research article modelling optimal control of cholera in …downloads.hindawi.com › journals ›...

Research ArticleModelling Optimal Control of Cholera inCommunities Linked by Migration

J B H Njagarah and F Nyabadza

Department of Mathematical Science University of Stellenbosch Private Bag X1 Matieland 7602 South Africa

Correspondence should be addressed to J B H Njagarah johnhatsonsunacza

Received 6 January 2015 Revised 30 May 2015 Accepted 1 June 2015

Academic Editor Chung-Min Liao

Copyright copy 2015 J B H Njagarah and F Nyabadza 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

A mathematical model for the dynamics of cholera transmission with permissible controls between two connected communitiesis developed and analysed The dynamics of the disease in the adjacent communities are assumed to be similar with the maindifferences only reflected in the transmission and disease related parameters This assumption is based on the fact that adjacentcommunities often have different living conditions and movement is inclined toward the community with better living conditionsCommunity specific reproduction numbers are given assuming movement of those susceptible infected and recovered betweencommunities We carry out sensitivity analysis of the model parameters using the Latin Hypercube Sampling scheme to ascertainthe degree of effect the parameters and controls have on progression of the infection Using principles from optimal controltheory a temporal relationship between the distribution of controls and severity of the infection is ascertained Our results indicatethat implementation of controls such as proper hygiene sanitation and vaccination across both affected communities is likely toannihilate the infection within half the time it would take through self-limitation In addition although an infectionmay still breakout in the presence of controls it may be up to 8 times less devastating when compared with the case when no controls are in place

1 Introduction

Cholera is a gastroenteritic infection contracted after con-suming the infectious dose or inoculum size of pathogenicVibrio cholerae The ability of V cholerae to invade thegastroenteritis proliferate and damage the host characterisestheir virulence [1] After invasion the capacity of the vibriosto cause symptoms in the host characterises their pathogenic-ity The transmission of vibrios from their natural habitatthe aquatic reservoirs is through consumption of Vibrioinfested water and seafood This route forms the primaryroute of transmission Cholera can also be transmitted fromone person to another through consumption of unhygienicor soiled food that may be infested with pathogenic vibriosfrom an infected person This route is characterised as thesecondary route of transmission For both of the mentionedroutes oral ingestion is the portal of entry of the pathogeninto the human body

Various controls for cholera have been recommended bytheWorldHealthOrganization (WHO)These controls range

from preventive measures to treatment protocols Preventivemeasures are aimed at averting new infections by prevent-ing the immunologically naive people from consuming orcoming into contact with the bacteria Treatment and controltarget the infected persons reducing the number of thoseinfected as well as reducing the case fatality rate Such preven-tive and control measures are explained in detail with regardto the metapopulation model of two connected communitiesas follows First an Oral Cholera Vaccine (OCV) recentlyrecommended by the World Health Organization (WHO)[2] is now in use Notably it was recently used duringthe cholera outbreak that affected Haiti after the 2010-2011earthquake [3] Secondly sanitation and hygiene reduce therate of pathogen ingestion Such controls include chemicaltreatment and boiling of water for cooking and drinkingand proper storage and preparation of foods to preventcontamination Thirdly for the infected case managementand treatment procedure which involves administering oralrehydration salts (ORS) to restore ion balance intravenous

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2015 Article ID 898264 12 pageshttpdxdoiorg1011552015898264

2 Computational and Mathematical Methods in Medicine

administration of fluids in serious cases and use of antibioticsare recommended

Mathematical modelling of cholera transmission dynam-ics dates back to the 1970s with the work done by Capassoand Paveri-Fontana [4] but more profound work has beendone in the last 1 to 2 decades Most of the work done so farfocuses on specific communities and not the metapopulationframework of the infection An extensive highlight of themetapopulation framework in epidemic models involvingboth cross community infections and exchange of popula-tions between communities is given in [5] but not a singlecase is specific to cholera Optimal control of the infectionhas only been studied recently in [6] when comparing choleratransmission between two separate Indian communities ofBogra and Calcutta It is also important to note that opti-mal control of cholera in single community settings wasstudied in [6 7] To the best of our knowledge no optimalcontrol study has been considered in the metapopulationframework for cholera transmission to dateHowever generalmodelling of control of epidemics in metapopulations wasrecently done using an SIS model [8] In this same workthe authors highlighted the likely difficulty andmathematicalintractability to be faced if a SIR or SIRS model is tobe used In this paper we consider a SIRS model withcontrols since cholera transmission typically follows suchdynamics In addition a model of cholera transmission in ametapopulation setting published recently [9] highlights thedetrimental effect of movement of epidemiologically naiveand infected individuals across communities In [9] howeverthe movement of recovered individuals was not consideredGiven the length of the modelling time we believe suchmovement is also possible and it is given consideration in thispaperWe aim to consider the dynamics of closed and isolatedcommunities in the presence of controls

The paper is organised as follows In Section 2 we presenta metapopulation model with permissible controls In thesame section vital mathematical analyses of the modelare given including the community specific reproductionnumbers followed by numerical results in Section 3 and inSection 5 we conclude the paper

2 Mathematical Model

In the model we consider two routes of transmission Theprimary route is characterised by consumption of Vibrioinfected water from aquatic sources The secondary route(also referred to as person-to-person transmission) is char-acterised by consumption of food that may be contaminatedwith vibrios from faecal matter The human population issubdivided into three compartments (119878 119868 and 119877) dependingon their status with respect to the infection The susceptible119878 are those who are at risk of contracting cholera eitherthrough contaminated water or by the secondary routeOnce infected individuals move into compartment 119868 Thosewho recover from the infection move into compartment 119877

at rates 1205741 and 1205742 for the first and second compartmentsrespectively Infected persons who show symptoms receiveoral rehydration salts to restore the ion balance and this

a1S1 a2S2 b1I1 b2I2 c1R1 c2R2

S2

S1 I1

I2

R1

R2

B1

B2

S1

S2

Λ1S1

Λ2S21205872

1205871

1205901I1

1205902I2

1205831S1

1205832S21205832R2

1205831R1

120596R1

120596R2

(1205832 + 1205752)I2

(1205831 + 1205751)I1

(1205741 + r)I1

(1205742 + r)I2

Figure 1 Flow diagram of disease dynamics in two communities

increases the recovery rate by a rate 119903This increased recoveryis assumed to be similar in both communities The infectionconfers some temporary immunity which wanes at a rate120596 In the infection dynamics the disease may be endemicor nonendemic In the former case the immunity of thoseinfectedwanes at a faster rate resulting in a SIRS type ofmodelcompared to a SIR model in the latter case It is assumedthat the time delay between consumption of Vibrio infectedfood or contaminated water and the commencement ofinfectiousness is negligible (see also [4]) The infectious doseor inoculum size 119870 must be consumed if an immunologicallynaive person is to be infected Given a high concentrationof the infectious dose (119870 = 106 cells per litre [10]) andthe relatively low probability of infection the dose-responserelationship is given by the Monod function 120573

119894119861119894(119870 + 119861

119894)

for 119894 = 1 2 where 119870 (the inoculum size) is concentrationof the pathogen required to cause 50 chance of infection1205731 and 1205732 are the specific contact rates of individuals inthe first and second communities with the aquatic reservoirsin their corresponding communities The use of a saturatedincidence function 120573

119894119861119894(119870 + 119861

119894) for 119870 gt 0 ensures

boundedness of the incidence rate and indicates that theincrease in incidence rate is more gradual than linear In themodel we denote vaccination at any time 119905 by V(119905) We notethat the immunologically naive individuals once vaccinatedmove straight to the compartment of those recovered Thelikelihood of infection is reduced by fractions 119906(119905) and 119898(119905)

related to contact with the aquatic reservoir and person-to-person contact respectively Here it is assumed that fraction119906(119905) uses clean treated water and fraction 119898(119905) practisesproper hygiene We assume that the implementation ofcontrols is by a government health organisation independentof the community specific health groups Therefore theimplementation of controls may not be greatly influenced bythe level of living condition in the adjacent communities butthe need to contain the infection

The flow diagram of disease progression in two commu-nities is given in Figure 1

Computational and Mathematical Methods in Medicine 3

The terms Λ 1 = (1minus 119906)(12057311198611(119870 + 1198611)) + (1minus 119898)12057211198681 andΛ 2 = (1 minus 119906)(12057321198612(119870 + 1198612)) + (1 minus 119898)12057221198682 are the incidencefunctions for the first and second communities respectivelyThese two terms describe the rate of apparition of new choleracases in the respective communities

The parameters 1198861 1198871 and 1198881 are the respective rates ofmovement of susceptible infected and recovered individualsfrom the first community to the second community Similarly1198862 1198872 and 1198882 are the rates of movement of the susceptibleinfected and recovered individuals from the second tothe first community The movement across communities isassumed to be instantaneous and therefore there is no changein epidemiological state during travel [5] When either thesusceptible infected or recovered individualsmove fromonecommunity to the other they follow the dynamics of thedestination community The secondary infections are onlygenerated from within a community and there is no crosscommunity infection

The populations in the first and second communities arereplenished through recruitments at rates 1205871 and 1205872 respec-tively Individuals in the two communities are howeversubjected to natural mortality at rates 1205831 and 1205832 respectivelyThe infected individuals in the two communities sufferdisease induced mortality at rates 1205751 and 1205752 respectivelyThe pathogen concentration in the aquatic environment isreplenished through shedding by the infected individuals atrates 1205901 and 1205902 for the two respective communities

The model system of equations for the two communitiesin presence of controls is given by

1198891198781119889119905

= 1205871 + 11988621198782 + 1205961198771 minus (1minus 119906)12057311198611

119870 + 11986111198781

minus (1minus 119898) 120572111986811198781 minus (1198861 + 1205831 + V) 1198781

(1a)

1198891198681119889119905

= (1minus 119906)12057311198611

119870 + 11986111198781 + (1minus 119898) 120572111986811198781 + 11988721198682

minus 11987611198681

(1b)

1198891198771119889119905

= V1198781 + (1205741 + 119903) 1198681 + 11988821198772 minus (1205831 + 120596 + 1198881) 1198771 (1c)

1198891198611119889119905

= 12059011198681 minus 11987621198611 (1d)

1198891198782119889119905

= 1205872 + 11988611198781 + 1205961198772 minus (1minus 119906)12057321198612

119870 + 11986121198782


(1e)

1198891198682119889119905

= (1minus 119906)12057321198612

119870 + 11986121198782 + (1minus 119898) 120572211986821198782 + 11988711198681

minus 11987631198682

(1f)

1198891198772119889119905

= V1198782 + (1205742 + 119903) 1198682 + 11988811198771 minus (1205832 + 120596 + 1198882) 1198772 (1g)

1198891198612119889119905

= 12059021198682 minus 11987641198612 (1h)

where 1198761 = (1205831 + 1205751 + 1205741 + 1198871 + 119903) 1198762 = (120583119901

minus 1198921)1198763 = (1205832 + 1205752 + 1205742 + 1198872 + 119903) and 1198764 = (120583

119901minus 1198922) Note

that all constants in the balance equations are nonnegativeIn addition1198762 and1198764 are positive indicating that in absenceof faecal contribution from infected individuals the bacteriacannot sustain themselves in the aquatic environment seealso [4] The initial conditions of the model are 11987810 11986810 1198771011986110 11987820 11986820 11987720 and 11986120 and are all nonnegative

21 Model Analysis The solutions to model system of(1a)ndash(1h) with nonnegative initial conditions are all non-negative and bounded Interested readers can investigatepositivity and boundedness of the solutions using themethodoutlined in ([5 Chapter 3])We note that if both communitiesare free of the infection no treatment control protocolmay be implemented However vaccination may still be inplace as will be indicated in the steady states Although thepermissible controls vary with time to analyse the steadystates we assume that the controls are constant therebyanalysing a nonautonomous system of differential equationssee also [7] Therefore the disease-free equilibrium E0 isgiven by

E0 = 119878lowast

1 0 0 0 119878lowast

2 0 0 0 (2)

where

1198781 =1205871 (1198862 + 1205832 + V) + 11988621205872

(1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1)

1198782 =1205872 (1198861 + 1205831 + V) + 11988611205871

(1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1)

(3)

The term Φ1 = 1198861(1198861 + 1205831 + V) sdot 1198862(1198862 + 1205832 + V) depictsthe proportion of susceptible individuals whomove back andforth in compartments 1198781 and 1198782 Therefore (1 minus Φ1) is thefraction of susceptible individuals who do not move fromtheir home compartments We note also that the proportionsof the susceptible 1198781 and 1198782 fall off as quadratic terms of thevaccination (in the denominator) and increase linearly (in thenumerator) Therefore the higher the vaccination coveragethe lower the fraction of the population that remains naive tothe infection

The community specific disease threshold numbers canbe obtained using the next generation matrix method out-lined in [20] When computing the disease thresholds how-ever we assume that the controls are constant so that themodel system of equations is nonautonomous

R01oc

=(1205871 (1198862 + 1205832 + V) + 11988621205872) ((1 minus 119898) 12057211198762119870 + (1 minus 119906) 12057311205901)

11987611198762 (1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1) 119870

(4)

for the first community and

R02oc

=(1205872 (1198861 + 1205831 + V) + 11988611205871) ((1 minus 119898) 12057221198764119870 + (1 minus 119906) 12057321205902)

11987631198764 (1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1) 119870

(5)


for the second community Then the model basic reproduc-tionR0oc is given by

R0oc = max R01ocR02oc (6)

Theorem 1 The disease-free steady state (2) of model of (1a)ndash(1h) is globally asymptotically stable whenever R0oc lt 1 andunstable otherwise

The proof of Theorem 1 can be given using the approachto the proof for Lemma 1 in [9] However in this case thecommunity specific reproduction numbers to be used are (4)and (5)

22 Optimal Control The general procedure of optimal con-trol process in an epidemiological model involves the follow-ing processes (1) identifying permissible controls applicableto the model (2) setting up the objective function withcontrols (3) constructing the Hamiltonian (4) evaluatingcostate variable (adjoint functions) and (5) identifying thethreshold controls that minimise the Hamiltonian Thisoptimal control minimisation procedure follows PontryaginrsquosMaximumMinimum principle [21]

The objective function for our minimisation problem isgiven by

J (119906 V 119898) = int

119879

0(12058511198681 + 12058521198682 + 120594119906

2+ 119910V2 + 119911119898

2) 119889119905 (7)

The coefficients 1205851 1205852 120594 119910 and 119911 are the coefficientsassociated with the costs over a finite period of time 119879 12058511198681and 12058521198682 indicate the cost associated with minimising theinfection in the first and second communities respectively120594 119910 and 119911 are relative cost weights for the respective controlmeasures We assume that the cost of controls is nonlinearhence the use of the quadratic terms The main goal is tominimise the number of those infected in both communitiesat the same time minimising the cost of implementing thecontrols In this respect we seek optimal controls 119906

lowast Vlowast and119898lowast such that

J (119906lowast

Vlowast 119898lowast

) = min119906V119898

J (119906 V 119898) | 119906 V 119898 isinU (8)

where

U = (119906 V 119898) | 119906 V 119898 [0 119879] 997888rarr [0 1] forall119906 V 119898 (9)

To evaluate the integral in (7) we use the Hamiltonianconstructed for the model system of equations with thenecessary adjoint functions Let 120582

1198781 1205821198681 1205821198771 1205821198611 1205821198782 1205821198682

1205821198772 and 120582

1198612be the adjoint functions or costate variables

associated with the evolution of the corresponding statevariables We multiply each of the adjoint functions with the

right side of the equation describing the evolution of eachstate variable

H = 12058511198681 + 12058521198682 + 1205941199062

+ 119910V2 + 1199111198982

+ 1205821198781

[1205871 + 11988621198782

+ 1205961198771 minus (1minus 119906) 12057311198611

119870 + 11986111198781 minus (1minus 119898) 120572111986811198781

minus (1198861 + 1205831 + V) 1198781] + 1205821198681

[(1minus 119906) 12057311198611

119870 + 11986111198781

+ (1minus 119898) 120572111986811198781 + 11988721198682 minus 11987611198681] + 1205821198771

[V1198781 + 12057411198681

+ 1199031198681 + 11988821198772 minus (1205831 + 120596 + 1198881) 1198771] + 1205821198611

[12059011198681

minus 11987621198611] + 1205821198782

[1205872 + 11988611198781 + 1205961198772

minus (1minus 119906) 12057321198612

119870 + 11986121198782 minus (1minus 119898) 120572211986821198782

minus (1198862 + 1205832) 1198782 minus V1198782] + 1205821198682

[(1minus 119906) 12057321198612

119870 + 11986121198782

+ (1minus 119898) 120572211986821198782 + 11988711198681 minus 11987631198682] + 1205821198772

[V1198782 + 12057421198682

+ 11988811198771 minus (1205832 + 120596 + 1198882) 1198772] + 1205821198772

[12059021198682 minus 11987641198612]

(10)

Theorem 2 (see [22]) Let 119906lowast Vlowast and 119898

lowast be the optimalcontrols for system (1a)ndash(1h) 119909

lowast the state space at equilibriumand 120582

(∙)positive semidefinite piecewise differentiable functions

for all 119905 If one supposes that for all 119905 isin [0 119879]

0 = H119906

(119905 119909lowast

119906lowast


120582 (119905))

= HV (119905 119909lowast

119906lowast


120582 (119905))

= H119898

(119905 119909lowast

119906lowast


120582 (119905))

(11)

then

H (119905 119909lowast

119906lowast

(119905) Vlowast (119905) 119898lowast

(119905) 120582 (119905))

le H (119905 119909lowast

119906 (119905) V (119905) 119898 (119905) 120582 (119905))

(12)

holds for the optimal controls 119906lowast Vlowast and 119898

lowast

To find the differential equations with respect to theassociated adjoint functions we differentiate theHamiltonianwith respect to each of the state variables and obtain

1198891205821198781

119889119905

= [(1minus 119906) 12057311198611

119870 + 1198611+ (1minus 119898) 12057211198681] (120582

1198781minus 1205821198681

)

+ (1198861 + 1205831 + V) 1205821198781

minus V1205821198771

minus 11988611205821198782


1198891205821198681

119889119905

= (1minus 119898) 12057211198781 (1205821198781

minus 1205821198681

) minus 1205851 + 11987611205821198681

minus (1205741 + 119903) 1205821198771

minus 12059011205821198611

minus 11988711205821198682

1198891205821198771

119889119905= (1205831 + 120596 + 1198881) 120582

1198771minus 1205961205821198781

minus 11988821205821198772

1198891205821198611

119889119905=

(1 minus 119906) 12057311198781119870

(119870 + 1198611)2 (120582

1198781minus 1205821198681

) + 11987621205821198611

1198891205821198782

119889119905

= [(1minus 119906) 12057311198612

119870 + 1198612+ (1minus 119898) 12057221198682] (120582

1198782minus 1205821198682

)

+ (1198862 + 1205832 + V) 1205821198782

minus V1205821198772

minus 11988621205821198781

1198891205821198682

119889119905

= (1minus 119898) 12057221198782 (1205821198782

minus 1205821198682

) minus 1205852 + 11987631205821198682

minus (1205742 + 119903) 1205821198772

minus 12059021205821198612

minus 11988721205821198681

1198891205821198772

119889119905= (1205832 + 120596 + 1198882) 120582

1198772minus 1205961205821198782

minus 11988811205821198771

1198891205821198611

119889119905=

(1 minus 119906) 12057321198782119870

(119870 + 1198612)2 (120582

1198782minus 1205821198682

) + 11987641205821198612

(13)

with transversality condition

1205821198781

(119879) = 1205821198681

(119879) = 1205821198771

(119879) = 1205821198611

(119879) = 1205821198782

(119879)

= 1205821198682

(119879) = 1205821198772

(119879) = 1205821198612

(119879) = 0

(14)

We note that for given transversality condition (14) thefollowing hold

1198891205821198781

119889119905= minus

120597H

1198891198781

1198891205821198612

119889119905= minus

120597H

1198891198612

(15)

The optimal controls are characterised by the followingexpressions

119906lowast

(119905) = max (0min ( (119905) 1))

Vlowast (119905) = max (0min (V (119905) 1))

119898lowast

(119905) = max (0min ( (119905) 1))

(16)

The standard control arguments on the controls [23] are suchthat

119906lowast

=

0 if le 0

if 0 lt lt 1

1 if ge 1

(17)

The upper bound of 119906lowast indicates that drinkable water is least

likely to be responsible for the oral faecal route of transmittingthe infection especially if water is chlorinated disinfected orboiled

Vlowast =

0 if V le 0

V if 0 lt V lt 1

1 if V ge 1

(18)

The value Vlowast = 1 is characteristic of a perfectly effectivevaccine Quite often cholera vaccines have low protectiveefficacy and have adverse effects associated with them Inone study by the Public Health Agency of Canada [24] theefficacy of the cholera and diarrhoeal vaccine was observedto range between 64 and 85 against Vibrio cholerae 01 ElTor Therefore the upper bound (Vlowast = 1) of this control maynot be necessarily attainable

119898lowast

=

0 if le 0

if 0 lt lt 1

1 if ge 1

(19)

Similarly 119898lowast

= 1 would signify no transmission of thepathogen through consumption of foods and that all food-stuffs are well prepared and hygienically stored and free fromcontamination Unfortunately this is not usually the case incommunities where cholera is endemic

Differentiating H with respect to each of the admissiblecontrols we obtain

120597H

120597119906= 2120594119906 + (120582

1198782minus 1205821198682

) 120573211986121198782

119870 + 1198612

+ (1205821198781

minus 1205821198681

) 120573111986111198781

119870 + 1198611

120597H

120597V= 2119910V+ (120582

1198771minus 1205821198781

) 1198781 + (1205821198772

minus 1205821198782

) 1198782

120597H

120597119898= 2119911119898 + (120582

1198781minus 1205821198681

) 120572111986811198781

+ (1205821198782

minus 1205821198682

) 120572211986821198782

(20)

The control characterisations V and of the optimalcontrols 119906

lowast Vlowast and 119898lowast are obtained from 120597H120597119906 =

120597H120597V = 120597H120597119898 = 0


=

(1205821198682

minus 1205821198782

) 1205732 (11986121198782 (119870 + 1198612)) + (1205821198681

minus 1205821198781

) 1205731 (11986111198781 (119870 + 1198611))

2120594

V =

(1205821198781

minus 1205821198771

) 1198781 + (1205821198782

minus 1205821198772

) 1198782

2119910

=

(1205821198681

minus 1205821198781

) 120572111986811198781 + (1205821198682

minus 1205821198782

) 120572211986821198782

2119911

(21)

It can be noted that the control characterisations are inverselyproportional to the associated costs The costs are thereforeinfluential in establishing the levels and trajectories of thecontrols

The optimal values of the controls are the ones anticipatedto contain the disease This is also in accordance with theother model parameters describing the dynamics of the dis-ease The incidence of the disease is also key in determiningthe trajectory of the state variables for the model withoutcontrols the model with controls and the control variablesover time

3 Numerical Simulations

To numerically solve the optimal control problem wemodifya code developed by Lenhart and Workman [22] to accom-modate the model system of (1a)ndash(1h) We note that thisoptimal control problem is classified as a quadratic program-ming problem since the controls are nonlinear and are of aquadratic form The state space variables are solved forwardin time and the adjoints associated with the state variablessolved backward in time The model system of equations isnumerically integrated using the ODE integration routine(ode45) a fourth-order Runge-Kutta Method in MATLAB

31 Parameter Estimation Due to lack of data the parametervalues used are estimated from published literature andsome intuitively selected from information related to choleratransmission dynamics The unit of the parameters is perday except for those where otherwise is indicated Nominalvalues are selected from within the specified ranges to caterfor the differences in the communities under study as wellas the disease transmission dynamics In the model weassume that the first community has better living conditionsand better facilities Therefore the transmission parametersare assumed to be relatively lower and the infection isassumed to be less devastating in the first community Thesecond community however is assumed to be slightly lessdisadvantaged and the infection more devastating Becauseof the difference in assumed living conditions migrationof individuals is assumed to be inclined towards the firstcommunity The parameters related to movement are setsuch that 1198862 gt 1198861 1198872 gt 1198871 and 1198882 gt 1198881 It is alsonecessary to determine the parameters related to the cost inthe objective function (7) The estimated cost of a vaccinedose per person from the study in [19] was about $5 This

gives an estimated cost of 50 rands within the South Africancontext We note that access to clean water and propersanitation are influenced by a number of factors ranging fromthe cost associated the standard of water treatment and thesophistication of the technology used According to the studyconducted in Johannesburg [18] if all households were toaccess clean in-house piped water with proper monitoringand in-house sewage connection with partial treatment itwould cost about US $1365 billion per year According toStatistics South Africa [25] the 2014 mid-year populationestimate in the country is 54 million This implies that theestimated cost of access to proper sanitation per personper day is approximately US $6925 This is equivalent toapproximately R80 In the case of infection the average costper percentage reduction in the infection in a communityis influenced by factors such as the cost of oral rehydrationsalts advertisement on social media transportation andremunerationGiven the difference of severity of the infectionin adjacent communities we assume that the cost is lower ina community with better facilitiesWe assume an average costof R200 in a more devastated community compared to R120in a community with improved facilities The estimated costsassociated with the controls per person are given in Table 2

4 Sensitivity Analysis

We analyse the sensitivity of the model community specificreproduction numbers (R01oc and R02oc) to the modelparameters using the Latin Hypercube Sampling (LHS)scheme with 1000 runs per simulation The LHS is a MonteCarlo stratified sampling method that allows us to simul-taneously obtain an unbiased estimate of the model outputfor a given set of input parameter values see [26] Thisstatistical sampling method for simultaneously carrying outsensitivity and uncertainty analysis was first applied in anepidemiological model by Blower and Dowlatabadi [27]The approach has since been employed in a number ofepidemiological models including [9 26]

The Partial Rank Correlation Coefficients (PRCCs) forthe full range of parameters are shown in the tornadoplots Figure 2(a) for the first community and Figure 2(b)for the second community The PRCCs for the communityspecific reproduction numbers in Figures 2(a) and 2(b) areproduced using the expressionsR01oc andR02ocThe processassociated with parameters with positive (negative) PRCCshas the potential of aggravating (reducing) the infection


minus08 minus06 minus04 minus02 0 02 04 06

1205731

1205721

1205741

1205901

a2a1

b1

120583p

g1

um

(a)


1205732

1205722

1205742

1205902

a1

a2

b2

120583p

g2

um

(b)

Figure 2 Partial RankCorrelationCoefficients (PRCCs) for the full range of parameters fromTable 1The control parameterswith the greatestpotential of containing the infection are improved hygiene and vaccination while the influx of susceptible individual in either communityincreases the likelihood of the disease outbreak

Table 1 Nominal values of estimated parameter values used in the simulations

Parameter Range Value Units Source1205871 120583 times 1198731 NA1205872 120583 times 1198732 NA1205731 0-1 000125 Assumed1205732 0-1 00125 Assumed119870 106ndash109 106 Cells Lminus1 [10]1205831 1205832 54 times 10

minus5 Dayminus1 [11 12]1205751 658 times 10minus4ndash00182 00125 Dayminus1 [6 13]1205752 658 times 10minus4ndash00182 0045 Dayminus1 [6 13]1205741 0031ndash0059 0045 Dayminus1 [6 14]1205742 0031ndash0059 0035 Dayminus1 [6 14]120583119901

1017ndash1083 106 Dayminus1 [10 15ndash17]1198921 1198922 073 Dayminus1 [10]1205721 0031ndash0059 0045 Dayminus1 Assumed1205722 0031ndash0059 0035 Dayminus1 Assumed1205901 1205902 10ndash100 500 CellsmLminus1 dayminus1 personminus1 [10]

Table 2 Costs associated with permissible controls

Coefficient Cost value Source1205851 R200 per percentage reduction in 1198681 Assumed1205852 R120 per percentage reduction in 1198682 Assumed119883 R80 per (level of water related treatment)2 [18]119884 R50 per (vaccination rate)2 [19]119885 R50 per (hand hygiene related infection reductions)2 [18]

when increased From the equations of the communityspecific reproduction numbersR01oc andR02oc the controlsparameters are evident From the tornado plots in Figure 2for community specific disease thresholds it is evident thatthe controls related to increasing access to clean waterand enhancing proper hand hygiene reducing the diseasethreshold are all associated with negative PRCCs Thereforewith enhancement of processes related to such controls

the subsequent numbers of those infected will be less thanthat of their predecessors hence reducing the severity ofthe infection It should be noted that of all the consideredcontrols vaccination and improvement in hygiene havethe greatest potential of easily containing the infection Inaddition emigration (immigration) of infected individualsout of (into) a community reduces (increases) the diseaseburden in that community


0

0002

0004

0006

0008

001

0012

0014

0016

0 20 40 60 80 100Time (days)

Infe

cted

gro

up in

the fi

rst c

omm

unity

Without controlsWith controls

(a)

0

002

004

006

008

01

012

014

Infe

cted

gro

up in

the s

econ

d co

mm

unity

0 20 40 60 80 100Time (days)


(b)

Figure 3 The infectious 1198681 (a) and 1198682 (b) in presence of controls (dashed line) and in absence of controls (solid line)

41 Numerical Results In the model simulations the pop-ulation is not constant and the initial conditions are set asproportions as follows 11987810 = 06 11986810 = 00002 11987710 = 000211986110 = 1 times 1011 11987820 = 08 11986820 = 0003 11987720 = 000211986120 = 1 times 1011

The algorithm for simulating the system using theldquoforward-backward sweep methodrdquo is adopted from [22]We outline the steps followed during the simulation forconvenience of the reader

Step 1 The time interval over which the system is to beintegrated is subdivided into 119873 equal subintervalsThe initialguesses of piecewise-constant controls 1199060(119905) V0(119905) and 1198980(119905)

on the interval 119905 isin [119905119894 119905119894+1] for 119894 = 0 1 119873 are assumed

Step 2 Using the assumed controls 1199060 V0 and 1198980 and theinitial conditions (0) = (11987810 11986810 11987710 11986110 11987820 11986820 11987720 11986120)

of the model the system is numerically integrated forwardin time [0 119879] according to its system of differential equationsin the optimality system

Step 3 Using the assumed controls 1199060(119905) V0(119905) and 1198980(119905)

and values of the costate system (119905) in accordance withits system of differential equations with the transversalitycondition is integrated backward in time [119879 0]

Step 4 The controls vector is updated by entering thenew state vector and the costate values in the optimalcharacterisations

Step 5 Convergence of the iterations is then checked byascertaining whether successive iterations are negligibly closefor some specified level of tolerance Once this is achievedthen optimal control is said to have been achieved theiteration is stopped and the output values are recorded assolutions Otherwise the algorithm is set back to Step 2

411 Isolated Communities in Presence of Controls The com-munities are assumed to be isolated in this case if there isno movement or exchange of individuals between adjacentcommunities However we allow for recruitment from otherareas and therefore the recruitment parameters are nonzeroIn the simulations we use the parameters in Table 1 andset all the parameters describing movement to zero Thetrajectories of the infected populations in both communitiesare given in Figure 3

When all the controls are implemented our resultssuggest that the time taken for the disease to be containedmay be approximately half of the time required to contain thedisease without controls that is by self-limitation Althoughthe infection is less devastating in the first community it staysrelatively longer in the community compared to the secondcommunity (cf the solid lines in Figures 3(a) and 3(b)) It isplausible that due to a high transmission in the second com-munity the disease depletes the susceptible pool at a muchfaster rate and in about 100 days there are virtually no moresusceptible people to infect for the chosen parameter valuesIn the model with controls however the disease is containedat approximately the same time in both communities Takinga closer look at dashed curves in Figures 3(a) and 3(b) oneobserves that the disease is contained slightly earlier in thefirst than the second community for the same level of effortThis scenario is a typical comparison of infections with highand low incidence

The trajectories of the state space variables for the modelwith control the adjoint and the control values convergearound relatively the same time The iteration process termi-nates when these different variables converge sufficiently

412 Connected Communities in Presence of Controls Whenthe communities are connected the parameters describingmovement are all nonzero and there is exchange of individu-als between the communities


0

001

002

003

004

005

006In

fect

ed g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)


(a) Infected group in the first community

0 20 40 60 80 100Time (days)


Infe

cted

gro

up in

the s

econ

d co

mm

unity

0

001

002

003

004

005

008

007

006

(b) Infected group in the second community

Figure 4 Infected groups in the two communities with and without controls

0 20 40 60 80 100Time (days)

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Profi

le o

f dom

estic

wat

er tr

eatm

entu

(a)

0 20 40 60 80 1000

001

002

003

004

005

006

007

008

009

01

Time (days)

Profi

le o

f vac

cina

tion

(b)

Figure 5 Profiles of the two most significant controls

Movement of susceptible and infected individuals isassociated with not only an increase in severity but also along duration of the infection in the two communities seeFigures 3 and 4 In presence of controls however there isno observed significant difference in the disease severity andduration whether in isolated or connected community casesFor comparison see dashed lines in Figures 3 and 4

The profiles of the controls investigated in the model areindicated in Figures 5 and 6The profiles indicate the need formore fundamental action at the beginning of the epidemicif the disease is to be contained All the three indicatedcontrols reduce the likelihood of susceptible individuals

getting infected The profiles for access to clean water andvaccination Figures 5(a) and 5(b) respectively indicate theimportance of the two controls in containing the infection

The profile for the control related to hygiene is deemedthe least significant see Figure 6 However we reiterate thatno effort is insignificant as long as it leads to reduction in thenumber of infections

The susceptible groups in both communities are initiallycharacterised by depletion followed by an increase Figure 7We note that the rapid depletion of the susceptible populationis due to vaccination which was assumed to be exponentialin the model As a result the recovered group is replenished


0 20 40 60 80 1000

1

2

3

4

5

6

7

8

Time (days)

times10minus4

Profi

le o

f hyg

iene

relat

ed co

ntro

lm

Figure 6 Profile of hygiene related control

01

02

03

04

05

06

07

08

09

Susc

eptib

le g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)


(a) Susceptible group in the first community

Susc

eptib

le g

roup

in th

e sec

ond

com

mun

ity

01

02

03

04

05

06

07

08

0 20 40 60 80 100Time (days)


(b) Susceptible group in the second community

Figure 7 Susceptible populations in the two connected communities

and reduces when the control is halted Figure 8 By the endof the control period the community is comprised of thosesusceptible and recovered Once the infection is contained itis assumed that the small group of people who could still becarrying the disease may pose a low threat of reintroducingit

Although the general trajectories of the susceptiblegroups in the two communities are similar there is a strikingdifference that ought to be highlighted Even though there isan observed initial increase in the trajectory of the susceptiblegroup in the first community that of the second communityonly decreases This trend is attributed to movement whichis inclined towards the first community due to the better

living conditions assumed as well as the high incidence ofthe disease in the second community

5 Conclusion

In this paper a metapopulation model for cholera trans-mission characterised by exchange of individuals betweencommunities is analysed The community specific diseasethresholds are given and their importance determining theexistence and stability of equilibria of the model is high-lighted The conditions for annihilating the infection basedon specific controls while keeping other parameters constanthave been indicated Our numerical results indicate that in


0

Reco

vere

d gr

oup

in th

e firs

t com

mun

ity

01

02

03

04

05

06

07

0 20 40 60 80 100Time (days)


(a) Recovered group in the first community

005

015

025

035

045

Reco

vere

d gr

oup

in th

e sec

ond

com

mun

ity

0

01

02

03

04

05

0 20 40 60 80 100Time (days)


(b) Recovered group in the second community

Figure 8 Recovered group in the two connected communities

presence of controls in case of an outbreak of the diseasethe infection may be 8 times less devastating when comparedto the case without controls In addition the duration ofthe infection in the community cannot be more than halfthe time it would in absence of controls From the modelresults we assert that implementation of preventive mea-sures and treatment control protocols reduces the likelihoodof recurrence of the infection in the communities Therecurrence of the infection is typical in the model withoutcontrols presented in [9] characterised by semiperiodic andsynchronous fluctuations of the involved subpopulations

The model presented in this paper is not short ofshortfalls We acknowledge the fact that controls acrosscommunities may not be uniform as assumed in the modelsControls efforts are often influenced by demographics ofthe community (including ageing which is associated withimmune dysfunction) as well as the behaviour of individualsinvolved in the disease transmission dynamics In additionpolitics greatly affects the minority groups as well as socialinequality which impacts the vulnerable groups urbanizationassociatedwith overcrowding inadequate infrastructure andeconomic stochasticity among others All these affect thebureaucracy associated with implementation of controlsputting up sustainable solutions resulting in heterogeneityThe model also assumes a negligible time difference betweeninfection and becoming infectious as well as apparitionof symptoms Although such simplifying assumptions maybe presumed feasible their investigation is necessary toascertain the effect on the epidemic size

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge with gratitude the support fromtheDepartment ofMathematical Sciences Faculty of ScienceUniversity of Stellenbosch and the African Institute forMathematical Sciences (AIMS)

References

[1] A Kramer M Kretzschmar and K Krickeberg Eds ModernInfectious Disease Epidemiology Concepts Methods Mathemat-ical Models and Public Health Springer 2010

[2] WHO ldquoPrevention and control of cholera outbreaksWHOpol-icy and recommendations vaccinesrdquo 2013 httpwwwwhointcholeratechnicalpreventioncontrolenindex5html

[3] K A Date A Vicari T B Hyde et al ldquoQualitative analysis of acholera bactriophage modelrdquo Emerging Infectious Diseases vol17 pp 2105ndash2112 2011

[4] V Capasso and S L Paveri-Fontana ldquoA Mathematical modelfor the 1973 cholera epidemic in the European Mediterraneanregionrdquo Revue drsquoepidemiologie et de Sante Publique vol 27 no2 pp 121ndash132 1979

[5] Z Ma Y Zhou and J Wu Eds Modelling and Dynamics ofInfectious Diseases Higher Eductation Press World Scientific2009

[6] R L Miller Neilan E Schaefer H Gaff K R Fister and SLenhart ldquoModeling optimal intervention strategies for cholerardquoBulletin of Mathematical Biology vol 72 no 8 pp 2004ndash20182010

[7] J Wang and C Modnak ldquoModeling cholera dynamics withcontrolsrdquo Canadian Applied Mathematics Quarterly vol 19 no3 pp 255ndash273 2011

[8] R E Rowthorn R Laxminarayan and C A Gilligan ldquoOptimalcontrol of epidemics in metapopulationsrdquo Journal of the RoyalSociety Interface vol 6 no 41 pp 1135ndash1144 2009


[9] J B H Njagarah and F Nyabadza ldquoAmetapopulationmodel forcholera transmission dynamics between communities linked bymigrationrdquoApplied Mathematics and Computation vol 241 pp317ndash331 2014

[10] C T Codeco ldquoEndemic and epidemic dynamics of cholera therole of the aquatic reservoirrdquo BMC Infectious Diseases vol 1article 1 2001

[11] World Bank Disease and Mortality in Sub-Saharan AfricaWorld Bank Washington DC USA 2006

[12] Statistic South Africa ldquoMid-year population estimates 2010rdquohttpbeta2statssagovzapage id=1854ampPPN=P0302ampSCH=4696

[13] J Sepulveda H Gomez-Dantes andM Bronfman ldquoCholera intheAmericas an overviewrdquo Infection vol 20 no 5 pp 243ndash2481992

[14] A A King E L Ionides M Pascual and M J BoumaldquoInapparent infections and cholera dynamicsrdquo Nature vol 454no 7206 pp 877ndash880 2008

[15] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[16] Z Mukandavire A Tripathi C Chiyaka G Musuka FNyabadza and H G Mwambi ldquoModelling and analysis of theintrinsic dynamics of cholerardquo Differential Equations and Dyn-amical Systems vol 19 no 3 pp 253ndash265 2011

[17] P M Munro and R R Colwell ldquoFate of Vibrio cholerae O1 inseawater microcosmsrdquoWater Research vol 30 no 1 pp 47ndash501996

[18] WHO Costs and Benefits of Water and Sanitation Improve-ments at the Global Level (Evaluation of The) World HealthOrganization Geneva Switzerland 2015 httpwwwwhointwater sanitation healthwsh0404summaryen

[19] S-Y Kim Y Choi P R Mason S Rusakaniko and S JGoldie ldquoPotential impact of reactive vaccination in controllingcholera outbreaks An exploratory analysis using a Zimbabweanexperiencerdquo South African Medical Journal vol 101 no 9 pp659ndash664 2011

[20] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[21] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[22] S Lenhart and J T Workman Optimal Control Applied toBiological Problems Chapman amp HallCRC Boca Raton FlaUSA 2007

[23] K O Okosun and O D Makinde ldquoModelling the impact ofdrug resistance in malaria transmission and its optimal controlanalysisrdquo International Journal of Physical Sciences vol 6 no 28pp 6479ndash6487 2011

[24] PHAC Canadian Immunization Guide Cholera and Entero-toxigenic Escherichia Coli (ETEC) Travellersrsquo Diarrhea Vac-cine 2006 httpwwwphac-aspcgccapublicatcig-gcip04-cho-engphpa3

[25] STATSSA Mid-Year Population Estimates Statistics SouthAfrica 2015 httpwwwstatssagovzapublicationsP0302P03022014pdf

[26] F Nyabadza J B Njagarah and R J Smith ldquoModelling thedynamics of crystal meth (lsquotikrsquo) abuse in the presence of drug-supply chains in SouthAfricardquoBulletin ofMathematical Biologyvol 75 no 1 pp 24ndash48 2013

[27] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of complex models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


administration of fluids in serious cases and use of antibioticsare recommended

Mathematical modelling of cholera transmission dynam-ics dates back to the 1970s with the work done by Capassoand Paveri-Fontana [4] but more profound work has beendone in the last 1 to 2 decades Most of the work done so farfocuses on specific communities and not the metapopulationframework of the infection An extensive highlight of themetapopulation framework in epidemic models involvingboth cross community infections and exchange of popula-tions between communities is given in [5] but not a singlecase is specific to cholera Optimal control of the infectionhas only been studied recently in [6] when comparing choleratransmission between two separate Indian communities ofBogra and Calcutta It is also important to note that opti-mal control of cholera in single community settings wasstudied in [6 7] To the best of our knowledge no optimalcontrol study has been considered in the metapopulationframework for cholera transmission to dateHowever generalmodelling of control of epidemics in metapopulations wasrecently done using an SIS model [8] In this same workthe authors highlighted the likely difficulty andmathematicalintractability to be faced if a SIR or SIRS model is tobe used In this paper we consider a SIRS model withcontrols since cholera transmission typically follows suchdynamics In addition a model of cholera transmission in ametapopulation setting published recently [9] highlights thedetrimental effect of movement of epidemiologically naiveand infected individuals across communities In [9] howeverthe movement of recovered individuals was not consideredGiven the length of the modelling time we believe suchmovement is also possible and it is given consideration in thispaperWe aim to consider the dynamics of closed and isolatedcommunities in the presence of controls

The paper is organised as follows In Section 2 we presenta metapopulation model with permissible controls In thesame section vital mathematical analyses of the modelare given including the community specific reproductionnumbers followed by numerical results in Section 3 and inSection 5 we conclude the paper

2 Mathematical Model

In the model we consider two routes of transmission Theprimary route is characterised by consumption of Vibrioinfected water from aquatic sources The secondary route(also referred to as person-to-person transmission) is char-acterised by consumption of food that may be contaminatedwith vibrios from faecal matter The human population issubdivided into three compartments (119878 119868 and 119877) dependingon their status with respect to the infection The susceptible119878 are those who are at risk of contracting cholera eitherthrough contaminated water or by the secondary routeOnce infected individuals move into compartment 119868 Thosewho recover from the infection move into compartment 119877

at rates 1205741 and 1205742 for the first and second compartmentsrespectively Infected persons who show symptoms receiveoral rehydration salts to restore the ion balance and this

a1S1 a2S2 b1I1 b2I2 c1R1 c2R2

S2

S1 I1

I2

R1

R2

B1

B2

S1

S2

Λ1S1

Λ2S21205872

1205871

1205901I1

1205902I2

1205831S1

1205832S21205832R2

1205831R1

120596R1

120596R2

(1205832 + 1205752)I2

(1205831 + 1205751)I1

(1205741 + r)I1

(1205742 + r)I2

Figure 1 Flow diagram of disease dynamics in two communities

increases the recovery rate by a rate 119903This increased recoveryis assumed to be similar in both communities The infectionconfers some temporary immunity which wanes at a rate120596 In the infection dynamics the disease may be endemicor nonendemic In the former case the immunity of thoseinfectedwanes at a faster rate resulting in a SIRS type ofmodelcompared to a SIR model in the latter case It is assumedthat the time delay between consumption of Vibrio infectedfood or contaminated water and the commencement ofinfectiousness is negligible (see also [4]) The infectious doseor inoculum size 119870 must be consumed if an immunologicallynaive person is to be infected Given a high concentrationof the infectious dose (119870 = 106 cells per litre [10]) andthe relatively low probability of infection the dose-responserelationship is given by the Monod function 120573

119894119861119894(119870 + 119861

119894)

for 119894 = 1 2 where 119870 (the inoculum size) is concentrationof the pathogen required to cause 50 chance of infection1205731 and 1205732 are the specific contact rates of individuals inthe first and second communities with the aquatic reservoirsin their corresponding communities The use of a saturatedincidence function 120573

119894119861119894(119870 + 119861

119894) for 119870 gt 0 ensures

boundedness of the incidence rate and indicates that theincrease in incidence rate is more gradual than linear In themodel we denote vaccination at any time 119905 by V(119905) We notethat the immunologically naive individuals once vaccinatedmove straight to the compartment of those recovered Thelikelihood of infection is reduced by fractions 119906(119905) and 119898(119905)

related to contact with the aquatic reservoir and person-to-person contact respectively Here it is assumed that fraction119906(119905) uses clean treated water and fraction 119898(119905) practisesproper hygiene We assume that the implementation ofcontrols is by a government health organisation independentof the community specific health groups Therefore theimplementation of controls may not be greatly influenced bythe level of living condition in the adjacent communities butthe need to contain the infection

The flow diagram of disease progression in two commu-nities is given in Figure 1






1198891198781119889119905

= 1205871 + 11988621198782 + 1205961198771 minus (1minus 119906)12057311198611

119870 + 11986111198781


(1a)

1198891198681119889119905

= (1minus 119906)12057311198611

119870 + 11986111198781 + (1minus 119898) 120572111986811198781 + 11988721198682

minus 11987611198681

(1b)

1198891198771119889119905

= V1198781 + (1205741 + 119903) 1198681 + 11988821198772 minus (1205831 + 120596 + 1198881) 1198771 (1c)

1198891198611119889119905

= 12059011198681 minus 11987621198611 (1d)

1198891198782119889119905

= 1205872 + 11988611198781 + 1205961198772 minus (1minus 119906)12057321198612

119870 + 11986121198782


(1e)

1198891198682119889119905

= (1minus 119906)12057321198612

119870 + 11986121198782 + (1minus 119898) 120572211986821198782 + 11988711198681

minus 11987631198682

(1f)

1198891198772119889119905

= V1198782 + (1205742 + 119903) 1198682 + 11988811198771 minus (1205832 + 120596 + 1198882) 1198772 (1g)

1198891198612119889119905

= 12059021198682 minus 11987641198612 (1h)

where 1198761 = (1205831 + 1205751 + 1205741 + 1198871 + 119903) 1198762 = (120583119901

minus 1198921)1198763 = (1205832 + 1205752 + 1205742 + 1198872 + 119903) and 1198764 = (120583

119901minus 1198922) Note



E0 = 119878lowast

1 0 0 0 119878lowast

2 0 0 0 (2)

where

1198781 =1205871 (1198862 + 1205832 + V) + 11988621205872

(1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1)

1198782 =1205872 (1198861 + 1205831 + V) + 11988611205871

(1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1)

(3)



R01oc

=(1205871 (1198862 + 1205832 + V) + 11988621205872) ((1 minus 119898) 12057211198762119870 + (1 minus 119906) 12057311205901)

11987611198762 (1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1) 119870

(4)


R02oc

=(1205872 (1198861 + 1205831 + V) + 11988611205871) ((1 minus 119898) 12057221198764119870 + (1 minus 119906) 12057321205902)

11987631198764 (1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1) 119870

(5)








J (119906 V 119898) = int

119879

0(12058511198681 + 12058521198682 + 120594119906

2+ 119910V2 + 119911119898

2) 119889119905 (7)



J (119906lowast


) = min119906V119898

J (119906 V 119898) | 119906 V 119898 isinU (8)

where

U = (119906 V 119898) | 119906 V 119898 [0 119879] 997888rarr [0 1] forall119906 V 119898 (9)


1198781 1205821198681 1205821198771 1205821198611 1205821198782 1205821198682

1205821198772 and 120582




H = 12058511198681 + 12058521198682 + 1205941199062

+ 119910V2 + 1199111198982

+ 1205821198781

[1205871 + 11988621198782

+ 1205961198771 minus (1minus 119906) 12057311198611

119870 + 11986111198781 minus (1minus 119898) 120572111986811198781

minus (1198861 + 1205831 + V) 1198781] + 1205821198681

[(1minus 119906) 12057311198611

119870 + 11986111198781

+ (1minus 119898) 120572111986811198781 + 11988721198682 minus 11987611198681] + 1205821198771

[V1198781 + 12057411198681

+ 1199031198681 + 11988821198772 minus (1205831 + 120596 + 1198881) 1198771] + 1205821198611

[12059011198681

minus 11987621198611] + 1205821198782

[1205872 + 11988611198781 + 1205961198772

minus (1minus 119906) 12057321198612

119870 + 11986121198782 minus (1minus 119898) 120572211986821198782

minus (1198862 + 1205832) 1198782 minus V1198782] + 1205821198682

[(1minus 119906) 12057321198612

119870 + 11986121198782

+ (1minus 119898) 120572211986821198782 + 11988711198681 minus 11987631198682] + 1205821198772

[V1198782 + 12057421198682

+ 11988811198771 minus (1205832 + 120596 + 1198882) 1198772] + 1205821198772

[12059021198682 minus 11987641198612]

(10)






0 = H119906

(119905 119909lowast

119906lowast


120582 (119905))

= HV (119905 119909lowast

119906lowast


120582 (119905))

= H119898

(119905 119909lowast

119906lowast


120582 (119905))

(11)

then

H (119905 119909lowast

119906lowast

(119905) Vlowast (119905) 119898lowast

(119905) 120582 (119905))

le H (119905 119909lowast

119906 (119905) V (119905) 119898 (119905) 120582 (119905))

(12)


lowast


1198891205821198781

119889119905

= [(1minus 119906) 12057311198611

119870 + 1198611+ (1minus 119898) 12057211198681] (120582

1198781minus 1205821198681

)

+ (1198861 + 1205831 + V) 1205821198781

minus V1205821198771

minus 11988611205821198782


1198891205821198681

119889119905

= (1minus 119898) 12057211198781 (1205821198781

minus 1205821198681

) minus 1205851 + 11987611205821198681

minus (1205741 + 119903) 1205821198771

minus 12059011205821198611

minus 11988711205821198682

1198891205821198771

119889119905= (1205831 + 120596 + 1198881) 120582

1198771minus 1205961205821198781

minus 11988821205821198772

1198891205821198611

119889119905=

(1 minus 119906) 12057311198781119870

(119870 + 1198611)2 (120582

1198781minus 1205821198681

) + 11987621205821198611

1198891205821198782

119889119905

= [(1minus 119906) 12057311198612

119870 + 1198612+ (1minus 119898) 12057221198682] (120582

1198782minus 1205821198682

)

+ (1198862 + 1205832 + V) 1205821198782

minus V1205821198772

minus 11988621205821198781

1198891205821198682

119889119905

= (1minus 119898) 12057221198782 (1205821198782

minus 1205821198682

) minus 1205852 + 11987631205821198682

minus (1205742 + 119903) 1205821198772

minus 12059021205821198612

minus 11988721205821198681

1198891205821198772

119889119905= (1205832 + 120596 + 1198882) 120582

1198772minus 1205961205821198782

minus 11988811205821198771

1198891205821198611

119889119905=

(1 minus 119906) 12057321198782119870

(119870 + 1198612)2 (120582

1198782minus 1205821198682

) + 11987641205821198612

(13)


1205821198781

(119879) = 1205821198681

(119879) = 1205821198771

(119879) = 1205821198611

(119879) = 1205821198782

(119879)

= 1205821198682

(119879) = 1205821198772

(119879) = 1205821198612

(119879) = 0

(14)


1198891205821198781

119889119905= minus

120597H

1198891198781

1198891205821198612

119889119905= minus

120597H

1198891198612

(15)


119906lowast

(119905) = max (0min ( (119905) 1))

Vlowast (119905) = max (0min (V (119905) 1))

119898lowast

(119905) = max (0min ( (119905) 1))

(16)


119906lowast

=

0 if le 0

if 0 lt lt 1

1 if ge 1

(17)



Vlowast =

0 if V le 0

V if 0 lt V lt 1

1 if V ge 1

(18)


119898lowast

=

0 if le 0

if 0 lt lt 1

1 if ge 1

(19)




120597H

120597119906= 2120594119906 + (120582

1198782minus 1205821198682

) 120573211986121198782

119870 + 1198612

+ (1205821198781

minus 1205821198681

) 120573111986111198781

119870 + 1198611

120597H

120597V= 2119910V+ (120582

1198771minus 1205821198781

) 1198781 + (1205821198772

minus 1205821198782

) 1198782

120597H

120597119898= 2119911119898 + (120582

1198781minus 1205821198681

) 120572111986811198781

+ (1205821198782

minus 1205821198682

) 120572211986821198782

(20)



120597H120597V = 120597H120597119898 = 0


=

(1205821198682

minus 1205821198782

) 1205732 (11986121198782 (119870 + 1198612)) + (1205821198681

minus 1205821198781

) 1205731 (11986111198781 (119870 + 1198611))

2120594

V =

(1205821198781

minus 1205821198771

) 1198781 + (1205821198782

minus 1205821198772

) 1198782

2119910

=

(1205821198681

minus 1205821198781

) 120572111986811198781 + (1205821198682

minus 1205821198782

) 120572211986821198782

2119911

(21)












1205731

1205721

1205741

1205901

a2a1

b1

120583p

g1

um

(a)


1205732

1205722

1205742

1205902

a1

a2

b2

120583p

g2

um

(b)











0

0002

0004

0006

0008

001

0012

0014

0016

0 20 40 60 80 100Time (days)

Infe

cted

gro

up in

the fi

rst c

omm

unity


(a)

0

002

004

006

008

01

012

014

Infe

cted

gro

up in

the s

econ

d co

mm

unity

0 20 40 60 80 100Time (days)


(b)

















0

001

002

003

004

005

006In

fect

ed g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



0 20 40 60 80 100Time (days)


Infe

cted

gro

up in

the s

econ

d co

mm

unity

0

001

002

003

004

005

008

007

006



0 20 40 60 80 100Time (days)

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Profi

le o

f dom

estic

wat

er tr

eatm

entu

(a)

0 20 40 60 80 1000

001

002

003

004

005

006

007

008

009

01

Time (days)

Profi

le o

f vac

cina

tion

(b)








0 20 40 60 80 1000

1

2

3

4

5

6

7

8

Time (days)

times10minus4

Profi

le o

f hyg

iene

relat

ed co

ntro

lm


01

02

03

04

05

06

07

08

09

Susc

eptib

le g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



Susc

eptib

le g

roup

in th

e sec

ond

com

mun

ity

01

02

03

04

05

06

07

08

0 20 40 60 80 100Time (days)







5 Conclusion



0

Reco

vere

d gr

oup

in th

e firs

t com

mun

ity

01

02

03

04

05

06

07

0 20 40 60 80 100Time (days)



005

015

025

035

045

Reco

vere

d gr

oup

in th

e sec

ond

com

mun

ity

0

01

02

03

04

05

0 20 40 60 80 100Time (days)








Acknowledgments


References


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















1198891198781119889119905

= 1205871 + 11988621198782 + 1205961198771 minus (1minus 119906)12057311198611

119870 + 11986111198781


(1a)

1198891198681119889119905

= (1minus 119906)12057311198611

119870 + 11986111198781 + (1minus 119898) 120572111986811198781 + 11988721198682

minus 11987611198681

(1b)

1198891198771119889119905

= V1198781 + (1205741 + 119903) 1198681 + 11988821198772 minus (1205831 + 120596 + 1198881) 1198771 (1c)

1198891198611119889119905

= 12059011198681 minus 11987621198611 (1d)

1198891198782119889119905

= 1205872 + 11988611198781 + 1205961198772 minus (1minus 119906)12057321198612

119870 + 11986121198782


(1e)

1198891198682119889119905

= (1minus 119906)12057321198612

119870 + 11986121198782 + (1minus 119898) 120572211986821198782 + 11988711198681

minus 11987631198682

(1f)

1198891198772119889119905

= V1198782 + (1205742 + 119903) 1198682 + 11988811198771 minus (1205832 + 120596 + 1198882) 1198772 (1g)

1198891198612119889119905

= 12059021198682 minus 11987641198612 (1h)

where 1198761 = (1205831 + 1205751 + 1205741 + 1198871 + 119903) 1198762 = (120583119901

minus 1198921)1198763 = (1205832 + 1205752 + 1205742 + 1198872 + 119903) and 1198764 = (120583

119901minus 1198922) Note



E0 = 119878lowast

1 0 0 0 119878lowast

2 0 0 0 (2)

where

1198781 =1205871 (1198862 + 1205832 + V) + 11988621205872

(1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1)

1198782 =1205872 (1198861 + 1205831 + V) + 11988611205871

(1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1)

(3)



R01oc

=(1205871 (1198862 + 1205832 + V) + 11988621205872) ((1 minus 119898) 12057211198762119870 + (1 minus 119906) 12057311205901)

11987611198762 (1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1) 119870

(4)


R02oc

=(1205872 (1198861 + 1205831 + V) + 11988611205871) ((1 minus 119898) 12057221198764119870 + (1 minus 119906) 12057321205902)

11987631198764 (1198861 + 1205831 + V) (1198862 + 1205832 + V) (1 minus Φ1) 119870

(5)








J (119906 V 119898) = int

119879

0(12058511198681 + 12058521198682 + 120594119906

2+ 119910V2 + 119911119898

2) 119889119905 (7)



J (119906lowast


) = min119906V119898

J (119906 V 119898) | 119906 V 119898 isinU (8)

where

U = (119906 V 119898) | 119906 V 119898 [0 119879] 997888rarr [0 1] forall119906 V 119898 (9)


1198781 1205821198681 1205821198771 1205821198611 1205821198782 1205821198682

1205821198772 and 120582




H = 12058511198681 + 12058521198682 + 1205941199062

+ 119910V2 + 1199111198982

+ 1205821198781

[1205871 + 11988621198782

+ 1205961198771 minus (1minus 119906) 12057311198611

119870 + 11986111198781 minus (1minus 119898) 120572111986811198781

minus (1198861 + 1205831 + V) 1198781] + 1205821198681

[(1minus 119906) 12057311198611

119870 + 11986111198781

+ (1minus 119898) 120572111986811198781 + 11988721198682 minus 11987611198681] + 1205821198771

[V1198781 + 12057411198681

+ 1199031198681 + 11988821198772 minus (1205831 + 120596 + 1198881) 1198771] + 1205821198611

[12059011198681

minus 11987621198611] + 1205821198782

[1205872 + 11988611198781 + 1205961198772

minus (1minus 119906) 12057321198612

119870 + 11986121198782 minus (1minus 119898) 120572211986821198782

minus (1198862 + 1205832) 1198782 minus V1198782] + 1205821198682

[(1minus 119906) 12057321198612

119870 + 11986121198782

+ (1minus 119898) 120572211986821198782 + 11988711198681 minus 11987631198682] + 1205821198772

[V1198782 + 12057421198682

+ 11988811198771 minus (1205832 + 120596 + 1198882) 1198772] + 1205821198772

[12059021198682 minus 11987641198612]

(10)






0 = H119906

(119905 119909lowast

119906lowast


120582 (119905))

= HV (119905 119909lowast

119906lowast


120582 (119905))

= H119898

(119905 119909lowast

119906lowast


120582 (119905))

(11)

then

H (119905 119909lowast

119906lowast

(119905) Vlowast (119905) 119898lowast

(119905) 120582 (119905))

le H (119905 119909lowast

119906 (119905) V (119905) 119898 (119905) 120582 (119905))

(12)


lowast


1198891205821198781

119889119905

= [(1minus 119906) 12057311198611

119870 + 1198611+ (1minus 119898) 12057211198681] (120582

1198781minus 1205821198681

)

+ (1198861 + 1205831 + V) 1205821198781

minus V1205821198771

minus 11988611205821198782


1198891205821198681

119889119905

= (1minus 119898) 12057211198781 (1205821198781

minus 1205821198681

) minus 1205851 + 11987611205821198681

minus (1205741 + 119903) 1205821198771

minus 12059011205821198611

minus 11988711205821198682

1198891205821198771

119889119905= (1205831 + 120596 + 1198881) 120582

1198771minus 1205961205821198781

minus 11988821205821198772

1198891205821198611

119889119905=

(1 minus 119906) 12057311198781119870

(119870 + 1198611)2 (120582

1198781minus 1205821198681

) + 11987621205821198611

1198891205821198782

119889119905

= [(1minus 119906) 12057311198612

119870 + 1198612+ (1minus 119898) 12057221198682] (120582

1198782minus 1205821198682

)

+ (1198862 + 1205832 + V) 1205821198782

minus V1205821198772

minus 11988621205821198781

1198891205821198682

119889119905

= (1minus 119898) 12057221198782 (1205821198782

minus 1205821198682

) minus 1205852 + 11987631205821198682

minus (1205742 + 119903) 1205821198772

minus 12059021205821198612

minus 11988721205821198681

1198891205821198772

119889119905= (1205832 + 120596 + 1198882) 120582

1198772minus 1205961205821198782

minus 11988811205821198771

1198891205821198611

119889119905=

(1 minus 119906) 12057321198782119870

(119870 + 1198612)2 (120582

1198782minus 1205821198682

) + 11987641205821198612

(13)


1205821198781

(119879) = 1205821198681

(119879) = 1205821198771

(119879) = 1205821198611

(119879) = 1205821198782

(119879)

= 1205821198682

(119879) = 1205821198772

(119879) = 1205821198612

(119879) = 0

(14)


1198891205821198781

119889119905= minus

120597H

1198891198781

1198891205821198612

119889119905= minus

120597H

1198891198612

(15)


119906lowast

(119905) = max (0min ( (119905) 1))

Vlowast (119905) = max (0min (V (119905) 1))

119898lowast

(119905) = max (0min ( (119905) 1))

(16)


119906lowast

=

0 if le 0

if 0 lt lt 1

1 if ge 1

(17)



Vlowast =

0 if V le 0

V if 0 lt V lt 1

1 if V ge 1

(18)


119898lowast

=

0 if le 0

if 0 lt lt 1

1 if ge 1

(19)




120597H

120597119906= 2120594119906 + (120582

1198782minus 1205821198682

) 120573211986121198782

119870 + 1198612

+ (1205821198781

minus 1205821198681

) 120573111986111198781

119870 + 1198611

120597H

120597V= 2119910V+ (120582

1198771minus 1205821198781

) 1198781 + (1205821198772

minus 1205821198782

) 1198782

120597H

120597119898= 2119911119898 + (120582

1198781minus 1205821198681

) 120572111986811198781

+ (1205821198782

minus 1205821198682

) 120572211986821198782

(20)



120597H120597V = 120597H120597119898 = 0


=

(1205821198682

minus 1205821198782

) 1205732 (11986121198782 (119870 + 1198612)) + (1205821198681

minus 1205821198781

) 1205731 (11986111198781 (119870 + 1198611))

2120594

V =

(1205821198781

minus 1205821198771

) 1198781 + (1205821198782

minus 1205821198772

) 1198782

2119910

=

(1205821198681

minus 1205821198781

) 120572111986811198781 + (1205821198682

minus 1205821198782

) 120572211986821198782

2119911

(21)












1205731

1205721

1205741

1205901

a2a1

b1

120583p

g1

um

(a)


1205732

1205722

1205742

1205902

a1

a2

b2

120583p

g2

um

(b)











0

0002

0004

0006

0008

001

0012

0014

0016

0 20 40 60 80 100Time (days)

Infe

cted

gro

up in

the fi

rst c

omm

unity


(a)

0

002

004

006

008

01

012

014

Infe

cted

gro

up in

the s

econ

d co

mm

unity

0 20 40 60 80 100Time (days)


(b)

















0

001

002

003

004

005

006In

fect

ed g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



0 20 40 60 80 100Time (days)


Infe

cted

gro

up in

the s

econ

d co

mm

unity

0

001

002

003

004

005

008

007

006



0 20 40 60 80 100Time (days)

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Profi

le o

f dom

estic

wat

er tr

eatm

entu

(a)

0 20 40 60 80 1000

001

002

003

004

005

006

007

008

009

01

Time (days)

Profi

le o

f vac

cina

tion

(b)








0 20 40 60 80 1000

1

2

3

4

5

6

7

8

Time (days)

times10minus4

Profi

le o

f hyg

iene

relat

ed co

ntro

lm


01

02

03

04

05

06

07

08

09

Susc

eptib

le g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



Susc

eptib

le g

roup

in th

e sec

ond

com

mun

ity

01

02

03

04

05

06

07

08

0 20 40 60 80 100Time (days)







5 Conclusion



0

Reco

vere

d gr

oup

in th

e firs

t com

mun

ity

01

02

03

04

05

06

07

0 20 40 60 80 100Time (days)



005

015

025

035

045

Reco

vere

d gr

oup

in th

e sec

ond

com

mun

ity

0

01

02

03

04

05

0 20 40 60 80 100Time (days)








Acknowledgments


References


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of























J (119906 V 119898) = int

119879

0(12058511198681 + 12058521198682 + 120594119906

2+ 119910V2 + 119911119898

2) 119889119905 (7)



J (119906lowast


) = min119906V119898

J (119906 V 119898) | 119906 V 119898 isinU (8)

where

U = (119906 V 119898) | 119906 V 119898 [0 119879] 997888rarr [0 1] forall119906 V 119898 (9)


1198781 1205821198681 1205821198771 1205821198611 1205821198782 1205821198682

1205821198772 and 120582




H = 12058511198681 + 12058521198682 + 1205941199062

+ 119910V2 + 1199111198982

+ 1205821198781

[1205871 + 11988621198782

+ 1205961198771 minus (1minus 119906) 12057311198611

119870 + 11986111198781 minus (1minus 119898) 120572111986811198781

minus (1198861 + 1205831 + V) 1198781] + 1205821198681

[(1minus 119906) 12057311198611

119870 + 11986111198781

+ (1minus 119898) 120572111986811198781 + 11988721198682 minus 11987611198681] + 1205821198771

[V1198781 + 12057411198681

+ 1199031198681 + 11988821198772 minus (1205831 + 120596 + 1198881) 1198771] + 1205821198611

[12059011198681

minus 11987621198611] + 1205821198782

[1205872 + 11988611198781 + 1205961198772

minus (1minus 119906) 12057321198612

119870 + 11986121198782 minus (1minus 119898) 120572211986821198782

minus (1198862 + 1205832) 1198782 minus V1198782] + 1205821198682

[(1minus 119906) 12057321198612

119870 + 11986121198782

+ (1minus 119898) 120572211986821198782 + 11988711198681 minus 11987631198682] + 1205821198772

[V1198782 + 12057421198682

+ 11988811198771 minus (1205832 + 120596 + 1198882) 1198772] + 1205821198772

[12059021198682 minus 11987641198612]

(10)






0 = H119906

(119905 119909lowast

119906lowast


120582 (119905))

= HV (119905 119909lowast

119906lowast


120582 (119905))

= H119898

(119905 119909lowast

119906lowast


120582 (119905))

(11)

then

H (119905 119909lowast

119906lowast

(119905) Vlowast (119905) 119898lowast

(119905) 120582 (119905))

le H (119905 119909lowast

119906 (119905) V (119905) 119898 (119905) 120582 (119905))

(12)


lowast


1198891205821198781

119889119905

= [(1minus 119906) 12057311198611

119870 + 1198611+ (1minus 119898) 12057211198681] (120582

1198781minus 1205821198681

)

+ (1198861 + 1205831 + V) 1205821198781

minus V1205821198771

minus 11988611205821198782


1198891205821198681

119889119905

= (1minus 119898) 12057211198781 (1205821198781

minus 1205821198681

) minus 1205851 + 11987611205821198681

minus (1205741 + 119903) 1205821198771

minus 12059011205821198611

minus 11988711205821198682

1198891205821198771

119889119905= (1205831 + 120596 + 1198881) 120582

1198771minus 1205961205821198781

minus 11988821205821198772

1198891205821198611

119889119905=

(1 minus 119906) 12057311198781119870

(119870 + 1198611)2 (120582

1198781minus 1205821198681

) + 11987621205821198611

1198891205821198782

119889119905

= [(1minus 119906) 12057311198612

119870 + 1198612+ (1minus 119898) 12057221198682] (120582

1198782minus 1205821198682

)

+ (1198862 + 1205832 + V) 1205821198782

minus V1205821198772

minus 11988621205821198781

1198891205821198682

119889119905

= (1minus 119898) 12057221198782 (1205821198782

minus 1205821198682

) minus 1205852 + 11987631205821198682

minus (1205742 + 119903) 1205821198772

minus 12059021205821198612

minus 11988721205821198681

1198891205821198772

119889119905= (1205832 + 120596 + 1198882) 120582

1198772minus 1205961205821198782

minus 11988811205821198771

1198891205821198611

119889119905=

(1 minus 119906) 12057321198782119870

(119870 + 1198612)2 (120582

1198782minus 1205821198682

) + 11987641205821198612

(13)


1205821198781

(119879) = 1205821198681

(119879) = 1205821198771

(119879) = 1205821198611

(119879) = 1205821198782

(119879)

= 1205821198682

(119879) = 1205821198772

(119879) = 1205821198612

(119879) = 0

(14)


1198891205821198781

119889119905= minus

120597H

1198891198781

1198891205821198612

119889119905= minus

120597H

1198891198612

(15)


119906lowast

(119905) = max (0min ( (119905) 1))

Vlowast (119905) = max (0min (V (119905) 1))

119898lowast

(119905) = max (0min ( (119905) 1))

(16)


119906lowast

=

0 if le 0

if 0 lt lt 1

1 if ge 1

(17)



Vlowast =

0 if V le 0

V if 0 lt V lt 1

1 if V ge 1

(18)


119898lowast

=

0 if le 0

if 0 lt lt 1

1 if ge 1

(19)




120597H

120597119906= 2120594119906 + (120582

1198782minus 1205821198682

) 120573211986121198782

119870 + 1198612

+ (1205821198781

minus 1205821198681

) 120573111986111198781

119870 + 1198611

120597H

120597V= 2119910V+ (120582

1198771minus 1205821198781

) 1198781 + (1205821198772

minus 1205821198782

) 1198782

120597H

120597119898= 2119911119898 + (120582

1198781minus 1205821198681

) 120572111986811198781

+ (1205821198782

minus 1205821198682

) 120572211986821198782

(20)



120597H120597V = 120597H120597119898 = 0


=

(1205821198682

minus 1205821198782

) 1205732 (11986121198782 (119870 + 1198612)) + (1205821198681

minus 1205821198781

) 1205731 (11986111198781 (119870 + 1198611))

2120594

V =

(1205821198781

minus 1205821198771

) 1198781 + (1205821198782

minus 1205821198772

) 1198782

2119910

=

(1205821198681

minus 1205821198781

) 120572111986811198781 + (1205821198682

minus 1205821198782

) 120572211986821198782

2119911

(21)












1205731

1205721

1205741

1205901

a2a1

b1

120583p

g1

um

(a)


1205732

1205722

1205742

1205902

a1

a2

b2

120583p

g2

um

(b)











0

0002

0004

0006

0008

001

0012

0014

0016

0 20 40 60 80 100Time (days)

Infe

cted

gro

up in

the fi

rst c

omm

unity


(a)

0

002

004

006

008

01

012

014

Infe

cted

gro

up in

the s

econ

d co

mm

unity

0 20 40 60 80 100Time (days)


(b)

















0

001

002

003

004

005

006In

fect

ed g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



0 20 40 60 80 100Time (days)


Infe

cted

gro

up in

the s

econ

d co

mm

unity

0

001

002

003

004

005

008

007

006



0 20 40 60 80 100Time (days)

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Profi

le o

f dom

estic

wat

er tr

eatm

entu

(a)

0 20 40 60 80 1000

001

002

003

004

005

006

007

008

009

01

Time (days)

Profi

le o

f vac

cina

tion

(b)








0 20 40 60 80 1000

1

2

3

4

5

6

7

8

Time (days)

times10minus4

Profi

le o

f hyg

iene

relat

ed co

ntro

lm


01

02

03

04

05

06

07

08

09

Susc

eptib

le g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



Susc

eptib

le g

roup

in th

e sec

ond

com

mun

ity

01

02

03

04

05

06

07

08

0 20 40 60 80 100Time (days)







5 Conclusion



0

Reco

vere

d gr

oup

in th

e firs

t com

mun

ity

01

02

03

04

05

06

07

0 20 40 60 80 100Time (days)



005

015

025

035

045

Reco

vere

d gr

oup

in th

e sec

ond

com

mun

ity

0

01

02

03

04

05

0 20 40 60 80 100Time (days)








Acknowledgments


References


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















1198891205821198681

119889119905

= (1minus 119898) 12057211198781 (1205821198781

minus 1205821198681

) minus 1205851 + 11987611205821198681

minus (1205741 + 119903) 1205821198771

minus 12059011205821198611

minus 11988711205821198682

1198891205821198771

119889119905= (1205831 + 120596 + 1198881) 120582

1198771minus 1205961205821198781

minus 11988821205821198772

1198891205821198611

119889119905=

(1 minus 119906) 12057311198781119870

(119870 + 1198611)2 (120582

1198781minus 1205821198681

) + 11987621205821198611

1198891205821198782

119889119905

= [(1minus 119906) 12057311198612

119870 + 1198612+ (1minus 119898) 12057221198682] (120582

1198782minus 1205821198682

)

+ (1198862 + 1205832 + V) 1205821198782

minus V1205821198772

minus 11988621205821198781

1198891205821198682

119889119905

= (1minus 119898) 12057221198782 (1205821198782

minus 1205821198682

) minus 1205852 + 11987631205821198682

minus (1205742 + 119903) 1205821198772

minus 12059021205821198612

minus 11988721205821198681

1198891205821198772

119889119905= (1205832 + 120596 + 1198882) 120582

1198772minus 1205961205821198782

minus 11988811205821198771

1198891205821198611

119889119905=

(1 minus 119906) 12057321198782119870

(119870 + 1198612)2 (120582

1198782minus 1205821198682

) + 11987641205821198612

(13)


1205821198781

(119879) = 1205821198681

(119879) = 1205821198771

(119879) = 1205821198611

(119879) = 1205821198782

(119879)

= 1205821198682

(119879) = 1205821198772

(119879) = 1205821198612

(119879) = 0

(14)


1198891205821198781

119889119905= minus

120597H

1198891198781

1198891205821198612

119889119905= minus

120597H

1198891198612

(15)


119906lowast

(119905) = max (0min ( (119905) 1))

Vlowast (119905) = max (0min (V (119905) 1))

119898lowast

(119905) = max (0min ( (119905) 1))

(16)


119906lowast

=

0 if le 0

if 0 lt lt 1

1 if ge 1

(17)



Vlowast =

0 if V le 0

V if 0 lt V lt 1

1 if V ge 1

(18)


119898lowast

=

0 if le 0

if 0 lt lt 1

1 if ge 1

(19)




120597H

120597119906= 2120594119906 + (120582

1198782minus 1205821198682

) 120573211986121198782

119870 + 1198612

+ (1205821198781

minus 1205821198681

) 120573111986111198781

119870 + 1198611

120597H

120597V= 2119910V+ (120582

1198771minus 1205821198781

) 1198781 + (1205821198772

minus 1205821198782

) 1198782

120597H

120597119898= 2119911119898 + (120582

1198781minus 1205821198681

) 120572111986811198781

+ (1205821198782

minus 1205821198682

) 120572211986821198782

(20)



120597H120597V = 120597H120597119898 = 0


=

(1205821198682

minus 1205821198782

) 1205732 (11986121198782 (119870 + 1198612)) + (1205821198681

minus 1205821198781

) 1205731 (11986111198781 (119870 + 1198611))

2120594

V =

(1205821198781

minus 1205821198771

) 1198781 + (1205821198782

minus 1205821198772

) 1198782

2119910

=

(1205821198681

minus 1205821198781

) 120572111986811198781 + (1205821198682

minus 1205821198782

) 120572211986821198782

2119911

(21)












1205731

1205721

1205741

1205901

a2a1

b1

120583p

g1

um

(a)


1205732

1205722

1205742

1205902

a1

a2

b2

120583p

g2

um

(b)











0

0002

0004

0006

0008

001

0012

0014

0016

0 20 40 60 80 100Time (days)

Infe

cted

gro

up in

the fi

rst c

omm

unity


(a)

0

002

004

006

008

01

012

014

Infe

cted

gro

up in

the s

econ

d co

mm

unity

0 20 40 60 80 100Time (days)


(b)

















0

001

002

003

004

005

006In

fect

ed g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



0 20 40 60 80 100Time (days)


Infe

cted

gro

up in

the s

econ

d co

mm

unity

0

001

002

003

004

005

008

007

006



0 20 40 60 80 100Time (days)

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Profi

le o

f dom

estic

wat

er tr

eatm

entu

(a)

0 20 40 60 80 1000

001

002

003

004

005

006

007

008

009

01

Time (days)

Profi

le o

f vac

cina

tion

(b)








0 20 40 60 80 1000

1

2

3

4

5

6

7

8

Time (days)

times10minus4

Profi

le o

f hyg

iene

relat

ed co

ntro

lm


01

02

03

04

05

06

07

08

09

Susc

eptib

le g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



Susc

eptib

le g

roup

in th

e sec

ond

com

mun

ity

01

02

03

04

05

06

07

08

0 20 40 60 80 100Time (days)







5 Conclusion



0

Reco

vere

d gr

oup

in th

e firs

t com

mun

ity

01

02

03

04

05

06

07

0 20 40 60 80 100Time (days)



005

015

025

035

045

Reco

vere

d gr

oup

in th

e sec

ond

com

mun

ity

0

01

02

03

04

05

0 20 40 60 80 100Time (days)








Acknowledgments


References


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















=

(1205821198682

minus 1205821198782

) 1205732 (11986121198782 (119870 + 1198612)) + (1205821198681

minus 1205821198781

) 1205731 (11986111198781 (119870 + 1198611))

2120594

V =

(1205821198781

minus 1205821198771

) 1198781 + (1205821198782

minus 1205821198772

) 1198782

2119910

=

(1205821198681

minus 1205821198781

) 120572111986811198781 + (1205821198682

minus 1205821198782

) 120572211986821198782

2119911

(21)












1205731

1205721

1205741

1205901

a2a1

b1

120583p

g1

um

(a)


1205732

1205722

1205742

1205902

a1

a2

b2

120583p

g2

um

(b)











0

0002

0004

0006

0008

001

0012

0014

0016

0 20 40 60 80 100Time (days)

Infe

cted

gro

up in

the fi

rst c

omm

unity


(a)

0

002

004

006

008

01

012

014

Infe

cted

gro

up in

the s

econ

d co

mm

unity

0 20 40 60 80 100Time (days)


(b)

















0

001

002

003

004

005

006In

fect

ed g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



0 20 40 60 80 100Time (days)


Infe

cted

gro

up in

the s

econ

d co

mm

unity

0

001

002

003

004

005

008

007

006



0 20 40 60 80 100Time (days)

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Profi

le o

f dom

estic

wat

er tr

eatm

entu

(a)

0 20 40 60 80 1000

001

002

003

004

005

006

007

008

009

01

Time (days)

Profi

le o

f vac

cina

tion

(b)








0 20 40 60 80 1000

1

2

3

4

5

6

7

8

Time (days)

times10minus4

Profi

le o

f hyg

iene

relat

ed co

ntro

lm


01

02

03

04

05

06

07

08

09

Susc

eptib

le g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



Susc

eptib

le g

roup

in th

e sec

ond

com

mun

ity

01

02

03

04

05

06

07

08

0 20 40 60 80 100Time (days)







5 Conclusion



0

Reco

vere

d gr

oup

in th

e firs

t com

mun

ity

01

02

03

04

05

06

07

0 20 40 60 80 100Time (days)



005

015

025

035

045

Reco

vere

d gr

oup

in th

e sec

ond

com

mun

ity

0

01

02

03

04

05

0 20 40 60 80 100Time (days)








Acknowledgments


References


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















1205731

1205721

1205741

1205901

a2a1

b1

120583p

g1

um

(a)


1205732

1205722

1205742

1205902

a1

a2

b2

120583p

g2

um

(b)











0

0002

0004

0006

0008

001

0012

0014

0016

0 20 40 60 80 100Time (days)

Infe

cted

gro

up in

the fi

rst c

omm

unity


(a)

0

002

004

006

008

01

012

014

Infe

cted

gro

up in

the s

econ

d co

mm

unity

0 20 40 60 80 100Time (days)


(b)

















0

001

002

003

004

005

006In

fect

ed g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



0 20 40 60 80 100Time (days)


Infe

cted

gro

up in

the s

econ

d co

mm

unity

0

001

002

003

004

005

008

007

006



0 20 40 60 80 100Time (days)

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Profi

le o

f dom

estic

wat

er tr

eatm

entu

(a)

0 20 40 60 80 1000

001

002

003

004

005

006

007

008

009

01

Time (days)

Profi

le o

f vac

cina

tion

(b)








0 20 40 60 80 1000

1

2

3

4

5

6

7

8

Time (days)

times10minus4

Profi

le o

f hyg

iene

relat

ed co

ntro

lm


01

02

03

04

05

06

07

08

09

Susc

eptib

le g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



Susc

eptib

le g

roup

in th

e sec

ond

com

mun

ity

01

02

03

04

05

06

07

08

0 20 40 60 80 100Time (days)







5 Conclusion



0

Reco

vere

d gr

oup

in th

e firs

t com

mun

ity

01

02

03

04

05

06

07

0 20 40 60 80 100Time (days)



005

015

025

035

045

Reco

vere

d gr

oup

in th

e sec

ond

com

mun

ity

0

01

02

03

04

05

0 20 40 60 80 100Time (days)








Acknowledgments


References


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0

0002

0004

0006

0008

001

0012

0014

0016

0 20 40 60 80 100Time (days)

Infe

cted

gro

up in

the fi

rst c

omm

unity


(a)

0

002

004

006

008

01

012

014

Infe

cted

gro

up in

the s

econ

d co

mm

unity

0 20 40 60 80 100Time (days)


(b)

















0

001

002

003

004

005

006In

fect

ed g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



0 20 40 60 80 100Time (days)


Infe

cted

gro

up in

the s

econ

d co

mm

unity

0

001

002

003

004

005

008

007

006



0 20 40 60 80 100Time (days)

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Profi

le o

f dom

estic

wat

er tr

eatm

entu

(a)

0 20 40 60 80 1000

001

002

003

004

005

006

007

008

009

01

Time (days)

Profi

le o

f vac

cina

tion

(b)








0 20 40 60 80 1000

1

2

3

4

5

6

7

8

Time (days)

times10minus4

Profi

le o

f hyg

iene

relat

ed co

ntro

lm


01

02

03

04

05

06

07

08

09

Susc

eptib

le g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



Susc

eptib

le g

roup

in th

e sec

ond

com

mun

ity

01

02

03

04

05

06

07

08

0 20 40 60 80 100Time (days)







5 Conclusion



0

Reco

vere

d gr

oup

in th

e firs

t com

mun

ity

01

02

03

04

05

06

07

0 20 40 60 80 100Time (days)



005

015

025

035

045

Reco

vere

d gr

oup

in th

e sec

ond

com

mun

ity

0

01

02

03

04

05

0 20 40 60 80 100Time (days)








Acknowledgments


References


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0

001

002

003

004

005

006In

fect

ed g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



0 20 40 60 80 100Time (days)


Infe

cted

gro

up in

the s

econ

d co

mm

unity

0

001

002

003

004

005

008

007

006



0 20 40 60 80 100Time (days)

0

0002

0004

0006

0008

001

0012

0014

0016

0018

Profi

le o

f dom

estic

wat

er tr

eatm

entu

(a)

0 20 40 60 80 1000

001

002

003

004

005

006

007

008

009

01

Time (days)

Profi

le o

f vac

cina

tion

(b)








0 20 40 60 80 1000

1

2

3

4

5

6

7

8

Time (days)

times10minus4

Profi

le o

f hyg

iene

relat

ed co

ntro

lm


01

02

03

04

05

06

07

08

09

Susc

eptib

le g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



Susc

eptib

le g

roup

in th

e sec

ond

com

mun

ity

01

02

03

04

05

06

07

08

0 20 40 60 80 100Time (days)







5 Conclusion



0

Reco

vere

d gr

oup

in th

e firs

t com

mun

ity

01

02

03

04

05

06

07

0 20 40 60 80 100Time (days)



005

015

025

035

045

Reco

vere

d gr

oup

in th

e sec

ond

com

mun

ity

0

01

02

03

04

05

0 20 40 60 80 100Time (days)








Acknowledgments


References


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 20 40 60 80 1000

1

2

3

4

5

6

7

8

Time (days)

times10minus4

Profi

le o

f hyg

iene

relat

ed co

ntro

lm


01

02

03

04

05

06

07

08

09

Susc

eptib

le g

roup

in th

e firs

t com

mun

ity

0 20 40 60 80 100Time (days)



Susc

eptib

le g

roup

in th

e sec

ond

com

mun

ity

01

02

03

04

05

06

07

08

0 20 40 60 80 100Time (days)







5 Conclusion



0

Reco

vere

d gr

oup

in th

e firs

t com

mun

ity

01

02

03

04

05

06

07

0 20 40 60 80 100Time (days)



005

015

025

035

045

Reco

vere

d gr

oup

in th

e sec

ond

com

mun

ity

0

01

02

03

04

05

0 20 40 60 80 100Time (days)








Acknowledgments


References


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0

Reco

vere

d gr

oup

in th

e firs

t com

mun

ity

01

02

03

04

05

06

07

0 20 40 60 80 100Time (days)



005

015

025

035

045

Reco

vere

d gr

oup

in th

e sec

ond

com

mun

ity

0

01

02

03

04

05

0 20 40 60 80 100Time (days)








Acknowledgments


References


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of









































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















research article modelling optimal control of cholera in …downloads.hindawi.com › journals ›...

Documents