# Modeling and Analysis of HIV and Cholera Direct Transmission with Optimal Control

Authors: Kumama Regassa Cheneke; Koya Purnachandra Rao; Geremew Kenassa Edessa
Journal: Discrete Dynamics in Nature and Society
 (2022)

---

## Abstract

In this study, a mathematical model of the human immunodeficiency virus (HIV) and cholera co infection is constructed and analyzed. The disease-free equilibrium of the co-infection model is both locally and globally asymptotically stable ifR0<1 and unstable if R0>1. The only cholera model and only the HIV model show forward bifurcation if the corresponding reproduction numbers attain a value one. The disease-free equilibria of only the cholera and only the HIV models is locally and globally asymptotically if R0<1, and the endemic equilibria of only the cholera model and only the HIV model are locally and globally asymptotically stable if the corresponding reproduction number is equal to one. The endemic equilibrium point of the HIV and cholera model is computed, and stability property is shown with numerical simulations. The computed partial derivatives ∂R0h/∂R0c>0 show that the increase of one infection contributes to the increase of other infection. Pontryagin’s maximum principle is applied to construct Hamiltonian function, and optimal controls are computed. The optimal system is solved numerically using forward and backward sweep method of Runge Kutta’s fourth-order methods. The numerical simulations are plotted using MATLAB.


---

## Body

## 1. Introduction

The history of HIV and cholera lasts for long periods of time. Human immunodeficiency virus (HIV) is a retrovirus that causes a fatal disease called Acquired Immune Deficiency Syndrome (AIDS) [1–8]. Antiretrovirus therapy (ART) is used for intervention of HIV progressions. However, cholera is a bacterial epidemic caused by Vibrio cholerae bacterium [9–12]. Since the discovery of the diseases, millions of people are killed over the world. The person infected with both Vibrio cholerae bacterium and human immunodeficiency virus faces poor health problem than a person infected with only HIV or Vibrio cholerae bacterium [3]. Moreover, the most transmission mode of HIV is through sexual intercourse practices among people whereas cholera is transmitted by drinking water and eating food contaminated Vibrio cholerae [3]. On the one hand, the infection caused by HIV is treatable but not curable. Antiretrovirus therapy (ART) is used for the intervention of HIV progressions. On the other hand, the infection caused by Vibrio cholerae is treatable and curable if treated as early as possible, if left untreated it can kill within an hour [10–12]. The co-infection model of HIV and cholera is developed and analyzed in [3]. Even though a few mathematical models are developed to study the codynamics of HIV and cholera, the model that considered optimal control measures is not developed. Thus, we are motivated to fill this gap, and we have modified the model developed in [3] and extend to the optimal mathematical model of HIV and Vibrio cholerae bacterium codynamics.


## 2. Mathematical Model Formulation

To develop our current model, we have modified four the co-infection model of HIV and cholera developed in [3]. We formulate an SCRHDE model of HIV and cholera coinfection by dividing the total population under consideration as classes that consists of (i) Susceptible individuals S. They are infection free individuals with possibility of acquiring HIV from HIV-infected individuals through sexual contact and acquiring cholera from cholera-infected individuals only through effective direct contacts, (ii) Cholera infected individuals C. They are individuals infected with cholera and capable of transmitting infection to susceptible population through direct contact, (iii) Cholera recovered individuals R. They are individuals recovered from cholera infection with temporary immunity, (iii) HIV/AIDS infected individuals H. They are individuals infected with only HIV/AIDS and capable of transmitting HIV to susceptible population through sexual contacts, (iv) Co-infected individuals D. They are individuals infected with HIV and cholera. They can transmit disease with effective contacts with susceptible population, and (v) Cholera recovered HIV/AIDS individuals E. They are HIV/AIDS individuals recovered from cholera with temporary immunity. The following assumptions are also stated:(i)
Total population size is not constant(ii)
New susceptible individuals are recruited at the rateτ(iii)
All populations die naturally at the rateμ(iv)
Cholera infected individuals die at the rateφ(v)
HIV/AIDS infected individuals die at the rateδ=ζ(vi)
Co-infected individuals die at the rateψ(vii)
Transmission rate of cholera isβc(viii)
Transmission rate of HIV/AIDS isβh(ix)
Recovery rate of only cholera infected with temporary immunity isω(x)
Recovery rate of co-infected person with temporary immunity isκ(xi)
γ is the immunity loss rate of cholera recovered R(xii)
η is the immunity loss rate of cholera recovered EThe total size of population under study is denoted byNt and defined as Nt=St+Ct+Rt+Ht+Dt+Et, where St is the size of susceptible population at time t, Ct is the size of cholera population at time t, Rt is the size of only cholera recovered population at time t, Ht is the size of HIV/AIDS population at time t, Dt is the size of co-infected population at time t, and Et is the size of cholera recovered HIV population at time t.Based on Figure1 and stated assumptions, the subsequent model is stated.(1)dSdt=τ−SβcC+DN−SβhH+D+EN+γR−μS,dCdt=SβcC+DN−CβhH+D+EN−φ+ω+μC,dRdt=ωC−RβhH+D+EN−γ+μR,dHdt=SβhH+D+EN−HβcC+DN+ηE−μ+δH,dDdt=HβcC+DN+CβhH+D+EN−ψ+κ+μD,dEdt=RβhH+D+EN+κD−ζ+η+μE,Figure 1
Flow diagram of HIV and cholera coinfection.with conditionsS0>0,C0≥0,R0≥0,H0≥0,D0≥0,E0≥0.


## 3. Mathematical Analysis of the Model

### 3.1. Invariant Region

Theorem 1.
The solutions of the model (1) are positively invariant in the regionΩ⊂ℜ+6.Proof. Using [1], and adding all expressions on the left and right of equality in the model (1), we obtain(2)dNdt=τ−μN−φC−δH−ψD−ζE,Logically ignoring the expressionsφC,δH,ψD, and ψD in the preceding equality, we get(3)dNdt≤τ−μN,Solving the preceding inequality over time interval0,t, we obtain(4)Nt≤τμ−τμ−N0e−μt,In the preceding inequality as timet⟶∞, the total population size Nt⟶τ/μ for all possible initial population size. Therefore, by the works presented in [13–20], the total population size is bounded for all time t.


### 3.2. Positivity Property

Theorem 2.
All solutions of the model are positive for all timet in the solution region Ω⊂ℜ+6.Proof. To show positivity property of solutions of model (1), we show that all solution variables are non-negative. Considering the first equation of model (1), we get(5)dSdt=τ−SβcC+DN−SβhH+D+EN+γR−μS.Ignoring the non-negative terms of the right expression in the preceding equality, we obtain(6)dSdt≥−SβcC+DN−SβhH+D+EN−μS.The solution of the foregoing inequality over time interval0,t is computed as(7)St≥S0e−μt−∫0tβcCξ+DξNξ+βhHξ+Dξ+EξNξdξ.Since the exponential expressione−μt−∫0tβcCξ+Dξ/Nξ+βhHξ+Dξ+Eξ/Nξdξ and S0 are positive, the state variable St is positive for all time t. Similarly, all state variables are non-negative for all time t. Therefore, all solutions of model (1) are non-negative in the solution region.


### 3.3. Existence and Uniqueness of Solutions

Theorem 3.
The solutions of model (1) exist and are unique in the solution region Ω⊂ℜ+6.Proof. Since all solution variables are bounded and all expression on the right of equality in the model (1) are continuous and differentiable by Chauchy–Lipschitz theorem stated in [3], the solutions of model (1) exist and unique.


### 3.4. Analysis of Submodels

Before analyzing the full model (1), we first analyze HIV only and cholera only models.


#### 3.4.1. HIV Only Model

We obtain the HIV only model from model (1) by setting the state variables C=R=D=E=0 so that N=S+H=Nh model (1) reduces to the HIV only model as(8)dSdt=τ−βhSHNh−μS,dHdt=βhSHNh−μ+δH,with non-negative initial conditions.


#### 3.4.2. Invariant Region

Theorem 4.
For HIV only model (2), the solution regionΩ1⊂ℜ+2 is defined as(9)Ω1=S,H∈ℜ+2:Nh=S+H≤τμ,,Proof. To prove Theorem4, we add expressions on the left and right sides of the HIV only model so that(10)dNhdt=τ−μNh−δH,≤τ−μNh.Solving the preceding inequality, we get(11)Nh≤τμ−τμ−Nh0e−μt,In the foregoing inequality,e−μt⟶0,Nh⟶τ/μ as t⟶∞. Therefore, the HIV only model is invariant in the region Ω1⊂ℜ+2 such that(12)Ω1=S,H∈ℜ+2:Nh=S+H≤τμ,


#### 3.4.3. Equilibrium of HIV Only Model

(1) Disease Free Equilibrium of HIV Only Model. We obtain disease-free equilibrium of the HIV only model by setting H=0 in the HIV only model by solving the following equation:(13)τ−μS=0.Solving for the state variableS from the preceding equation, we get(14)S=τμ.Therefore, the disease-free equilibriumE0h of the HIV only model is given by(15)E0h=τμ,0.(2) Endemic Equilibrium of HIV Only Model. The endemic equilibrium Eh=Sh,Hh1 of the HIV only model is obtained by solving the equations:(16)τ−βhSHNh−μS=0,βhSHNh−μ+δH=0.Solving the preceding equation, we obtain(17)S=τβh−δ=Sh∗,H=τR0h−1βh−δ=H∗.By taking this,R0h=βh/δ+μ and the endemic equilibrium of the HIV only model exist if βh>δ and given by(18)Eh=τR0hδ+μ−δ,τR0h−1βh−δ.


#### 3.4.4. Reproduction Number

In this section, we computed the reproduction number from the HIV only model using the next-generation method described and applied in [8]. Thus, considering the infected variable, the newly and transition individuals are given, respectively, by f=βhSH/Nh,v=μ+δH. The Jacobian matrix J evaluated at disease-free equilibrium is given by F=βh,v=μ+δ. Hence, R0 is defined by R0=FV−1 and given by(19)R0=βhμ+δ.


#### 3.4.5. Bifurcation Analysis

Theorem 5.
The HIV only model exhibits forward bifurcation atR0=1.Proof. The prove of this theorem follows from the forward bifurcation diagram plotted in Figure2.Figure 2
Forward bifurcation diagram of the HIV only model.


#### 3.4.6. Stability of Equilibriums of HIV Only Model

Theorem 6.
The disease-free equilibrium of the HIV only modelE0h=τ/μ,0 is locally asymptotically stable if R0h<1 and unstable if R0h>1.Proof. To determine the local stability, we computed the Jacobian matrixJ from the HIV only model at disease free-equilibrium E0h as(20)J=−μ−β0−μ+δ.The eigenvalues of the computed preceding matrixJ are(21)λ1=−μ,λ2=−μ+δ.Since both foregoing eigenvalues are negative, by stability theory, the disease-free equilibrium of the HIV only model is locally asymptotically stable ifR0h<1 and unstable if R0h>1.Theorem 7.
The disease-free equilibrium of the HIV only modelE0h=τ/μ,0 is globally asymptotically stable if R0h<1 and unstable if R0h>1.Proof. We follow from Figure2.Theorem 8.
The endemic equilibrium of the HIV only modelEh is locally asymptotically stable if R0h>1 and unstable if R0h<1.Proof. The Jacobian matrix at endemic equilibrium is given by(22)J=−μ−βhR0h−12R0h2−βhR0h2βhR0h−12R0h2βhR0h2−μ+δ.TraceJ=−μ−βhR0h−12/R0h2+βh/R0h2−μ+δ(23)=−μ−βhR0h−12R0h2+μ+δ1R0h−1<0,⇔R0h>1.Again,detJ=−μ−βhR0h−12/R0h2μ+δ1/R0h−1+βh/R0h2βhR0h−12/R0h2>0⇔R0h>1.Therefore, by trace-determinant rule, the eigenvalues of the preceding Jacobian matrix are negative. Hence, by stability theory of differential equations, the endemic equilibrium of the HIV only model is locally asymptotically stable ifR0h>1.Theorem 9.
The endemic equilibrium of HIV only modelEh is globally asymptotically stable if R0h>1 and unstable if R0h<1.Proof. Based on center manifold theory applied in [11] and works done in [7, 18], stability condition holds true. In general, the model that exhibit forward bifurcation at R0=1 has both locally and globally endemic equilibrium.


### 3.5. Cholera Only Model

The cholera only model is obtained from model (1) by setting H=D=E=0 so that Nc=S+C+R and given by(24)dSdt=τ−βcSCNc+γR−μS,dCdt=βcSCNc−φ+ω+μC,dRdt=ωC−γ+μR,with non-negative initial conditions.


#### 3.5.1. Invariant Region

Theorem 10.
For cholera only model (2), the solution region Ω2⊂ℜ+3 is defined as(25)Ω2=S,C,R∈ℜ+3:Nc=S+C+R≤τμ,,Proof. To prove Theorem4, we add expressions on the left and right sides of the HIV only model so that(26)dNcdt=τ−μNc−φC,≤τ−μNc.Solving the preceding inequality, we get(27)Nc≤τμ−τμ−Nc0e−μt.In the foregoing inequality,e−μt⟶0,Nc⟶τ/μ as t⟶∞. Thus, the cholera only model is invariant in the region Ω2⊂ℜ+2 such that(28)Ω1=S,H∈ℜ+2:Nc=S+C+R≤τμ.


#### 3.5.2. Equilibria of Cholera Only Model

In this section, we compute the disease-free equilibriumE0c and endemic equilibrium Ec of the cholera only model.(1) Disease Free Equilibrium. The disease-free equilibrium E0c of the cholera only model is computed by setting C=0, in the cholera only model at the equilibrium and given by(29)E0c=τμ,0,0.(2) Endemic Equilibrium. The endemic equilibrium Ec=S1,C1,R1 of the cholera only model is computed by setting the rate of change state variables equal to zero. That is,(30)τ−βcSCNc+γR−μS=0,βcSCNc−φ+ω+μC=0,ωC−γ+μR=0.Adding first and second equation and using fourth equation such thatR=ωC/γ+μ, we get the following mathematical equations:(31)τ−μS=φ+ω+μ−ωγγ+μC.Lettingd=φ+ω+μ−ωγ/γ+μ,R0c=βc/φ+ω+μ, and using R0cS=Nc=S+C+R, the preceding equation reduces to the form:(32)SR0c−1=τ−μSdγ+μ+ωγ+μ.Furthermore, lettinga=γ+μ+ω/γ+μ, the preceding equations reduce to(33)SR0c−1=aτ−μaSd.This gives(34)S=τadR0c−1+μa.Furthermore, solving forC and R, we get(35)C=τ−μSd=τR0c−1dR0c−1+μa.And(36)R=ωCγ+μ=ωτR0c−1γ+μdR0c−1+μa.Hence,(37)S=τadR0c−1+μa=S∗,C=τR0c−1dR0c−1+μa=C∗,R=ωτR0c−1γ+μdR0c−1+μa=R∗.Thus, the endemic equilibrium is given by(38)Ec=τadR0c−1+μa,τR0c−1dR0c−1+μa,ωτR0c−1γ+μdR0c−1+μa.


#### 3.5.3. Reproduction Number

The basic reproduction numberR0c is computed using the next-generation method. Following [3], the next-generation matrix FV−1, computed from the cholera only model is given by(39)FV−1=βcφ+ω+μ−1=βcφ+ω+μ.Therefore, the reproduction numberR0c=FV−1 is given by(40)R0c=βcφ+ω+μ.


#### 3.5.4. Bifurcation Analysis

Theorem 11.
The cholera only model exhibits forward bifurcation atR0c=1.Proof. The prove of this theorem follows from forward bifurcation diagram of the cholera only model plotted in Figure3.Figure 3
Forward bifurcation of cholera only model atR0=1.


#### 3.5.5. Stability of Equilibria of Cholera Only Model

Theorem 12.
The disease-free equilibrium (DFE) of the cholera only model is locally asymptotically stable ifβc>ω and R0c<1, unstable if R0c>1.Proof.
It is corollary of forward bifurcation [Ref. 7, 18] Ded:a0>0,a1>0,a2>0, this condition is satisfied if, βc>ω and R0c<1a1a2>a0 is satisfied as βc2−bcR0c−1−μcR0c−1μ−cR0c−1+b>βc2b−βcωγ−bcμR0c−1 where b=γ+μ,c=φ+ω+μTherefore, by stability theory, the disease-free equilibriumE0 is locally asymptotically stable if R0c<1.Theorem 13.
The disease-free equilibrium of the cholera only model is globally asymptotically stable ifR0c<1 and unstable if R0c>1.Proof. We follow from Figure3.Theorem 14.
The endemic equilibrium of cholera only model is locally asymptotically stable ifR0c>1 and unstable if R0c<1.


### 3.6. Equilibria of the Full Model

The equilibrium point is a steady state where the system does not change with time. In this section, we compute disease-free equilibrium.


#### 3.6.1. Disease-Free Equilibrium (DFE)

The disease-free equilibrium pointE0 is a point where there is no disease in the population. The disease-free equilibrium computed from model (1) is given by(41)E0=τμ,0,0,0,0,0.


#### 3.6.2. Endemic Equilibrium (EE)

The endemic equilibrium pointE1 of model (1) is a state where the disease persists in the population with the zero rate of change of population. Hence, we obtain(42)τ−SβcC+DN−SβhH+D+EN+γR−μS=0,SβcC+DN−CβhH+D+EN−φ+ω+μC=0,ωC−RβhH+D+EN−γ+μR=0,SβhH+D+EN−HβcC+DN+ηE−μ+δH=0,HβcC+DN+CβhH+D+EN−ψ+κ+μD=0,RβhH+D+EN+κD−ζ+η+μE=0,with positive initial conditions.Moreover, logically taking the parameter values and initial population size, the endemic equilibriumE1 of model (1) can be written as(43)200−0.000002SC+DS+C+R+H+D+E−0.034SH+D+ES+C+R+H+D+E+0.025R−0.0000548S=0,0.000002SC+DS+C+R+H+D+E−0.034CH+D+ES+C+R+H+D+E−0.015+0.04+0.0000548C=0,0.04C−0.034RH+D+ES+C+R+H+D+E−0.025+0.0000548R=0,0.034SH+D+ES+C+R+H+D+E−0.000002HC+DS+C+R+H+D+E+0.025E−0.00000548+0.009H=0,0.000002HC+DS+C+R+H+D+E+0.034CH+D+ES+C+R+H+D+E−0.1+0.001+0.0000548D=0,0.034RH+D+ES+C+R+H+D+E+0.001D−0.009+0.025+0.0000548E=0,with initial conditions100000,5000,3000,10000,4000,1000.The solution of the preceding equations is computed using MATLAB and the numerical simulation of model (1) is performed using ode45 solver as illustrated in Figure 4 and fits with the obtained solutions. Hence, the endemic equilibrium of co-infection is given by(44)E1=8000,0,0,2039,0,0.Figure 4
Numerical simulations of population dynamics with no infection.


### 3.7. Reproduction Number

The reproduction number of the model (1) is computed using next-generation method defined and described in [8, 20–22]. Let fx be a vector consists of newly infected individuals arriving at compartments, and vx be a vector consists of the remaining terms in the infected compartments, where x=S,C,R,H,D,E. That is,(45)fx=SβcC+DNSβhH+D+ENHβcC+DN+CβhH+D+ENRβhH+D+EN,vx=φ+ω+μC−ηE+μ+δHψ+κ+μD−κD+ζ+η+μE.LetF be the Jacobian matrix computed from vector fx and V be the Jacobian matrix computed from vector vx and evaluated at disease-free equilibrium E0. The computed matrices are(46)F=βc0βc00βhβhβh00000000,V=φ+ω+μ0000μ+δ0−η00ψ+κ+μ000−κζ+η+μ.Leta=φ+ω+μ,b=ψ+κ+μ,c=ζ+η+μ.The next-generation matrixFV−1 is computed from matrices F and V as(47)FV−1=βca0βcb00βhμ+δβhb+βhκbc+βhκηbcδ+μβhc+ηβhcδ+μ00000000.The eigenvalues computed from the preceding next-generation matrix are(48)λ1=βca,λ2=βhμ+δ,λ3=0,λ4=0.Since reproduction numberR0 is the spectral radius of next-generation matrix, we have(49)R0=ρFV−1=maxβca,βhμ+δ.Hence, in this study,(50)R0=maxR0c,R0h=R0h.


### 3.8. Stability of Equilibria

Theorem 15.
The disease-free equilibrium of model (1) is locally asymptotically stable ifR0<1.Proof. To determine the local stability of disease-free equilibrium, similar to works presented in [23], we compute eigenvalues from the Jacobian matrix of model (1) evaluated at the disease-free equilibrium. Thus, the constructed Jacobian matrix J of model (1) evaluated at E0 is given by(51)J=−μ−βcγ−βh−βc−βh−βh0βc−ω+φ+μ00βc00ω−γ+μ000000βh−μ+δβhη0000−κ+ψ+μ00000κ−η+ζ+μ.The computed eigenvalues of the foregoing matrixJ are(52)λ1=−μ,λ2=aR0c−1,λ3=−γ−μ,λ4=μ+δR0h−1,λ5=−b,λ6=−c,where a=ω+φ+μ,b=κ+ψ+μ,c=η+ζ+μ,R0c=βc/a,R0h=βh/μ+δ.Since all computed eigenvalues are negative ifR0c<1 and R0h<1, by stability theory, the disease-free equilibrium of model (1) is locally asymptotically stable if R0<1.Theorem 16.
The disease-free equilibrium of co-infection model (1) is globally asymptotically stability if R0<1.Proof. Following from Figure2, forward bifurcation of model (1) at R0=maxR0c,R0h=R0h, (see [7, 18]).Theorem 17.
The endemic equilibrium of the constructed mode (1) is both locally and globally asymptotically stable if R0=R0h>1.Proof. The prove of this theorem follows from forward bifurcation behavior illustrated in Figure2 and stability behavior shown in Figure 4 (see [7, 18]).


### 3.9. Impact of Cholera on HIV-Infected Individuals

Based on the works done in [3], we analyze the impact of outbreak of cholera using partial derivative. Accordingly, μ is the common parameter of both R0h and R0c. Hence, we compute partial derivative ∂R0h/∂R0c=∂R0h/∂μ∂R0c/∂μ. Taking the computed results, R0c=βc/η+ζ+μ, and R0h=βh/μ+δ, we have(53)∂R0h∂R0c=∂R0h/∂μ∂R0c/∂μ=−βh/μ+δ2−βc/η+ζ+μ2=βhη+ζ+μ2βcμ+δ2>0.The positivity of the preceding partial derivative results shows that the existence of cholera infection increases the risk of HIV and vice versa.


## 3.1. Invariant Region

Theorem 1.
The solutions of the model (1) are positively invariant in the regionΩ⊂ℜ+6.Proof. Using [1], and adding all expressions on the left and right of equality in the model (1), we obtain(2)dNdt=τ−μN−φC−δH−ψD−ζE,Logically ignoring the expressionsφC,δH,ψD, and ψD in the preceding equality, we get(3)dNdt≤τ−μN,Solving the preceding inequality over time interval0,t, we obtain(4)Nt≤τμ−τμ−N0e−μt,In the preceding inequality as timet⟶∞, the total population size Nt⟶τ/μ for all possible initial population size. Therefore, by the works presented in [13–20], the total population size is bounded for all time t.


## 3.2. Positivity Property

Theorem 2.
All solutions of the model are positive for all timet in the solution region Ω⊂ℜ+6.Proof. To show positivity property of solutions of model (1), we show that all solution variables are non-negative. Considering the first equation of model (1), we get(5)dSdt=τ−SβcC+DN−SβhH+D+EN+γR−μS.Ignoring the non-negative terms of the right expression in the preceding equality, we obtain(6)dSdt≥−SβcC+DN−SβhH+D+EN−μS.The solution of the foregoing inequality over time interval0,t is computed as(7)St≥S0e−μt−∫0tβcCξ+DξNξ+βhHξ+Dξ+EξNξdξ.Since the exponential expressione−μt−∫0tβcCξ+Dξ/Nξ+βhHξ+Dξ+Eξ/Nξdξ and S0 are positive, the state variable St is positive for all time t. Similarly, all state variables are non-negative for all time t. Therefore, all solutions of model (1) are non-negative in the solution region.


## 3.3. Existence and Uniqueness of Solutions

Theorem 3.
The solutions of model (1) exist and are unique in the solution region Ω⊂ℜ+6.Proof. Since all solution variables are bounded and all expression on the right of equality in the model (1) are continuous and differentiable by Chauchy–Lipschitz theorem stated in [3], the solutions of model (1) exist and unique.


## 3.4. Analysis of Submodels

Before analyzing the full model (1), we first analyze HIV only and cholera only models.


### 3.4.1. HIV Only Model

We obtain the HIV only model from model (1) by setting the state variables C=R=D=E=0 so that N=S+H=Nh model (1) reduces to the HIV only model as(8)dSdt=τ−βhSHNh−μS,dHdt=βhSHNh−μ+δH,with non-negative initial conditions.


### 3.4.2. Invariant Region

Theorem 4.
For HIV only model (2), the solution regionΩ1⊂ℜ+2 is defined as(9)Ω1=S,H∈ℜ+2:Nh=S+H≤τμ,,Proof. To prove Theorem4, we add expressions on the left and right sides of the HIV only model so that(10)dNhdt=τ−μNh−δH,≤τ−μNh.Solving the preceding inequality, we get(11)Nh≤τμ−τμ−Nh0e−μt,In the foregoing inequality,e−μt⟶0,Nh⟶τ/μ as t⟶∞. Therefore, the HIV only model is invariant in the region Ω1⊂ℜ+2 such that(12)Ω1=S,H∈ℜ+2:Nh=S+H≤τμ,


### 3.4.3. Equilibrium of HIV Only Model

(1) Disease Free Equilibrium of HIV Only Model. We obtain disease-free equilibrium of the HIV only model by setting H=0 in the HIV only model by solving the following equation:(13)τ−μS=0.Solving for the state variableS from the preceding equation, we get(14)S=τμ.Therefore, the disease-free equilibriumE0h of the HIV only model is given by(15)E0h=τμ,0.(2) Endemic Equilibrium of HIV Only Model. The endemic equilibrium Eh=Sh,Hh1 of the HIV only model is obtained by solving the equations:(16)τ−βhSHNh−μS=0,βhSHNh−μ+δH=0.Solving the preceding equation, we obtain(17)S=τβh−δ=Sh∗,H=τR0h−1βh−δ=H∗.By taking this,R0h=βh/δ+μ and the endemic equilibrium of the HIV only model exist if βh>δ and given by(18)Eh=τR0hδ+μ−δ,τR0h−1βh−δ.


### 3.4.4. Reproduction Number

In this section, we computed the reproduction number from the HIV only model using the next-generation method described and applied in [8]. Thus, considering the infected variable, the newly and transition individuals are given, respectively, by f=βhSH/Nh,v=μ+δH. The Jacobian matrix J evaluated at disease-free equilibrium is given by F=βh,v=μ+δ. Hence, R0 is defined by R0=FV−1 and given by(19)R0=βhμ+δ.


### 3.4.5. Bifurcation Analysis

Theorem 5.
The HIV only model exhibits forward bifurcation atR0=1.Proof. The prove of this theorem follows from the forward bifurcation diagram plotted in Figure2.Figure 2
Forward bifurcation diagram of the HIV only model.


### 3.4.6. Stability of Equilibriums of HIV Only Model

Theorem 6.
The disease-free equilibrium of the HIV only modelE0h=τ/μ,0 is locally asymptotically stable if R0h<1 and unstable if R0h>1.Proof. To determine the local stability, we computed the Jacobian matrixJ from the HIV only model at disease free-equilibrium E0h as(20)J=−μ−β0−μ+δ.The eigenvalues of the computed preceding matrixJ are(21)λ1=−μ,λ2=−μ+δ.Since both foregoing eigenvalues are negative, by stability theory, the disease-free equilibrium of the HIV only model is locally asymptotically stable ifR0h<1 and unstable if R0h>1.Theorem 7.
The disease-free equilibrium of the HIV only modelE0h=τ/μ,0 is globally asymptotically stable if R0h<1 and unstable if R0h>1.Proof. We follow from Figure2.Theorem 8.
The endemic equilibrium of the HIV only modelEh is locally asymptotically stable if R0h>1 and unstable if R0h<1.Proof. The Jacobian matrix at endemic equilibrium is given by(22)J=−μ−βhR0h−12R0h2−βhR0h2βhR0h−12R0h2βhR0h2−μ+δ.TraceJ=−μ−βhR0h−12/R0h2+βh/R0h2−μ+δ(23)=−μ−βhR0h−12R0h2+μ+δ1R0h−1<0,⇔R0h>1.Again,detJ=−μ−βhR0h−12/R0h2μ+δ1/R0h−1+βh/R0h2βhR0h−12/R0h2>0⇔R0h>1.Therefore, by trace-determinant rule, the eigenvalues of the preceding Jacobian matrix are negative. Hence, by stability theory of differential equations, the endemic equilibrium of the HIV only model is locally asymptotically stable ifR0h>1.Theorem 9.
The endemic equilibrium of HIV only modelEh is globally asymptotically stable if R0h>1 and unstable if R0h<1.Proof. Based on center manifold theory applied in [11] and works done in [7, 18], stability condition holds true. In general, the model that exhibit forward bifurcation at R0=1 has both locally and globally endemic equilibrium.


## 3.4.1. HIV Only Model

We obtain the HIV only model from model (1) by setting the state variables C=R=D=E=0 so that N=S+H=Nh model (1) reduces to the HIV only model as(8)dSdt=τ−βhSHNh−μS,dHdt=βhSHNh−μ+δH,with non-negative initial conditions.


## 3.4.2. Invariant Region

Theorem 4.
For HIV only model (2), the solution regionΩ1⊂ℜ+2 is defined as(9)Ω1=S,H∈ℜ+2:Nh=S+H≤τμ,,Proof. To prove Theorem4, we add expressions on the left and right sides of the HIV only model so that(10)dNhdt=τ−μNh−δH,≤τ−μNh.Solving the preceding inequality, we get(11)Nh≤τμ−τμ−Nh0e−μt,In the foregoing inequality,e−μt⟶0,Nh⟶τ/μ as t⟶∞. Therefore, the HIV only model is invariant in the region Ω1⊂ℜ+2 such that(12)Ω1=S,H∈ℜ+2:Nh=S+H≤τμ,


## 3.4.3. Equilibrium of HIV Only Model

(1) Disease Free Equilibrium of HIV Only Model. We obtain disease-free equilibrium of the HIV only model by setting H=0 in the HIV only model by solving the following equation:(13)τ−μS=0.Solving for the state variableS from the preceding equation, we get(14)S=τμ.Therefore, the disease-free equilibriumE0h of the HIV only model is given by(15)E0h=τμ,0.(2) Endemic Equilibrium of HIV Only Model. The endemic equilibrium Eh=Sh,Hh1 of the HIV only model is obtained by solving the equations:(16)τ−βhSHNh−μS=0,βhSHNh−μ+δH=0.Solving the preceding equation, we obtain(17)S=τβh−δ=Sh∗,H=τR0h−1βh−δ=H∗.By taking this,R0h=βh/δ+μ and the endemic equilibrium of the HIV only model exist if βh>δ and given by(18)Eh=τR0hδ+μ−δ,τR0h−1βh−δ.


## 3.4.4. Reproduction Number

In this section, we computed the reproduction number from the HIV only model using the next-generation method described and applied in [8]. Thus, considering the infected variable, the newly and transition individuals are given, respectively, by f=βhSH/Nh,v=μ+δH. The Jacobian matrix J evaluated at disease-free equilibrium is given by F=βh,v=μ+δ. Hence, R0 is defined by R0=FV−1 and given by(19)R0=βhμ+δ.


## 3.4.5. Bifurcation Analysis

Theorem 5.
The HIV only model exhibits forward bifurcation atR0=1.Proof. The prove of this theorem follows from the forward bifurcation diagram plotted in Figure2.Figure 2
Forward bifurcation diagram of the HIV only model.


## 3.4.6. Stability of Equilibriums of HIV Only Model

Theorem 6.
The disease-free equilibrium of the HIV only modelE0h=τ/μ,0 is locally asymptotically stable if R0h<1 and unstable if R0h>1.Proof. To determine the local stability, we computed the Jacobian matrixJ from the HIV only model at disease free-equilibrium E0h as(20)J=−μ−β0−μ+δ.The eigenvalues of the computed preceding matrixJ are(21)λ1=−μ,λ2=−μ+δ.Since both foregoing eigenvalues are negative, by stability theory, the disease-free equilibrium of the HIV only model is locally asymptotically stable ifR0h<1 and unstable if R0h>1.Theorem 7.
The disease-free equilibrium of the HIV only modelE0h=τ/μ,0 is globally asymptotically stable if R0h<1 and unstable if R0h>1.Proof. We follow from Figure2.Theorem 8.
The endemic equilibrium of the HIV only modelEh is locally asymptotically stable if R0h>1 and unstable if R0h<1.Proof. The Jacobian matrix at endemic equilibrium is given by(22)J=−μ−βhR0h−12R0h2−βhR0h2βhR0h−12R0h2βhR0h2−μ+δ.TraceJ=−μ−βhR0h−12/R0h2+βh/R0h2−μ+δ(23)=−μ−βhR0h−12R0h2+μ+δ1R0h−1<0,⇔R0h>1.Again,detJ=−μ−βhR0h−12/R0h2μ+δ1/R0h−1+βh/R0h2βhR0h−12/R0h2>0⇔R0h>1.Therefore, by trace-determinant rule, the eigenvalues of the preceding Jacobian matrix are negative. Hence, by stability theory of differential equations, the endemic equilibrium of the HIV only model is locally asymptotically stable ifR0h>1.Theorem 9.
The endemic equilibrium of HIV only modelEh is globally asymptotically stable if R0h>1 and unstable if R0h<1.Proof. Based on center manifold theory applied in [11] and works done in [7, 18], stability condition holds true. In general, the model that exhibit forward bifurcation at R0=1 has both locally and globally endemic equilibrium.


## 3.5. Cholera Only Model

The cholera only model is obtained from model (1) by setting H=D=E=0 so that Nc=S+C+R and given by(24)dSdt=τ−βcSCNc+γR−μS,dCdt=βcSCNc−φ+ω+μC,dRdt=ωC−γ+μR,with non-negative initial conditions.


### 3.5.1. Invariant Region

Theorem 10.
For cholera only model (2), the solution region Ω2⊂ℜ+3 is defined as(25)Ω2=S,C,R∈ℜ+3:Nc=S+C+R≤τμ,,Proof. To prove Theorem4, we add expressions on the left and right sides of the HIV only model so that(26)dNcdt=τ−μNc−φC,≤τ−μNc.Solving the preceding inequality, we get(27)Nc≤τμ−τμ−Nc0e−μt.In the foregoing inequality,e−μt⟶0,Nc⟶τ/μ as t⟶∞. Thus, the cholera only model is invariant in the region Ω2⊂ℜ+2 such that(28)Ω1=S,H∈ℜ+2:Nc=S+C+R≤τμ.


### 3.5.2. Equilibria of Cholera Only Model

In this section, we compute the disease-free equilibriumE0c and endemic equilibrium Ec of the cholera only model.(1) Disease Free Equilibrium. The disease-free equilibrium E0c of the cholera only model is computed by setting C=0, in the cholera only model at the equilibrium and given by(29)E0c=τμ,0,0.(2) Endemic Equilibrium. The endemic equilibrium Ec=S1,C1,R1 of the cholera only model is computed by setting the rate of change state variables equal to zero. That is,(30)τ−βcSCNc+γR−μS=0,βcSCNc−φ+ω+μC=0,ωC−γ+μR=0.Adding first and second equation and using fourth equation such thatR=ωC/γ+μ, we get the following mathematical equations:(31)τ−μS=φ+ω+μ−ωγγ+μC.Lettingd=φ+ω+μ−ωγ/γ+μ,R0c=βc/φ+ω+μ, and using R0cS=Nc=S+C+R, the preceding equation reduces to the form:(32)SR0c−1=τ−μSdγ+μ+ωγ+μ.Furthermore, lettinga=γ+μ+ω/γ+μ, the preceding equations reduce to(33)SR0c−1=aτ−μaSd.This gives(34)S=τadR0c−1+μa.Furthermore, solving forC and R, we get(35)C=τ−μSd=τR0c−1dR0c−1+μa.And(36)R=ωCγ+μ=ωτR0c−1γ+μdR0c−1+μa.Hence,(37)S=τadR0c−1+μa=S∗,C=τR0c−1dR0c−1+μa=C∗,R=ωτR0c−1γ+μdR0c−1+μa=R∗.Thus, the endemic equilibrium is given by(38)Ec=τadR0c−1+μa,τR0c−1dR0c−1+μa,ωτR0c−1γ+μdR0c−1+μa.


### 3.5.3. Reproduction Number

The basic reproduction numberR0c is computed using the next-generation method. Following [3], the next-generation matrix FV−1, computed from the cholera only model is given by(39)FV−1=βcφ+ω+μ−1=βcφ+ω+μ.Therefore, the reproduction numberR0c=FV−1 is given by(40)R0c=βcφ+ω+μ.


### 3.5.4. Bifurcation Analysis

Theorem 11.
The cholera only model exhibits forward bifurcation atR0c=1.Proof. The prove of this theorem follows from forward bifurcation diagram of the cholera only model plotted in Figure3.Figure 3
Forward bifurcation of cholera only model atR0=1.


### 3.5.5. Stability of Equilibria of Cholera Only Model

Theorem 12.
The disease-free equilibrium (DFE) of the cholera only model is locally asymptotically stable ifβc>ω and R0c<1, unstable if R0c>1.Proof.
It is corollary of forward bifurcation [Ref. 7, 18] Ded:a0>0,a1>0,a2>0, this condition is satisfied if, βc>ω and R0c<1a1a2>a0 is satisfied as βc2−bcR0c−1−μcR0c−1μ−cR0c−1+b>βc2b−βcωγ−bcμR0c−1 where b=γ+μ,c=φ+ω+μTherefore, by stability theory, the disease-free equilibriumE0 is locally asymptotically stable if R0c<1.Theorem 13.
The disease-free equilibrium of the cholera only model is globally asymptotically stable ifR0c<1 and unstable if R0c>1.Proof. We follow from Figure3.Theorem 14.
The endemic equilibrium of cholera only model is locally asymptotically stable ifR0c>1 and unstable if R0c<1.


## 3.5.1. Invariant Region

Theorem 10.
For cholera only model (2), the solution region Ω2⊂ℜ+3 is defined as(25)Ω2=S,C,R∈ℜ+3:Nc=S+C+R≤τμ,,Proof. To prove Theorem4, we add expressions on the left and right sides of the HIV only model so that(26)dNcdt=τ−μNc−φC,≤τ−μNc.Solving the preceding inequality, we get(27)Nc≤τμ−τμ−Nc0e−μt.In the foregoing inequality,e−μt⟶0,Nc⟶τ/μ as t⟶∞. Thus, the cholera only model is invariant in the region Ω2⊂ℜ+2 such that(28)Ω1=S,H∈ℜ+2:Nc=S+C+R≤τμ.


## 3.5.2. Equilibria of Cholera Only Model

In this section, we compute the disease-free equilibriumE0c and endemic equilibrium Ec of the cholera only model.(1) Disease Free Equilibrium. The disease-free equilibrium E0c of the cholera only model is computed by setting C=0, in the cholera only model at the equilibrium and given by(29)E0c=τμ,0,0.(2) Endemic Equilibrium. The endemic equilibrium Ec=S1,C1,R1 of the cholera only model is computed by setting the rate of change state variables equal to zero. That is,(30)τ−βcSCNc+γR−μS=0,βcSCNc−φ+ω+μC=0,ωC−γ+μR=0.Adding first and second equation and using fourth equation such thatR=ωC/γ+μ, we get the following mathematical equations:(31)τ−μS=φ+ω+μ−ωγγ+μC.Lettingd=φ+ω+μ−ωγ/γ+μ,R0c=βc/φ+ω+μ, and using R0cS=Nc=S+C+R, the preceding equation reduces to the form:(32)SR0c−1=τ−μSdγ+μ+ωγ+μ.Furthermore, lettinga=γ+μ+ω/γ+μ, the preceding equations reduce to(33)SR0c−1=aτ−μaSd.This gives(34)S=τadR0c−1+μa.Furthermore, solving forC and R, we get(35)C=τ−μSd=τR0c−1dR0c−1+μa.And(36)R=ωCγ+μ=ωτR0c−1γ+μdR0c−1+μa.Hence,(37)S=τadR0c−1+μa=S∗,C=τR0c−1dR0c−1+μa=C∗,R=ωτR0c−1γ+μdR0c−1+μa=R∗.Thus, the endemic equilibrium is given by(38)Ec=τadR0c−1+μa,τR0c−1dR0c−1+μa,ωτR0c−1γ+μdR0c−1+μa.


## 3.5.3. Reproduction Number

The basic reproduction numberR0c is computed using the next-generation method. Following [3], the next-generation matrix FV−1, computed from the cholera only model is given by(39)FV−1=βcφ+ω+μ−1=βcφ+ω+μ.Therefore, the reproduction numberR0c=FV−1 is given by(40)R0c=βcφ+ω+μ.


## 3.5.4. Bifurcation Analysis

Theorem 11.
The cholera only model exhibits forward bifurcation atR0c=1.Proof. The prove of this theorem follows from forward bifurcation diagram of the cholera only model plotted in Figure3.Figure 3
Forward bifurcation of cholera only model atR0=1.


## 3.5.5. Stability of Equilibria of Cholera Only Model

Theorem 12.
The disease-free equilibrium (DFE) of the cholera only model is locally asymptotically stable ifβc>ω and R0c<1, unstable if R0c>1.Proof.
It is corollary of forward bifurcation [Ref. 7, 18] Ded:a0>0,a1>0,a2>0, this condition is satisfied if, βc>ω and R0c<1a1a2>a0 is satisfied as βc2−bcR0c−1−μcR0c−1μ−cR0c−1+b>βc2b−βcωγ−bcμR0c−1 where b=γ+μ,c=φ+ω+μTherefore, by stability theory, the disease-free equilibriumE0 is locally asymptotically stable if R0c<1.Theorem 13.
The disease-free equilibrium of the cholera only model is globally asymptotically stable ifR0c<1 and unstable if R0c>1.Proof. We follow from Figure3.Theorem 14.
The endemic equilibrium of cholera only model is locally asymptotically stable ifR0c>1 and unstable if R0c<1.


## 3.6. Equilibria of the Full Model

The equilibrium point is a steady state where the system does not change with time. In this section, we compute disease-free equilibrium.


### 3.6.1. Disease-Free Equilibrium (DFE)

The disease-free equilibrium pointE0 is a point where there is no disease in the population. The disease-free equilibrium computed from model (1) is given by(41)E0=τμ,0,0,0,0,0.


### 3.6.2. Endemic Equilibrium (EE)

The endemic equilibrium pointE1 of model (1) is a state where the disease persists in the population with the zero rate of change of population. Hence, we obtain(42)τ−SβcC+DN−SβhH+D+EN+γR−μS=0,SβcC+DN−CβhH+D+EN−φ+ω+μC=0,ωC−RβhH+D+EN−γ+μR=0,SβhH+D+EN−HβcC+DN+ηE−μ+δH=0,HβcC+DN+CβhH+D+EN−ψ+κ+μD=0,RβhH+D+EN+κD−ζ+η+μE=0,with positive initial conditions.Moreover, logically taking the parameter values and initial population size, the endemic equilibriumE1 of model (1) can be written as(43)200−0.000002SC+DS+C+R+H+D+E−0.034SH+D+ES+C+R+H+D+E+0.025R−0.0000548S=0,0.000002SC+DS+C+R+H+D+E−0.034CH+D+ES+C+R+H+D+E−0.015+0.04+0.0000548C=0,0.04C−0.034RH+D+ES+C+R+H+D+E−0.025+0.0000548R=0,0.034SH+D+ES+C+R+H+D+E−0.000002HC+DS+C+R+H+D+E+0.025E−0.00000548+0.009H=0,0.000002HC+DS+C+R+H+D+E+0.034CH+D+ES+C+R+H+D+E−0.1+0.001+0.0000548D=0,0.034RH+D+ES+C+R+H+D+E+0.001D−0.009+0.025+0.0000548E=0,with initial conditions100000,5000,3000,10000,4000,1000.The solution of the preceding equations is computed using MATLAB and the numerical simulation of model (1) is performed using ode45 solver as illustrated in Figure 4 and fits with the obtained solutions. Hence, the endemic equilibrium of co-infection is given by(44)E1=8000,0,0,2039,0,0.Figure 4
Numerical simulations of population dynamics with no infection.


## 3.6.1. Disease-Free Equilibrium (DFE)

The disease-free equilibrium pointE0 is a point where there is no disease in the population. The disease-free equilibrium computed from model (1) is given by(41)E0=τμ,0,0,0,0,0.


## 3.6.2. Endemic Equilibrium (EE)

The endemic equilibrium pointE1 of model (1) is a state where the disease persists in the population with the zero rate of change of population. Hence, we obtain(42)τ−SβcC+DN−SβhH+D+EN+γR−μS=0,SβcC+DN−CβhH+D+EN−φ+ω+μC=0,ωC−RβhH+D+EN−γ+μR=0,SβhH+D+EN−HβcC+DN+ηE−μ+δH=0,HβcC+DN+CβhH+D+EN−ψ+κ+μD=0,RβhH+D+EN+κD−ζ+η+μE=0,with positive initial conditions.Moreover, logically taking the parameter values and initial population size, the endemic equilibriumE1 of model (1) can be written as(43)200−0.000002SC+DS+C+R+H+D+E−0.034SH+D+ES+C+R+H+D+E+0.025R−0.0000548S=0,0.000002SC+DS+C+R+H+D+E−0.034CH+D+ES+C+R+H+D+E−0.015+0.04+0.0000548C=0,0.04C−0.034RH+D+ES+C+R+H+D+E−0.025+0.0000548R=0,0.034SH+D+ES+C+R+H+D+E−0.000002HC+DS+C+R+H+D+E+0.025E−0.00000548+0.009H=0,0.000002HC+DS+C+R+H+D+E+0.034CH+D+ES+C+R+H+D+E−0.1+0.001+0.0000548D=0,0.034RH+D+ES+C+R+H+D+E+0.001D−0.009+0.025+0.0000548E=0,with initial conditions100000,5000,3000,10000,4000,1000.The solution of the preceding equations is computed using MATLAB and the numerical simulation of model (1) is performed using ode45 solver as illustrated in Figure 4 and fits with the obtained solutions. Hence, the endemic equilibrium of co-infection is given by(44)E1=8000,0,0,2039,0,0.Figure 4
Numerical simulations of population dynamics with no infection.


## 3.7. Reproduction Number

The reproduction number of the model (1) is computed using next-generation method defined and described in [8, 20–22]. Let fx be a vector consists of newly infected individuals arriving at compartments, and vx be a vector consists of the remaining terms in the infected compartments, where x=S,C,R,H,D,E. That is,(45)fx=SβcC+DNSβhH+D+ENHβcC+DN+CβhH+D+ENRβhH+D+EN,vx=φ+ω+μC−ηE+μ+δHψ+κ+μD−κD+ζ+η+μE.LetF be the Jacobian matrix computed from vector fx and V be the Jacobian matrix computed from vector vx and evaluated at disease-free equilibrium E0. The computed matrices are(46)F=βc0βc00βhβhβh00000000,V=φ+ω+μ0000μ+δ0−η00ψ+κ+μ000−κζ+η+μ.Leta=φ+ω+μ,b=ψ+κ+μ,c=ζ+η+μ.The next-generation matrixFV−1 is computed from matrices F and V as(47)FV−1=βca0βcb00βhμ+δβhb+βhκbc+βhκηbcδ+μβhc+ηβhcδ+μ00000000.The eigenvalues computed from the preceding next-generation matrix are(48)λ1=βca,λ2=βhμ+δ,λ3=0,λ4=0.Since reproduction numberR0 is the spectral radius of next-generation matrix, we have(49)R0=ρFV−1=maxβca,βhμ+δ.Hence, in this study,(50)R0=maxR0c,R0h=R0h.


## 3.8. Stability of Equilibria

Theorem 15.
The disease-free equilibrium of model (1) is locally asymptotically stable ifR0<1.Proof. To determine the local stability of disease-free equilibrium, similar to works presented in [23], we compute eigenvalues from the Jacobian matrix of model (1) evaluated at the disease-free equilibrium. Thus, the constructed Jacobian matrix J of model (1) evaluated at E0 is given by(51)J=−μ−βcγ−βh−βc−βh−βh0βc−ω+φ+μ00βc00ω−γ+μ000000βh−μ+δβhη0000−κ+ψ+μ00000κ−η+ζ+μ.The computed eigenvalues of the foregoing matrixJ are(52)λ1=−μ,λ2=aR0c−1,λ3=−γ−μ,λ4=μ+δR0h−1,λ5=−b,λ6=−c,where a=ω+φ+μ,b=κ+ψ+μ,c=η+ζ+μ,R0c=βc/a,R0h=βh/μ+δ.Since all computed eigenvalues are negative ifR0c<1 and R0h<1, by stability theory, the disease-free equilibrium of model (1) is locally asymptotically stable if R0<1.Theorem 16.
The disease-free equilibrium of co-infection model (1) is globally asymptotically stability if R0<1.Proof. Following from Figure2, forward bifurcation of model (1) at R0=maxR0c,R0h=R0h, (see [7, 18]).Theorem 17.
The endemic equilibrium of the constructed mode (1) is both locally and globally asymptotically stable if R0=R0h>1.Proof. The prove of this theorem follows from forward bifurcation behavior illustrated in Figure2 and stability behavior shown in Figure 4 (see [7, 18]).


## 3.9. Impact of Cholera on HIV-Infected Individuals

Based on the works done in [3], we analyze the impact of outbreak of cholera using partial derivative. Accordingly, μ is the common parameter of both R0h and R0c. Hence, we compute partial derivative ∂R0h/∂R0c=∂R0h/∂μ∂R0c/∂μ. Taking the computed results, R0c=βc/η+ζ+μ, and R0h=βh/μ+δ, we have(53)∂R0h∂R0c=∂R0h/∂μ∂R0c/∂μ=−βh/μ+δ2−βc/η+ζ+μ2=βhη+ζ+μ2βcμ+δ2>0.The positivity of the preceding partial derivative results shows that the existence of cholera infection increases the risk of HIV and vice versa.


## 4. Extension of the Model to Optimal Control

In this section, similar to [1, 14, 16, 17], Pontryagin’s maximum principle is used to determine optimal controls of the HIV-cholera co-infection model. Also, we extend model (1) to optimal control problem by incorporating optimal control measures: cholera prevention control (u1), HIV prevention control u2, and immunity control u3. In this model, we assume that cholera vaccination is administered for cholera recovered population. Furthermore, we normalized the state variables such that(54)τ=μN,s=SN,c=CN,r=RN,h=HN,d=DN,e=EN.Therefore, incorporating the above suggested assumptions, model (1) can be written as(55)dsdt=μ1−s−1−u1sβcc+d−1−u2sβhh+d+e+1−u3γr+sφc+δsh+ψsd+ζse,dcdt=1−u1sβcc+d−1−u2cβhh+d+e−φ+ω+μc+φc2+δch+ψcd+ζce,drdt=ωc−1−u2rβhh+d+e−1−u3γ+μr+φrc+δrh+ψrd+ζre,dhdt=1−u2sβhh+d+e−1−u1hβcc+d+1−u3ηe−μ+δh+φhc+δh2+ψhd+ζhe,dddt=1−u1hβcc+d+1−u2cβhh+d+e−ψ+κ+μd+φdc+δdh+ψd2+ζde,dedt=1−u2rβhh+d+e+κd−ζ+1−u3η+μe+φec+δeh+ψed+ζe2.The objective function:(56)J=max∫0tfc1s−c2c−c3h−c4d−c5e−12w1u12+w2u22+w3u32.The Hamiltonian function:Similar to the work done in [1, 14, 16], to solve the optimal problem, we defined Hamiltonian function H as(57)H=c1s−c2c−c3h−c4d−c5e−12w1u12+w2u22+w3u32+λ1dsdt+λ2dcdt+λ3drdt+λ4dhdt+λ5dddt+λ6dedt,dλ1dt=−∂H∂s=−c1+1−u1βcc+dλ1−λ2+1−u2βhh+d+eλ1−λ4−φc+δh+ψd+ζeλ1+μλ1dλ2dt=−∂H∂c=c2+1−u1sβcλ1−λ2+1−u2βhh+d+eλ2−λ5+1−u1hβcλ4−λ5−sφλ1+φ+ω+μλ2−2φc+δh+ψd+ζeλ2−λ3ω+φr−φhλ4−φdλ5−φeλ6,dλ3dt=−∂H∂r=1−u3γλ3−λ1+1−u2βhh+d+eλ3−λ6−φc+δh+ψd+ζeλ3+μλ3,dλ4dt=−∂H∂h=c3+1−u2sβhλ1−λ4+1−u1βcc+dλ4−λ5+1−u2cβhλ2−λ5+1−u2rβhλ3−λ6−δsλ1−δcλ2−δrλ3−φc+2δh+ψd+ζeλ4−δdλ5−δeλ6+μ+δλ4,dλ5dt=−∂H∂d=c4+1−u1sβcλ1−λ2+1−u2sβcλ1−λ4+1−u2cβhλ2−λ5+1−u2rβhλ3−λ6+1−u1hβcλ4−λ5−−ψsλ1−ψcλ2−ψrλ3−ψhλ4+ψ+k+μλ5−φc+δh+2ψd+ζeλ5−k+ψeλ6,dλ6dt=−∂H∂e=c5+1−u2sβhλ1−λ4+1−u2cβhλ2−λ5+1−u2rβhλ3−λ6+1−u3ηλ6−λ4−ζsλ1−ζcλ2−ζrλ3−ζhλ4−ζdλ5−−ζ+μ+φc+δh+ψd+2ζeλ6.(58)H=c1s−c2c−c3h−c4d−c5e−12w1u12+w2u22+w3u32+λ1dsdt+λ2dcdt+λ3drdt+λ4dhdt+λ5dddt+λ6dedt.where λitf=0,i=1,2,3,4,5,6 are adjoint variables corresponding to state variables S,C,R,H,D,E, respectively.Optimal controls:Let us consider the next Hamiltonian function as defined earlier:(59)H=c1s−c2c−c3h−c4d−c5e+12w1u12+w2u22+w3u32+λ1dsdt+λ2dcdt+λ3drdt+λ4dhdt+λ15dddt+λ16dedt.The optimal controls are obtained by solving the following equations:(60)∂H∂u1=0,∂H∂u2=0,∂H∂u3=0,The first equation of (61) gives(61)u1=sβcc+dλ2−λ1+hβcc+dλ5−λ4w1.second equation of (61) gives(62)w2u2+λ1sβhh+d+e+λ2cβhh+d+e+λ3rβhh+d+e−λ4sβhh+d+e−λ5cβhh+d+e−λ6rβhh+d+e=0.Also, from the second equation, we get(63)⇒u2=sβhh+d+eλ4−λ1+cβhh+d+eλ5−λ2+rβhh+d+eλ6−λ3w2.Also, solving foru2, we get(64)−w3u3−λ1γr+λ3γr−λ4ηe+λ6ηe=0.Thus,(65)u3=γrλ1−λ3+ηeλ4−λ6w3.Solving foru3 from the preceding equation, we get(66)u1∗=minmax0,sβcc+dλ2−λ1+hβcc+dλ5−λ4w1,1,Therefore, the maximum controlsu1∗,u2∗, and u3∗ are given by(67)u1∗=minmax0,sβcc+dλ2−λ1+hβcc+dλ15−λ4w1,1,u2∗=minmax0,sβhh+d+eλ4−λ1+cβhh+d+eλ5−λ2+rβhh+d+eλ6−λ3w2,1,u3∗=minmax0,γrλ1−λ3+ηeλ4−λ6w3,1.


## 5. Numerical Simulations

The parameters and initial conditions used in this study are taken logically and given byμ=0.0000548,βc=0.000002,βh=0.034or0.0034,γ=0.025,ϕ=0.015,δ=0.009,ζ=0.009,ω=0.04,ψ=0.1,η=0.025,κ=0.001,c1=100,c2=100,c3=1,c4=40;c5=200;0≤u1≤1;0≤u2≤1;0≤u3≤1;w1=w2=w3=1;s0=0.5;c0=0.2;r0=0.05;h0=0.15;d0=0.05;e0=0.05.


## 6. Results and Discussion

In this study, we have constructed co-infection of the HIV and cholera model. From plotted bifurcation diagram, it is observed that the full model and submodels exhibit forward bifurcation atR0=1. Furthermore, the stability analysis results show that the disease-free equilibrium points of the full model and submodels are locally asymptotically stable if the corresponding reproduction number is less than one. Also, the computed eigenvalues and the bifurcation diagram results show that the endemic equilibrium of the co-infection model and submodels is locally asymptotically stable if the corresponding reproduction number is greater than unity. The bifurcation diagram shows that R0=1 is the threshold point where bifurcation occurs. In Figure 4, in the absence of infection, the population dynamics is simulated. The numerical simulation results show that if there is no infection in the population, the population size attains the maximum possible population size where the birth balances the death. In Figure 5, the dynamics of population size, in the persistence of HIV infection and extinction of cholera infection, is simulated. Moreover, the numerical simulation results indicate cholera infection can be controlled with the effort of control measures, while HIV infection can persist in the population. In Figure 6, the simulations show that optimal control reduces the number of cholera-infected individuals. In Figure 7, the dynamics of all population are plotted for R0c<1 and R0h>1. In Figure 8, the coinfection of HIV and cholera with and without controls is plotted. The number of individuals infected both with HIV and cholera infections is reduced. Figure 9 shows that the number of HIV-infected population decreases with control interventions. Figure 10 depicts that the more HIV populations become cholera recovered and less infected with control interventions. Figure 11 shows that more cholera-infected individuals become cholera free with control interventions.Figure 5
Numerical simulations of dynamics of populations with extinction of infections in the populationR0<1.Figure 6
The simulations of cholera infected population with and without controls.Figure 7
Simulation of population dynamics forR0c<1 and R0h>1.Figure 8
Simulations of co-infected populations with and without controls.Figure 9
Simulation of HIV-only model with and without controls.Figure 10
Simulation of cholera recovered HIV population.Figure 11
Simulation of cholera recovered and HIV-free population.


## 7. Conclusion

The HIV and cholera co-infection model is developed and analyzed both as submodels and full model. The most important behavior exhibited by the model at bifurcation point is used to analyze all stability property of the equilibrium points. Submodels and full model analysis and numerical simulation results depict that the controlling or clearing of infections depends on the ability of controlling HIV infection. Hence, the more we manage the HIV infections, the less we are expected to be exposed to cholera infection. HIV-infected people are more vulnerable than healthy people. Based on simulations of Figures2–8, HIV and cholera that can be controlled for basic reproduction number is less than unity. The co-infected model exhibits forward bifurcation at R0=1, and the optimal control reduces the cholera-infected individuals. Since the model exhibits forward bifurcation at R0=1, the disease extinct R0<1 and persists in the population if R0>1. Optimal control can reduce the disease in the population. The study depicts that HIV/AIDS and cholera co infection may increase, through direct interaction of co-infected individuals.



---

*Source: 5460337-2022-05-23.xml*