrates of an individual usually have the same Arrhenius temperature [10],so we will assume this is the case for µ and σ in the miracidium model.So constructed, the model’s predictions of miracidial lifespans at differingtemperatures match data from Al-Habbib with an Arrhenius temperature of2757 K.

30

10 12 14 16 18 20 22 24 26 28 300

20

40

60

80

100

120

days

temperature in C


10 12 14 16 18 20 22 24 26 28 300.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

days

temperature in C


Figure 13: Effect of temperature on the development time of a F. hepatica embryo(left) and lifespan of a miracidium (right). As temperature increases, developmentand survival times fall. Data points are from [1]. Relative error: 0.180 (embryo),0.031 (miracidium). Arrhenius temperatures are 12990 K (embryo) and 2757 K(miracidium).

6 Discussion

6.1 Application of the model and further study

Now that we have formulated models for the various life stages of F. hep-atica, we have a complete picture of its energy dynamics over its life cycle.In particular, we have information on how development is affected by envi-ronmental temperature and intra-host crowding. The models can be coupledby the values of state variables at the endpoints of their timescales, to inferhow changing conditions in one life stage affects the life cycle as a whole. Ofparticular interest is the vulnerability of miracidia to temperature changes:once environmental temperatrure rises above 25 degrees C, miracidial lifes-pans begin to become very short, which inhibits them from finding a snailhost and continuing the life cycle.

In the snail-trematode system, the high percentage of snail energy usedby the parasite, and the negative effects of sharing a host, provide a quanti-tative rationale for the evolution of competitive exclusion and the dominanceheirarchy in trematodes.

31

Lastly, the model provides information about the effect of multiple infec-tions in a single host. With the right parameters, it may be useful to estimatethe effect of a given mean parasite burden on a herd of livestock, a contextin which F. hepatica is economically significant.

References

[1] Al-Habbib, W.M.S. The effect of constant and changing temperatures onthe development of the larval stages of Fasciola hepatica. Ph. D thesis,University of Dublin. 1974.

[2] Augot et al. Fasciola hepatica: an unusual development of redial gen-erations in an isolate of Lymnaea truncatula. Journal of Helminthology73:27-30. 1999.

[3] Rondelaud, D., Vignoles, P. Dreyfuss, G. Fasciola hepatica: the de-velopmental patterns of redial generations in naturally infected Galbatruncatula. Parisatol. Res. 94:183-187. 2004.

[4] Boniecka, B. Guttowa, A. The influence of pesticides on the oxy-gen uptake by miracidia of Fasciola hepatica (Trematoda). Bulletin del’Academie polonaise des sciences (Sciences biologiques). 23: 463-7. 1975.

[5] Einarsson, B., Birnir, B., Sigurdsson, S. A Dynamic Energy Budget(DEB) model for the en ergy usage and reproduction of the Icelandiccapelin (mallotus villosus). Journal of Theoretical Biology, vol. 281, issue1, p. 1-8. 2011.

[6] Hall, Spencer R., Becker, Claes., Caceres, Carla E. Parasitic castration:a perspective from a model of dynamic energy budgets. Integrative andComparative Biology, 2007.

[7] Hayes, T.J., Bailer, J., Mitrovic, M. Immunity in rats to superinfectionwith Fasciola hepatica. The Journal of Parasitology, Vol. 58, No. 6, p.1103-1105. 1972.

[8] Hechinger, et al. How big is the hand in the puppet? Ecological andevolutionary factors affecting the body mass of 15 trematode parasiticcastrators in their snail host. Evol. Ecol. 2008.

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[9] Horstmann, H. J. Sauerstoffveibrauch und glykogenhalt der eier vonFasciola hepatica witrend der entwicklung der miracidien Z. f. Para-sitenkunde 21, 437-445 1962.

[10] Kooijman, S.A.L.M. Dynamic Energy and Mass Budgets in BiologicalSystems 2nd ed. Cambridge University Press, 2000.

[11] Lafferty, K.D. and Kuris, A.M. Parasitic castration: the evolution andecology of body snatchers. Trends in Parasitology Vol. 25 No. 12. 2009.

[12] Schmidt, Gerald D., Roberts, Larry S. Foundations of Parasitology.McGraw-Hill Companies, Inc. 2000.

[13] Smyth, J.D. Halton, D.W. The Physiology of Trematodes 2nd ed. Cam-bridge University Press, 1983.

[14] Valero et al. Crowding effect on adult growth, pre-patent period and eggshedding of Fasciola hepatica. 2006.

[15] Valero, M.A., Marcos, M.D., Mas-Coma, S. A Mathematical model forthe ontogeny of Fasciola hepatica in the definitive host. Research andReviews in Parasitology. 56(1). 13-20 1996.

[16] Wilson, R.A. A physiological study of the development of the egg ofFasciola hepatica L., the common liver fluke. Comp. Biochem. Physiol.,Vol. 21, pp. 307 to 320. 1967.

[17] Zonneveld, C., Kooijman, S. A. L. M., Application of a Dynamic EnergyBudget Model to Lymnaea stagnalis (L.) Functional Ecology, Vol. 3, No.3, pp. 269-278. 1989.

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a theoretical model for the energy dynamics of the

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