mathematical modelling of bacterial meningitis...

Research ArticleMathematical Modelling of Bacterial MeningitisTransmission Dynamics with Control Measures

Joshua Kiddy K Asamoah 12 Farai Nyabadza 3 Baba Seidu 4

Mehar Chand5 and Hemen Dutta6

1Department of Mathematics Kwame Nkrumah University of Science and Technology Kumasi Ghana2African Institute for Mathematical Sciences Biriwa Ghana3Division of Mathematics Stellenbosch University Western Cape South Africa4Department of Mathematics University for Development Studies Navrongo Ghana5Department of Applied Sciences Guru Kashi University Bathinda India6Department of Mathematics Gauhati University Guwahati 781014 India

Correspondence should be addressed to Joshua Kiddy K Asamoah jasamoahaimsedugh

Received 23 September 2017 Accepted 4 January 2018 Published 27 March 2018

Academic Editor Konstantin Blyuss

Copyright copy 2018 Joshua Kiddy K Asamoah et al 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

Vaccination and treatment are themost effective ways of controlling the transmission ofmost infectious diseasesWhile vaccinationhelps susceptible individuals to build either a long-term immunity or short-term immunity treatment reduces the number ofdisease-induced deaths and the number of infectious individuals in a communitynation In this paper a nonlinear deterministicmodel with time-dependent controls has been proposed to describe the dynamics of bacterialmeningitis in a populationThemodelis shown to exhibit a unique globally asymptotically stable disease-free equilibrium E0 when the effective reproduction numberRVT le 1 and a globally asymptotically stable endemic equilibriumE1 whenRVT gt 1 and it exhibits a transcritical bifurcation atRVT = 1 Carriers have been shown (by Tornado plot) to have a higher chance of spreading the infection than those with clinicalsymptoms who will sometimes be bound to bed during the acute phase of the infection In order to find the best strategy forminimizing the number of carriers and ill individuals and the cost of control implementation an optimal control problem is set upby defining a Lagrangian function 119871 to be minimized subject to the proposed model Numerical simulation of the optimal problemdemonstrates that the best strategy to control bacterial meningitis is to combine vaccination with other interventions (such astreatment and public health education) Additionally this research suggests that stakeholders should press hard for the productionof existingnew vaccines and antibiotics and their disbursement to areas that are most affected by bacterial meningitis especiallySub-Saharan Africa furthermore individuals who live in communities where the environment is relatively warm (hotmoisture)are advised to go for vaccination against bacterial meningitis

1 Introduction

Meningitis is an inflammation of the meninges which aremembranes that surround the spinal cord and the brain [1]It is often caused by viruses bacteria and protozoa Bacterialmeningitis is common in children and young adults Thisdisease mostly spreads in communitiessocieties that livein close quarters (eg police staff police cells college stu-dents military staff and prisons) [2] Bacterial meningitis isgenerally caused by germs such as Listeria monocytogenesStreptococcus pneumoniae Group B Streptococcus Neisseria

meningitidis and Haemophilus influenzae which spreadsfrom one person to another [3] This infection varies byage groups Group B Streptococcus Streptococcus pneumo-niae Listeria monocytogenes and Escherichia coli are mostlyfound in newborn babies Streptococcus pneumoniae Neisse-ria meningitidis Haemophilus influenzae type b (Hib) andGroup B Streptococcus are common in babies and childrenNeisseria meningitidis and Streptococcus pneumoniae are pre-dominant in teens and young adults and Streptococcus pneu-moniae Neisseria meningitidis Haemophilus influenzae typeb (Hib) Group B Streptococcus and Listeria monocytogenes

HindawiComputational and Mathematical Methods in MedicineVolume 2018 Article ID 2657461 21 pageshttpsdoiorg10115520182657461

2 Computational and Mathematical Methods in Medicine

are commonly found in older adults [3] Bacterial meningitisis characterized by intense headache and fever vomitingsensitivity to light and stiff neck which result in convulsiondelirium and death

It is estimated that meningococcal meningitis causes over10000 deaths annually in Sub-Saharan Africa [4] About4100 cases of bacterial meningitis occurred between 2003and 2007 in the United States [3 5] Between 5 to 40 ofchildren and 20 to 50 of adults with this condition die [6]Infections from bacterial meningitis can cause permanentdisabilities such as brain damage hearing loss and learningdisabilities [3] The illness of bacterial meningitis becomesworse when symptoms are not detected early enough evenwith proper treatment the individual could die [2]

Prevention of bacterial meningitis can be achievedthrough vaccination andor preventing contact with infec-tious individuals Vaccination is the most effective wayof protecting children against certain types of bacterialmeningitis [3] Vaccines that can prevent meningitis includeHaemophilus influenza type B (Hib) pneumococcal conju-gate andmeningococcal vaccine [6 7]The conjugatemenin-gitis A vaccine MenAfrivac is recommended to protectpeople in Sub-Saharan Africa against the most commontype serotype A [8] In the United States the primarymeans of preventing meningococcal meningitis is antimi-crobial chemoprophylaxis [9] Empirical therapy includesceftriaxone or cefotaxime and vancomycin for Streptococcuspneumoniae [2] There is a vaccine against meningococcaldisease which is 85ndash100 effective in preventing four kindsof bacteria (serogroups A C Y and W-135) that cause about70 of the disease in the United States [2]

Trotter andRamsay [10] outlined some recommendationson the use of conjugate vaccines in Europe based on the expe-rience with meningococcal C conjugate (MCC) vaccines Inareas with limited health infrastructure and resources thereare a number of antibiotics including penicillin ampicillinand chloramphenicol that can be used to treat the infectionmeningitis

Mathematical models have been shown to help increasethe understanding of the spread and control of infectious dis-eases Martınez et al [2] studied the spread of meningococcalmeningitis with the use of a discrete mathematical modelbased on cellular automata where the population was dividedinto five classes susceptible asymptomatic infected infectedwith symptoms carriers recovered and died classes Broutinet al [11] studied the dynamics of meningococcal meningitisin nine African countries by adopting some mathematicaltools to time series analysis andwaveletmethod the results oftheir studies suggest that ldquointernational cooperation in PublicHealth and cross disciplines studies are highly recommendedto help in controlling this infectious diseaserdquo Miller andShahab [12] studied the cost effectiveness of immunisationstrategies for the control of epidemic meningococcal menin-gitis The research work in [13] gives a detailed descriptionof the use of antibiotics for the prevention and treatmentof meningitis infection Irving et al [14] used deterministiccompartmental models to investigate how well simple modelstructureswith seasonal forcingwere able to qualitatively cap-ture the patterns of meningitis infection They demonstrated

that the complex and irregular timing of epidemics could becaused by the interaction of temporary immunity conferredby carriage of the bacteria together with seasonal changes inthe transmissibility of infection Actually there have been asignificant number of studies of various types ofMeningitis inAfrica and Europe without the use of optimal control analysis(see [15ndash28])

It is obvious that mathematical modelling has becomecrucial in investigating the epidemiological behaviour ofmeningitis Furthermore mathematical modelling helps toidentify the risk factors for diseases so as to find out whyeveryone does not have the same infection uniformly [29]

The application of optimal control in disease modellinggives valuable information on how to apply control measuresThrough vaccination treatment public education and soforth many infectious diseases have been controlled [29]Since the introduction of optimal control theory in diseasemodelling there have been a considerable number of studiesof infectious diseases using optimal control analysis (see [30ndash41]) With the significant influence of optimal control theoryin disease modelling this paper presents an optimal controlmodel for bacterial meningitis in the presence of vaccinationand treatment due to public health education The model isqualitatively analyzed and numerically simulated in order tohelp give policy direction on how to control the spread of thedisease

The rest of the paper is organized as follows Section 2presents the model formulation and analysis Section 3presents the analysis of the optimal control problem leadingto the existence and characterization of the control measuresSection 4 contains the numerical simulations and discussionSection 5 presents the conclusion of the study

2 Model Formulation and Analysis

Adopting the epidemiological studies of a meningitis modelas presented in [4] we consider four mutually exclusivecompartments to indicate individuals with unique natures(ie susceptibles 119878(119905) carriers 119862(119905) ill individuals 119868(119905) andrecovered individuals 119877(119905)) in relation to the disease It isassumed that the susceptible compartment 119878(119905) is populatedthrough recruitment at the rate 120587 (thus migration andorbirth rate) and 120573 is the rate of effective contact of carriersandor infected (ill) individuals in the susceptible populationThe carrier compartment consists of individuals that havethe infection and do not show any clinical symptoms butcontribute to the spread of the disease When a susceptibleindividual is exposed to this infection that individual canharbour the bacterium for weeks or even months [42] butin a normal circumstance an individual develops symptomsof the infection within 3 to 7 days after exposure [3] Carriersare assumed to develop clinical symptoms (ie move to illindividuals compartment 119868(119905)) at rate 120572 Ill individuals whoare seriously infected are assumed to have no natural recoveryexcept when given treatment on time From an epidemi-ological perspective individuals in the removedrecoveredcompartment 119877(119905) do not attain permanent immunityAfter vaccination immunity develops within 7ndash10 days andremains effective for approximately 3ndash5 years [2] Therefore

Computational and Mathematical Methods in Medicine 3

it is assumed that the immunity acquired from developingthe diseases or carrying the bacteria or through vaccinationis of the same intensity (they all lead to the recoveredcompartment from which people return to the susceptiblecompartment at a given unique rate 120579) Research indicatesthat carriersmay recover naturally from the infectionwithouttreatment and we denote such natural recovery rate as 120596Infected individuals are assumed to die from disease at rate120575 Since natural death is inevitable 120583 is assumed to be thenatural death rate of individuals in all the compartmentsVaccination and treatment due to public health educationhave been shown to be strategies of control of diseasesThere-fore we introduced this two control measures in the modelas 1199061(119905) and 1199062(119905) respectively here 1199061 thus vaccinationis comprised of both reactive vaccination and preventativevaccination The effectiveness of both control measures inminimizing the disease is denoted by 120590 and 120574 respectivelyIf 120590 = 120574 = 0 it signifies that vaccination and treatment haveno effect on the model if 120590 = 120574 = 1 it also signifies thatvaccination and treatment are perfectly effective (ie 100effectiveness) (see [43]) If 0 lt 120590 and 120574 lt 1 it signifiesthat both vaccine and treatment are imperfect [44] In viewof this it is assumed that administering treatment to the illindividuals leads to recovery at rate 1205741199062 It is also assumedthat the vaccinated individuals develop partial immunity atrate 1205901199061 From an epidemiological perspective treatment isnot given to carriers in real life (since we do not know whocarries the bacteria or not) but it is assumed that carriers ofmeningitis are just like people with the HIV virus who do notknow their status unless they go for medical test hence thispaper seeks to encourage individuals to go for regular test ofthis bacterial disease therefore it is assumed that a certainportion of carriers could be treated before any symptoms ofthe infection show up which results in 1205741199062119862 as shown inFigure 1The vaccine is assumed to be imperfect and thus hasa failure rate of (1minus1199061120590)Therefore the force of new infectionsis given by

120582 = 120573 (1 minus 1199061120590) (120578119862 + 1205781119868)119873 (1)

Equation (1) is often referred to as standard incidence rate ofnew infections which is normalized by the total population119873 = 119878+119862+119868+119877 Table 1 gives a full description of parametersused in the model

The set of differential equations and flow diagram corre-sponding to the bacterial meningitis dynamics and diseasepathwaywith control terms is given in system (2) andFigure 1

119889119878119889119905 = 120587 minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873 minus (120583 + 1199061120590) 119878+ 120579119877

119889119862119889119905 = (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862119889119868119889119905 = 120572119862 minus (120575 + 120583 + 1199062120574) 119868

u1S

u2CCS

I

RRu2IICCSS

I

C

R

Figure 1 The flowchart diagram describing bacterial meningitistransmission dynamics within the population The four circlesrepresent the four compartments of individuals the movementbetween the compartments is indicated by the continuous arrows1199061 is a control measure (vaccination) and 1199062 is the second controlmeasure (treatment) with the consideration that both controlmeasures lies in 0 lt 1199061 and 1199062 le 1

119889119877119889119905 = 1199062120574119868 + (120596 + 1199062120574) 119862 + 1199061120590119878 minus (120579 + 120583) 119877119878 (0) gt 0 119862 (0) ge 0 119868 (0) ge 0 119877 (0) ge 0

(2)

Adding the equations in model (2) gives the rate of change oftotal population as

119889119873119889119905 = 120587 minus 120583119873 minus 120575I le 120587 minus 120583119873 (3)

Let

Ω = (119878 119862 119868 119877) isin R4+ 119873 le 120587120583 (4)

System (2) is well-posed with all solutions in Ω remaining inΩ if initial conditions are positive It can easily be shown thatif the initial conditions start outside Ω the solutions tend toΩ

21 Equilibrium Points To obtain the equilibrium points ofsystem (2) we assume that the control measures are time-independent (see [45] for similar analysis)

211 Disease-Free EquilibriumE0 To obtain the disease-freeequilibrium 119862(119905) 119868(119905) and the right-hand-side of system(2) are set to zero If susceptible individuals are assumed toreceive vaccination against the disease at a constant rate thenthe disease-free equilibrium will be given by

E0 (1198780 1198620 1198680 1198770)= ( 120587 (120579 + 120583)

120583 (1199061120590 + 120579 + 120583) 0 01205871199061120590120583 (1199061120590 + 120579 + 120583)) (5)

212 Effective Reproduction Number RVT Using the nextgeneration matrix method [46] the effective reproduction


Table1Descriptio

nof

parameter

values

used

inthem

odel

ldquoDue

tolack

ofrelevant

datam

osto

fthe

parameter

values

area

ssum

edwith

inrealisticranges

fora

typicalscenario

inarural

commun

itysocietyforthe

purposeo

fillu

stratio

nrdquo[44]w

ithmosto

fthe

parameter

values

where

appreciablep

eryear

Parameter

Descriptio

nVa

lueRa

nge

Reference

120587Re

cruitm

entrate(migratio

nandor

birthrate)

100ndash

1000

00Assum

ed120573

Effectiv

econ

tactrate

088[28]

120579Lo

ssof

immun

ity004ndash2

[1447]

120583Naturaldeathrate

002[14

]120575

Dise

ase-indu

cedmortality

005ndash0

5[48]

120572Ra

teof

progressionfro

m119862to

11986801ndash052

[14]

120596Naturalrecovery

rate

006ndash0

2Assum

ed120578 1

Perc

apita

infectionrateby

illindividu

als

02ndash095

Assum

ed120578

Perc

apita

infectionrateby

Carriers

02ndash085

Assum

ed119906 1

Vaccinationrateof

susceptib

leindividu

alsa

gainstMeningitis

[01]

Assum

ed119906 2

Treatm

entratefor

carrierswith

outn

aturalrecovery

andill

individu

als

[01]

Assum

ed120590

Effectiv

enesso

fvaccinatio

n085ndash1

[2]

120574Eff

ectiv

enesso

fTreatment

01ndash09

Assum

ed


01 02 03 04 050

05

1

Trea

tmen

tu2

Vaccination u1

ℛ６４ gt 1

ℛ６４ lt 1

Figure 2 Region whereRVT lt 1 andRVT gt 1 in the 1199061 minus 1199062 parameter space with the parameter values 120573 = 088 120578 = 02 120596 = 006 120575 =003 120579 = 00839 120572 = 005 120574 = 02 and 120590 = 07

number of the bacterial meningitis model with vaccinationand treatment is obtained as

RVT = R119862VT + R119868VT (6)

whereR119862VT = 120573120578 (1 minus 1199061120590) (120579 + 120583)

(120572 + 120596 + 120583 + 1199062120574) (120579 + 120583 + 1199061120590) R119868VT

= 1205731205721205781 (1 minus 1199061120590) (120579 + 120583)(120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574) (120579 + 120583 + 1199061120590)

(7)

To determine how the two control measures impact on thereproduction number we make a plot ofRVT on the 1199061 minus 1199062plane for arbitrary constant values of model parameters inFigure 2 From the figure it is shown that 1199062 decreases withincreasing 1199061 So an increase in vaccination levels decreasesthe need for treatment Figure 2 also shows the region inwhich the vaccination and treatment values should lie for thedisease control

213 Endemic Equilibrium E1 System (2) can be shownto have a unique endemic equilibrium of the form(119878lowast 119862lowast 119868lowast 119877lowast) where

119878lowast = 120572120587 (120583 + 120579) + [(120596 + 1199062120574) (120575 + 120583 + 1199062120574) 120579 + 1205791205721199062120574 minus (120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574) (120579 + 120583)] 120587120572 (RVT minus 1)120572120583 (120579 + 120583 + 1199061120590) [A (RVT minus 1) + B]

119862lowast = (120575 + 120583 + 1199062120574) [120587120572 (RVT minus 1)]120572 [A (RVT minus 1) + B]

119877lowast= 1205721205871199061120590 + [(120575 + 120583 + 1199062120574) (120596 + 1199062120574) (120583 + 1199061120590) + 1205721199062120574 (120583 + 1199061120590) minus (120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574) 1199061120590] [120587120572 (RVT minus 1)]

120572120583 (120579 + 120583 + 1199061120590) [A (RVT minus 1) + B]

119868lowast = 120587120572 (RVT minus 1)A (RVT minus 1) + B

(8)

with A = (120575 + 120583 + 1199062120574)(120572 + 120596 + 120583 + 1199062120574) and B = (120575 + 120583 +1199062120574)[120596 + 1199062120574 minus 1199061120590) + 120572(1199062120574 + 120583)]Remark 1 (i) IfRVT lt 1 then system (2) will have only oneequilibrium the disease-free equilibrium

(ii) If RVT gt 1 then system (2) will have two equilibriathe disease-free equilibrium E0 and the endemic equilib-rium E1

(iii) The caseRVT = 1 is a critical threshold point wherethe disease-free equilibrium E0 loses its local asymptoticstability Thus RVT = 1 gives the idea of transcriticalbifurcation where the stability of system (2) moves betweenE0 and E1 [37]

22 Stability Analysis In analyzing the local stability ofthe disease-free equilibrium the Routh-Hurwitz criteria areused and for the global stability of the two equilibria thedirect Lyapunov technique is employed

221 Local Stability Analysis of E0

Theorem 2 The disease-free equilibrium of system (2) islocally asymptotically stable ifRVT lt 1 and unstable ifRVT gt1Proof Evaluating the Jacobian matrix of system (2) at thedisease-free equilibrium gives


119869 (E0) =[[[[[[[[[

minus (120583 + 1199061120590) minus(1 minus 1199061120590) 120573120578 (120579 + 120583)(1199061120590 + 120579 + 120583) minus(1 minus 1199061120590) 1205731205781 (120579 + 120583)

(1199061120590 + 120579 + 120583) 1205790 (1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) minus (120572 + 120596 + 120583 + 1199062120574) (1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) 0

0 120572 minus (120575 + 120583 + 1199062120574) 01199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]]]]

(9)

The characteristic polynomial of the Jacobian matrix 119869(E0) isgiven by

119875 (120582) = 1205824 + DT1205823 + DT11205822 + DT2120582 + DT3 (10)

whereT = 119860 + 119861 + 119862 minus R119862VT T1

= (119860 + 119861 minus R119862VT) (2120583 + 1199061120590 + 120579)+ (120583 + 1199061120590) (120579 + 120583) + 120583120579 (1 minus RVT)

T2

= (120579 + 120583) [(1 minus RVT) + (119860 minus R119862VT)] minus R119862VT

+ (120583 + 1199061120590) [(1 minus R119868VT) + (120575 + 120583 + 1199062120574) (120579 + 120583)]+ (119860 + 119861 minus R119862VT) 120583120579

T3 = 120579120583 (R119868VT minus 1) + (120583 + 1199061120590) (120579 + 120583) (R119868VT minus 1) 119860 = 1

(120575 + 120583 + 1199062120574) 119861 = 1

(120572 + 120596 + 120583 + 1199062120574)

119862 = (120583 + 1199061120590) + (120579 + 120583)(120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

D = (120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

(11)

The Routh-Hurwitz conditions [46] that guarantee that theeigenvalues of the characteristic polynomial in (10) havenegative real parts are given by

DT gt 0DT1 gt 0DT2 gt 0DT3 gt 0

D3TT1T2 gt D2T22 + D3T2T3

(12)

These conditions are easily seen to be satisfiedwhenRVT lt 1Thus the disease-free equilibrium of system (2) is locally

asymptotically stable when RVT lt 1 and unstable whenRVT gt 1 This completes the proof

222 Global Stability of E0

Theorem 3 The disease-free equilibrium E0 of system (2) isglobally asymptotically stable if RVT le 1 and unstable ifRVT gt 1Proof LetV(119878 119862 119868 119877) with positive constantsK1 andK2be a Lyapunov function defined as

V (S 119862 119868 119877) = (119878 minus 1198780 minus 1198780 ln 1198781198780) + K1119862 + K2119868+ (119877 minus 1198770 minus 1198770 ln 1198771198770)

(13)

Taking the time derivative of the Lyapunov function weobtain

119889V119889119905 = (1 minus 1198780119878 ) 119889119878119889119905 + K1119889119862119889119905 + K2

119889119868119889119905+ (1 minus 1198770119877 ) 119889119877119889119905 where K1 ge 0 K2 ge 0

(14)

Substituting 119889119878119889119905 119889119862119889119905 119889119868119889119905 and 119889119877119889119905 in (2) into (14)gives

119889V119889119905 = (1 minus 1198781198780)[120587 minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120583 + 1205901199061) 119878 + 120579119877]

+ K1 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + K2 [120572119862minus (120575 + 120583 + 1199062120574) 119868] + (1 minus 1198771198770) [1199062120574119868


+ (120596 + 1199062120574)119862 + 1205901199061119878 minus (120579 + 120583) 119877]le K1 [((1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) )119862

+ ((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) ) 119868

minus (120572 + 120596 + 120583 + 1199062120574) 119862] + K2 [120572119862minus (120575 + 120583 + 1199062120574) 119868]

(15)

Since

119878 le 1198780 = 120587 (120579 + 120583)120583 (1199061120590 + 120579 + 120583)

119877 le 1198770 = 1205871199061120590120583 (1199061120590 + 120579 + 120583) 119873 le 120587120583

on Ω

(16)

this implies that

119889V119889119905 le [((1 minus 1199061120590) 120573120578 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1 + 120572K1

minus (120572 + 120596 + 120583 + 1199062120574)K2]119862

+ [((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1

minus (120575 + 120583 + 1199062120574)K2] 119868

(17)

Equating the coefficient of 119868 in (17) to zero gives

(1 minus 1199061120590) 120573 (120579 + 120583)K1= (120575 + 120583 + 1199062120574) (1199061120590 + 120579 + 120583)K2 (18)

Choosing K1 = (120575 + 120583 + 1199062120574)(1199061120590 + 120579 + 120583) and K2 = (1 minus1199061120590)120573(120579 + 120583) and pluggingK1 andK2 into (17) we have

119889V119889119905 le (120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574) (1199061120590 + 120579 + 120583)sdot (RVT minus 1)119862 le 0 if RVT le 1

(19)

Additionally 119889V119889119905 = 0 if and only if 119862 = 0 Hencethe largest compact invariant set in (119878 119862 119868 119877) isin Ω 119889V119889119905 le 0 is the singleton set E0 Therefore fromLaSallersquos invariance principle we conclude thatE0 is globallyasymptotically stable in Ω ifRVT le 1 [37 49]


Theorem 4 The endemic equilibrium E1 of system (2) isglobally asymptotically stable wheneverRVT gt 1Proof Suppose RVT gt 1 and then the existence of theendemic equilibrium point is assured Using the commonquadratic Lyapunov function

119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (20)

as illustrated in [50] we consider the following candidateLyapunov function

V (119878 119862 119868 119877)= 12 [(119878 minus 119878lowast) + (119862 minus 119862lowast) + (119868 minus 119868lowast) + (119877 minus 119877lowast)]2 (21)

The time derivative ofV(119878 119862 119868 119877) in (21) is given by

119889V119889119905 (119878 119862 119868 119877) = [(119878 minus 119878lowast) + (119862 minus 119862lowast) + (119868 minus 119868lowast)+ (119877 minus 119877lowast)] 119889 (119878 + 119862 + 119868 + 119877)119889119905

(22)

Plugging the equations in system (2) into (22) yields

119889V119889119905 = [(119878 minus 119878lowast) + (119862 minus 119862lowast) + (119868 minus 119868lowast) + (119877 minus 119877lowast)]sdot [120587 minus 120583 (119878 + 119862 + 119868 + 119877) minus 120575119868]

(23)

Now setting

120587 = 120583 (119878lowast + 119862lowast + 119868lowast + 119877lowast) + 120575119868lowast (24)

we have

119889V119889119905 = [(119878 minus 119878lowast) + (119862 minus 119862lowast) + (119868 minus 119868lowast) + (119877 minus 119877lowast)]sdot [120583 (119878lowast + 119862lowast + 119868lowast + 119877lowast) + 120575119868lowastminus 120583 (119878 + 119862 + 119868 + 119877) minus 120575119868] = [(119878 minus 119878lowast) + (119862 minus 119862lowast)+ (119868 minus 119868lowast) + (119877 minus 119877lowast)] [minus120583 (119878 minus 119878lowast) minus 120583 (119862 minus 119862lowast)minus 120583 (119868 minus 119868lowast) minus 120583 (119877 minus 119877lowast) minus 120575 (119868 minus 119868lowast)]

(25)


Further simplification gives

119889V119889119905 = minus120583 (119878 minus 119878lowast)2 minus 120583 (119878 minus 119878lowast)sdot [(119862 minus 119862lowast) + (119868 minus 119868lowast) + (119877 minus 119877lowast) + 120575 (119868 minus 119868lowast)]minus 120583 (119862 minus 119862lowast)2 minus 120583 (119862 minus 119862lowast)sdot [(119878 minus 119878lowast) + (119868 minus 119868lowast) + (119877 minus 119877lowast) + 120575 (119868 minus 119868lowast)]minus (120583 + 120575) (119868 minus 119868lowast)2 minus 120583 (119868 minus 119868lowast)sdot [(119862 minus 119862lowast) + (119877 minus 119877lowast)] minus 120583 (119877 minus 119877lowast)2minus 120583 (119877 minus 119877lowast)sdot [(119878 minus 119878lowast) + (119862 minus 119862lowast) + (119868 minus 119868lowast) + 120575 (119868 minus 119868lowast)]

(26)

It has therefore been shown that 119889V119889119905 is negative andadditionally atE1 (ie if 119878 = 119878lowast119862 = 119862lowast 119868 = 119868lowast and119877 = 119877lowast)119889V119889119905 = 0 It follows from LaSallersquos invariant principle [51]that all solutions of system (2) approach E1 as 119905 rarr infin ifRVT gt 1 Therefore the endemic equilibrium E1 is globallyasymptotically stable in Ω whenever RVT gt 1 [37 49] Thiscompletes the proof

23 Sensitivity Analysis Sensitivity analysis is used to deter-mine the response of a model to variations in its parametervalues In the present case the focus is given to determininghow changes in the model parameters impact the effectivereproduction number This is done through the normalizedforward-sensitivity index We also use the Latin hypercubesampling and the partial rank correlation coefficients (PRCC)to plot scatter diagrams and Tornado plots to determine therelative importance of the parameters inRVT for the diseasetransmission and prevalence (see also [52])

Definition 5 The normalized forward-sensitivity index ofRVT to any parameter say 120588 as given in [46] can be definedas

Γ120588RVT

= 120597RVT120597120588120588

RVT (27)

The sensitivity indexes ofRVT with respect to its parametersare computed as follows

Γ120573RVT

= 120597RVT120597120573120573

RVT= 1

Γ120578RVT

= 120597RVT120597120578120578

RVT= (120575 + 120583 + 1199062120574) 120578120578 (120575 + 120583 + 1199062120574) + 1205781120572

Γ1205781RVT

= 120597RVT12059712057811205781

RVT= 1205721205781(1205721205781 + (120575 + 120583 + 1199062120574))

Γ120596RVT= 120597RVT120597120596 120596

RVT= minus 120596

(120572 + 120596 + 120583 + 1199062120574) Γ120590RVT

= 120597RVT120597120590 120590RVT

= minus 1199061120590(1199061120590 + 120579 + 120583)

(28)

Similarly we can compute the sensitivity indexes ofRVT withrespect to the remaining parameters in RVT in the samemanner Using the parameter values 120574 = 04 120590 = 1 120578 = 035120596 = 02 120575 = 01 120573 = 088 120572 = 02 120579 = 00839 and 120583 = 002with 1199061 = 05 and 1199062 = 05 the sensitivity indexes ofRVT areshown in Table 2

The corresponding Tornado plots based on a randomsample of 1000 points for the twelve parameters in RVT areshown in Figure 3(a) The positive values in Table 2 showa promotion of the propagation of the disease Thereforean increase in the values of 120573 120578 1205781 120579 and 120572 will have anincrease in the spread of the disease For example Γ120573

RVT= 1

indicates that increasing the effective contact rate by 10increases the number of secondary infections by 10 Thenegative values in Table 2 indicate a reduction in the effectivereproduction numberRVT if the values of the correspondingparameters are increased Thus a reduction in the values ofvaccination 1199061 treatment 1199062 and natural recovery120596will leadto an increase in the number of secondary infections in thepopulation

Figure 3(a) shows the Tornado plots for the twelve param-eters in RVT It can be seen that in controlling the spreadof bacterial meningitis in a population more susceptibleindividuals should be given vaccination Figure 3(a) alsosuggests that carriers are likely to have more contacts withthe susceptible population than the ill individuals who willtypically be bound to their beds during the acute phaseof the disease Therefore the probability of ill individualstransmitting the infections to susceptibles may be lowerthan that of carriers who are able to mix well with otherswithin the population Figures 3(b) 3(c) and 3(d) show theregression plots of effective contact rate (120573) vaccination rate(1199061) and treatment rate (1199062) respectively Figure 3(b) showsthat transmission rate has a positive correlation in the spreadof bacterialmeningitis Figure 3(c) shows that vaccination hasa negative correlation in the spread of bacterial meningitisand hence vaccination increases the immunity of individ-uals against the meningitis infection thereby reducing thespread of the infection Figure 3(d) shows that individualswho receive treatment after being infected with bacterialmeningitis have a higher chance of recovery and that reducesthe spread of the infection and deaths due to bacterialmeningitis

3 Optimal Control Problem

Since the goal of this paper is to find the best ways to controlthe spread of meningitis we define the following optimalcontrol problem

119869 = min1199061120590119906

2

int1198790

[1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)

+ 11986122 11990622 (119905)] 119889119905(29)

subject to model (2)


Table 2 Sensitivity indexes ofRVT to the parameters in (6)

Parameter Description Sensitivity index120573 Effective contact rate 1120578 Per capita infection rate by Carriers 061541205781 Per capita infection rate by ill individuals 03846120579 Loss of immunity 06686120572 Rate of falling ill 00620120583 Natural death rate 01031120575 Disease-induced death rate minus01202120596 Natural recovery rate minus032261199061 Vaccination rate minus18280120590 Effectiveness of vaccination minus082801199062 Treatment rate minus05630120574 Effectiveness of treatment minus05230

minus1 minus05 0 05 1PRCC Values

BetaEta

eta_1Thetau_1u_2mu

DeltaAlpha

OmegaGamma

Sigma

(a) The Tornado plots for the elven parameters inRVT

0 02 04 06 08 1minus15

minus10

minus5

0

Effective contact rate

log(ℛ

６４)

(b) The effect of effective contact rate onRVT

02 04 06 08 1minus15

minus10

minus5

0

Vaccination rate u1

log(ℛ

６４)

(c) The effect of vaccination onRVT

02 04 06 08 1minus15

minus10

minus5

0

Treatment rate u2

log(ℛ

６４)

(d) The effect of treatment onRVT

Figure 3 Sensitivity plots

The admissible control set 119880 is Lebesgue measurablewhich is defined by

119880 = (1199061 (119905) 1199062 (119905)) | 0 le 1199061 le 1199061max le 1 0 le 1199062le 1199062max le 1 119905 isin [0 119879] (30)

Our objective is to find (119906lowast1 119906lowast2 ) isin 119880 which minimizes theassociated cost of the vaccination and the associated costof the treatment over the specified time interval as wellas minimizing the number of infections at a terminal time(see also [37]) The coefficients 1198601 gt 0 and 1198602 gt 0 areconstants that are introduced to maintain a balance in thesize of 119862(119905) and 119868(119905) respectively 1198611 gt 0 and 1198612 gt 0

are the corresponding weights associated with the cost ofvaccination (1199061) and treatment (1199062) respectively The higherbounds (maximum) attainable for the control measures 1199061120590and 1199062 are 1199061max and 1199062max respectively We fix the controlmeasures 1199061 and 1199062 to lie between 0 and 1 so that 1199061max = 1and 1199062max = 1 Therefore the attainment of 1199061max and 1199062maxdepends on the number of resources available [37] Theseresources may include the human effort material resourcescost of producing vaccine and disbursement infrastructuralresources the number of health facilities in the communityand the number of hospital beds at the health facilities Thecost of hospitalization medical test diagnosis drug cost andso forth (see [38ndash40]) can be associated with treatment The


cost of vaccination may include the cost of the vaccine thecost of production the cost of disbursement the vaccinestorage cost and other related overheads [37] The severityof the side effects and overdoses of the vaccination andtreatment is taken care of by squaring the control measuresand 119879 is the final time during the optimal simulation

31 Existence of the Optimal Control Model (2) can bewritten as

119866119905 = 119870 (119866) + 119865 (119866) (31)

where

119866 =[[[[[[

119878 (119905)119862 (119905)119868 (119905)119877 (119905)

]]]]]]

119870 =[[[[[[

minus (120583 + 1199061120590) 0 0 1205790 minus (120572 + 120596 + 120583 + 1199062120574) 0 00 120572 minus (120575 + 120583 + 1199061120590) 0

1199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]

119865 (119866) =[[[[[[[[[

120587 minus (1 minus 1199061120590) 120573119878 (119905) (120578119862 (119905) + 1205781119868 (119905))119873 (119905)(1 minus 1199061120590) 120573119878 (119905) (120578119862 (119905) + 1205781119868 (119905))

119873 (119905)00

]]]]]]]]]

(32)

and119866119905 is the times derivative of119866(119905) System (31) is nonlinearwith a bounded coefficient

Setting 1198661 = (1198781(119905) 1198621(119905) 1198681(119905) 1198771(119905)) and 1198662 = (1198782(119905)1198622(119905) 1198682(119905) 1198772(119905)) gives

119865 (1198661) minus 119865 (1198662) =[[[[[[[[[

minus (1 minus 1199061120590) 120573120578(1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) ) minus (1 minus 1199061120590) 120573(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )(1 minus 1199061120590) 120573120578(1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) ) minus (1 minus 1199061120590) 120573(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )

00

]]]]]]]]]

(33)

Therefore

1003816100381610038161003816119865 (1198661) minus 119865 (1198662)1003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816minus (1 minus 1199061120590) 120573120578(1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816+ 10038161003816100381610038161003816100381610038161003816minus (1 minus 1199061120590) 1205731205781 (1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816+ 10038161003816100381610038161003816100381610038161003816(1 minus 1199061120590) 120573120578(1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816+ 10038161003816100381610038161003816100381610038161003816minus (1 minus 1199061120590) 1205731205781 (1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816le 2 (1 minus 1199061120590) 120573120578 10038161003816100381610038161003816100381610038161003816(

1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816

+ 2 (1 minus 1199061120590) 1205731205781 10038161003816100381610038161003816100381610038161003816(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816

le 2 (1 minus 1199061120590) 120573120578 100381610038161003816100381611987811198621 minus 119878211986221003816100381610038161003816 + 2 (1 minus 1199061120590)sdot 1205731205781 100381610038161003816100381611987811198681 minus 119878211986821003816100381610038161003816 le 2 (1 minus 1199061120590)sdot 120573120578 10038161003816100381610038161198621 (1198781 minus 1198782) + 1198782 (1198621 minus 1198622)1003816100381610038161003816 + 2 (1 minus 1199061120590)sdot 1205731205781 10038161003816100381610038161198681 (1198781 minus 1198782) + 1198782 (1198681 minus 1198682)1003816100381610038161003816 le 10038161003816100381610038161198781 minus 11987821003816100381610038161003816sdot (2 (1 minus 1199061120590) 120573120578 100381610038161003816100381611986211003816100381610038161003816 + 2 (1 minus 1199061120590) 1205731205781 100381610038161003816100381611986811003816100381610038161003816)+ 2 (1 minus 1199061120590) 120573120578 100381610038161003816100381611987821003816100381610038161003816 10038161003816100381610038161198621 minus 11986221003816100381610038161003816 + 2 (1 minus 1199061120590)sdot 1205731205781 100381610038161003816100381611987821003816100381610038161003816 10038161003816100381610038161198681 minus 11986821003816100381610038161003816 le 2 (1 minus 1199061120590) 120573120587

120583 (120578 + 1)


sdot 10038161003816100381610038161198781 minus 11987821003816100381610038161003816 + 2 (1 minus 1199061120590) 120573120578120587120583 10038161003816100381610038161198621 minus 11986221003816100381610038161003816

+ 2 (1 minus 1199061120590) 1205731205781120587120583 10038161003816100381610038161198681 minus 11986821003816100381610038161003816le 119872 (10038161003816100381610038161198781 minus 11987821003816100381610038161003816 + 10038161003816100381610038161198621 minus 11986221003816100381610038161003816 + 10038161003816100381610038161198681 minus 11986821003816100381610038161003816)

(34)

where

119872 = max(2 (1 minus 1199061120590) 120573120587120583 (120578 + 1) 2 (1 minus 1199061120590) 120573120578120587

120583 2 (1 minus 1199061120590) 1205731205781120587120583 )

(35)

so that

1003816100381610038161003816119863 (1198661) minus 119863 (1198662)1003816100381610038161003816 le 119881 10038161003816100381610038161198661 minus 11986621003816100381610038161003816 where 119881 = max (119872 119870) lt infin (36)

The function 119863 is therefore uniformly Lipschitz continuousFrom the definition of the control measures 1199061(119905) and 1199062(119905)and the constraint on the state variables such that 119878(119905) gt0 119862(119905) ge 0 119868(119905) ge 0 and 119877(119905) ge 0 we observe that asolution of system (31) exists [40 53 54] From the objectivefunctional and its associated constraints in model (2) we canfind the optimal solution for our model Firstly we find theLagrangian (119871) and Hamiltonian (119867) for the control problem[55] The Lagrangian of the optimal problem is given by

119871 (119878 119862 119868 119877 1199061 1199062) = 1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)+ 11986122 11990622 (119905)

(37)

Our focus is to find the minimal value of the Lagrangianfunction which is done by a pointwise minimization ofthe Hamiltonian (119867) defined as follows (using Pontryaginrsquosmaximum principle)

119867(119878 119862 119868 119877 1199061 1199062 119905) = 1198601119862 + 1198602119868 + 11986112 11990621 + 11986122 11990622+ 1205821 [120587 minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873 minus (120583 + 1199061120590) 119878

+ 120579119877] + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + 1205823 [120572119862

minus (120575 + 120583 + 1199062120574) 119868] + 1205824 [1199062120574119868 + (120596 + 1199062120574)119862+ 1199061120590119878 minus (120579 + 120583) 119877]

(38)

where 120582119894 119894 = 1 2 3 4 are the adjoint variables associatedwith 119878(119905) 119862(119905) 119868(119905) and 119877(119905) defined by

1198891205821119889119905 = minus120597119867120597119878 1198891205822119889119905 = minus120597119867120597119862 1198891205823119889119905 = minus120597119867120597119868 1198891205824119889119905 = minus120597119867120597119877

(39)

Theorem 6 There exists an optimal control pair 119906lowast1 (119905) 119906lowast2 (119905)such that

119869 (119906lowast1 (119905) 119906lowast2 (119905)) = min1199061119906

2isin119880

119869 (1199061 (119905) 1199062 (119905)) (40)

Proof We start our proof by considering the properties of theexistence of the optimal control (see [56]) Following [57]the set of controlmeasures with corresponding state variablesare positive The set 119880 is convex and closed by definitionTherefore our optimal system is closed and bounded whichascertains the compactness required for the existence of theoptimal control Additionally the integrand in the objectivefunctional (29) 1198601119862(119905) + 1198602119868(119905) + (11986112)11990621(119905) + (11986122)11990622(119905)is convex on the control set119880 Furthermore we can state thatthere exists a positive constant 120588 gt 1 [58] and nonnegativenumbers ]1 and ]2 such that the objective functional has alower bound of ]1(|1199061|2 + |1199062|2)1205882 minus ]2 so that

119869 (1199061 (119905) 1199062 (119905)) ge ]1 (1003816100381610038161003816119906110038161003816100381610038162 + 1003816100381610038161003816119906210038161003816100381610038162)1205882 minus ]2 (41)

since the control measures and the state variables arebounded this leads us to a compact proof of existence of theoptimal control

32 Characterization of the Optimal Control We will applyPontryaginrsquosmaximumprinciple to theHamiltonian functionabove to derive the necessary condition of optimality for ourcontrol problem


Theorem 7 Let 119878 119862 119868 and 119877 be optimal state solutionswith corresponding optimal control variables 119906lowast1 and 119906lowast2 forthe objective functional and its constraints in model (2) with119873 = 119878 + 119862 + 119868 + 119877 Then there exist four adjoint variables 12058211205822 1205823 and 1205824 that satisfy

1198891205821119889119905 = 1205821 [(1 minus 1199061120590) 120573 (120578119862 + 1205781119868)119873minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 + (120583 + 1199061120590)]

+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732minus (1 minus 1199061120590) 120573 (120578119862 + 1205781119868)119873 ] minus 12058241199061120590

1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583

+ 1199062120574) + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732minus (1 minus 1199061120590) 120573120578119878

119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)

1198891205823119889119905 = minus1198602 + 1205821 [(1 minus 1199061120590) 1205731205781119878119873minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ]

+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732

minus (1 minus 1199061120590) 1205731205781119878119873 ] + 1205823 (120575 + 120583 + 1199062120574) minus 120582411990621205741198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583)

(42)

with transversality conditions

120582119894 (119879) = 0 119894 = 1 2 3 4for the control set 119880 = (1199061 1199062) such that 120597119867120597119906119894 = 0

where 119894 = 1 2(43)

Therefore the optimal control pair (119906lowast1 119906lowast2 ) is given by

119906lowast1 = minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max

119906lowast2 = minmax(0 120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 ) 1199062max (44)

Proof We use the Hamiltonian function in (38) in order toobtain the adjoint relations and the transversality conditionsWe set the state variables in the Hamiltonian function to119878 119862 119868 and 119877 and differentiating the Hamiltonian (119867) withrespect to 119878 119862 119868 and 119877 respectively yields (42) Also dif-ferentiating the Hamiltonian (119867) with respect to the controlmeasures 1199061120590 and 1199062 in the interior of 119880 we obtain theoptimality conditions below

1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)

Plugging 1199061 = 119906lowast1 and 1199062 = 119906lowast2 into (45) and solving theoptimal control pair (119906lowast1 119906lowast2 ) we have

119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]

119906lowast2 = 120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 (46)

The two control measures which are bounded with lowerbounds zero and upper bounds 119906119894max = 1 where 119894 = 1 2give


119906lowast1 isin 119880 997904rArr119906lowast1 (119905)

=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] lt 1199061max1199061max if 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] ge 1199061max119906lowast2 isin 119880 997904rArr

119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0

120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 if 0 lt 120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 lt 1199062max1199062max if

120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 ge 1199062max

(47)

Using (47) the optimal control measures are characterized as(44) completing the proof

Therefore our optimality system is given by

119889119878119889119905 = 120587 minus (1 minus minmax(0 11198611 [120573119878 (1205781119868 + 120578119862)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max) 120573119878 (120578119862 + 1205781119868)119873

minus (120583 + (minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max)120590)119878 + 120579119877

119889119862119889119905 = (1 minus minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max) 120573119878 (120578119862 + 1205781119868)119873

minus (120572 + 120596 + 120583 + (minmax(0 120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 ) 1199062max)120574)119862119889119868119889119905 = 120572119862 minus (120575 + 120583 + (minmax(0 120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 ) 1199062max)120574) 119868119889119877119889119905 = ((minmax(0 120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 ) 1199062max)120574) 119868

+ (120596 + (minmax(0 120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 ) 1199062max)120574)119862 minus (120579 + 120583) 119877

+ (minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max) 120590119878

(48)

1198891205821119889119905 = 1205821 [(1 minus (minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max)) 120573 (120578119862 + 1205781119868)119873 ]

minus 1205821 [(1 minus (minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max)) 120573119878 (120578119862 + 1205781119868)1198732 ]


+ 1205821 [(120583 + (minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max)120590)]

+ 1205822 [(1 minus (minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max)) 120573119878 (120578119862 + 1205781119868)1198732 ]


minus 1205824 (minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max)120590

(49)

1198891205822119889119905 = minus1198601 + 1205821 [(1 minus (minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max)) 120573120578119878

119873 ]minus 1205821 [(1 minus (minmax(0 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max)) 120573119878 (120578119862 + 1205781119868)1198732 ]+ 1205822 (120572 + 120596 + 120583 + (minmax(0 120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 ) 1199062max)120574)


minus 1205822 [(1 minus (minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max)) 120573120578119878

119873 ] minus 1205823120572minus 1205824 (120596 + (minmax(0 120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 ) 1199062max)120574)

1198891205823119889119905 = minus1198602 + 1205821 [(1 minus (minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max)) 1205731205781119878119873 ]



minus 1205822 [(1 minus (minmax(0 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]) 1199061max)) 1205731205781119878119873 ]

+ 1205823 (120575 + 120583 + (minmax(0 120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 ) 1199062max)120574)minus 1205824 (minmax(0 120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 ) 1199062max)120574

1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)

4 Numerical Simulations of theOptimal Control

The optimality system consisting of the state equations (2)and the adjoint equations (45) is solved using the forward-backward sweep schemeThe solution is startedwith an initial

guess for the controlmeasures and the final time set to119879 = 30days and later varied to 119879 = 60 days and 119879 = 80 days Thestate system is solved forward in timewhile the costate systemis solved backward in time The current solutions of the statesystem togetherwith the initial guess for the controlmeasures1199061 and 1199062 are used to solve the costate systemThe controls 1199061


0 10 15 20 25 30Time

0

200

400

Carr

iers

5

Without controlsWith vaccination only

ℛ６４ = 4225

(a) The optimal effect on 119862(119905) with vaccination control

0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225

(b) The optimal effect on 119868(119905) with vaccination control

Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225

(c) The optimal effect on 119862(119905) with an increase in time

Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225

(d) The optimal effect on 119868(119905) with an increase in time

Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225

(e) Vaccination control profile

Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60

(f) Optimal control profile with an increase in time

Figure 4 (a) (b) (c) and (d) show that administering vaccination on susceptible individuals reduces the number of secondary infectionsin the carrier and the ill population (e) and (f) show the optimal control profile of vaccination Initial conditions 119878(0) = 700 119862(0) =250 119868(0) = 40 and 119877(0) = 10 and the parameter values 120587 = 100 120578 = 035 120596 = 02 120575 = 01 120590 = 1 120572 = 02 and 120579 = 00839 were used inthis simulation

and 1199062 are updated using the characterizations in (47) Withthe current cosystem solution and updated controls the statesystem is solved and the whole solution process continuesuntil convergence is achieved Since bacterial meningitis is anendemic disease especially in Sub-Sahara Africa parametervalues that make effective reproduction number RVT morethan unity are considered Taking the associated costs oncarriers and ill individuals as 1198601 = 1198602 = 1 1198611 = 2 and1198612 = 4 the graphical results of the model are as followsFigures 4 and 5 show the dynamics of the infection in thepresences of control measures and without control measuresin a more localised form Figure 5(f) shows the effect ofvarying the cost associated with the treatment control It is

seen that if the weight of treatment cost 1198612 is continuouslyincreased the upper bound time reduces to about 9 daysas indicated by the red line in Figure 5(f) This is truebecause a higher cost of treatment causes the use of thetreatment control to be less Figure 6 shows the spreadingrate of meningitis with and without control measures in ahighly dense population thus in colleges prisons cities andso forth which suggests that when carriers become morein a highly populated settings there is a likelihood of theinfection spreading faster thereby leading to a correspondingincrease in the number of ill (sick) individuals hence thetrajectories in Figures 6(a) and 6(b) show that there is asignificant difference in the number of carriers and infected


Time

0

200

400Ca

rrie

rs

Without controlsWith treatment only

0 5 10 15 20 25 30

ℛ６４ = 4225

(a) The optimal effect on 119862(119905) with treatment control

Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225

(b) The optimal effect on 119868(119905) with treatment control

Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225

(e) Treatment control effect

Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2

(f) Optimal control plot of treatment with different cost weights

Figure 5 (a) and (b) show that giving treatment to carriers and the ill individuals reduces the number of secondary infections in the carrierand the ill population but the disease may not be eradicated if the use of only treatment is stuck to Furthermore we increased time so as tosee whether the disease could be eradicated when using treatment only We see in (c) and (d) that there is a possibility of recurrence of thedisease using only treatment control strategy as indicated by the curved green line in (c) and the linear flow of the green line in (d) (e) showsthat to minimize bacterial meningitis outbreak in 30 days the treatment control should be held intensively for 30 days at a constant rate (f)shows the optimal control plot of treatment with different cost weights

individuals with and without control measures Thereforeapplying both vaccination and treatment has a higher rateof controlling bacterial meningitis than depending only onone control method Figures 6(a) and 6(b) also show thatas infections increase in the population there is a need forintroducing vaccination and treatment at the earlier stage soas to minimize the spread of the infection in the population

This further suggests that college students and prisonersshould be given regular vaccination against meningitis sincebacterial meningitis is more likely to affect college studentsand prisoners than other people due to the close proximateof beds in the dormitories and prison cells especially inSub-Saharan Africa Figure 6(c) shows that the trajectory fortreatment moves to an upper bound at about 119905 = 35 days and


Time

0

2

4Ca

rrie

rs

Without optimal controlWith optimal control

times105

0 20 40 60 80

ℛ６４ = 42254

(a) The optimal effect on 119862(119905) with both control measures

Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254


(b) The optimal effect on 119868(119905) with both control measures

Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254

(c) Simulation diagram for control measures


Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d

(d) Immunity of recovered individuals

Figure 6 Further simulation trajectories (d) shows the immunity levels when the two control measures are applied with the followingparameter values 120587 = 100000 120578 = 035 120596 = 02 120575 = 01 120590 = 1 120572 = 02 and 120579 = 00839

0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05

Without controls = 05 With controls = 02

Without controls = 02

5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520

(a) Optimal simulation effect in varying 120572 on 119862(119905)

0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520

(b) Optimal simulation effect in varying 120572 on 119868(119905)

Figure 7 Simulation results of the optimal control model with 120572 = 02-05 on carriers and the ill compartment

slowly decreases to a lower bound which implies that duringthe period before 35 days substantial amount of the control1199062 should be applied while administering the vaccinationcontrol in the susceptible population so as to reduce thenumber of carriers and ill individuals until the 50th dayAfterwards less amount of treatment can be used since thenumber of carriers and ill individuals will be considerablyreduced by the earlier investment in treatment within theperiod before 35 days

Figures 7(a) 7(b) 8(c) and 8(d) show the impact ofvarying 120572 and 120578 It can be seen that an increase in 120572 and120578 without control measures has a corresponding decreasein the number of carriers and a considerable increase inthe number of infected individuals Therefore applying thecontrol measures gives a drastic decrease in the number ofinfections in both carriers and the ill compartment at a higherrate of 120572 = 05 and 120578 = 065 Furthermore in Figures 7(a)and 7(b) authors kept the natural recovery rate (120596) fixed at


0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030

Absence of treatment with = 02

= 07

= 06

= 05

= 04

= 03

= 02

(a) Numerical simulation without control measures on 119877(119905)

0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02

Absence of vaccination and treatment with = 05

(b) Numerical simulation without control measures on 119877(119905)

Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854

(c) Optimal simulation effect in varying 120578 on 119862(119905)

0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854

(d) Optimal simulation effect in varying 120578 on 119868(119905)

Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02

(e) Effectiveness of treatment 120574 on 119868(119905)

0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02

(f) Effectiveness of treatment 120574 on 119862(119905)

Figure 8 Simulation results of the optimal control model with 120578 = 035-065 The solutions for the carriers and ill individuals with 120572 = 02and 120572 = 05 give RVT = 4225 and RVT = 5250 respectively The case where 120578 = 035 and 120578 = 065 gives RVT = 4225 and RVT = 4854respectively which indicates the existence of an unstable disease-free equilibrium and a stable endemic equilibrium In (e) and (f) vaccinationand treatment rates are set to 05 and the effectiveness of the treatment control 120574 varied (e) and (f) show that as the effectiveness of thetreatment control increases the number of ill individuals reduces faster as compared to the number of carriers

02 while assessing the effect of varying the rate of fallingill on carriers and on ill individuals respectively Henceconsidering an increased rate from 02 to 07 of the rate offalling ill as shown in Figures 8(a) and 8(b) demonstrateshow serious the infection is in the absence of vaccination ortreatment which indicates that ill individuals have a lowermeans of recovering from the disease not even through a

higher rate of natural defences of 120596 = 05 as indicatedin Figure 8(b) without vaccination and treatment whichclearly shows that vaccination and treatment are essential incontrollingmeningitis irrespective of the age group thereforewe suggest that health authorities should increase the rateat which susceptible individuals get vaccination throughpublic education and also encourage individuals with and


05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity


(a) The contour plot of 120579 and 120573 toRVT

05 4

2

2

3

0 0

05

1

15

2

25

Effective contact rate Loss of immunity

Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４

(b) The 3D plot ofRVT to 120579 and 120573

Figure 9 The interepidemic period of the SCIRS model depending on parameters 120579 and 120573

02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４


Figure 10 The interepidemic period of the SCIRSmodel depending on parameters 120578 and 1205781

without symptoms of the infection to visit health centers for aquick check-up and if the infection is detected an immediatetreatmentvaccination should be given on time to avert thespreading of the infection to community members relativesor close friends

41 Vaccination Control Only Figures 4(a)ndash4(f) show simu-lation results of implementing only control strategy 1199061 with1199062 set to zero42 Treatment Control Only Considering treatment control1199062 by setting the vaccination control 1199061 to zero thesimulation results are as shown in Figures 5(a)ndash5(f)

43 Applying Both Control Measures Finally both controlmeasures are implemented using the following initial popula-tion size of 119878(0) = 700000 119862(0) = 250000 119868(0) = 40000 and119877(0) = 10000 and the results are plotted in Figures 6(a)ndash6(d)

Figures 9(a) 9(b) 10(a) and 10(b) show the interepidemicnature of some selected parameters in the 119878119862119868119877119878 modelpresented above Figures 9(a) and 9(b) indicate that asindividuals lose immunity and become susceptible againthere is a corresponding increase in the rate of infectionand this suggests that an individual who receives vaccinationandor treatment and remains immune for 3ndash5 years shouldimmediately go for another vaccination against the infectionso as to avert transmission of the infection to that individualFigures 10(a) and 10(b) depict that the per capita infection

rate by ill individuals 1205781 and the per capita infection rate bycarriers 120578 are interlinked and so have a corresponding influ-ence in the spread of the infection Figure 10(b) also indicatesthat as the infectivity rate of carriers and ill individuals goesbeyond 06 there could be a high spread of the infection inthe communitysociety should there be an outbreak since theeffective reproductionnumberwill be getting closer to one (1)with a possibility of going beyond a unit

5 Conclusion

We presented a mathematical framework of vaccination andtreatment on 119878119862119868119877119878 bacterial meningitis model The inves-tigation of the stability of the model shows that the disease-free equilibrium is locally and globally asymptotically stableand the endemic equilibrium depicts a global stability Scatterplots and the Tornado plots of the twelve parameters inRVTshow that the effective contact rate 120573 has a major impact intransmitting the disease followed by the infectivity potentialof carriers 120578This supports the fact that asymptomatic carriersare likely to havemore contacts in a communitynation whenthere is an outbreak of bacterial meningitis compared toill individuals who will typically be bound to their bedsduring the acute phase of the disease thereby lowering theprobability of infecting susceptible individuals Finally thenumerical simulation shows that the optimal (best) way ofcontrolling the transmission of meningitis in Sub-SaharanAfrica and the world at large is to encourage susceptible


individuals to go for vaccination against meningitis at thehealth centers and also report any suspected symptoms ofmeningitis to health practitioners for early detection andimmediate care (treatment)

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W P Howlett Neurology in Africa Clinical Skills and Neurolog-ical Disorders Cambridge University Press 2015

[2] M J F Martınez E G Merino E G Sanchez J E G SanchezA M Del Rey and G R Sanchez ldquoA mathematical modelto study the meningococcal meningitisrdquo in Proceedings of the13th Annual International Conference on Computational ScienceICCS 2013 pp 2492ndash2495 Spain June 2013

[3] CDC Centers for Disease Control Bacterial Meningitis 2017httpwwwwhointghoepidemic diseasesmeningitissuspectedcases deaths texten

[4] K B Blyuss ldquoMathematical modelling of the dynamics of men-ingococcalmeningitis inAfricardquoUKSuccess Stories in IndustrialMathematics pp 221ndash226 2016

[5] MCThigpen CGWhitneyN EMessonnier et al ldquoBacterialmeningitis in the United States 1998ndash2007rdquo The New EnglandJournal of Medicine vol 364 no 21 pp 2016ndash2025 2011

[6] CDC Centers for Disease Control Non-polio enterovirus Infec-tions 2010 httpwwwcdcgovncidoddvrdrevb

[7] B Shmaefsky and H BabcockMMeningitis Infobase Publish-ing 2010

[8] WorldbankMeningitis vaccine provides hope to people inGhana2012 httpwwwwhointfeatures2012meningitis ghanaen

[9] CDC Centers for Disease Control Control and Preventionof Meningococcal Disease Recommendations of the AdvisoryCommittee on Immunization Practices (ACIP) 2017 httpswwwcdcgovmmwrpreviewmmwrhtml00046263htm

[10] C L Trotter and M E Ramsay ldquoVaccination against meningo-coccal disease in Europe review and recommendations for theuse of conjugate vaccinesrdquo FEMS Microbiology Reviews vol 31no 1 pp 101ndash107 2007

[11] H Broutin S PhilipponG Constantin deMagnyM-F CourelB Sultan and J-F Guegan ldquoComparative study of meningitisdynamics across nine African countries A global perspectiverdquoInternational Journal of Health Geographics vol 6 article no 292007

[12] M A Miller and C K Shahab ldquoReview of the cost effective-ness of immunisation strategies for the control of epidemicmeningococcal meningitisrdquo PharmacoEconomics vol 23 no 4pp 333ndash343 2005

[13] National Collaborating Centre for Womenrsquos and ChildrenrsquosHealth Antibiotics for Early-Onset Neonatal Infection Antibi-otics for The Prevention And Treatment of Early-Onset NeonatalInfection RCOG Press 2012

[14] T J Irving K B Blyuss C Colijn and C L Trotter ldquoModellingmeningococcal meningitis in the African meningitis beltrdquoEpidemiology and Infection vol 140 no 5 pp 897ndash905 2012

[15] B Greenwood ldquoPriorities for research on meningococcal dis-ease and the impact of serogroup A vaccination in the Africanmeningitis beltrdquo Vaccine vol 31 no 11 pp 1453ndash1457 2013

[16] A C Cohn and L H Harrison ldquoMeningococcal vaccinesCurrent issues and future strategiesrdquo Drugs vol 73 no 11 pp1147ndash1155 2013

[17] R Yogev and T Tan ldquoMeningococcal disease the advancesand challenges of meningococcal disease preventionrdquo HumanVaccines amp Immunotherapeutics vol 7 no 8 pp 828ndash837 2011

[18] S B Gordon S Kanyanda A L Walsh et al ldquoPoor poten-tial coverage for 7-valent pneumococcal conjugate vaccineMalawirdquo Emerging Infectious Diseases vol 9 no 6 pp 747ndash7492003

[19] A J Pollard J OchnioMHoM CallaghanM Bigham and SDobson ldquoDisease Susceptibility to ST11 complex meningococcibearing serogroup C or W135 polysaccharide capsules NorthAmericardquo Emerging Infectious Diseases vol 10 no 10 pp 1812ndash1815 2004

[20] W P Hanage ldquoSerotype-specific problems associated withpneumococcal conjugate vaccinationrdquo FutureMicrobiology vol3 no 1 pp 23ndash30 2008

[21] G B Coulson A Von Gottberg M Du Plessis A M SmithL De Gouveia and K P Klugman ldquoMeningococcal disease inSouth Africa 1999-2002rdquo Emerging Infectious Diseases vol 13no 2 pp 273ndash281 2007

[22] MC Thomson E Firth M Jancloes et al ldquoA climate andhealth partnership to inform the prevention and controlof meningoccocal meningitis in sub-Saharan Africardquo in theMERIT initiative in Climate Science for Serving Society pp459ndash484 Springer Climate Science for Serving Society 2013

[23] J E Mueller L Sangare B-M Njanpop-Lafourcade et alldquoMolecular characteristics and epidemiology of meningococcalcarriage Burkina Faso 2003rdquo Emerging Infectious Diseases vol13 no 6 pp 847ndash854 2007

[24] V Dukic M Hayden A A Forgor and et al ldquoThe role ofweather in meningitis outbreaks in Navrongo Ghana a gen-eralized additive modeling approachrdquo Journal of AgriculturalBiological and Environmental Statistics vol 17 no 3 pp 442ndash460 2012

[25] A Beresniak E Bertherat W Perea et al ldquoA Bayesian networkapproach to the study of historical epidemiological databasesModelling meningitis outbreaks in the Nigerrdquo Bulletin of theWorld Health Organization vol 90 no 6 pp 412ndash416 2012

[26] S Tartof A Cohn F Tarbangdo et al ldquoIdentifying OptimalVaccination Strategies for Serogroup A Neisseria meningitidisConjugate Vaccine in the African Meningitis Beltrdquo PLoS ONEvol 8 no 5 Article ID e63605 2013

[27] L Agier H Broutin E Bertherat et al ldquoTimely detection ofbacterial meningitis epidemics at district level A study in threecountries of the African Meningitis Beltrdquo Transactions of theRoyal Society of Tropical Medicine and Hygiene vol 107 no 1Article ID trs010 pp 30ndash36 2013

[28] K Vereen An SCIRModel of Meningococcal Meningitis 2008[29] H W Hethcote ldquoThree basic epidemiological modelsrdquo in

Applied Mathematical Ecology S A Levin T G Hallam and LJ Gross Eds vol 18 of Biomathematics pp 119ndash144 SpringerBerlin Germany 1989

[30] R L Miller Neilan Optimal Control Applied to Population andDisease Models (Doctoral Dissertations) 2009

[31] A O Isere J E Osemwenkhae and D Okuonghae ldquoOptimalcontrol model for the outbreak of cholera in Nigeriardquo AfricanJournal of Mathematics and Computer Science Research vol 7no 2 pp 24ndash30 2014


[32] A A Lashari Mathematical Modeling And Optimal ControlOf A Vector Borne Disease National University of ModernLanguages Islamabad Pakistan 2012

[33] A Abdelrazec S Lenhart andH Zhu ldquoTransmission dynamicsofWest Nile virus in mosquitoes and corvids and non-corvidsrdquoJournal of Mathematical Biology vol 68 no 6 pp 1553ndash15822014

[34] A StashkoThe effects of prevention and treatment interventionsin a microeconomic model of HIV transmission Duke UniversityDurham 2012

[35] B Seidu and O D Makinde ldquoOptimal control of HIVAIDSin the workplace in the presence of careless individualsrdquoComputational and Mathematical Methods in Medicine vol2014 Article ID 831506 2014

[36] S D Djiomba Njankou and F Nyabadza ldquoAn optimal controlmodel for Ebola virus diseaserdquo Journal of Biological Systems vol24 no 1 pp 29ndash49 2016

[37] T T Yusuf and F Benyah ldquoOptimal control of vaccination andtreatment for an SIR epidemiological modelrdquo World Journal ofModelling and Simulation vol 8 no 3 pp 194ndash204 2012

[38] H Gaff and E Schaefer ldquoOptimal control applied to vaccinationand treatment strategies for various epidemiological modelsrdquoMathematical Biosciences and Engineering vol 6 no 3 pp 469ndash492 2009

[39] S Nanda H Moore and S Lenhart ldquoOptimal control oftreatment in a mathematical model of chronic myelogenousleukemiardquoMathematical Biosciences vol 210 no 1 pp 143ndash1562007

[40] G Zaman Y Han Kang and I H Jung ldquoStability analysis andoptimal vaccination of an SIR epidemicmodelrdquo BioSystems vol93 no 3 pp 240ndash249 2008

[41] J K K Asamoah F T Oduro E Bonyah and B SeiduldquoModelling of Rabies Transmission Dynamics Using OptimalControl Analysisrdquo Journal of Applied Mathematics vol 2017Article ID 2451237 2017

[42] U PrincetonEmergencyGuidelines for the Campus CommunityMeningitis FAQGeneral Information 2015 httpwebprincetonedusitesemergencymeningitisFAQ-Generalhtml

[43] C Sun and Y-H Hsieh ldquoGlobal analysis of an SEIRmodel withvarying population size and vaccinationrdquoAppliedMathematicalModelling vol 34 no 10 pp 2685ndash2697 2010

[44] S M Moghadas ldquoModelling the effect of imperfect vaccineson disease epidemiologyrdquo Discrete and Continuous DynamicalSystems - Series B vol 4 no 4 pp 999ndash1012 2004

[45] D Aldila N Rarasati N Nuraini and E Soewono ldquoOptimalcontrol problem of treatment for obesity in a closed popula-tionrdquo International Journal of Mathematics and MathematicalSciences vol 2014 Article ID 273037 2014

[46] M Martcheva An Introduction to Mathematical Epidemiologyvol 61 of Texts in AppliedMathematics Springer New York NYUSA 2015

[47] A Karachaliou A J K ConlanM-P Preziosi andC L TrotterldquoModeling long-term vaccination strategies with MenAfriVacin the African Meningitis Beltrdquo Clinical Infectious Diseases vol61 pp S594ndashS600 2015

[48] WHO World Health Organization Global Health Observatory(GHO) data Number of suspected meningitis cases and deathsreported 2010 httpwwwwhointemergenciesdiseasesen

[49] S D Hove-Musekwa F Nyabadza and H Mambili-Mamboundou ldquoModelling hospitalization home-based careand individual withdrawal for people living with HIVAIDS in

high prevalence settingsrdquo Bulletin of Mathematical Biology vol73 no 12 pp 2888ndash2915 2011

[50] C V De Le and C V De Leon ldquoConstructions of Lyapunovfunctions for classics SIS SIR and SIRS epidemic model withvariable population sizerdquo Foro-Red-Mat Revista electronica decontenido matematico vol 26 no 5 p 1 2009

[51] J LaSalle ldquoThe stability of dynamical systemsrdquo in Proceedings ofthe CBMS-NSF regional conference series in appliedmathematics25 Philadelphia Pennsylvania 1976

[52] S Marino I B Hogue C J Ray and D E Kirschner ldquoAmethodology for performing global uncertainty and sensitivityanalysis in systems biologyrdquo Journal of Theoretical Biology vol254 no 1 pp 178ndash196 2008

[53] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn 1982

[54] D Iacoviello and N Stasio ldquoOptimal control for SIRC epidemicoutbreakrdquoComputerMethods and Programs in Biomedicine vol110 no 3 pp 333ndash342 2013

[55] H Laarabi M Rachik O El Kahlaoui and E H Labriji ldquoOpti-mal vaccination strategies of an SIR epidemic model with asaturated treatmentrdquo Universal Journal of Applied Mathematicsvol 1 no 3 pp 185ndash191 2013

[56] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014

[57] D L Lukes Differential Equations Classical to ControlledMathematics in Science and Engineering Academic Press NewYork NY USA 1982

[58] A A Lashari K Hattaf G Zaman and X Li ldquoBackwardbifurcation and optimal control of a vector borne diseaserdquoApplied Mathematics amp Information Sciences vol 7 no 1 pp301ndash309 2013

Stem Cells International

Hindawiwwwhindawicom Volume 2018


MEDIATORSINFLAMMATION

of

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Immunology ResearchHindawiwwwhindawicom Volume 2018

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine


Behavioural Neurology

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom


are commonly found in older adults [3] Bacterial meningitisis characterized by intense headache and fever vomitingsensitivity to light and stiff neck which result in convulsiondelirium and death

It is estimated that meningococcal meningitis causes over10000 deaths annually in Sub-Saharan Africa [4] About4100 cases of bacterial meningitis occurred between 2003and 2007 in the United States [3 5] Between 5 to 40 ofchildren and 20 to 50 of adults with this condition die [6]Infections from bacterial meningitis can cause permanentdisabilities such as brain damage hearing loss and learningdisabilities [3] The illness of bacterial meningitis becomesworse when symptoms are not detected early enough evenwith proper treatment the individual could die [2]

Prevention of bacterial meningitis can be achievedthrough vaccination andor preventing contact with infec-tious individuals Vaccination is the most effective wayof protecting children against certain types of bacterialmeningitis [3] Vaccines that can prevent meningitis includeHaemophilus influenza type B (Hib) pneumococcal conju-gate andmeningococcal vaccine [6 7]The conjugatemenin-gitis A vaccine MenAfrivac is recommended to protectpeople in Sub-Saharan Africa against the most commontype serotype A [8] In the United States the primarymeans of preventing meningococcal meningitis is antimi-crobial chemoprophylaxis [9] Empirical therapy includesceftriaxone or cefotaxime and vancomycin for Streptococcuspneumoniae [2] There is a vaccine against meningococcaldisease which is 85ndash100 effective in preventing four kindsof bacteria (serogroups A C Y and W-135) that cause about70 of the disease in the United States [2]

Trotter andRamsay [10] outlined some recommendationson the use of conjugate vaccines in Europe based on the expe-rience with meningococcal C conjugate (MCC) vaccines Inareas with limited health infrastructure and resources thereare a number of antibiotics including penicillin ampicillinand chloramphenicol that can be used to treat the infectionmeningitis

Mathematical models have been shown to help increasethe understanding of the spread and control of infectious dis-eases Martınez et al [2] studied the spread of meningococcalmeningitis with the use of a discrete mathematical modelbased on cellular automata where the population was dividedinto five classes susceptible asymptomatic infected infectedwith symptoms carriers recovered and died classes Broutinet al [11] studied the dynamics of meningococcal meningitisin nine African countries by adopting some mathematicaltools to time series analysis andwaveletmethod the results oftheir studies suggest that ldquointernational cooperation in PublicHealth and cross disciplines studies are highly recommendedto help in controlling this infectious diseaserdquo Miller andShahab [12] studied the cost effectiveness of immunisationstrategies for the control of epidemic meningococcal menin-gitis The research work in [13] gives a detailed descriptionof the use of antibiotics for the prevention and treatmentof meningitis infection Irving et al [14] used deterministiccompartmental models to investigate how well simple modelstructureswith seasonal forcingwere able to qualitatively cap-ture the patterns of meningitis infection They demonstrated

that the complex and irregular timing of epidemics could becaused by the interaction of temporary immunity conferredby carriage of the bacteria together with seasonal changes inthe transmissibility of infection Actually there have been asignificant number of studies of various types ofMeningitis inAfrica and Europe without the use of optimal control analysis(see [15ndash28])

It is obvious that mathematical modelling has becomecrucial in investigating the epidemiological behaviour ofmeningitis Furthermore mathematical modelling helps toidentify the risk factors for diseases so as to find out whyeveryone does not have the same infection uniformly [29]

The application of optimal control in disease modellinggives valuable information on how to apply control measuresThrough vaccination treatment public education and soforth many infectious diseases have been controlled [29]Since the introduction of optimal control theory in diseasemodelling there have been a considerable number of studiesof infectious diseases using optimal control analysis (see [30ndash41]) With the significant influence of optimal control theoryin disease modelling this paper presents an optimal controlmodel for bacterial meningitis in the presence of vaccinationand treatment due to public health education The model isqualitatively analyzed and numerically simulated in order tohelp give policy direction on how to control the spread of thedisease

The rest of the paper is organized as follows Section 2presents the model formulation and analysis Section 3presents the analysis of the optimal control problem leadingto the existence and characterization of the control measuresSection 4 contains the numerical simulations and discussionSection 5 presents the conclusion of the study

2 Model Formulation and Analysis

Adopting the epidemiological studies of a meningitis modelas presented in [4] we consider four mutually exclusivecompartments to indicate individuals with unique natures(ie susceptibles 119878(119905) carriers 119862(119905) ill individuals 119868(119905) andrecovered individuals 119877(119905)) in relation to the disease It isassumed that the susceptible compartment 119878(119905) is populatedthrough recruitment at the rate 120587 (thus migration andorbirth rate) and 120573 is the rate of effective contact of carriersandor infected (ill) individuals in the susceptible populationThe carrier compartment consists of individuals that havethe infection and do not show any clinical symptoms butcontribute to the spread of the disease When a susceptibleindividual is exposed to this infection that individual canharbour the bacterium for weeks or even months [42] butin a normal circumstance an individual develops symptomsof the infection within 3 to 7 days after exposure [3] Carriersare assumed to develop clinical symptoms (ie move to illindividuals compartment 119868(119905)) at rate 120572 Ill individuals whoare seriously infected are assumed to have no natural recoveryexcept when given treatment on time From an epidemi-ological perspective individuals in the removedrecoveredcompartment 119877(119905) do not attain permanent immunityAfter vaccination immunity develops within 7ndash10 days andremains effective for approximately 3ndash5 years [2] Therefore



120582 = 120573 (1 minus 1199061120590) (120578119862 + 1205781119868)119873 (1)





u1S

u2CCS

I

RRu2IICCSS

I

C

R


119889119877119889119905 = 1199062120574119868 + (120596 + 1199062120574) 119862 + 1199061120590119878 minus (120579 + 120583) 119877119878 (0) gt 0 119862 (0) ge 0 119868 (0) ge 0 119877 (0) ge 0

(2)



Let

Ω = (119878 119862 119868 119877) isin R4+ 119873 le 120587120583 (4)




E0 (1198780 1198620 1198680 1198770)= ( 120587 (120579 + 120583)

120583 (1199061120590 + 120579 + 120583) 0 01205871199061120590120583 (1199061120590 + 120579 + 120583)) (5)



Table1Descriptio

nof

parameter

values

used

inthem

odel

ldquoDue

tolack

ofrelevant

datam

osto

fthe

parameter

values

area

ssum

edwith

inrealisticranges

fora

typicalscenario

inarural

commun

itysocietyforthe

purposeo

fillu

stratio

nrdquo[44]w

ithmosto

fthe

parameter

values

where

appreciablep

eryear

Parameter

Descriptio

nVa

lueRa

nge

Reference

120587Re

cruitm

entrate(migratio

nandor

birthrate)

100ndash

1000

00Assum

ed120573

Effectiv

econ

tactrate

088[28]

120579Lo

ssof

immun

ity004ndash2

[1447]


002[14

]120575

Dise

ase-indu

cedmortality

005ndash0

5[48]

120572Ra

teof

progressionfro

m119862to

11986801ndash052

[14]


rate

006ndash0

2Assum

ed120578 1

Perc

apita

infectionrateby

illindividu

als

02ndash095

Assum

ed120578

Perc

apita

infectionrateby

Carriers

02ndash085

Assum

ed119906 1

Vaccinationrateof

susceptib

leindividu

alsa

gainstMeningitis

[01]

Assum

ed119906 2

Treatm

entratefor

carrierswith

outn

aturalrecovery

andill

individu

als

[01]

Assum

ed120590

Effectiv

enesso

fvaccinatio

n085ndash1

[2]

120574Eff

ectiv

enesso

fTreatment

01ndash09

Assum

ed


01 02 03 04 050

05

1

Trea

tmen

tu2

Vaccination u1

ℛ６４ gt 1

ℛ６４ lt 1



RVT = R119862VT + R119868VT (6)

whereR119862VT = 120573120578 (1 minus 1199061120590) (120579 + 120583)

(120572 + 120596 + 120583 + 1199062120574) (120579 + 120583 + 1199061120590) R119868VT

= 1205731205721205781 (1 minus 1199061120590) (120579 + 120583)(120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574) (120579 + 120583 + 1199061120590)

(7)






120572120583 (120579 + 120583 + 1199061120590) [A (RVT minus 1) + B]


(8)








119869 (E0) =[[[[[[[[[


(1199061120590 + 120579 + 120583) 1205790 (1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) minus (120572 + 120596 + 120583 + 1199062120574) (1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) 0

0 120572 minus (120575 + 120583 + 1199062120574) 01199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]]]]

(9)


119875 (120582) = 1205824 + DT1205823 + DT11205822 + DT2120582 + DT3 (10)



T2




(120575 + 120583 + 1199062120574) 119861 = 1

(120572 + 120596 + 120583 + 1199062120574)

119862 = (120583 + 1199061120590) + (120579 + 120583)(120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

D = (120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

(11)




(12)






(13)


119889V119889119905 = (1 minus 1198780119878 ) 119889119878119889119905 + K1119889119862119889119905 + K2


(14)





+ (120596 + 1199062120574)119862 + 1205901199061119878 minus (120579 + 120583) 119877]le K1 [((1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) )119862

+ ((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) ) 119868

minus (120572 + 120596 + 120583 + 1199062120574) 119862] + K2 [120572119862minus (120575 + 120583 + 1199062120574) 119868]

(15)

Since

119878 le 1198780 = 120587 (120579 + 120583)120583 (1199061120590 + 120579 + 120583)

119877 le 1198770 = 1205871199061120590120583 (1199061120590 + 120579 + 120583) 119873 le 120587120583

on Ω

(16)

this implies that

119889V119889119905 le [((1 minus 1199061120590) 120573120578 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1 + 120572K1

minus (120572 + 120596 + 120583 + 1199062120574)K2]119862

+ [((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1

minus (120575 + 120583 + 1199062120574)K2] 119868

(17)


(1 minus 1199061120590) 120573 (120579 + 120583)K1= (120575 + 120583 + 1199062120574) (1199061120590 + 120579 + 120583)K2 (18)



(19)




119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (20)





(22)



(23)

Now setting


we have


(25)




(26)




Γ120588RVT

= 120597RVT120597120588120588

RVT (27)


Γ120573RVT

= 120597RVT120597120573120573

RVT= 1

Γ120578RVT

= 120597RVT120597120578120578

RVT= (120575 + 120583 + 1199062120574) 120578120578 (120575 + 120583 + 1199062120574) + 1205781120572

Γ1205781RVT

= 120597RVT12059712057811205781

RVT= 1205721205781(1205721205781 + (120575 + 120583 + 1199062120574))

Γ120596RVT= 120597RVT120597120596 120596

RVT= minus 120596

(120572 + 120596 + 120583 + 1199062120574) Γ120590RVT

= 120597RVT120597120590 120590RVT

= minus 1199061120590(1199061120590 + 120579 + 120583)

(28)



RVT= 1





119869 = min1199061120590119906

2

int1198790

[1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)

+ 11986122 11990622 (119905)] 119889119905(29)






BetaEta

eta_1Thetau_1u_2mu

DeltaAlpha

OmegaGamma

Sigma


0 02 04 06 08 1minus15

minus10

minus5

0


log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Vaccination rate u1

log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Treatment rate u2

log(ℛ

６４)










119866119905 = 119870 (119866) + 119865 (119866) (31)

where

119866 =[[[[[[

119878 (119905)119862 (119905)119868 (119905)119877 (119905)

]]]]]]

119870 =[[[[[[


1199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]

119865 (119866) =[[[[[[[[[


119873 (119905)00

]]]]]]]]]

(32)



119865 (1198661) minus 119865 (1198662) =[[[[[[[[[


00

]]]]]]]]]

(33)

Therefore


1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816

+ 2 (1 minus 1199061120590) 1205731205781 10038161003816100381610038161003816100381610038161003816(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816


120583 (120578 + 1)




(34)

where

119872 = max(2 (1 minus 1199061120590) 120573120587120583 (120578 + 1) 2 (1 minus 1199061120590) 120573120578120587

120583 2 (1 minus 1199061120590) 1205731205781120587120583 )

(35)

so that



119871 (119878 119862 119868 119877 1199061 1199062) = 1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)+ 11986122 11990622 (119905)

(37)



+ 120579119877] + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + 1205823 [120572119862

minus (120575 + 120583 + 1199062120574) 119868] + 1205824 [1199062120574119868 + (120596 + 1199062120574)119862+ 1199061120590119878 minus (120579 + 120583) 119877]

(38)



(39)



2isin119880

119869 (1199061 (119905) 1199062 (119905)) (40)


119869 (1199061 (119905) 1199062 (119905)) ge ]1 (1003816100381610038161003816119906110038161003816100381610038162 + 1003816100381610038161003816119906210038161003816100381610038162)1205882 minus ]2 (41)







1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583


119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)


+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732


(42)



where 119894 = 1 2(43)





1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)


119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]





=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















120582 = 120573 (1 minus 1199061120590) (120578119862 + 1205781119868)119873 (1)





u1S

u2CCS

I

RRu2IICCSS

I

C

R


119889119877119889119905 = 1199062120574119868 + (120596 + 1199062120574) 119862 + 1199061120590119878 minus (120579 + 120583) 119877119878 (0) gt 0 119862 (0) ge 0 119868 (0) ge 0 119877 (0) ge 0

(2)



Let

Ω = (119878 119862 119868 119877) isin R4+ 119873 le 120587120583 (4)




E0 (1198780 1198620 1198680 1198770)= ( 120587 (120579 + 120583)

120583 (1199061120590 + 120579 + 120583) 0 01205871199061120590120583 (1199061120590 + 120579 + 120583)) (5)



Table1Descriptio

nof

parameter

values

used

inthem

odel

ldquoDue

tolack

ofrelevant

datam

osto

fthe

parameter

values

area

ssum

edwith

inrealisticranges

fora

typicalscenario

inarural

commun

itysocietyforthe

purposeo

fillu

stratio

nrdquo[44]w

ithmosto

fthe

parameter

values

where

appreciablep

eryear

Parameter

Descriptio

nVa

lueRa

nge

Reference

120587Re

cruitm

entrate(migratio

nandor

birthrate)

100ndash

1000

00Assum

ed120573

Effectiv

econ

tactrate

088[28]

120579Lo

ssof

immun

ity004ndash2

[1447]


002[14

]120575

Dise

ase-indu

cedmortality

005ndash0

5[48]

120572Ra

teof

progressionfro

m119862to

11986801ndash052

[14]


rate

006ndash0

2Assum

ed120578 1

Perc

apita

infectionrateby

illindividu

als

02ndash095

Assum

ed120578

Perc

apita

infectionrateby

Carriers

02ndash085

Assum

ed119906 1

Vaccinationrateof

susceptib

leindividu

alsa

gainstMeningitis

[01]

Assum

ed119906 2

Treatm

entratefor

carrierswith

outn

aturalrecovery

andill

individu

als

[01]

Assum

ed120590

Effectiv

enesso

fvaccinatio

n085ndash1

[2]

120574Eff

ectiv

enesso

fTreatment

01ndash09

Assum

ed


01 02 03 04 050

05

1

Trea

tmen

tu2

Vaccination u1

ℛ６４ gt 1

ℛ６４ lt 1



RVT = R119862VT + R119868VT (6)

whereR119862VT = 120573120578 (1 minus 1199061120590) (120579 + 120583)

(120572 + 120596 + 120583 + 1199062120574) (120579 + 120583 + 1199061120590) R119868VT

= 1205731205721205781 (1 minus 1199061120590) (120579 + 120583)(120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574) (120579 + 120583 + 1199061120590)

(7)






120572120583 (120579 + 120583 + 1199061120590) [A (RVT minus 1) + B]


(8)








119869 (E0) =[[[[[[[[[


(1199061120590 + 120579 + 120583) 1205790 (1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) minus (120572 + 120596 + 120583 + 1199062120574) (1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) 0

0 120572 minus (120575 + 120583 + 1199062120574) 01199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]]]]

(9)


119875 (120582) = 1205824 + DT1205823 + DT11205822 + DT2120582 + DT3 (10)



T2




(120575 + 120583 + 1199062120574) 119861 = 1

(120572 + 120596 + 120583 + 1199062120574)

119862 = (120583 + 1199061120590) + (120579 + 120583)(120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

D = (120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

(11)




(12)






(13)


119889V119889119905 = (1 minus 1198780119878 ) 119889119878119889119905 + K1119889119862119889119905 + K2


(14)





+ (120596 + 1199062120574)119862 + 1205901199061119878 minus (120579 + 120583) 119877]le K1 [((1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) )119862

+ ((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) ) 119868

minus (120572 + 120596 + 120583 + 1199062120574) 119862] + K2 [120572119862minus (120575 + 120583 + 1199062120574) 119868]

(15)

Since

119878 le 1198780 = 120587 (120579 + 120583)120583 (1199061120590 + 120579 + 120583)

119877 le 1198770 = 1205871199061120590120583 (1199061120590 + 120579 + 120583) 119873 le 120587120583

on Ω

(16)

this implies that

119889V119889119905 le [((1 minus 1199061120590) 120573120578 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1 + 120572K1

minus (120572 + 120596 + 120583 + 1199062120574)K2]119862

+ [((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1

minus (120575 + 120583 + 1199062120574)K2] 119868

(17)


(1 minus 1199061120590) 120573 (120579 + 120583)K1= (120575 + 120583 + 1199062120574) (1199061120590 + 120579 + 120583)K2 (18)



(19)




119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (20)





(22)



(23)

Now setting


we have


(25)




(26)




Γ120588RVT

= 120597RVT120597120588120588

RVT (27)


Γ120573RVT

= 120597RVT120597120573120573

RVT= 1

Γ120578RVT

= 120597RVT120597120578120578

RVT= (120575 + 120583 + 1199062120574) 120578120578 (120575 + 120583 + 1199062120574) + 1205781120572

Γ1205781RVT

= 120597RVT12059712057811205781

RVT= 1205721205781(1205721205781 + (120575 + 120583 + 1199062120574))

Γ120596RVT= 120597RVT120597120596 120596

RVT= minus 120596

(120572 + 120596 + 120583 + 1199062120574) Γ120590RVT

= 120597RVT120597120590 120590RVT

= minus 1199061120590(1199061120590 + 120579 + 120583)

(28)



RVT= 1





119869 = min1199061120590119906

2

int1198790

[1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)

+ 11986122 11990622 (119905)] 119889119905(29)






BetaEta

eta_1Thetau_1u_2mu

DeltaAlpha

OmegaGamma

Sigma


0 02 04 06 08 1minus15

minus10

minus5

0


log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Vaccination rate u1

log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Treatment rate u2

log(ℛ

６４)










119866119905 = 119870 (119866) + 119865 (119866) (31)

where

119866 =[[[[[[

119878 (119905)119862 (119905)119868 (119905)119877 (119905)

]]]]]]

119870 =[[[[[[


1199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]

119865 (119866) =[[[[[[[[[


119873 (119905)00

]]]]]]]]]

(32)



119865 (1198661) minus 119865 (1198662) =[[[[[[[[[


00

]]]]]]]]]

(33)

Therefore


1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816

+ 2 (1 minus 1199061120590) 1205731205781 10038161003816100381610038161003816100381610038161003816(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816


120583 (120578 + 1)




(34)

where

119872 = max(2 (1 minus 1199061120590) 120573120587120583 (120578 + 1) 2 (1 minus 1199061120590) 120573120578120587

120583 2 (1 minus 1199061120590) 1205731205781120587120583 )

(35)

so that



119871 (119878 119862 119868 119877 1199061 1199062) = 1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)+ 11986122 11990622 (119905)

(37)



+ 120579119877] + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + 1205823 [120572119862

minus (120575 + 120583 + 1199062120574) 119868] + 1205824 [1199062120574119868 + (120596 + 1199062120574)119862+ 1199061120590119878 minus (120579 + 120583) 119877]

(38)



(39)



2isin119880

119869 (1199061 (119905) 1199062 (119905)) (40)


119869 (1199061 (119905) 1199062 (119905)) ge ]1 (1003816100381610038161003816119906110038161003816100381610038162 + 1003816100381610038161003816119906210038161003816100381610038162)1205882 minus ]2 (41)







1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583


119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)


+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732


(42)



where 119894 = 1 2(43)





1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)


119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]





=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















Table1Descriptio

nof

parameter

values

used

inthem

odel

ldquoDue

tolack

ofrelevant

datam

osto

fthe

parameter

values

area

ssum

edwith

inrealisticranges

fora

typicalscenario

inarural

commun

itysocietyforthe

purposeo

fillu

stratio

nrdquo[44]w

ithmosto

fthe

parameter

values

where

appreciablep

eryear

Parameter

Descriptio

nVa

lueRa

nge

Reference

120587Re

cruitm

entrate(migratio

nandor

birthrate)

100ndash

1000

00Assum

ed120573

Effectiv

econ

tactrate

088[28]

120579Lo

ssof

immun

ity004ndash2

[1447]


002[14

]120575

Dise

ase-indu

cedmortality

005ndash0

5[48]

120572Ra

teof

progressionfro

m119862to

11986801ndash052

[14]


rate

006ndash0

2Assum

ed120578 1

Perc

apita

infectionrateby

illindividu

als

02ndash095

Assum

ed120578

Perc

apita

infectionrateby

Carriers

02ndash085

Assum

ed119906 1

Vaccinationrateof

susceptib

leindividu

alsa

gainstMeningitis

[01]

Assum

ed119906 2

Treatm

entratefor

carrierswith

outn

aturalrecovery

andill

individu

als

[01]

Assum

ed120590

Effectiv

enesso

fvaccinatio

n085ndash1

[2]

120574Eff

ectiv

enesso

fTreatment

01ndash09

Assum

ed


01 02 03 04 050

05

1

Trea

tmen

tu2

Vaccination u1

ℛ６４ gt 1

ℛ６４ lt 1



RVT = R119862VT + R119868VT (6)

whereR119862VT = 120573120578 (1 minus 1199061120590) (120579 + 120583)

(120572 + 120596 + 120583 + 1199062120574) (120579 + 120583 + 1199061120590) R119868VT

= 1205731205721205781 (1 minus 1199061120590) (120579 + 120583)(120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574) (120579 + 120583 + 1199061120590)

(7)






120572120583 (120579 + 120583 + 1199061120590) [A (RVT minus 1) + B]


(8)








119869 (E0) =[[[[[[[[[


(1199061120590 + 120579 + 120583) 1205790 (1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) minus (120572 + 120596 + 120583 + 1199062120574) (1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) 0

0 120572 minus (120575 + 120583 + 1199062120574) 01199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]]]]

(9)


119875 (120582) = 1205824 + DT1205823 + DT11205822 + DT2120582 + DT3 (10)



T2




(120575 + 120583 + 1199062120574) 119861 = 1

(120572 + 120596 + 120583 + 1199062120574)

119862 = (120583 + 1199061120590) + (120579 + 120583)(120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

D = (120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

(11)




(12)






(13)


119889V119889119905 = (1 minus 1198780119878 ) 119889119878119889119905 + K1119889119862119889119905 + K2


(14)





+ (120596 + 1199062120574)119862 + 1205901199061119878 minus (120579 + 120583) 119877]le K1 [((1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) )119862

+ ((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) ) 119868

minus (120572 + 120596 + 120583 + 1199062120574) 119862] + K2 [120572119862minus (120575 + 120583 + 1199062120574) 119868]

(15)

Since

119878 le 1198780 = 120587 (120579 + 120583)120583 (1199061120590 + 120579 + 120583)

119877 le 1198770 = 1205871199061120590120583 (1199061120590 + 120579 + 120583) 119873 le 120587120583

on Ω

(16)

this implies that

119889V119889119905 le [((1 minus 1199061120590) 120573120578 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1 + 120572K1

minus (120572 + 120596 + 120583 + 1199062120574)K2]119862

+ [((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1

minus (120575 + 120583 + 1199062120574)K2] 119868

(17)


(1 minus 1199061120590) 120573 (120579 + 120583)K1= (120575 + 120583 + 1199062120574) (1199061120590 + 120579 + 120583)K2 (18)



(19)




119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (20)





(22)



(23)

Now setting


we have


(25)




(26)




Γ120588RVT

= 120597RVT120597120588120588

RVT (27)


Γ120573RVT

= 120597RVT120597120573120573

RVT= 1

Γ120578RVT

= 120597RVT120597120578120578

RVT= (120575 + 120583 + 1199062120574) 120578120578 (120575 + 120583 + 1199062120574) + 1205781120572

Γ1205781RVT

= 120597RVT12059712057811205781

RVT= 1205721205781(1205721205781 + (120575 + 120583 + 1199062120574))

Γ120596RVT= 120597RVT120597120596 120596

RVT= minus 120596

(120572 + 120596 + 120583 + 1199062120574) Γ120590RVT

= 120597RVT120597120590 120590RVT

= minus 1199061120590(1199061120590 + 120579 + 120583)

(28)



RVT= 1





119869 = min1199061120590119906

2

int1198790

[1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)

+ 11986122 11990622 (119905)] 119889119905(29)






BetaEta

eta_1Thetau_1u_2mu

DeltaAlpha

OmegaGamma

Sigma


0 02 04 06 08 1minus15

minus10

minus5

0


log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Vaccination rate u1

log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Treatment rate u2

log(ℛ

６４)










119866119905 = 119870 (119866) + 119865 (119866) (31)

where

119866 =[[[[[[

119878 (119905)119862 (119905)119868 (119905)119877 (119905)

]]]]]]

119870 =[[[[[[


1199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]

119865 (119866) =[[[[[[[[[


119873 (119905)00

]]]]]]]]]

(32)



119865 (1198661) minus 119865 (1198662) =[[[[[[[[[


00

]]]]]]]]]

(33)

Therefore


1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816

+ 2 (1 minus 1199061120590) 1205731205781 10038161003816100381610038161003816100381610038161003816(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816


120583 (120578 + 1)




(34)

where

119872 = max(2 (1 minus 1199061120590) 120573120587120583 (120578 + 1) 2 (1 minus 1199061120590) 120573120578120587

120583 2 (1 minus 1199061120590) 1205731205781120587120583 )

(35)

so that



119871 (119878 119862 119868 119877 1199061 1199062) = 1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)+ 11986122 11990622 (119905)

(37)



+ 120579119877] + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + 1205823 [120572119862

minus (120575 + 120583 + 1199062120574) 119868] + 1205824 [1199062120574119868 + (120596 + 1199062120574)119862+ 1199061120590119878 minus (120579 + 120583) 119877]

(38)



(39)



2isin119880

119869 (1199061 (119905) 1199062 (119905)) (40)


119869 (1199061 (119905) 1199062 (119905)) ge ]1 (1003816100381610038161003816119906110038161003816100381610038162 + 1003816100381610038161003816119906210038161003816100381610038162)1205882 minus ]2 (41)







1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583


119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)


+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732


(42)



where 119894 = 1 2(43)





1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)


119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]





=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















01 02 03 04 050

05

1

Trea

tmen

tu2

Vaccination u1

ℛ６４ gt 1

ℛ６４ lt 1



RVT = R119862VT + R119868VT (6)

whereR119862VT = 120573120578 (1 minus 1199061120590) (120579 + 120583)

(120572 + 120596 + 120583 + 1199062120574) (120579 + 120583 + 1199061120590) R119868VT

= 1205731205721205781 (1 minus 1199061120590) (120579 + 120583)(120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574) (120579 + 120583 + 1199061120590)

(7)






120572120583 (120579 + 120583 + 1199061120590) [A (RVT minus 1) + B]


(8)








119869 (E0) =[[[[[[[[[


(1199061120590 + 120579 + 120583) 1205790 (1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) minus (120572 + 120596 + 120583 + 1199062120574) (1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) 0

0 120572 minus (120575 + 120583 + 1199062120574) 01199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]]]]

(9)


119875 (120582) = 1205824 + DT1205823 + DT11205822 + DT2120582 + DT3 (10)



T2




(120575 + 120583 + 1199062120574) 119861 = 1

(120572 + 120596 + 120583 + 1199062120574)

119862 = (120583 + 1199061120590) + (120579 + 120583)(120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

D = (120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

(11)




(12)






(13)


119889V119889119905 = (1 minus 1198780119878 ) 119889119878119889119905 + K1119889119862119889119905 + K2


(14)





+ (120596 + 1199062120574)119862 + 1205901199061119878 minus (120579 + 120583) 119877]le K1 [((1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) )119862

+ ((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) ) 119868

minus (120572 + 120596 + 120583 + 1199062120574) 119862] + K2 [120572119862minus (120575 + 120583 + 1199062120574) 119868]

(15)

Since

119878 le 1198780 = 120587 (120579 + 120583)120583 (1199061120590 + 120579 + 120583)

119877 le 1198770 = 1205871199061120590120583 (1199061120590 + 120579 + 120583) 119873 le 120587120583

on Ω

(16)

this implies that

119889V119889119905 le [((1 minus 1199061120590) 120573120578 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1 + 120572K1

minus (120572 + 120596 + 120583 + 1199062120574)K2]119862

+ [((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1

minus (120575 + 120583 + 1199062120574)K2] 119868

(17)


(1 minus 1199061120590) 120573 (120579 + 120583)K1= (120575 + 120583 + 1199062120574) (1199061120590 + 120579 + 120583)K2 (18)



(19)




119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (20)





(22)



(23)

Now setting


we have


(25)




(26)




Γ120588RVT

= 120597RVT120597120588120588

RVT (27)


Γ120573RVT

= 120597RVT120597120573120573

RVT= 1

Γ120578RVT

= 120597RVT120597120578120578

RVT= (120575 + 120583 + 1199062120574) 120578120578 (120575 + 120583 + 1199062120574) + 1205781120572

Γ1205781RVT

= 120597RVT12059712057811205781

RVT= 1205721205781(1205721205781 + (120575 + 120583 + 1199062120574))

Γ120596RVT= 120597RVT120597120596 120596

RVT= minus 120596

(120572 + 120596 + 120583 + 1199062120574) Γ120590RVT

= 120597RVT120597120590 120590RVT

= minus 1199061120590(1199061120590 + 120579 + 120583)

(28)



RVT= 1





119869 = min1199061120590119906

2

int1198790

[1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)

+ 11986122 11990622 (119905)] 119889119905(29)






BetaEta

eta_1Thetau_1u_2mu

DeltaAlpha

OmegaGamma

Sigma


0 02 04 06 08 1minus15

minus10

minus5

0


log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Vaccination rate u1

log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Treatment rate u2

log(ℛ

６４)










119866119905 = 119870 (119866) + 119865 (119866) (31)

where

119866 =[[[[[[

119878 (119905)119862 (119905)119868 (119905)119877 (119905)

]]]]]]

119870 =[[[[[[


1199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]

119865 (119866) =[[[[[[[[[


119873 (119905)00

]]]]]]]]]

(32)



119865 (1198661) minus 119865 (1198662) =[[[[[[[[[


00

]]]]]]]]]

(33)

Therefore


1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816

+ 2 (1 minus 1199061120590) 1205731205781 10038161003816100381610038161003816100381610038161003816(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816


120583 (120578 + 1)




(34)

where

119872 = max(2 (1 minus 1199061120590) 120573120587120583 (120578 + 1) 2 (1 minus 1199061120590) 120573120578120587

120583 2 (1 minus 1199061120590) 1205731205781120587120583 )

(35)

so that



119871 (119878 119862 119868 119877 1199061 1199062) = 1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)+ 11986122 11990622 (119905)

(37)



+ 120579119877] + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + 1205823 [120572119862

minus (120575 + 120583 + 1199062120574) 119868] + 1205824 [1199062120574119868 + (120596 + 1199062120574)119862+ 1199061120590119878 minus (120579 + 120583) 119877]

(38)



(39)



2isin119880

119869 (1199061 (119905) 1199062 (119905)) (40)


119869 (1199061 (119905) 1199062 (119905)) ge ]1 (1003816100381610038161003816119906110038161003816100381610038162 + 1003816100381610038161003816119906210038161003816100381610038162)1205882 minus ]2 (41)







1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583


119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)


+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732


(42)



where 119894 = 1 2(43)





1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)


119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]





=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















119869 (E0) =[[[[[[[[[


(1199061120590 + 120579 + 120583) 1205790 (1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) minus (120572 + 120596 + 120583 + 1199062120574) (1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) 0

0 120572 minus (120575 + 120583 + 1199062120574) 01199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]]]]

(9)


119875 (120582) = 1205824 + DT1205823 + DT11205822 + DT2120582 + DT3 (10)



T2




(120575 + 120583 + 1199062120574) 119861 = 1

(120572 + 120596 + 120583 + 1199062120574)

119862 = (120583 + 1199061120590) + (120579 + 120583)(120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

D = (120572 + 120596 + 120583 + 1199062120574) (120575 + 120583 + 1199062120574)

(11)




(12)






(13)


119889V119889119905 = (1 minus 1198780119878 ) 119889119878119889119905 + K1119889119862119889119905 + K2


(14)





+ (120596 + 1199062120574)119862 + 1205901199061119878 minus (120579 + 120583) 119877]le K1 [((1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) )119862

+ ((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) ) 119868

minus (120572 + 120596 + 120583 + 1199062120574) 119862] + K2 [120572119862minus (120575 + 120583 + 1199062120574) 119868]

(15)

Since

119878 le 1198780 = 120587 (120579 + 120583)120583 (1199061120590 + 120579 + 120583)

119877 le 1198770 = 1205871199061120590120583 (1199061120590 + 120579 + 120583) 119873 le 120587120583

on Ω

(16)

this implies that

119889V119889119905 le [((1 minus 1199061120590) 120573120578 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1 + 120572K1

minus (120572 + 120596 + 120583 + 1199062120574)K2]119862

+ [((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1

minus (120575 + 120583 + 1199062120574)K2] 119868

(17)


(1 minus 1199061120590) 120573 (120579 + 120583)K1= (120575 + 120583 + 1199062120574) (1199061120590 + 120579 + 120583)K2 (18)



(19)




119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (20)





(22)



(23)

Now setting


we have


(25)




(26)




Γ120588RVT

= 120597RVT120597120588120588

RVT (27)


Γ120573RVT

= 120597RVT120597120573120573

RVT= 1

Γ120578RVT

= 120597RVT120597120578120578

RVT= (120575 + 120583 + 1199062120574) 120578120578 (120575 + 120583 + 1199062120574) + 1205781120572

Γ1205781RVT

= 120597RVT12059712057811205781

RVT= 1205721205781(1205721205781 + (120575 + 120583 + 1199062120574))

Γ120596RVT= 120597RVT120597120596 120596

RVT= minus 120596

(120572 + 120596 + 120583 + 1199062120574) Γ120590RVT

= 120597RVT120597120590 120590RVT

= minus 1199061120590(1199061120590 + 120579 + 120583)

(28)



RVT= 1





119869 = min1199061120590119906

2

int1198790

[1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)

+ 11986122 11990622 (119905)] 119889119905(29)






BetaEta

eta_1Thetau_1u_2mu

DeltaAlpha

OmegaGamma

Sigma


0 02 04 06 08 1minus15

minus10

minus5

0


log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Vaccination rate u1

log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Treatment rate u2

log(ℛ

６４)










119866119905 = 119870 (119866) + 119865 (119866) (31)

where

119866 =[[[[[[

119878 (119905)119862 (119905)119868 (119905)119877 (119905)

]]]]]]

119870 =[[[[[[


1199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]

119865 (119866) =[[[[[[[[[


119873 (119905)00

]]]]]]]]]

(32)



119865 (1198661) minus 119865 (1198662) =[[[[[[[[[


00

]]]]]]]]]

(33)

Therefore


1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816

+ 2 (1 minus 1199061120590) 1205731205781 10038161003816100381610038161003816100381610038161003816(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816


120583 (120578 + 1)




(34)

where

119872 = max(2 (1 minus 1199061120590) 120573120587120583 (120578 + 1) 2 (1 minus 1199061120590) 120573120578120587

120583 2 (1 minus 1199061120590) 1205731205781120587120583 )

(35)

so that



119871 (119878 119862 119868 119877 1199061 1199062) = 1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)+ 11986122 11990622 (119905)

(37)



+ 120579119877] + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + 1205823 [120572119862

minus (120575 + 120583 + 1199062120574) 119868] + 1205824 [1199062120574119868 + (120596 + 1199062120574)119862+ 1199061120590119878 minus (120579 + 120583) 119877]

(38)



(39)



2isin119880

119869 (1199061 (119905) 1199062 (119905)) (40)


119869 (1199061 (119905) 1199062 (119905)) ge ]1 (1003816100381610038161003816119906110038161003816100381610038162 + 1003816100381610038161003816119906210038161003816100381610038162)1205882 minus ]2 (41)







1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583


119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)


+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732


(42)



where 119894 = 1 2(43)





1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)


119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]





=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















+ (120596 + 1199062120574)119862 + 1205901199061119878 minus (120579 + 120583) 119877]le K1 [((1 minus 1199061120590) 120573120578 (120579 + 120583)

(1199061120590 + 120579 + 120583) )119862

+ ((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) ) 119868

minus (120572 + 120596 + 120583 + 1199062120574) 119862] + K2 [120572119862minus (120575 + 120583 + 1199062120574) 119868]

(15)

Since

119878 le 1198780 = 120587 (120579 + 120583)120583 (1199061120590 + 120579 + 120583)

119877 le 1198770 = 1205871199061120590120583 (1199061120590 + 120579 + 120583) 119873 le 120587120583

on Ω

(16)

this implies that

119889V119889119905 le [((1 minus 1199061120590) 120573120578 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1 + 120572K1

minus (120572 + 120596 + 120583 + 1199062120574)K2]119862

+ [((1 minus 1199061120590) 1205731205781 (120579 + 120583)(1199061120590 + 120579 + 120583) )K1

minus (120575 + 120583 + 1199062120574)K2] 119868

(17)


(1 minus 1199061120590) 120573 (120579 + 120583)K1= (120575 + 120583 + 1199062120574) (1199061120590 + 120579 + 120583)K2 (18)



(19)




119881 (1199091 1199092 119909119899) = 119899sum119894=1

1198881198942 (119909119894 minus 119909lowast119894 )2 (20)





(22)



(23)

Now setting


we have


(25)




(26)




Γ120588RVT

= 120597RVT120597120588120588

RVT (27)


Γ120573RVT

= 120597RVT120597120573120573

RVT= 1

Γ120578RVT

= 120597RVT120597120578120578

RVT= (120575 + 120583 + 1199062120574) 120578120578 (120575 + 120583 + 1199062120574) + 1205781120572

Γ1205781RVT

= 120597RVT12059712057811205781

RVT= 1205721205781(1205721205781 + (120575 + 120583 + 1199062120574))

Γ120596RVT= 120597RVT120597120596 120596

RVT= minus 120596

(120572 + 120596 + 120583 + 1199062120574) Γ120590RVT

= 120597RVT120597120590 120590RVT

= minus 1199061120590(1199061120590 + 120579 + 120583)

(28)



RVT= 1





119869 = min1199061120590119906

2

int1198790

[1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)

+ 11986122 11990622 (119905)] 119889119905(29)






BetaEta

eta_1Thetau_1u_2mu

DeltaAlpha

OmegaGamma

Sigma


0 02 04 06 08 1minus15

minus10

minus5

0


log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Vaccination rate u1

log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Treatment rate u2

log(ℛ

６４)










119866119905 = 119870 (119866) + 119865 (119866) (31)

where

119866 =[[[[[[

119878 (119905)119862 (119905)119868 (119905)119877 (119905)

]]]]]]

119870 =[[[[[[


1199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]

119865 (119866) =[[[[[[[[[


119873 (119905)00

]]]]]]]]]

(32)



119865 (1198661) minus 119865 (1198662) =[[[[[[[[[


00

]]]]]]]]]

(33)

Therefore


1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816

+ 2 (1 minus 1199061120590) 1205731205781 10038161003816100381610038161003816100381610038161003816(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816


120583 (120578 + 1)




(34)

where

119872 = max(2 (1 minus 1199061120590) 120573120587120583 (120578 + 1) 2 (1 minus 1199061120590) 120573120578120587

120583 2 (1 minus 1199061120590) 1205731205781120587120583 )

(35)

so that



119871 (119878 119862 119868 119877 1199061 1199062) = 1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)+ 11986122 11990622 (119905)

(37)



+ 120579119877] + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + 1205823 [120572119862

minus (120575 + 120583 + 1199062120574) 119868] + 1205824 [1199062120574119868 + (120596 + 1199062120574)119862+ 1199061120590119878 minus (120579 + 120583) 119877]

(38)



(39)



2isin119880

119869 (1199061 (119905) 1199062 (119905)) (40)


119869 (1199061 (119905) 1199062 (119905)) ge ]1 (1003816100381610038161003816119906110038161003816100381610038162 + 1003816100381610038161003816119906210038161003816100381610038162)1205882 minus ]2 (41)







1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583


119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)


+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732


(42)



where 119894 = 1 2(43)





1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)


119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]





=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






















(26)




Γ120588RVT

= 120597RVT120597120588120588

RVT (27)


Γ120573RVT

= 120597RVT120597120573120573

RVT= 1

Γ120578RVT

= 120597RVT120597120578120578

RVT= (120575 + 120583 + 1199062120574) 120578120578 (120575 + 120583 + 1199062120574) + 1205781120572

Γ1205781RVT

= 120597RVT12059712057811205781

RVT= 1205721205781(1205721205781 + (120575 + 120583 + 1199062120574))

Γ120596RVT= 120597RVT120597120596 120596

RVT= minus 120596

(120572 + 120596 + 120583 + 1199062120574) Γ120590RVT

= 120597RVT120597120590 120590RVT

= minus 1199061120590(1199061120590 + 120579 + 120583)

(28)



RVT= 1





119869 = min1199061120590119906

2

int1198790

[1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)

+ 11986122 11990622 (119905)] 119889119905(29)






BetaEta

eta_1Thetau_1u_2mu

DeltaAlpha

OmegaGamma

Sigma


0 02 04 06 08 1minus15

minus10

minus5

0


log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Vaccination rate u1

log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Treatment rate u2

log(ℛ

６４)










119866119905 = 119870 (119866) + 119865 (119866) (31)

where

119866 =[[[[[[

119878 (119905)119862 (119905)119868 (119905)119877 (119905)

]]]]]]

119870 =[[[[[[


1199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]

119865 (119866) =[[[[[[[[[


119873 (119905)00

]]]]]]]]]

(32)



119865 (1198661) minus 119865 (1198662) =[[[[[[[[[


00

]]]]]]]]]

(33)

Therefore


1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816

+ 2 (1 minus 1199061120590) 1205731205781 10038161003816100381610038161003816100381610038161003816(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816


120583 (120578 + 1)




(34)

where

119872 = max(2 (1 minus 1199061120590) 120573120587120583 (120578 + 1) 2 (1 minus 1199061120590) 120573120578120587

120583 2 (1 minus 1199061120590) 1205731205781120587120583 )

(35)

so that



119871 (119878 119862 119868 119877 1199061 1199062) = 1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)+ 11986122 11990622 (119905)

(37)



+ 120579119877] + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + 1205823 [120572119862

minus (120575 + 120583 + 1199062120574) 119868] + 1205824 [1199062120574119868 + (120596 + 1199062120574)119862+ 1199061120590119878 minus (120579 + 120583) 119877]

(38)



(39)



2isin119880

119869 (1199061 (119905) 1199062 (119905)) (40)


119869 (1199061 (119905) 1199062 (119905)) ge ]1 (1003816100381610038161003816119906110038161003816100381610038162 + 1003816100381610038161003816119906210038161003816100381610038162)1205882 minus ]2 (41)







1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583


119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)


+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732


(42)



where 119894 = 1 2(43)





1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)


119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]





=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























BetaEta

eta_1Thetau_1u_2mu

DeltaAlpha

OmegaGamma

Sigma


0 02 04 06 08 1minus15

minus10

minus5

0


log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Vaccination rate u1

log(ℛ

６４)


02 04 06 08 1minus15

minus10

minus5

0

Treatment rate u2

log(ℛ

６４)










119866119905 = 119870 (119866) + 119865 (119866) (31)

where

119866 =[[[[[[

119878 (119905)119862 (119905)119868 (119905)119877 (119905)

]]]]]]

119870 =[[[[[[


1199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]

119865 (119866) =[[[[[[[[[


119873 (119905)00

]]]]]]]]]

(32)



119865 (1198661) minus 119865 (1198662) =[[[[[[[[[


00

]]]]]]]]]

(33)

Therefore


1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816

+ 2 (1 minus 1199061120590) 1205731205781 10038161003816100381610038161003816100381610038161003816(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816


120583 (120578 + 1)




(34)

where

119872 = max(2 (1 minus 1199061120590) 120573120587120583 (120578 + 1) 2 (1 minus 1199061120590) 120573120578120587

120583 2 (1 minus 1199061120590) 1205731205781120587120583 )

(35)

so that



119871 (119878 119862 119868 119877 1199061 1199062) = 1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)+ 11986122 11990622 (119905)

(37)



+ 120579119877] + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + 1205823 [120572119862

minus (120575 + 120583 + 1199062120574) 119868] + 1205824 [1199062120574119868 + (120596 + 1199062120574)119862+ 1199061120590119878 minus (120579 + 120583) 119877]

(38)



(39)



2isin119880

119869 (1199061 (119905) 1199062 (119905)) (40)


119869 (1199061 (119905) 1199062 (119905)) ge ]1 (1003816100381610038161003816119906110038161003816100381610038162 + 1003816100381610038161003816119906210038161003816100381610038162)1205882 minus ]2 (41)







1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583


119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)


+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732


(42)



where 119894 = 1 2(43)





1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)


119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]





=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






















119866119905 = 119870 (119866) + 119865 (119866) (31)

where

119866 =[[[[[[

119878 (119905)119862 (119905)119868 (119905)119877 (119905)

]]]]]]

119870 =[[[[[[


1199061120590 (120596 + 1199062120574) 1199062120574 minus (120579 + 120583)

]]]]]]

119865 (119866) =[[[[[[[[[


119873 (119905)00

]]]]]]]]]

(32)



119865 (1198661) minus 119865 (1198662) =[[[[[[[[[


00

]]]]]]]]]

(33)

Therefore


1198781 (119905) 1198621 (119905)1198731 (119905) minus 1198782 (119905) 1198622 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816

+ 2 (1 minus 1199061120590) 1205731205781 10038161003816100381610038161003816100381610038161003816(1198781 (119905) 1198681 (119905)1198731 (119905) minus 1198782 (119905) 1198682 (119905)1198732 (119905) )10038161003816100381610038161003816100381610038161003816


120583 (120578 + 1)




(34)

where

119872 = max(2 (1 minus 1199061120590) 120573120587120583 (120578 + 1) 2 (1 minus 1199061120590) 120573120578120587

120583 2 (1 minus 1199061120590) 1205731205781120587120583 )

(35)

so that



119871 (119878 119862 119868 119877 1199061 1199062) = 1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)+ 11986122 11990622 (119905)

(37)



+ 120579119877] + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + 1205823 [120572119862

minus (120575 + 120583 + 1199062120574) 119868] + 1205824 [1199062120574119868 + (120596 + 1199062120574)119862+ 1199061120590119878 minus (120579 + 120583) 119877]

(38)



(39)



2isin119880

119869 (1199061 (119905) 1199062 (119905)) (40)


119869 (1199061 (119905) 1199062 (119905)) ge ]1 (1003816100381610038161003816119906110038161003816100381610038162 + 1003816100381610038161003816119906210038161003816100381610038162)1205882 minus ]2 (41)







1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583


119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)


+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732


(42)



where 119894 = 1 2(43)





1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)


119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]





=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






















(34)

where

119872 = max(2 (1 minus 1199061120590) 120573120587120583 (120578 + 1) 2 (1 minus 1199061120590) 120573120578120587

120583 2 (1 minus 1199061120590) 1205731205781120587120583 )

(35)

so that



119871 (119878 119862 119868 119877 1199061 1199062) = 1198601119862 (119905) + 1198602119868 (119905) + 11986112 11990621 (119905)+ 11986122 11990622 (119905)

(37)



+ 120579119877] + 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)119873minus (120572 + 120596 + 120583 + 1199062120574) 119862] + 1205823 [120572119862

minus (120575 + 120583 + 1199062120574) 119868] + 1205824 [1199062120574119868 + (120596 + 1199062120574)119862+ 1199061120590119878 minus (120579 + 120583) 119877]

(38)



(39)



2isin119880

119869 (1199061 (119905) 1199062 (119905)) (40)


119869 (1199061 (119905) 1199062 (119905)) ge ]1 (1003816100381610038161003816119906110038161003816100381610038162 + 1003816100381610038161003816119906210038161003816100381610038162)1205882 minus ]2 (41)







1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583


119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)


+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732


(42)



where 119894 = 1 2(43)





1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)


119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]





=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























1198891205822119889119905 = minus1198601 + 1205821 [(1 minus 1199061120590) 120573120578119878119873

minus (1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732 ] + 1205822 (120572 + 120596 + 120583


119873 ] minus 1205823120572 minus 1205824 (120596 + 1199062120574)


+ 1205822 [(1 minus 1199061120590) 120573119878 (120578119862 + 1205781119868)1198732


(42)



where 119894 = 1 2(43)





1205971198671205971199061 = 11986111199061 minus 1205821120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 minus 1205821120590119878+ 1205822120573119878 (120578119862 + 1205781119868)119878 + 119862 + 119868 + 119877 + 1205824120590119878 = 0

1205971198671205971199062 = 11986121199062 minus 1205822120574119862 minus 1205823120574119868 + 1205824120574119862 + 1205824120574119868 = 0(45)


119906lowast1= 11198611 [

120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878]





=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















=

0 if 11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] le 0

11198611 [120573119878 (120578119862 + 1205781119868)(119878 + 119862 + 119868 + 119877) (1205821 minus 1205822) + (1205821 minus 1205824) 120590119878] if 0 lt 11198611 [



119906lowast2 (119905) =

0 if120574 (1205822 minus 1205824) 119862 + 120574 (1205823 minus 1205824) 1198681198612 le 0



(47)









(48)








(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of
























(49)












1198891205824119889119905 = minus1205821120579 + 1205824 (120579 + 120583) with 120582119894 (119879) = 0 119894 = 1 2 3 4

(50)





0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















0 10 15 20 25 30Time

0

200

400

Carr

iers

5


ℛ６４ = 4225


0 10 15 20 25 30Time

0

200

400

600

ILL

indi

vidu

als

5


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

1

2

Vacc

inat

ion

cont

rol

profi

le

minus1

u1

0 5 10 15 20 25 30

ℛ６４ = 4225


Vacc

inat

ion

cont

rol

profi

le

Time

0

1

2

minus1

ℛ６４ = 4225

u1

0 10 20 30 40 50 60






Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















Time

0

200

400Ca

rrie

rs


0 5 10 15 20 25 30

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als

0 5 10 15 20 25 30


ℛ６４ = 4225


Time

0

200

400

Carr

iers


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

0

200

400

600

ILL

indi

vidu

als


0 10 20 30 40 50 60

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 10 20 30 40 50 60

u2

ℛ６４ = 4225


Time

02

04

06

08

Trea

tmen

t con

trol p

rofil

e

0 5 10 15 20 25 30

ℛ６４ = 4225

A1 = A2 = 1 B1 = 2 B2 = 16

A1 = A2 = 1 B1 = 2 B2 = 12

A1 = A2 = 1 B1 = 2 B2 = 8

A1 = A2 = 1 B1 = 2 B2 = 4

A1 = A2 = 1 B1 = 2 B2 = 2






Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















Time

0

2

4Ca

rrie

rs


times105

0 20 40 60 80

ℛ６４ = 42254


Time

0

2

4

6

ILL

indi

vidu

als

0 20 40 60 80

times105 ℛ６４ = 42254



Time

02

04

06

08

Con

trol p

rofil

e

0 20 40 60 80

Treatment u2

Vaccination u1

ℛ６４ = 42254



Time

times106

0 10 20 30 40 50 60 70 80

ℛ６４ = 4225435

3

25

2

15

1

05

0

Reco

vere

d



0

100

200

300

400

Carr

iers

0 10 15 20 25 30Time

With controls = 05



5

ℛ６４ = 4225

ℛ６４ = 4225ℛ６４ = 5520

ℛ６４ = 5520


0

200

400

600

800

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 05



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 5520

ℛ６４ = 5520






0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















0

50

100

150Re

cove

red

indi

vidu

als

Time0 10 20 4030


= 07

= 06

= 05

= 04

= 03

= 02


0

100

200

Reco

vere

d in

divi

dual

s

Time0 10 20 4030

= 07

= 06

= 05

= 04

= 03

= 02



Time

0

100

200

300

400

Carr

iers

0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4854


0

200

400

600

ILL

indi

vidu

als

Time0 10 15 20 25 305

With controls = 065



ℛ６４ = 4225

ℛ６４ = 4854

ℛ６４ = 4255ℛ６４ = 4854


Time

0

20

40

60

80

ILL

indi

vidu

als

0 10 15 20 25 305

= 1

= 09

= 08

= 07

= 06

= 05

= 01 = 04

= 03

= 02


0

100

200

300

Carr

iers

Decreasing order

10 15 20 25 30Time

0 5

= 1

= 09

= 08

= 07

= 06

= 05

= 04

= 03

= 01

= 02






05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















05

1

15

2 25

0 1 2 3 41

2

3

4

5

05

1

15

2

25

Loss

of i

mm

unity



05 4

2

2

3

0 0

05

1

15

2

25


Effec

tive r

epro

duct

ion

2

25

152

1

05

num

berℛ

６４



02

0304

05

06

07 0809

0 1 2 3 41

2

3

4

5

02

04

06

08

1


05 4

05

2

1

0 002

04

06

08

1

ℛ６４








5 Conclusion






References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























References

































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















mathematical modelling of bacterial meningitis...

Documents