# Optimal Control Analysis of Treatment Strategies of the Dynamics of Cholera

Authors: Sani Fakai Abubakar; M. O. Ibrahim
Journal: Journal of Optimization
 (2022)

---

## Abstract

A nine-compartment deterministic cholera model was formulated, and the model describes interactions between human,Vibrio cholerae bacteria, and the enviroment that warrant the interaction. Realities and socioeconomic burden influence spread and control mechanism of the disease. The model investigated some effective ways of hindering cholera outbreak and spread. The existence and uniqueness of solution of the system of equations that the model comprises were ascertained. The basic reproduction number R0 of the model was obtained using “next-generation matrix” method, and the most sensitive parameters were identified using “normalised forward sensitivity index” method. Three controls, hygiene consciousness denoted by X1, cholera vaccine X2, and cholera awareness programme X3, were chosen. Optimal control theory is applied to ascertain the level of effect of the controls in reducing susceptible, exposed, infected individuals and causative pathogen population. Pontryagin’s maximum principle is used to prove the optimal solution of the model, and the optimal system was derived and numerically solved. Simulations were made with graphs that show the effects of the controls on susceptible, exposed, infected, and Vibrio cholerae population. The findings are that simultaneous application of the three controls can be one of the fast and effective ways of controlling cholera. If two controls are to be selected, hygiene consciousness and vaccine are the best combination.


---

## Body

## 1. Introduction

Cholera is a profuse diarrhoeal disease that can lead to death within short period of onset, if prompt treatment measures are not taken. It was estimated in [1], to be annually causing 21,000 to 143,000 deaths from 1,300,000 to 4,000,000 cholera cases worldwide, representing 1.62% to 3.58% of the reported cases. Natural disasters, climate change, social conflicts (terrorism), and economic meltdown usually force people to live in slums and refugee camps. The camps usually have very poor or insufficient infrastructures, especially good drinking water, and this serves as igniting factors of cholera outbreaks [2].Severe cholera symptoms include vomiting, profuse rice-water stool, cramps, sunken eyes, high dehydration, and shock. Individuals that ingest incomplete cholera-causing dose do not usually manifest any cholera symptom and are usually referred asVibrio cholerae carriers [3]. Severe cholera usually leads to death within short period of time that ranges between hours and three days. Chance of exposed susceptible individuals catching cholera will be half, if the concentration of Vibrio cholerae is 105 cells per millilitre (i.e., B=k) and the least daily consumption of untreated water peg is a minimum of 1 litre per day [4].Pontryagin’s maximum principle is one of the few methods of obtaining analytical solutions to optimal control problems. [5] derived optimal control model SIB, with two controls, and [6] derived an optimal intervention strategies model. [7] presented and analysed optimal control model for cholera disease with three controls, and [8] derived SEIRVIVE cholera model and introduced an exposed individual compartment. They all used Pontryagin’s maximum principle to analyse their models.Big risk in cholera disease and its outbreaks is that about 75% ofVibrio cholerae carriers do not have symptoms of the disease, but can spread the bacteria in the community through their faeces for one to two weeks after infection [9]. Cholera infection can be asymptomatic, mild or moderate, and severe; among the infected individuals that manifest symptoms, only 20% develop severe watery diarrhoea, while 80% have mild or moderate symptoms [10].This calls for the formulation of cholera model that will analyse dynamics ofVibrio cholerae carriers during and after cholera outbreak, to optimise application of hygiene, vaccination, and cholera awareness programme controls. The paper is organised as follows: Section 2 describes the model formulation, which comprises the basic assumptions, description of state variables and parameters, system of differential equations, existence and uniqueness of solution, positivity of solution, and disease-free and endemic equilibria. Section 3 discusses the calculation of basic reproduction number and sensitivity of its parameters. Section 4 defines optimal control. Section 5 illustrates numerical simulation, and Section 6 explains summary, conclusion, and recommendations.


## 2. Model Formulation

We derive a mathematical model of the dynamics of cholera disease, with homogeneous total number of humans denoted byPH representing combined human population at time t. This human population of the model is divided into eight mutually exclusive compartments and one nonhuman compartment.


### 2.1. Basic Assumptions of the Model

(i)
Exposed individuals imply a very short period at which individuals have ingestedVibrio cholerae bacteria but yet to develop symptoms and cannot spread cholera pathogens(ii)
Infected individuals cannot be vaccinated(iii)
Infection can be as follows:(a)
Moderate cholera infection, individuals that have the pathogen in their system but did not develop into cholera disease(b)
Severe cholera infection, infected individuals that have manifested symptoms(iv)
Moderate infected individuals can recover without treatment(v)
Severe cholera infection must be treated before recovery(vi)
Recovered individuals have no permanent cholera immunity


### 2.2. State Variables and Parameters of the Model

Description of the model’s state variables and parameters is shown in Tables1 and 2.Table 1
State variables of the model.
VariableDescriptionSSusceptible individualsVThe vaccinated individualsEExposed individualsIInfected individualsCHygiene conscious individualsUUntreatedVibrio cholerae carriesTInfected individuals under treatmentRRecovered individualsBConcentration ofVibrio cholerae bacteria in food and water consumed by the human populationTable 2
Description of parameters of the model, their values, units, and references (source).
ParameterSymbolValueUnitSourceHuman recruitment rateq9.49day−1EstimatedNon-cholera human death rateδ0.01177day/1000CalculatedRate of ingestingVibrio cholerae bacteria from contaminated food and water consumed by the human populationv10.5day−1[11]Rate of gettingVibrio cholerae bacteria through human-to-human interactionsv20.03day−1[11]Rate of recovery of hygiene conscious individualsθ0.38day−1[11]Hygiene consciousness of individualsh0.604day−1EstimatedRate ofVibrio cholerae decreaseη20.033day−1[12]Rate at which infected individuals increaseη110Cells/ml/day[13]Vibrio cholerae bacteria in the environmentRate of generatingβ0.073day−1[7]Vibrio cholerae bacteria in the environmentRate of freedom fromVibrio cholerae by the hygiene conscious individualsf0.65day−1EstimatedConcentration of cholera pathogen that yields 50% chance of individual developing cholera diseasek105Cells/ml[14]Progression of recovery of treated individuals to untreatedVibrio cholerae carriersg10.08day−1EstimatedRate of interruption of treatmentg20.2day−1[15]Recovery rate of untreatedVibrio cholerae carriersr10.8day−1EstimatedRecovery of treated cholera patientsr20.63day−1EstimatedDisease induced death rate for individuals inI classϵ0.105day−1EstimatedDisease induced death rate for individuals inU classμ0.1day−1EstimatedDisease induced death rate for individuals inT classα0.08day−1EstimatedRate of movement of the infected classm0.24day−1EstimatedRate of vaccineψ0.3day−1[16]Rate of waning out of the vaccine induced immunityω0.5day−1[16]Rate at which the vaccine reduces infectionσ0.2day−1[16]Cholera awareness programmeχ0.66day−1EstimatedRate of exposure to infectionξ0.23day−1EstimatedThe fractionB/k+Bϕ0.000599DimensionlessCalculatedA fraction of early infected individualsτ0-1Dimensionless[15]


### 2.3. The Model’s Equations

Schematic diagram of the flow between different classes of the state variablesS, V, E, I, C, U, T, R, and B of the model is shown in Figure 1.Figure 1
Flow diagram of theSVEICUTRB dynamic model.A system of differential equations (1)–(9) was obtained from the model’s assumptions, descriptions of variables and parameters, and the compartmental flow diagram.(1)dSdt=q−v11−χϕS+1−χ1−hv2SI+ωV+ωR−ψ+δS,(2)dVdt=ψS−σv2VI−ω+δV,(3)dEdt=v11−χϕS+1−χ1−hv2SI+σv2VI−ξ+h+δE,(4)dIdt=ξE−τm+1−τm+ϵ+δI,(5)dCdt=hE−f+θ+δC,(6)dUdt=fC+1−τmI+g2T−g1+r1+μ+δU,(7)dTdt=τmI+g1U−g1+r2+g2+α+δT,(8)dRdt=g1+r2T+r1U+θC−ω+δR,(9)dBdt=1−hη1I+η2−βB.


### 2.4. Properties of the Model

Properties that ensure the correctness of the system of equations of the model, such as existence, uniqueness, boundedness, positivity, and equilibria, are ascertained.


#### 2.4.1. Existence and Uniqueness of Solution

The existence and uniqueness of solution in formulated mathematical models need to undergo surety test to ascertain whether the solution exists, and if it exists, it is unique. Using the Lipschitz criteria, it is established as follows.Let(10)u1=q−v11−χϕS+1−χ1−hv2SI+ωV+ωR−ψ+δS,u2=ψS−σv2VI−ω+δV,u3=v11−χϕS+1−χ1−hv2SI+σv2VI−ξ+h+δE,u4=ξE−m+ε+δI,u5=hE−f+θ+δC,u6=fC+1−τmI+g2T−g1+r1+μ+δU,u7=τmI+g1U−g1+r2+g2+α+δT,u8=g1+r2T+r1U+θC−ω+δR,u9=1−hη1I+η2−βB.Considering the region0≤P≤ℜ, bounded solution of system of equations (1)–(9) will be looked for in the same region whose partial derivatives satisfy δ1≤P≤ℜ, where δ1 and P are positive constants.Theorem 1.
Let the region0≤P≤ℜ be denoted by Λ, and then, the system of equations (1)–(9) has a unique solution provided that it is established that(11)∂ui∂tj,i=1,2,3,4,5,6,7,8,9,is continuous and bounded in Λ.Proof.
Partial derivatives with respect to each state variables of the system of equations (1)–(9) are obtained as follows: for u1,(12)∂u1∂S=−v11−χϕ−1−χ1−hv2I−ψ−δ<∞,∂u1∂V=ω<∞,∂u1∂I=−1−χ1−hv2S<∞,∂u1∂B=v11−χBSk+B2−v11−χSk+B<∞,∂u1∂R=ω<∞,∂u1∂E=∂u1∂C=∂u1∂U=∂u1∂T=0<∞.
Similarly, foru2, u3, u4, u5, u7, u8, and u9, the system of equations (1)–(9) exists and has a unique solution in ℜ9 since all the partial derivatives are <∞ and therefore bounded and defined.


### 2.5. Positivity of Solutions

Let equations (1)–(9) be subject to initial conditions:(13)S0,V0,E0,I0,C0,U0,T0,R0,B0.For theSVEICUTRB cholera model to be meaningfully epidemiological, it is mandatory to prove that all the state variables are nonnegative at time t>0, since the population of human is non negative but positive. This implies that the solutions of the system of equations (1)–(9) have nonnegative initial values and will remain nonnegative for all time t>0.Let(14)St,Vt,Et,It,Ct,Ut,Tt,Rt,Bt,be solutions to the system of the equations (1)–(9).Theorem 2.
(a) Solution (11) is positive for all t>0 if S0>0, V0>0, E0>0, I0>0, C0>0, U0>0, T0>0, R0>0, and B0>0. (b) Solution (11) is nonnegative for all t>0 with respect to initial conditions (10).Proof.
(a) Suppose that (11) is defined for all t∈0,M, where M>0, to prove that (11) is positive for all t≥0.
Considering (1), it is obvious that(15)dStdt≥−v11−χϕ+1−χ1−hv2I+ψ+δS.
Integrating both sides of (15) with respect to p, with the range p=0 to p=t, and applying initial condition, it gives(16)St=S0exp∫0t−v11−χϕ+1−χ1−hv2I+ψ+δdp.
SinceS0>0, ∀t∈0,M.
Therefore,(17)S0exp−v11−χϕ+1−χ1−hv2I+ψ+δt,must be greater than zero.
Also, from (2)(18)dVtdt≥−σv2I+ω+δV,∀0≤t<M.
Integrating both sides within the interval0,t and applying initial conditions, it gives(19)Vt=V0exp−σv2I+ω+δt,and since V0>0, V0exp−σv2I+ω+δt must be greater than zero.
Similarly, from equations (3)–(9), their respective results are as follows:(20)Et≥E0exp−ξ+h+δt>0,∀0≤t<M,It≥I0exp−τm+1−τm+ε+δt>0,∀0≤t<M,Ct≥C0exp−f+θ+δt>0,∀0≤t<M∈,Ut≥U0exp−g1+r1+μ+δt>0,∀0≤t<M,Tt≥T0exp−g1+g2+r2+α+δ,∀0≤t<M,Rt≥R0exp−ω+δt>0,∀0≤t<M,Bt≥B0exp−βt>0,∀0≤t<M.
(b) With the aid of proof of (a) and since the solution of the system of equations (1)–(9) continuously depends on the respective initial conditions in neighbourhood of zero, obviously it implies that (14) is nonnegative for all 0≤t<M.
The model is dealing with population of living things, their number cannot be negative, and hence they are positive.Theorem 3.
Solutions of cholera modelSVEICUTRB are bounded in ℜ+9 with initial conditions (10), and hence, dynamics of the model is a dynamical system in the biologically feasible compact set.Proof.
LetΛ=S,V,E,I,C,U,T,R,B∈ℜ+9 be the solutions of the model with nonnegative initial conditions. Total human population is as follows:(21)PH=S+V+E+I+C+U+T+R,dPHdt=dSdt+dVdt+dEdt+dIdt+dCdt+dUdt+dTdt+dRdt,dPHdt=q−δS+V+E+I+C+U+T+R−ϵI−αT−μU.
Without cholera infection, all the cholera-related deaths will be zero; therefore,ϵI=αT=μU=0.(22)dPHdt=q−δPH.
Using differential inequality theorem from [17] and the Birkhoff and Rota version as used in [11, 18],(23)dPHdt≤q−δPH.
Multiplying both sides of (23) by the integrating factor expδt and integrating, it gives(24)PHexpδt≤qδ+C.
Substituting initial conditions are as follows:t=0 and PH0=P0.
This further givesPH≤q/δ+P0−q/δexp−δt
Ast⟶∞, P0−q/δexp−δt⟶0, which gives PH≤q/δ, but PH≥0; therefore, for all t>0,(25)0≤PHt≤qδ.
Hence,PHt is bounded.
The solution set of human population of the system of equations (1)–(9) belongs to the feasible region:(26)ΛH=S,V,E,I,C,U,T,R∈ℜ+8,PH≤qδ.
Also, let(27)PHt≤PH∗=maxP0,qδ.
On the other hand, the equation for the dynamics of the pathogen is (9) as follows:(28)dBtdt=1−hη1I+η2−βBt≤η2−βBt+1−hη1PH.
SinceI can be less than q/δ, then from (27),(29)dPBtdt≤η2−βBt+1−hη1PH∗.
Integrating (29) gives(30)PBt≤1−hη1PH∗η2−βBt1−exp−η2−βt+B0expη2−βt.
Ast⟶∞, then expη2−βt⟶0.
The population ofVibrio cholerae bacteria is as follows:(31)PBt≤1−hη1PH∗η2−β,provided η2≠β. Hence, the feasible solution set of Vibrio cholerae population of (9) enters the region(32)ΛB=PBt∈ℜ+|0≤PBt≤1−hη1PH∗η2−β.
This shows that the cholera model governed by the system of equations (1)–(9) is epidemiological well posed in a feasible region Λ, defined by Λ∈ℜ+9 and ΛH∪ΛB⊂ℜ+8∗ℜ+.
It then implies that the entire human and nonhuman population of the model enter the feasible regionΛ. Also, since PHt>0 and PBt≥0, for all t≥0, then PHt and PBt cannot blow up to infinity in the finite time. Consequently, the model system is dissipative; that is, its solutions are bounded and exist for all t>0 in the invariant and compact set Λ.(33)Λ=S,V,E,I,C,U,T,R∈ℜ+8,B∈ℜ+,S,V,E,I,C,U,T,R,B≥0,PH≤qδ,PH≤PH∗,PB≤1−hη1PH∗η2−β.
The regionΛ is positively invariant and attracts all solutions in ℜ+9.


#### 2.5.1. Disease-Free Equilibrium

Analysis of cholera model governed by a system of equations (1)–(9) is qualitatively carried out to obtain features of the dynamics and impact of control strategies, spread, and transmission dynamics of the cholera causative bacteria.Disease-free equilibrium of the model governed by system of equations (1)–(9) is algebraically calculated by setting dS/dt=dV/dV=dE/dt=dI/dt=dC/dt=dU/dt=dT/dt=dR/dt=dB/dt=0.In the absence of cholera disease, it is assumed thatS=S0,V=V0,E=0,I=0,C=0,U=0,T=0,R=0, and B=0.(34)S0=qω+δψ+δω+δ−ωψ,V0=qψψ+δω+δ−ωψ.The DFE point is as follows:(35)Q0=S0,V0,E,I,C,U,T,R,B=qω+δψ+δω+δ−ωψ,qψψ+δω+δ−ωψ,0,0,0,0,0,0,0.


#### 2.5.2. Existence of Endemic Equilibrium State (EES)Ξ∗

This is a state where the disease cannot be completely eliminated, but stay in the concern region. The state variables of the model with exception ofR∗ are not zero.LetΞ∗=S∗,V∗,E∗,I∗,C∗,U∗,T∗,R∗,B∗ be the endemic equilibrium point, and let(36)Q1=ξ+h+δ,Q2=m+ε+δ,Q3=g1+r1+μ+δ,Q4=g1+g2+r2+α+δ,Q5=ψ+δ,Q6=1−χ,Q7=1−h,Q8=Bk+B,Q9=ω+δ,Q10=f+θ+δ,Q11=g1+r2,Q12=η2−β.Substituting (37) into the system of equations (1)–(9), the following is obtained:(37)q−v1Q6Q8S∗−Q6Q7v2S∗I∗+ωV∗+ωR∗−Q5S∗=0,(38)ψS∗−σv2V∗I∗−Q9V∗=0,(39)v1Q6Q8S∗+Q6Q7v2S∗I∗+σv2V∗I∗−Q1E∗=0,(40)ξE∗−Q2I∗=0,(41)hE∗−Q10C∗=0,(42)fC∗+1−τmI∗+g2T∗−Q3U∗=0,(43)τmI∗+g1U∗−Q4T∗=0,(44)Q11T∗+r1U∗+θC∗−Q9R∗=0,(45)Q7η1I∗+Q12B∗=0,Adding (19) and (21),(46)q+ωR∗+ω+δv2I∗V∗−Q5S∗−Q1E∗=0.From (22),(47)E∗=Q2I∗ξ,and from (23),(48)C∗=hE∗Q10.Putting (40) into (41),(49)C∗=hQ2I∗ξQ10.From (25),(50)T∗=τmI∗+g1U∗Q4.Substituting (42) and (43) into (24) gives(51)U∗=fhQ2Q4I∗+ξQ4Q101−τmI∗+ξQ10g2τmI∗ξQ10Q3Q4−g1.Let(52)H1=fhQ2Q4,H2=ξQ4Q101−τm,H3=ξQ10g2τm,H4=ξQ10Q3Q4−g1,H5=Q10Q11ξτmH4+g1H1+H2+H3,H6=ξr1Q4Q10H1+H2+H3,H7=θhQ2Q4H4,H8=ξH4Q4Q9Q10.Substituting related terms of (45) into (44) gives(53)U∗=H1+H2+H3H4I∗,and from (43),(54)T∗=τmH4+g1H1+H2+H3H4Q4I∗.From (20),(55)V∗=ψS∗σv2I∗−Q9.Putting (31), (35), and (36) in (26), we obtain(56)R∗=H5+H6+H7I∗H8.Also let(57)H10=H8ξq+ξωH5+H6+H7−H8Q1Q2,H11=σv2Q5−ψ,H12=Q5Q9−ψω..Substituting (29), (37), (38), and related terms of (50) in (39) gives(58)S∗=H10I∗σv2I∗−Q9H11I∗−H12.Substituting (51) into (48) and simplifying, (48) becomes(59)V∗=ψH10I∗H11−H12.Substituting (29), (40), and (41) into (21) gives the following:(60)ξv22Q6Q7H10σI∗3+ξH10σv1v2Q6Q8−ξH10v2Q6Q7Q9+ξψH10σv2−Q1Q2H11I∗2+H12Q1Q2−ξH10v1Q6Q8Q9I∗=0,setting(61)B1=ξv22Q6Q7H10σ,B2=ξH10σv1v2Q6Q8−ξH10v2Q6Q7Q9+ξψH10σv2−Q1Q2H11,B3=H12Q1Q2−ξH10v1Q6Q8Q9..This gives(62)B1I∗3+B2I∗2+B3I∗=0,B1I∗2+B2I∗+B3I∗=0,which means either(63)B1I∗2+B2I∗+B3=0,or(64)I∗=0.ForI∗=0, it implies a disease-free equilibrium; when H12Q1Q2<ξH10v1Q6Q8Q9, it implies B1>0 and B3<0; hence, product of the roots I1∗I2∗=B3/B1<0, which means either I1∗>0 or I2∗>0, and the positive root will then be(65)I∗=−B2+B22−4B1B32B1.Therefore, there exists an endemic equilibrium stateΞ∗ defined as follows:(66)Ξ∗=H10σv2Δ2−H10Q9ΔH11Δ−H12,ψH10ΔH11Δ−H12,Q2Δξ,Δ,hQ2ΔξQ10,H1+H2+H3ΔH4,τmH4+g1H1+H2+H3ΔH4Q4,H5+H6+H7ΔH8,where(67)Δ=−B2+B22−4B1B32B1.Note thatη2−β must be negative for endemic.


## 2.1. Basic Assumptions of the Model

(i)
Exposed individuals imply a very short period at which individuals have ingestedVibrio cholerae bacteria but yet to develop symptoms and cannot spread cholera pathogens(ii)
Infected individuals cannot be vaccinated(iii)
Infection can be as follows:(a)
Moderate cholera infection, individuals that have the pathogen in their system but did not develop into cholera disease(b)
Severe cholera infection, infected individuals that have manifested symptoms(iv)
Moderate infected individuals can recover without treatment(v)
Severe cholera infection must be treated before recovery(vi)
Recovered individuals have no permanent cholera immunity


## 2.2. State Variables and Parameters of the Model

Description of the model’s state variables and parameters is shown in Tables1 and 2.Table 1
State variables of the model.
VariableDescriptionSSusceptible individualsVThe vaccinated individualsEExposed individualsIInfected individualsCHygiene conscious individualsUUntreatedVibrio cholerae carriesTInfected individuals under treatmentRRecovered individualsBConcentration ofVibrio cholerae bacteria in food and water consumed by the human populationTable 2
Description of parameters of the model, their values, units, and references (source).
ParameterSymbolValueUnitSourceHuman recruitment rateq9.49day−1EstimatedNon-cholera human death rateδ0.01177day/1000CalculatedRate of ingestingVibrio cholerae bacteria from contaminated food and water consumed by the human populationv10.5day−1[11]Rate of gettingVibrio cholerae bacteria through human-to-human interactionsv20.03day−1[11]Rate of recovery of hygiene conscious individualsθ0.38day−1[11]Hygiene consciousness of individualsh0.604day−1EstimatedRate ofVibrio cholerae decreaseη20.033day−1[12]Rate at which infected individuals increaseη110Cells/ml/day[13]Vibrio cholerae bacteria in the environmentRate of generatingβ0.073day−1[7]Vibrio cholerae bacteria in the environmentRate of freedom fromVibrio cholerae by the hygiene conscious individualsf0.65day−1EstimatedConcentration of cholera pathogen that yields 50% chance of individual developing cholera diseasek105Cells/ml[14]Progression of recovery of treated individuals to untreatedVibrio cholerae carriersg10.08day−1EstimatedRate of interruption of treatmentg20.2day−1[15]Recovery rate of untreatedVibrio cholerae carriersr10.8day−1EstimatedRecovery of treated cholera patientsr20.63day−1EstimatedDisease induced death rate for individuals inI classϵ0.105day−1EstimatedDisease induced death rate for individuals inU classμ0.1day−1EstimatedDisease induced death rate for individuals inT classα0.08day−1EstimatedRate of movement of the infected classm0.24day−1EstimatedRate of vaccineψ0.3day−1[16]Rate of waning out of the vaccine induced immunityω0.5day−1[16]Rate at which the vaccine reduces infectionσ0.2day−1[16]Cholera awareness programmeχ0.66day−1EstimatedRate of exposure to infectionξ0.23day−1EstimatedThe fractionB/k+Bϕ0.000599DimensionlessCalculatedA fraction of early infected individualsτ0-1Dimensionless[15]


## 2.3. The Model’s Equations

Schematic diagram of the flow between different classes of the state variablesS, V, E, I, C, U, T, R, and B of the model is shown in Figure 1.Figure 1
Flow diagram of theSVEICUTRB dynamic model.A system of differential equations (1)–(9) was obtained from the model’s assumptions, descriptions of variables and parameters, and the compartmental flow diagram.(1)dSdt=q−v11−χϕS+1−χ1−hv2SI+ωV+ωR−ψ+δS,(2)dVdt=ψS−σv2VI−ω+δV,(3)dEdt=v11−χϕS+1−χ1−hv2SI+σv2VI−ξ+h+δE,(4)dIdt=ξE−τm+1−τm+ϵ+δI,(5)dCdt=hE−f+θ+δC,(6)dUdt=fC+1−τmI+g2T−g1+r1+μ+δU,(7)dTdt=τmI+g1U−g1+r2+g2+α+δT,(8)dRdt=g1+r2T+r1U+θC−ω+δR,(9)dBdt=1−hη1I+η2−βB.


## 2.4. Properties of the Model

Properties that ensure the correctness of the system of equations of the model, such as existence, uniqueness, boundedness, positivity, and equilibria, are ascertained.


### 2.4.1. Existence and Uniqueness of Solution

The existence and uniqueness of solution in formulated mathematical models need to undergo surety test to ascertain whether the solution exists, and if it exists, it is unique. Using the Lipschitz criteria, it is established as follows.Let(10)u1=q−v11−χϕS+1−χ1−hv2SI+ωV+ωR−ψ+δS,u2=ψS−σv2VI−ω+δV,u3=v11−χϕS+1−χ1−hv2SI+σv2VI−ξ+h+δE,u4=ξE−m+ε+δI,u5=hE−f+θ+δC,u6=fC+1−τmI+g2T−g1+r1+μ+δU,u7=τmI+g1U−g1+r2+g2+α+δT,u8=g1+r2T+r1U+θC−ω+δR,u9=1−hη1I+η2−βB.Considering the region0≤P≤ℜ, bounded solution of system of equations (1)–(9) will be looked for in the same region whose partial derivatives satisfy δ1≤P≤ℜ, where δ1 and P are positive constants.Theorem 1.
Let the region0≤P≤ℜ be denoted by Λ, and then, the system of equations (1)–(9) has a unique solution provided that it is established that(11)∂ui∂tj,i=1,2,3,4,5,6,7,8,9,is continuous and bounded in Λ.Proof.
Partial derivatives with respect to each state variables of the system of equations (1)–(9) are obtained as follows: for u1,(12)∂u1∂S=−v11−χϕ−1−χ1−hv2I−ψ−δ<∞,∂u1∂V=ω<∞,∂u1∂I=−1−χ1−hv2S<∞,∂u1∂B=v11−χBSk+B2−v11−χSk+B<∞,∂u1∂R=ω<∞,∂u1∂E=∂u1∂C=∂u1∂U=∂u1∂T=0<∞.
Similarly, foru2, u3, u4, u5, u7, u8, and u9, the system of equations (1)–(9) exists and has a unique solution in ℜ9 since all the partial derivatives are <∞ and therefore bounded and defined.


## 2.4.1. Existence and Uniqueness of Solution

The existence and uniqueness of solution in formulated mathematical models need to undergo surety test to ascertain whether the solution exists, and if it exists, it is unique. Using the Lipschitz criteria, it is established as follows.Let(10)u1=q−v11−χϕS+1−χ1−hv2SI+ωV+ωR−ψ+δS,u2=ψS−σv2VI−ω+δV,u3=v11−χϕS+1−χ1−hv2SI+σv2VI−ξ+h+δE,u4=ξE−m+ε+δI,u5=hE−f+θ+δC,u6=fC+1−τmI+g2T−g1+r1+μ+δU,u7=τmI+g1U−g1+r2+g2+α+δT,u8=g1+r2T+r1U+θC−ω+δR,u9=1−hη1I+η2−βB.Considering the region0≤P≤ℜ, bounded solution of system of equations (1)–(9) will be looked for in the same region whose partial derivatives satisfy δ1≤P≤ℜ, where δ1 and P are positive constants.Theorem 1.
Let the region0≤P≤ℜ be denoted by Λ, and then, the system of equations (1)–(9) has a unique solution provided that it is established that(11)∂ui∂tj,i=1,2,3,4,5,6,7,8,9,is continuous and bounded in Λ.Proof.
Partial derivatives with respect to each state variables of the system of equations (1)–(9) are obtained as follows: for u1,(12)∂u1∂S=−v11−χϕ−1−χ1−hv2I−ψ−δ<∞,∂u1∂V=ω<∞,∂u1∂I=−1−χ1−hv2S<∞,∂u1∂B=v11−χBSk+B2−v11−χSk+B<∞,∂u1∂R=ω<∞,∂u1∂E=∂u1∂C=∂u1∂U=∂u1∂T=0<∞.
Similarly, foru2, u3, u4, u5, u7, u8, and u9, the system of equations (1)–(9) exists and has a unique solution in ℜ9 since all the partial derivatives are <∞ and therefore bounded and defined.


## 2.5. Positivity of Solutions

Let equations (1)–(9) be subject to initial conditions:(13)S0,V0,E0,I0,C0,U0,T0,R0,B0.For theSVEICUTRB cholera model to be meaningfully epidemiological, it is mandatory to prove that all the state variables are nonnegative at time t>0, since the population of human is non negative but positive. This implies that the solutions of the system of equations (1)–(9) have nonnegative initial values and will remain nonnegative for all time t>0.Let(14)St,Vt,Et,It,Ct,Ut,Tt,Rt,Bt,be solutions to the system of the equations (1)–(9).Theorem 2.
(a) Solution (11) is positive for all t>0 if S0>0, V0>0, E0>0, I0>0, C0>0, U0>0, T0>0, R0>0, and B0>0. (b) Solution (11) is nonnegative for all t>0 with respect to initial conditions (10).Proof.
(a) Suppose that (11) is defined for all t∈0,M, where M>0, to prove that (11) is positive for all t≥0.
Considering (1), it is obvious that(15)dStdt≥−v11−χϕ+1−χ1−hv2I+ψ+δS.
Integrating both sides of (15) with respect to p, with the range p=0 to p=t, and applying initial condition, it gives(16)St=S0exp∫0t−v11−χϕ+1−χ1−hv2I+ψ+δdp.
SinceS0>0, ∀t∈0,M.
Therefore,(17)S0exp−v11−χϕ+1−χ1−hv2I+ψ+δt,must be greater than zero.
Also, from (2)(18)dVtdt≥−σv2I+ω+δV,∀0≤t<M.
Integrating both sides within the interval0,t and applying initial conditions, it gives(19)Vt=V0exp−σv2I+ω+δt,and since V0>0, V0exp−σv2I+ω+δt must be greater than zero.
Similarly, from equations (3)–(9), their respective results are as follows:(20)Et≥E0exp−ξ+h+δt>0,∀0≤t<M,It≥I0exp−τm+1−τm+ε+δt>0,∀0≤t<M,Ct≥C0exp−f+θ+δt>0,∀0≤t<M∈,Ut≥U0exp−g1+r1+μ+δt>0,∀0≤t<M,Tt≥T0exp−g1+g2+r2+α+δ,∀0≤t<M,Rt≥R0exp−ω+δt>0,∀0≤t<M,Bt≥B0exp−βt>0,∀0≤t<M.
(b) With the aid of proof of (a) and since the solution of the system of equations (1)–(9) continuously depends on the respective initial conditions in neighbourhood of zero, obviously it implies that (14) is nonnegative for all 0≤t<M.
The model is dealing with population of living things, their number cannot be negative, and hence they are positive.Theorem 3.
Solutions of cholera modelSVEICUTRB are bounded in ℜ+9 with initial conditions (10), and hence, dynamics of the model is a dynamical system in the biologically feasible compact set.Proof.
LetΛ=S,V,E,I,C,U,T,R,B∈ℜ+9 be the solutions of the model with nonnegative initial conditions. Total human population is as follows:(21)PH=S+V+E+I+C+U+T+R,dPHdt=dSdt+dVdt+dEdt+dIdt+dCdt+dUdt+dTdt+dRdt,dPHdt=q−δS+V+E+I+C+U+T+R−ϵI−αT−μU.
Without cholera infection, all the cholera-related deaths will be zero; therefore,ϵI=αT=μU=0.(22)dPHdt=q−δPH.
Using differential inequality theorem from [17] and the Birkhoff and Rota version as used in [11, 18],(23)dPHdt≤q−δPH.
Multiplying both sides of (23) by the integrating factor expδt and integrating, it gives(24)PHexpδt≤qδ+C.
Substituting initial conditions are as follows:t=0 and PH0=P0.
This further givesPH≤q/δ+P0−q/δexp−δt
Ast⟶∞, P0−q/δexp−δt⟶0, which gives PH≤q/δ, but PH≥0; therefore, for all t>0,(25)0≤PHt≤qδ.
Hence,PHt is bounded.
The solution set of human population of the system of equations (1)–(9) belongs to the feasible region:(26)ΛH=S,V,E,I,C,U,T,R∈ℜ+8,PH≤qδ.
Also, let(27)PHt≤PH∗=maxP0,qδ.
On the other hand, the equation for the dynamics of the pathogen is (9) as follows:(28)dBtdt=1−hη1I+η2−βBt≤η2−βBt+1−hη1PH.
SinceI can be less than q/δ, then from (27),(29)dPBtdt≤η2−βBt+1−hη1PH∗.
Integrating (29) gives(30)PBt≤1−hη1PH∗η2−βBt1−exp−η2−βt+B0expη2−βt.
Ast⟶∞, then expη2−βt⟶0.
The population ofVibrio cholerae bacteria is as follows:(31)PBt≤1−hη1PH∗η2−β,provided η2≠β. Hence, the feasible solution set of Vibrio cholerae population of (9) enters the region(32)ΛB=PBt∈ℜ+|0≤PBt≤1−hη1PH∗η2−β.
This shows that the cholera model governed by the system of equations (1)–(9) is epidemiological well posed in a feasible region Λ, defined by Λ∈ℜ+9 and ΛH∪ΛB⊂ℜ+8∗ℜ+.
It then implies that the entire human and nonhuman population of the model enter the feasible regionΛ. Also, since PHt>0 and PBt≥0, for all t≥0, then PHt and PBt cannot blow up to infinity in the finite time. Consequently, the model system is dissipative; that is, its solutions are bounded and exist for all t>0 in the invariant and compact set Λ.(33)Λ=S,V,E,I,C,U,T,R∈ℜ+8,B∈ℜ+,S,V,E,I,C,U,T,R,B≥0,PH≤qδ,PH≤PH∗,PB≤1−hη1PH∗η2−β.
The regionΛ is positively invariant and attracts all solutions in ℜ+9.


### 2.5.1. Disease-Free Equilibrium

Analysis of cholera model governed by a system of equations (1)–(9) is qualitatively carried out to obtain features of the dynamics and impact of control strategies, spread, and transmission dynamics of the cholera causative bacteria.Disease-free equilibrium of the model governed by system of equations (1)–(9) is algebraically calculated by setting dS/dt=dV/dV=dE/dt=dI/dt=dC/dt=dU/dt=dT/dt=dR/dt=dB/dt=0.In the absence of cholera disease, it is assumed thatS=S0,V=V0,E=0,I=0,C=0,U=0,T=0,R=0, and B=0.(34)S0=qω+δψ+δω+δ−ωψ,V0=qψψ+δω+δ−ωψ.The DFE point is as follows:(35)Q0=S0,V0,E,I,C,U,T,R,B=qω+δψ+δω+δ−ωψ,qψψ+δω+δ−ωψ,0,0,0,0,0,0,0.


### 2.5.2. Existence of Endemic Equilibrium State (EES)Ξ∗

This is a state where the disease cannot be completely eliminated, but stay in the concern region. The state variables of the model with exception ofR∗ are not zero.LetΞ∗=S∗,V∗,E∗,I∗,C∗,U∗,T∗,R∗,B∗ be the endemic equilibrium point, and let(36)Q1=ξ+h+δ,Q2=m+ε+δ,Q3=g1+r1+μ+δ,Q4=g1+g2+r2+α+δ,Q5=ψ+δ,Q6=1−χ,Q7=1−h,Q8=Bk+B,Q9=ω+δ,Q10=f+θ+δ,Q11=g1+r2,Q12=η2−β.Substituting (37) into the system of equations (1)–(9), the following is obtained:(37)q−v1Q6Q8S∗−Q6Q7v2S∗I∗+ωV∗+ωR∗−Q5S∗=0,(38)ψS∗−σv2V∗I∗−Q9V∗=0,(39)v1Q6Q8S∗+Q6Q7v2S∗I∗+σv2V∗I∗−Q1E∗=0,(40)ξE∗−Q2I∗=0,(41)hE∗−Q10C∗=0,(42)fC∗+1−τmI∗+g2T∗−Q3U∗=0,(43)τmI∗+g1U∗−Q4T∗=0,(44)Q11T∗+r1U∗+θC∗−Q9R∗=0,(45)Q7η1I∗+Q12B∗=0,Adding (19) and (21),(46)q+ωR∗+ω+δv2I∗V∗−Q5S∗−Q1E∗=0.From (22),(47)E∗=Q2I∗ξ,and from (23),(48)C∗=hE∗Q10.Putting (40) into (41),(49)C∗=hQ2I∗ξQ10.From (25),(50)T∗=τmI∗+g1U∗Q4.Substituting (42) and (43) into (24) gives(51)U∗=fhQ2Q4I∗+ξQ4Q101−τmI∗+ξQ10g2τmI∗ξQ10Q3Q4−g1.Let(52)H1=fhQ2Q4,H2=ξQ4Q101−τm,H3=ξQ10g2τm,H4=ξQ10Q3Q4−g1,H5=Q10Q11ξτmH4+g1H1+H2+H3,H6=ξr1Q4Q10H1+H2+H3,H7=θhQ2Q4H4,H8=ξH4Q4Q9Q10.Substituting related terms of (45) into (44) gives(53)U∗=H1+H2+H3H4I∗,and from (43),(54)T∗=τmH4+g1H1+H2+H3H4Q4I∗.From (20),(55)V∗=ψS∗σv2I∗−Q9.Putting (31), (35), and (36) in (26), we obtain(56)R∗=H5+H6+H7I∗H8.Also let(57)H10=H8ξq+ξωH5+H6+H7−H8Q1Q2,H11=σv2Q5−ψ,H12=Q5Q9−ψω..Substituting (29), (37), (38), and related terms of (50) in (39) gives(58)S∗=H10I∗σv2I∗−Q9H11I∗−H12.Substituting (51) into (48) and simplifying, (48) becomes(59)V∗=ψH10I∗H11−H12.Substituting (29), (40), and (41) into (21) gives the following:(60)ξv22Q6Q7H10σI∗3+ξH10σv1v2Q6Q8−ξH10v2Q6Q7Q9+ξψH10σv2−Q1Q2H11I∗2+H12Q1Q2−ξH10v1Q6Q8Q9I∗=0,setting(61)B1=ξv22Q6Q7H10σ,B2=ξH10σv1v2Q6Q8−ξH10v2Q6Q7Q9+ξψH10σv2−Q1Q2H11,B3=H12Q1Q2−ξH10v1Q6Q8Q9..This gives(62)B1I∗3+B2I∗2+B3I∗=0,B1I∗2+B2I∗+B3I∗=0,which means either(63)B1I∗2+B2I∗+B3=0,or(64)I∗=0.ForI∗=0, it implies a disease-free equilibrium; when H12Q1Q2<ξH10v1Q6Q8Q9, it implies B1>0 and B3<0; hence, product of the roots I1∗I2∗=B3/B1<0, which means either I1∗>0 or I2∗>0, and the positive root will then be(65)I∗=−B2+B22−4B1B32B1.Therefore, there exists an endemic equilibrium stateΞ∗ defined as follows:(66)Ξ∗=H10σv2Δ2−H10Q9ΔH11Δ−H12,ψH10ΔH11Δ−H12,Q2Δξ,Δ,hQ2ΔξQ10,H1+H2+H3ΔH4,τmH4+g1H1+H2+H3ΔH4Q4,H5+H6+H7ΔH8,where(67)Δ=−B2+B22−4B1B32B1.Note thatη2−β must be negative for endemic.


## 2.5.1. Disease-Free Equilibrium

Analysis of cholera model governed by a system of equations (1)–(9) is qualitatively carried out to obtain features of the dynamics and impact of control strategies, spread, and transmission dynamics of the cholera causative bacteria.Disease-free equilibrium of the model governed by system of equations (1)–(9) is algebraically calculated by setting dS/dt=dV/dV=dE/dt=dI/dt=dC/dt=dU/dt=dT/dt=dR/dt=dB/dt=0.In the absence of cholera disease, it is assumed thatS=S0,V=V0,E=0,I=0,C=0,U=0,T=0,R=0, and B=0.(34)S0=qω+δψ+δω+δ−ωψ,V0=qψψ+δω+δ−ωψ.The DFE point is as follows:(35)Q0=S0,V0,E,I,C,U,T,R,B=qω+δψ+δω+δ−ωψ,qψψ+δω+δ−ωψ,0,0,0,0,0,0,0.


## 2.5.2. Existence of Endemic Equilibrium State (EES)Ξ∗

This is a state where the disease cannot be completely eliminated, but stay in the concern region. The state variables of the model with exception ofR∗ are not zero.LetΞ∗=S∗,V∗,E∗,I∗,C∗,U∗,T∗,R∗,B∗ be the endemic equilibrium point, and let(36)Q1=ξ+h+δ,Q2=m+ε+δ,Q3=g1+r1+μ+δ,Q4=g1+g2+r2+α+δ,Q5=ψ+δ,Q6=1−χ,Q7=1−h,Q8=Bk+B,Q9=ω+δ,Q10=f+θ+δ,Q11=g1+r2,Q12=η2−β.Substituting (37) into the system of equations (1)–(9), the following is obtained:(37)q−v1Q6Q8S∗−Q6Q7v2S∗I∗+ωV∗+ωR∗−Q5S∗=0,(38)ψS∗−σv2V∗I∗−Q9V∗=0,(39)v1Q6Q8S∗+Q6Q7v2S∗I∗+σv2V∗I∗−Q1E∗=0,(40)ξE∗−Q2I∗=0,(41)hE∗−Q10C∗=0,(42)fC∗+1−τmI∗+g2T∗−Q3U∗=0,(43)τmI∗+g1U∗−Q4T∗=0,(44)Q11T∗+r1U∗+θC∗−Q9R∗=0,(45)Q7η1I∗+Q12B∗=0,Adding (19) and (21),(46)q+ωR∗+ω+δv2I∗V∗−Q5S∗−Q1E∗=0.From (22),(47)E∗=Q2I∗ξ,and from (23),(48)C∗=hE∗Q10.Putting (40) into (41),(49)C∗=hQ2I∗ξQ10.From (25),(50)T∗=τmI∗+g1U∗Q4.Substituting (42) and (43) into (24) gives(51)U∗=fhQ2Q4I∗+ξQ4Q101−τmI∗+ξQ10g2τmI∗ξQ10Q3Q4−g1.Let(52)H1=fhQ2Q4,H2=ξQ4Q101−τm,H3=ξQ10g2τm,H4=ξQ10Q3Q4−g1,H5=Q10Q11ξτmH4+g1H1+H2+H3,H6=ξr1Q4Q10H1+H2+H3,H7=θhQ2Q4H4,H8=ξH4Q4Q9Q10.Substituting related terms of (45) into (44) gives(53)U∗=H1+H2+H3H4I∗,and from (43),(54)T∗=τmH4+g1H1+H2+H3H4Q4I∗.From (20),(55)V∗=ψS∗σv2I∗−Q9.Putting (31), (35), and (36) in (26), we obtain(56)R∗=H5+H6+H7I∗H8.Also let(57)H10=H8ξq+ξωH5+H6+H7−H8Q1Q2,H11=σv2Q5−ψ,H12=Q5Q9−ψω..Substituting (29), (37), (38), and related terms of (50) in (39) gives(58)S∗=H10I∗σv2I∗−Q9H11I∗−H12.Substituting (51) into (48) and simplifying, (48) becomes(59)V∗=ψH10I∗H11−H12.Substituting (29), (40), and (41) into (21) gives the following:(60)ξv22Q6Q7H10σI∗3+ξH10σv1v2Q6Q8−ξH10v2Q6Q7Q9+ξψH10σv2−Q1Q2H11I∗2+H12Q1Q2−ξH10v1Q6Q8Q9I∗=0,setting(61)B1=ξv22Q6Q7H10σ,B2=ξH10σv1v2Q6Q8−ξH10v2Q6Q7Q9+ξψH10σv2−Q1Q2H11,B3=H12Q1Q2−ξH10v1Q6Q8Q9..This gives(62)B1I∗3+B2I∗2+B3I∗=0,B1I∗2+B2I∗+B3I∗=0,which means either(63)B1I∗2+B2I∗+B3=0,or(64)I∗=0.ForI∗=0, it implies a disease-free equilibrium; when H12Q1Q2<ξH10v1Q6Q8Q9, it implies B1>0 and B3<0; hence, product of the roots I1∗I2∗=B3/B1<0, which means either I1∗>0 or I2∗>0, and the positive root will then be(65)I∗=−B2+B22−4B1B32B1.Therefore, there exists an endemic equilibrium stateΞ∗ defined as follows:(66)Ξ∗=H10σv2Δ2−H10Q9ΔH11Δ−H12,ψH10ΔH11Δ−H12,Q2Δξ,Δ,hQ2ΔξQ10,H1+H2+H3ΔH4,τmH4+g1H1+H2+H3ΔH4Q4,H5+H6+H7ΔH8,where(67)Δ=−B2+B22−4B1B32B1.Note thatη2−β must be negative for endemic.


## 3. Basic Reproduction NumberR0

Basic reproduction number denoted byR0 is the average number of secondary infections produced by a single infective introduced into the whole susceptible population.R0 is a threshold quantity that can be used to predict infection; when R0<1, it implies that only less than one person can be infected, which means the disease dies out. When R0>1 implies that more than one person can be infected, this means the disease will persist.There are many methods of obtainingR0 for mathematical models, and “the next-generation matrix method,” a technique introduced by [19], which was also reported by [20], was used in this work, simply because the model is divided into compartments.TakingFi to be the rate of appearance of new infection in compartment i, Vi is the transfer of individuals out of compartment i by other means and substituting values of state variables at disease-free equilibrium point Q0. R0 will then be the spectral radius of (68)∂FiQ0∂Qj⋅∂ViQ0∂Qj−1.Note thatF is nonnegative, V is a nonsingular M matrix, V−1 is nonnegative, and FV−1 is also nonnegative as in [19]. The matrix FV−1 was referred as the next-generation matrix as in [20] that gave the method its name. Finally, R0=ρFV−1, which is the spectral radius of FV−1, considered as maximum real part of the matrix FV−1. Rearranging the equations of the model as Ft, V+t, and V−t, the R0 is calculated and obtained as follows:(69)Ft=v11−χBk+BS+1−χ1−hv2SI+σv2VI00000,V+t=ξEfC+1−τmI+g2TτmI+g1U0q+ωV+ωR1−hη1I+η2B,V−t=τm+1−τm+ε+δIg1+r1+μ+δUg1+g2+r2+α+δTξ+h+δEv11−χBk+BS+1−χ1−hv2SI+ψ+δSβB,with Vt=V−t−V+t.(70)Vt=τm+1−τm+ε+δI−ξEg1+r1+μ+δU−fC−1−τmI−g2Tg1+g2+r2+α+δT−τmI−g1Uξ+h+δEv11−χBk+BS+1−χ1−hv2SI+ψ+δS−q−ωV−ωRβB−1−hη1I−η2B.Substituting terms given by (37), it gives(71)Ft=v1Q6Q8S+Q6Q7v2SI+σv2VI00000,V1t=Q2I−ξEQ3U−fC−1−τmI−g2TQ4T−τmI−g1UQ1Ev1Q6Q8S+Q6Q7v2SI+Q5S−q−ωV−ωRβB−Q7η1I−η2B.The Jacobian matrices ofFt and V1t about the disease-free equilibrium point Q0 are obtained by partially differentiating Ft and V1t with respect to I,U,T,E,S, and B to get matrices FJ and V1J.(72)FJ=Q6Q7v2S+σv2V0000Q6v1Sk000000000000000000000000000000,V1J=Q200−ξ00−1−τmQ3−g2000−τm−g1Q4000000Q100Q6Q7v2S000Q5Q6v1Sk−Q7η10000β−η2.LetF−1=FJ−1 and V1−1=V1J−1.(73)V1−1=1Q200ξQ1Q200−mQ4τ−g2τ−Q4Q2Q3Q4−g1g2Q4Q3Q4−g1g2g2Q3Q4−g1g2−mξQ4τ−g2τ−Q4Q2Q1Q3Q4−g1g200mQ3τ−g1τ+g1Q2Q3Q4−g1g2g1Q3Q4−g1g2Q3Q3Q4−g1g2mξQ3τ−g1τ+g1Q2Q1Q3Q4−g1g2000001Q100−Q6Q7Sβkv2−η2kv2+η1v1kQ2β−η2Q500−Q6Q7ξSβkv2−η2kv2+η1v1kQ2Q1β−η2Q51Q5−Q6v1Skβ−η2Q5Q7η1Q2β−η200Q7η1ξQ2Q1β−η201β−η2.LetZ=FV1−1 and R0=ρZ; hence, R0=qQ6Q7Q9βkv2−Q6Q7Q9η2kv2+Q6Q7Q9η1v1+βkψσv2−η2kψσv2/Q5Q9−ωψkQ2β−η2.(74)R0=q1−χ1−hω+δβkv2−1−χ1−hω+δη2kv2+1−χ1−hω+δη1v1+βkψσv2−η2kψσv2ψ+δω+δ−ωψkm+ε+δβ−η2.


### 3.1. Sensitivity Analysis

Sensitivity analysis is determining the level of effectiveness of parameters. The sensitivity analysis was carried out on model’sR0. The most sensitive parameters in this model are h and v2; increasing h reduces R0, thereby reducing infection. On the other hand, increasing v2 increases R0, thereby increasing infection. The normalised forward sensitivity indexes were performed using the following formula:(75)ζΩR0=ΩR0⋅∂R0∂Ω,where Ω=q,h,δ,m,ε,v1,v2,χ,ω,ψ,σ,β,η1,η2,k.Using (68), the level of sensitivity of the R0 parameters was obtained as given in Table 3. Level of sensitivity of each parameter in the obtained R0 is shown in Figure 2.Table 3
Level of sensitivity ofR0 parameters.
ParameterSensitivity valueParameterSensitivity valueq1.0000ω−0.0512h−0.8671ψ0.0524δ−1.0390σ0.4219m−0.6415β−0.0026ϵ−0.3208η10.0023v10.0023η20.0004v20.9978k−0.0023χ−0.8671Figure 2
Graph of sensitivity analysis of parameters in the model’s “Basic reproduction number”R0.


## 3.1. Sensitivity Analysis

Sensitivity analysis is determining the level of effectiveness of parameters. The sensitivity analysis was carried out on model’sR0. The most sensitive parameters in this model are h and v2; increasing h reduces R0, thereby reducing infection. On the other hand, increasing v2 increases R0, thereby increasing infection. The normalised forward sensitivity indexes were performed using the following formula:(75)ζΩR0=ΩR0⋅∂R0∂Ω,where Ω=q,h,δ,m,ε,v1,v2,χ,ω,ψ,σ,β,η1,η2,k.Using (68), the level of sensitivity of the R0 parameters was obtained as given in Table 3. Level of sensitivity of each parameter in the obtained R0 is shown in Figure 2.Table 3
Level of sensitivity ofR0 parameters.
ParameterSensitivity valueParameterSensitivity valueq1.0000ω−0.0512h−0.8671ψ0.0524δ−1.0390σ0.4219m−0.6415β−0.0026ϵ−0.3208η10.0023v10.0023η20.0004v20.9978k−0.0023χ−0.8671Figure 2
Graph of sensitivity analysis of parameters in the model’s “Basic reproduction number”R0.


## 4. Optimal Control Problem of the Model

Minimising susceptible, exposed, and infected individuals through hygiene consciousness, vaccine coverage, and cholera awareness strategies is the goal set to achieve using optimisation. The objectives of optimizing the model are: to reduce as much as possible, cholera susceptible individuals, exposed individuals, and cholera infection through three controls, hygiene consciousness of human, vaccinating susceptible individuals and awareness of measures to be taken to avoid cholera infection. The model is reformulated to identify permissible time-dependent controlsχ1t, χ2t, and χ3t, which are now functions of time t, and are, respectively, replacing parameters h, ψ, and χ of the system of equations (1)–(9) that is reformulated as follows:(76)dSdt=q−v11−χ3tBtk+BtSt+1−χ1t1−χ3tv2StIt+ωVt+ωRt−χ2t+δSt,dVdt=χ2tSt−σv2VtIt−ω+δVt,dEdt=v11−χ3tBtk+BtSt+1−χ1t1−χ3tv2StdtIt+σv2VtIt−ξ+χ1t+δEt,dIdt=ξEt−m+ϵ+δIt,dCdt=χ1tEt−f+θ+δCt,dUdt=fCt+1−τmIt+g2Tt−g1+r1+μ+δUt,dTdt=τmIt+g1Ut−g1+r2+g2+α+δTt,dRdt=g1+r2Tt+r1Ut+θCt−ω+δRt,dBdt=1−χ1tη1It+η2−βBt.Considering biological reasons, initial conditions of the state variables are assumed to be as follows: letS0=S0 similarly for the remaining state variables I0≥0,S0>0,V0≥0,E0≥0,C0≥0,U0≥0,T0≥0,R0≥0,andB0≥0.Let the terminal time beTf.The objective functional or performance index of the model is as follows:(77)Jχ1,χ2,χ3=∫0TfM1St+M2Et+M3It+12N1χ12+N2χ22+N3χ32dt.The set of controlsw=χ1t,χ2t,χ3t∈L is Lebesgue measurable where(78)L=χ1,χ2,χ3|χ1,χ2,χ3:0,Tf⟶0,1,∀χ1,χ2,χ3.M1, M2, and M3, respectively, denote the weight constant of susceptible, exposed, and infected, while N1, N2, and N3, respectively, denote the constant relative cost weight of hygiene consciousness, rate of vaccine, and public awareness programme on cholera. The controls take their values in a closed interval 0,1.χ1=1 implies inhibition of cholera outbreak and spread, while χ1=0 means no cholera outbreak and spread inhibition. χ2=1 implies 100%vaccination coverage and effective cholera immune population. From general experience in vaccinations, as in [21], about 90% of susceptible is usually the maximum to be covered in vaccination programme. χ2=0 means the susceptible have only natural cholera immunity (which might be weak) and are not complimented with vaccine, and this in turn increases chance of catching cholera. χ3=1 implies full cholera awareness in the population, which is assumed to alert the community to observe preventive measures to contain and curtail cholera outbreak and spread. χ3=0 will render the population unaware of the best ways to avoid and prevent the disease outbreak and spread.The main aim is to minimise susceptible, exposed, and infected individuals to a level that will reduce cholera disease escalation through these three controls,χ1∗,χ2∗,χ3∗ such that (79)Jχ1∗,χ2∗,χ3∗=minχ1,χ2,χ3Jχ1,χ2,χ3|χ1,χ2,χ3∈L.


### 4.1. Existence of an Optimal Control Set

From (33), all feasible solutions of system (69) enter into the region Λ. The system (69) can be written in the form(80)dpdt=Qp+Zp,where(81)pt=St,Vt,Et,It,Ct,Ut,Tt,Rt,BtT.(82)Q=−B1000000000−B2000000000−B3000000000−B4000000000−B5000000000−B6000000000−B7000000000−B8000000000−B8,Zp=q−B9St−B10St+ωVt+ωRtχ2tSt−B11B9St+B10St+B11ξEtχ1tEtfCt+1−τmIt+g2TtτmIt+g1Utg1+r2Tt+r1Ut+θCt1−χ1tη1It,where(83)B1=χ1t+δ,B2=ω+δ,B3=ξ+χ1t+δ,B4=m+ε+δ,B5=f+θ+δ,B6=g1+r1+μ+δ,B7=g1+r2+g2+α+δ,B8=η2+β,B9=v11−χ3tBtk+Bt,B10=1−χ1t1−χ3tv2It,B11=σv2VtIt.Zp satisfies(84)Zp1−Zp2≤ρ1S1t−S2t+ρ2V1t−V2t+ρ3E1t−E2t+ρ4I1t−I2t+ρ5C1t−C2t+ρ6U1t−U2t+ρ7T1t−T2t+ρ8R1t−R2t+ρ9B1t−B2t≤ρnS1t−S2t+V1t−V2t+E1t−E2t+I1t−I2t+C1t−C2t+U1t−U2t+T1t−T2t+R1t−R2t+B1t−B2t,with the positive constant ρn=maxρifori=1,2,…,9, independent to the state variables St, Vt, Et, It, Ct, Ut, Tt, Rt, and Bt.It follows that optimality system and the optimal controls exist and the solution of system is bounded, since(85)Zp1−Zp2≤ρnZ1−Z2,where(86)ρn=∑i=19ρi+K<∞.This follows that the functionZ is uniformly Lipchitz continuous. Hence, the control variables and nonnegative initial conditions reveal that a solution of the system (69) exists (see [22, 23]).Theorem 4.
Given the objective functional (48) with set L defined in (49) subject to the system (46) with initial conditions (47), then there exists an optimal control set w1=χ1∗,χ2∗,χ3∗ such that (50) is true if the following conditions hold (KS,E,I,χ1,χ2,χ3≤μ2−μ1χ1t2+χ2t2+χ3t2α1/2):(i)
The set of controls and corresponding state variables is nonempty(ii)
The control setL is closed and convex(iii)
Each right-hand side of (46) is continuous bounded above by a sum of the bounded control and state and can be written as a linear function w with coefficients depending on time and state(iv)
The integrand of the objective functional is convex onL(v)
There exist nonnegative constantsμ1 and μ2 and α1>1 satisfying the following expressionProof.
Using established approach as in [6, 24, 25], the existence of solution of (46) is satisfied, the bounded coefficients and all solutions are bounded on a finite time interval, and first condition is satisfied.
It suffices to state thatL=La×Lb, where La and Lb are closed and convex sets defined as follows: (87)La=Lb=ωmeasurable:χ1t∈0,1,χ2t∈0,0.95,χ3t∈0,1,∀t∈t0,Tf.
Hence, by definition of convex, the control set is convex and closed, satisfying second condition.
By definition, each right-hand side of the system (69) is continuous and can be written as a linear function of ω with coefficients depending on time and state. All variables S,V,E,I,C,U,T,R,B,χ1,χ2, and χ3 are bounded on t0,Tf. Using the boundedness of the solution and its positivity, as (69) is bilinear in χ1,χ2, and χ3, its right-hand side satisfies condition 3. The integrand in the objective functional (71) is convex on L, and the fourth condition is satisfied.
Finally, it can be easily seen that there exist a constantα1>1 and positive numbers μ1 and μ2 satisfying(88)KS,E,I,χ1,χ2,χ3≤μ2−μ1χ12+χ22+χ32α1/2.
The Hamiltonian equation is formed accordingly by allowing each adjoint variable to correspond to each state variable, all combined with the objective functional.
Applying Pontryagin’s maximum principle, system (69) together with equations (48) and (49) is converted into problem of minimising the Hamiltonian H defined below.
Since necessary conditions that an optimal control and state must satisfy can be generated from HamiltonianH, see [26].
Hence, the Hamiltonian functionH of the control problem of the model will be(89)HS,V,E,I,C,U,T,R,B,χ1,χ2,χ3,λS,λV,λE,λI,λC,λU,λT,λR,λB=Θ+λSdSdt+λVdVdt+λEdEdt+λIdIdt+λCdCdt+λUdUdt+λTdTdt+λRdRdt+λBdBdt,where λS,λV,λE,λI,λC,λU,λT,λR, and λB are the adjoint functions, which can also be the co-state functions with respect to H and(90)Θ=M1St+M2Et+M3It+12N1χ1t2+N2χ2t2+N3χ3t2,H=M1St+M2Et+M3It+12N1χ1t2+N2χ2t2+N3χ3t2+λSq−v11−χ3tBtk+BtSt+1−χ1t1−χ3tv2StIt+ωVt+ωRt−χ2t+δSt+λVχ2tSt−σv2VtIt−ω+δVt+λEv11−χ3tBtk+BtSt+1−χ1t1−χ3tv2StIt+σv2VtIt−ξ+χ1t+δEt+λIξEt−m+ε+δIt+λCχ1tEt−f+θ+δCt+λUfCt+1−τmIt+g2Tt−g1+r1+μ+δUt+λTτmIt+g1Ut−g1+r2+g2+α+δTt+λRg1+r2Tt+r1Ut+θCt−ω+δRt+λB1−χ1tη1It+η2−βBt.
LetS∗∗,V∗∗,E∗∗,I∗∗,C∗∗,U∗∗,T∗∗,R∗∗, and B∗∗ be the optimal state solutions with associated optimal control variables χ1∗,χ2∗,χ3∗ for the optimal control of system (69).
To achieve optimal control, the adjoint or co-state functionsλS,λV,λE,λI,λC,λU,λT,λR, and λB must satisfy(91)dλSdt=−∂H∂S,dλVdt=−∂H∂V,dλEdt=−∂H∂E,dλIdt=−∂H∂I,dλCdt=−∂H∂C,dλUdt=−∂H∂U,dλTdt=−∂H∂T,dλRdt=−∂H∂R,dλBdt=−∂H∂B.(92)dλSdt=−M1−λS−v11−χ3Bk+B−1−χ11−χ3v2I−χ2−δ−λVχ2−λEv11−χ3Bk+B+1−χ11−χ3v2I,dλVdt=−λSω−λV−σv2I−δ−ω−λEσv2I,dλEdt=−M2−λE−ξ−χ1−δ−λIξ−λCχ1,dλIdt=−M3−λS1−χ11−χ3v2S+λVσv2V−λE1−χ11−χ3v2S+σv2V−λI−m−ε−δ−λU1−τm−λTτm−λB1−χ1η1,dλCdt=−λC−f−θ−δ−λUf−λRθ,dλUdt=−λU−g1−r1−μ−δ−λTg1−λRr1,dλTdt=−λUg2−λT−g1−r2−g2−α−δ−λRg1+r2,dλRdt=−λSω−λR−ω−δ,dλBdt=−λS−v11−χ3Sk+B+v11−χ3BSk+B2−λEv11−χ3Sk+B−v11−χ3BSk+B2−λBη2−β.


### 4.2. Optimality Equations of the Model’s Controls

Partial derivative of the Hamiltonian with respect to each admissible control is all equal to zero, ease solving the optimal controls; thus,(93)⇒∂H∂χ1=N1χ1+λS1−χ3v2SI+λE−1−χ3v2SI−E−λBη1I=0,⇒χ1∗=χ3λSv2S∗∗I∗∗−χ3λEv2S∗∗I∗∗−λSv2S∗∗I∗∗+λVv2S∗∗I∗∗+η1λBI∗∗+λEE∗∗N1,⇒∂H∂χ2=N2χ2−λSS+λVS,⇒χ2∗=S∗∗λS−λVN2,⇒∂H∂χ3=N3χ3+λSv1BSk+B+1−χ1v2SI+λCE+λE−v1BSk+B−1−χ1v2SI,⇒χ3∗=1k+BN3χ1kv2S∗∗I∗∗λS−χ1kv2S∗∗I∗∗λEχ1v2S∗∗I∗∗B∗∗λS−χ1v2S∗∗I∗∗B∗∗λEkv2S∗∗I∗∗λS+kv2S∗∗I∗∗λE−v2S∗∗I∗∗B∗∗λS+v2S∗∗I∗∗B∗∗λE−v1S∗∗B∗∗λS+v1S∗∗B∗∗λE−kE∗∗λC−E∗∗B∗∗λC.Now,a1≤χ1≤b1, a2≤χ2≤b2, and a3≤χ3≤b3.The characterisation is as follows:(94)χ1∗=a1,whenχ1¯≤a1,χ1¯,whena1<χ1¯<b1,b1,whenχ1¯≥b1.χ2∗=a2,whenχ2¯≤a2,χ2¯,whena2<χ2¯<b2,b2,whenχ2¯≥b2.χ3∗=a3,whenχ3¯≤a3,χ3¯,whena3<χ3¯<b3,b3,whenχ3¯≥b3,where(95)χ1¯=χ3−1v2S∗∗I∗∗λS+E∗∗−χ3S∗∗I∗∗λE+v2S∗∗I∗∗λV+η1I∗∗λBN1,χ2¯=S∗∗λS−λVN2,χ3¯=χ1k+χ1B∗∗−k−B∗∗v2S∗∗I∗∗−v1S∗∗B∗∗λS+B∗∗−χ1k−χ1B∗∗−kv2S∗∗I∗∗λEk+B∗∗N3, (96)χ1∗t=max0,minχ1¯t,1,χ2∗t=max0,minχ2¯t,0.95,χ3∗t=max0,minχ3¯t,1.


## 4.1. Existence of an Optimal Control Set

From (33), all feasible solutions of system (69) enter into the region Λ. The system (69) can be written in the form(80)dpdt=Qp+Zp,where(81)pt=St,Vt,Et,It,Ct,Ut,Tt,Rt,BtT.(82)Q=−B1000000000−B2000000000−B3000000000−B4000000000−B5000000000−B6000000000−B7000000000−B8000000000−B8,Zp=q−B9St−B10St+ωVt+ωRtχ2tSt−B11B9St+B10St+B11ξEtχ1tEtfCt+1−τmIt+g2TtτmIt+g1Utg1+r2Tt+r1Ut+θCt1−χ1tη1It,where(83)B1=χ1t+δ,B2=ω+δ,B3=ξ+χ1t+δ,B4=m+ε+δ,B5=f+θ+δ,B6=g1+r1+μ+δ,B7=g1+r2+g2+α+δ,B8=η2+β,B9=v11−χ3tBtk+Bt,B10=1−χ1t1−χ3tv2It,B11=σv2VtIt.Zp satisfies(84)Zp1−Zp2≤ρ1S1t−S2t+ρ2V1t−V2t+ρ3E1t−E2t+ρ4I1t−I2t+ρ5C1t−C2t+ρ6U1t−U2t+ρ7T1t−T2t+ρ8R1t−R2t+ρ9B1t−B2t≤ρnS1t−S2t+V1t−V2t+E1t−E2t+I1t−I2t+C1t−C2t+U1t−U2t+T1t−T2t+R1t−R2t+B1t−B2t,with the positive constant ρn=maxρifori=1,2,…,9, independent to the state variables St, Vt, Et, It, Ct, Ut, Tt, Rt, and Bt.It follows that optimality system and the optimal controls exist and the solution of system is bounded, since(85)Zp1−Zp2≤ρnZ1−Z2,where(86)ρn=∑i=19ρi+K<∞.This follows that the functionZ is uniformly Lipchitz continuous. Hence, the control variables and nonnegative initial conditions reveal that a solution of the system (69) exists (see [22, 23]).Theorem 4.
Given the objective functional (48) with set L defined in (49) subject to the system (46) with initial conditions (47), then there exists an optimal control set w1=χ1∗,χ2∗,χ3∗ such that (50) is true if the following conditions hold (KS,E,I,χ1,χ2,χ3≤μ2−μ1χ1t2+χ2t2+χ3t2α1/2):(i)
The set of controls and corresponding state variables is nonempty(ii)
The control setL is closed and convex(iii)
Each right-hand side of (46) is continuous bounded above by a sum of the bounded control and state and can be written as a linear function w with coefficients depending on time and state(iv)
The integrand of the objective functional is convex onL(v)
There exist nonnegative constantsμ1 and μ2 and α1>1 satisfying the following expressionProof.
Using established approach as in [6, 24, 25], the existence of solution of (46) is satisfied, the bounded coefficients and all solutions are bounded on a finite time interval, and first condition is satisfied.
It suffices to state thatL=La×Lb, where La and Lb are closed and convex sets defined as follows: (87)La=Lb=ωmeasurable:χ1t∈0,1,χ2t∈0,0.95,χ3t∈0,1,∀t∈t0,Tf.
Hence, by definition of convex, the control set is convex and closed, satisfying second condition.
By definition, each right-hand side of the system (69) is continuous and can be written as a linear function of ω with coefficients depending on time and state. All variables S,V,E,I,C,U,T,R,B,χ1,χ2, and χ3 are bounded on t0,Tf. Using the boundedness of the solution and its positivity, as (69) is bilinear in χ1,χ2, and χ3, its right-hand side satisfies condition 3. The integrand in the objective functional (71) is convex on L, and the fourth condition is satisfied.
Finally, it can be easily seen that there exist a constantα1>1 and positive numbers μ1 and μ2 satisfying(88)KS,E,I,χ1,χ2,χ3≤μ2−μ1χ12+χ22+χ32α1/2.
The Hamiltonian equation is formed accordingly by allowing each adjoint variable to correspond to each state variable, all combined with the objective functional.
Applying Pontryagin’s maximum principle, system (69) together with equations (48) and (49) is converted into problem of minimising the Hamiltonian H defined below.
Since necessary conditions that an optimal control and state must satisfy can be generated from HamiltonianH, see [26].
Hence, the Hamiltonian functionH of the control problem of the model will be(89)HS,V,E,I,C,U,T,R,B,χ1,χ2,χ3,λS,λV,λE,λI,λC,λU,λT,λR,λB=Θ+λSdSdt+λVdVdt+λEdEdt+λIdIdt+λCdCdt+λUdUdt+λTdTdt+λRdRdt+λBdBdt,where λS,λV,λE,λI,λC,λU,λT,λR, and λB are the adjoint functions, which can also be the co-state functions with respect to H and(90)Θ=M1St+M2Et+M3It+12N1χ1t2+N2χ2t2+N3χ3t2,H=M1St+M2Et+M3It+12N1χ1t2+N2χ2t2+N3χ3t2+λSq−v11−χ3tBtk+BtSt+1−χ1t1−χ3tv2StIt+ωVt+ωRt−χ2t+δSt+λVχ2tSt−σv2VtIt−ω+δVt+λEv11−χ3tBtk+BtSt+1−χ1t1−χ3tv2StIt+σv2VtIt−ξ+χ1t+δEt+λIξEt−m+ε+δIt+λCχ1tEt−f+θ+δCt+λUfCt+1−τmIt+g2Tt−g1+r1+μ+δUt+λTτmIt+g1Ut−g1+r2+g2+α+δTt+λRg1+r2Tt+r1Ut+θCt−ω+δRt+λB1−χ1tη1It+η2−βBt.
LetS∗∗,V∗∗,E∗∗,I∗∗,C∗∗,U∗∗,T∗∗,R∗∗, and B∗∗ be the optimal state solutions with associated optimal control variables χ1∗,χ2∗,χ3∗ for the optimal control of system (69).
To achieve optimal control, the adjoint or co-state functionsλS,λV,λE,λI,λC,λU,λT,λR, and λB must satisfy(91)dλSdt=−∂H∂S,dλVdt=−∂H∂V,dλEdt=−∂H∂E,dλIdt=−∂H∂I,dλCdt=−∂H∂C,dλUdt=−∂H∂U,dλTdt=−∂H∂T,dλRdt=−∂H∂R,dλBdt=−∂H∂B.(92)dλSdt=−M1−λS−v11−χ3Bk+B−1−χ11−χ3v2I−χ2−δ−λVχ2−λEv11−χ3Bk+B+1−χ11−χ3v2I,dλVdt=−λSω−λV−σv2I−δ−ω−λEσv2I,dλEdt=−M2−λE−ξ−χ1−δ−λIξ−λCχ1,dλIdt=−M3−λS1−χ11−χ3v2S+λVσv2V−λE1−χ11−χ3v2S+σv2V−λI−m−ε−δ−λU1−τm−λTτm−λB1−χ1η1,dλCdt=−λC−f−θ−δ−λUf−λRθ,dλUdt=−λU−g1−r1−μ−δ−λTg1−λRr1,dλTdt=−λUg2−λT−g1−r2−g2−α−δ−λRg1+r2,dλRdt=−λSω−λR−ω−δ,dλBdt=−λS−v11−χ3Sk+B+v11−χ3BSk+B2−λEv11−χ3Sk+B−v11−χ3BSk+B2−λBη2−β.


## 4.2. Optimality Equations of the Model’s Controls

Partial derivative of the Hamiltonian with respect to each admissible control is all equal to zero, ease solving the optimal controls; thus,(93)⇒∂H∂χ1=N1χ1+λS1−χ3v2SI+λE−1−χ3v2SI−E−λBη1I=0,⇒χ1∗=χ3λSv2S∗∗I∗∗−χ3λEv2S∗∗I∗∗−λSv2S∗∗I∗∗+λVv2S∗∗I∗∗+η1λBI∗∗+λEE∗∗N1,⇒∂H∂χ2=N2χ2−λSS+λVS,⇒χ2∗=S∗∗λS−λVN2,⇒∂H∂χ3=N3χ3+λSv1BSk+B+1−χ1v2SI+λCE+λE−v1BSk+B−1−χ1v2SI,⇒χ3∗=1k+BN3χ1kv2S∗∗I∗∗λS−χ1kv2S∗∗I∗∗λEχ1v2S∗∗I∗∗B∗∗λS−χ1v2S∗∗I∗∗B∗∗λEkv2S∗∗I∗∗λS+kv2S∗∗I∗∗λE−v2S∗∗I∗∗B∗∗λS+v2S∗∗I∗∗B∗∗λE−v1S∗∗B∗∗λS+v1S∗∗B∗∗λE−kE∗∗λC−E∗∗B∗∗λC.Now,a1≤χ1≤b1, a2≤χ2≤b2, and a3≤χ3≤b3.The characterisation is as follows:(94)χ1∗=a1,whenχ1¯≤a1,χ1¯,whena1<χ1¯<b1,b1,whenχ1¯≥b1.χ2∗=a2,whenχ2¯≤a2,χ2¯,whena2<χ2¯<b2,b2,whenχ2¯≥b2.χ3∗=a3,whenχ3¯≤a3,χ3¯,whena3<χ3¯<b3,b3,whenχ3¯≥b3,where(95)χ1¯=χ3−1v2S∗∗I∗∗λS+E∗∗−χ3S∗∗I∗∗λE+v2S∗∗I∗∗λV+η1I∗∗λBN1,χ2¯=S∗∗λS−λVN2,χ3¯=χ1k+χ1B∗∗−k−B∗∗v2S∗∗I∗∗−v1S∗∗B∗∗λS+B∗∗−χ1k−χ1B∗∗−kv2S∗∗I∗∗λEk+B∗∗N3, (96)χ1∗t=max0,minχ1¯t,1,χ2∗t=max0,minχ2¯t,0.95,χ3∗t=max0,minχ3¯t,1.


## 5. Numerical Simulations of the Model

The numerical algorithm that simulates the optimal control of system (69) is presented below using a semi-implicit finite-difference method. This method is known as improved Gauss–Seidel-like implicit finite-difference method, and it was introduced by [27] and denoted it as GSS1. It was successfully applied by [5, 28].Time intervalt0,Tf where t0=0 is discretely calibrated into points ti=ih+t0i=0,1,…,n, where h is the step. State and adjoint variables together with controls are defined in terms of nodal points.State variablesSt,Vt,Et,It,Ct,Ut,Tt,Rt, and Bthave nodal points Si,Vi,Ei,Ii,Ci,Ui,Ti,Ri, and Bi.Adjoint variablesλ1t,λ2t,λ3t,λ4t,λ5t,λ6t,λ7t,λ8t, and λ9t have nodal points λ1,i,λ2,i,λ3,i,λ4,i,λ5,i,λ6,i,λ7,i,λ8,i,andλ9,i, and controls χ1t,χ2t,andχ3t have nodal points χ1,i,χ2,i,andχ3,i with Tf as the final time. The approximation of first-order time derivative dSt/dt can be numerically given as follows:(97)dStdt=limh⟶∞St+h−Sth.Using the GSS1 method, the forward-difference approximation of this optimal control problem is as follows:(98)Si+1=Si+hq+ωVi+ωRi1+hv11−χ3iBi/k+Bi−1−χ1i1−χ3iv2Ii+χ2i+δ,Vi+1=Vi+hχ2iSi+11+hσv2Ii+1+ω+δ,Ei+1=Ei+hv11−χ3iBiSi+1/k+Bi+1−χ1i1−χ3iv2IiSi+1+σv2Vi+1Ii1+hξ+χ1i+δ,Ii+1=Ii+hξEi+11+hm+ε+δ,Ci+1=Ci+h1−χ1iEi+11+hf+θ+δ,Ui+1=Ui+hfCi+1+1−τmIi+1+g2Ti1+hg1+r1+μ+δ,Ti+1=Ti+hτmIi+1+g1Ui+11+hg1+r2+g2+α+δ,Ri+1=Ri+hg1+r2Ti+1+r1Ui+1+θCi+11+hω+δ,Bi+1=Bi+h1−χ1iη1Ii+11−hη2−β.The backward-difference is obtained using similar technique as follows:(99)λ1,n−i−1=λ1,n−i+h−M1−λ2,n−iχ2i−λ3,n−iv11−χ3iBi/k+Bi+1−χ1i1−χ3iv2Ii1+h−v11−χ3iBi/k+Bi−1−χ1i1−χ3iv2Ii−χ2i−δ,λ2,n−i−1=λ2,n−i−hλ1,n−iω−λ3,n−iσv2Ii1−hσv2Ii+ω+δ,λ3,n−i−1=λ3,n−i+h−M2−λ4ξ−λ5χ1i1+h−ξ−χ1i−δ,λ4,n−i−1=λ4,n−i+h−M3+λ1,n−i−1−λ3,n−i−11−χ1i1−χ3iv2Si+11+h−m−ε−δ+hσv2Vi+1λ2,n−i−λ3,n−i−1−λ6,n−i1−τm−λ7,n−iτm1+h−m−ε−δ−hλ9,n−i1−χ1iη11+h−m−ε−δ,λ5,n−i−1=λ5,n−i−hλ6,n−if+λ8,n−iθ1−hf+θ+δ,λ6,n−i−1=λ6,n−i−hλ7,n−ig1+λ8,n−ir11−hg1+r1+μ+δ,λ7,n−i−1=λ7,n−i−hλ6,n−ig2+λ8,n−ig1+r21−hg1+r2+g2+α+δ,λ8,n−i−1=λ8,n−i−λ1,n−ihω1−hω+δ,λ9,n−i−1=λ9,n−i1+hη2−β+λ1,n−ih−v11−χ3iSi/k+Bi+v11−χ3iBiSi/k+Bi21+hη2−β−λ3,n−ihv11−χ3iSi/k+Bi−v11−χ3iBiSi/k+Bi21+hη2−β.The initial values of the state variables of system (46) used in the simulation are as follows:S0=0.3,V0=0.3333,E0=0.0833,I0=0.0167,C0=0.1667,U0=0.0127,T0=0.0037,R0=0.0163,B0=0.0006.Values of the parameters used in the simulations are in Table2, with exception of the values of controls χ1=h,χ2=ψ, and χ3=χ. Time Tf used in the model is fixed at 35 days in the simulation of optimal control graphs except Figure 3(a), whose maximum value is 25 days. Values of weighting constants are as follows: M1=900,M2=100,M3=38,N1=145,N2=−1,andN3=32. The graphs from simulations of the model of system of equation (69) here are used to display the impact of the control measures on four state variables: susceptible, exposed, infected, and population of Vibrio cholerae.Figure 3
Effect of application of vaccine and cholera awareness programme as controls on susceptible, exposed, infected, andVibrio cholerae population.
(a)(b)(c)(d)Figure 4
Template of the optimised controls, a combination of the controls for cholera.See Figure4 shows the template of the three controls. Figure 5 displays impact of the three controls on susceptible, and it shows how the controls reduce population of susceptible to below 20 throughout the period of 35 days, but without the controls, population of the susceptible rise up to 900 in less than 10 days of the onset of the disease.Figure 5
Effect of all the three controls on susceptible, exposed, infected, andVibrio cholerae population.
(a)(b)(c)(d)Figures6–8 show how the three controls reduced population of the exposed, infected, and Vibrio cholerae population. On the other hand without controls, the population of the exposed, infected, and the Vibrio cholerae rises 150000, 60000, and 100000, respectively.Figure 6
Effect of application of only hygiene control on susceptible, exposed, infected, andVibrio cholerae population.
(a)(b)(c)(d)Figure 7
Effect of application of only vaccine control on susceptible, exposed, infected, andVibrio cholerae population.
(a)(b)(c)(d)Figure 8
Effect of application of only cholera awareness programme as control on susceptible, exposed, infected, andVibrio cholerae population.
(a)(b)(c)(d)


## 6. Summary and Conclusion

The study has succeeded in obtaining the model's threshold R0 and the most sensitive parameters in it; The interpretation of the figures and conclusion shows the final result.


### 6.1. Summary

The model’s basic reproduction number is as follows:(100)R0=q1−χ1−hω+δβkv2−1−χ1−hω+δη2kv2+1−χ1−hω+δη1v1+βkψσv2−η2kψσv2ψ+δω+δ−ωψkm+ε+δβ−η2.The most sensitive parameters in this obtainedR0 are h and v2, increasing h, which denotes hygiene consciousness of individuals will definitely reduce the spread of Vibrio cholerae, which will in turn reduce rate of contracting cholera disease. Since reducing R0 implies making it less than unity, this is interpreted as “the disease reduces and dies out.” Increasing R0 is increasing Vibrio cholerae, thereby spreading it to many individuals. On the other hand, v2 denotes the rate of getting Vibrio cholerae through human-to-human interaction and its sensitivity index value is a positive, that is, 0.9978. Increasing it by 10% will increase R0 by 9.978%. Literally increasing rate of ingesting Vibrio cholerae will definitely increase R0, the increment that will make it greater than unity, which implies escalation of the disease and persistence.Simulations ofSVEICUTRB model displayed in graphs show the impact of the control measures on four state variables: susceptible, exposed, infected, and population of Vibrio cholerae. Though [29] suggested the need to eradicate epidemics and its spread using appropriate measures in approximately 40 days, Figure 4 shows a template of the three controls the model use: hygiene consciousness, cholera vaccine, and cholera awareness programme. Impact of the three controls on four of the nine state variables of the model was presented as follows. Figure 5(a) shows how the controls reduce population of susceptible to zero throughout the period of 35 days, but without the controls, population of the susceptible rises up to 900 in less than 10 days of the onset of the disease. Figures 5(b)–5(d) show how the three controls reduced population of the exposed, infected, and Vibrio cholerae population. On the other hand, without controls, the population of the exposed, infected, and the Vibrio cholerae bacteria rises infinitely.Figures6(a)–6(d) show results of applying only one control “hygiene consciousness,” compared with when no control is used. The outcome shows that using the single control alone will increase the population of susceptible from 1000 when there is no control applied to 4000. Applying hygiene consciousness alone can suppress the population of the exposed, infected, and cholera pathogens (Vibrio cholerae bacteria).Using single control “cholera vaccine” alone, results in Figures7(a)–7(d) were obtained. Administering cholera vaccine as the only control for cholera can reduce the number of susceptible, exposed, rate of infection, and population of Vibrio cholerae bacteria to zero. It is hence an effective measure of eradicating cholera. Cholera awareness programme administered alone yields the results in Figures 8(a)–8(d). Application of single control 'cholera awareness' alone and not applying any controlat all seem to give the same result. The graphs show that population of the susceptible, exposed, infected, and Vibrio cholerae bacteria will increase the same way, when no controls were adopted. The population of susceptible will even increase the more, because cholera awareness programme is a preventive measure, not curative.Application of two controls, hygiene measures and cholera awareness programme, gives result in Figures9(a)–9(d). Figure 9(a) shows how susceptible individuals increase up to 5000 individuals, which is five times more than when there is no administered control. Application of two controls generally reduced exposed and infected individuals and population of Vibrio cholerae bacteria. In particular, two controls, hygiene measures and cholera vaccine, as shown in Figures 10(a)–10(d) show how the two controls were able to reduce number of susceptible, exposed, and infected individuals and population of Vibrio cholerae bacteria to a very small rate that is less than when no control is administered. Figures 3(a)–3(d) also display two controls: cholera vaccine and awareness programme; they show how the two controls reduced number of susceptible, exposed, and infected individuals and reduced the number of Vibrio cholerae bacteria population.Figure 9
Effect of application of hygiene and cholera awareness programme as controls on susceptible, exposed, infected, andVibrio cholerae population.
(a)(b)(c)(d)Figure 10
Effect of application of hygiene and vaccine as controls on susceptible, exposed, infected, andVibrio cholerae population.
(a)(b)(c)(d)


### 6.2. Conclusion

Effects of the controls on susceptible, exposed, and infected individuals together with population of cholera pathogens were obtained and plotted. The effects show that the fastest way to control cholera quickly and effectively is application of all the three controls, at least above average rates.


### 6.3. Recommendations

After treatment measures, application of cholera vaccines, hygiene, and cholera awareness programmes are the best measures of preventing the spread of cholera and entire full control of the disease. Keeping hygiene consciousness rate always high and hindering ingestion ofVibrio cholerae bacteria by any means to keep its rate zero will surely prevent cholera disease. This has been justified by the model’s R0.


## 6.1. Summary

The model’s basic reproduction number is as follows:(100)R0=q1−χ1−hω+δβkv2−1−χ1−hω+δη2kv2+1−χ1−hω+δη1v1+βkψσv2−η2kψσv2ψ+δω+δ−ωψkm+ε+δβ−η2.The most sensitive parameters in this obtainedR0 are h and v2, increasing h, which denotes hygiene consciousness of individuals will definitely reduce the spread of Vibrio cholerae, which will in turn reduce rate of contracting cholera disease. Since reducing R0 implies making it less than unity, this is interpreted as “the disease reduces and dies out.” Increasing R0 is increasing Vibrio cholerae, thereby spreading it to many individuals. On the other hand, v2 denotes the rate of getting Vibrio cholerae through human-to-human interaction and its sensitivity index value is a positive, that is, 0.9978. Increasing it by 10% will increase R0 by 9.978%. Literally increasing rate of ingesting Vibrio cholerae will definitely increase R0, the increment that will make it greater than unity, which implies escalation of the disease and persistence.Simulations ofSVEICUTRB model displayed in graphs show the impact of the control measures on four state variables: susceptible, exposed, infected, and population of Vibrio cholerae. Though [29] suggested the need to eradicate epidemics and its spread using appropriate measures in approximately 40 days, Figure 4 shows a template of the three controls the model use: hygiene consciousness, cholera vaccine, and cholera awareness programme. Impact of the three controls on four of the nine state variables of the model was presented as follows. Figure 5(a) shows how the controls reduce population of susceptible to zero throughout the period of 35 days, but without the controls, population of the susceptible rises up to 900 in less than 10 days of the onset of the disease. Figures 5(b)–5(d) show how the three controls reduced population of the exposed, infected, and Vibrio cholerae population. On the other hand, without controls, the population of the exposed, infected, and the Vibrio cholerae bacteria rises infinitely.Figures6(a)–6(d) show results of applying only one control “hygiene consciousness,” compared with when no control is used. The outcome shows that using the single control alone will increase the population of susceptible from 1000 when there is no control applied to 4000. Applying hygiene consciousness alone can suppress the population of the exposed, infected, and cholera pathogens (Vibrio cholerae bacteria).Using single control “cholera vaccine” alone, results in Figures7(a)–7(d) were obtained. Administering cholera vaccine as the only control for cholera can reduce the number of susceptible, exposed, rate of infection, and population of Vibrio cholerae bacteria to zero. It is hence an effective measure of eradicating cholera. Cholera awareness programme administered alone yields the results in Figures 8(a)–8(d). Application of single control 'cholera awareness' alone and not applying any controlat all seem to give the same result. The graphs show that population of the susceptible, exposed, infected, and Vibrio cholerae bacteria will increase the same way, when no controls were adopted. The population of susceptible will even increase the more, because cholera awareness programme is a preventive measure, not curative.Application of two controls, hygiene measures and cholera awareness programme, gives result in Figures9(a)–9(d). Figure 9(a) shows how susceptible individuals increase up to 5000 individuals, which is five times more than when there is no administered control. Application of two controls generally reduced exposed and infected individuals and population of Vibrio cholerae bacteria. In particular, two controls, hygiene measures and cholera vaccine, as shown in Figures 10(a)–10(d) show how the two controls were able to reduce number of susceptible, exposed, and infected individuals and population of Vibrio cholerae bacteria to a very small rate that is less than when no control is administered. Figures 3(a)–3(d) also display two controls: cholera vaccine and awareness programme; they show how the two controls reduced number of susceptible, exposed, and infected individuals and reduced the number of Vibrio cholerae bacteria population.Figure 9
Effect of application of hygiene and cholera awareness programme as controls on susceptible, exposed, infected, andVibrio cholerae population.
(a)(b)(c)(d)Figure 10
Effect of application of hygiene and vaccine as controls on susceptible, exposed, infected, andVibrio cholerae population.
(a)(b)(c)(d)


## 6.2. Conclusion

Effects of the controls on susceptible, exposed, and infected individuals together with population of cholera pathogens were obtained and plotted. The effects show that the fastest way to control cholera quickly and effectively is application of all the three controls, at least above average rates.


## 6.3. Recommendations

After treatment measures, application of cholera vaccines, hygiene, and cholera awareness programmes are the best measures of preventing the spread of cholera and entire full control of the disease. Keeping hygiene consciousness rate always high and hindering ingestion ofVibrio cholerae bacteria by any means to keep its rate zero will surely prevent cholera disease. This has been justified by the model’s R0.



---

*Source: 2314104-2022-08-17.xml*