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

Cheneke, Kumama Regassa; Rao, Koya Purnachandra; Edessa, Geremew Kenassa

doi:https://doi.org/10.1155/2022/5460337

Discrete Dynamics in Nature and Society

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Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5460337 | https://doi.org/10.1155/2022/5460337

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

Kumama Regassa Cheneke,¹Koya Purnachandra Rao,¹and Geremew Kenassa Edessa¹

Academic Editor: Fahad Al Basir

Received10 Nov 2021

Revised13 Jan 2022

Accepted15 Feb 2022

Published23 May 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 if and unstable if . 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 , 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 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.

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 . 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 . They are individuals infected with cholera and capable of transmitting infection to susceptible population through direct contact, (iii) Cholera recovered individuals . They are individuals recovered from cholera infection with temporary immunity, (iii) HIV/AIDS infected individuals . They are individuals infected with only HIV/AIDS and capable of transmitting HIV to susceptible population through sexual contacts, (iv) Co-infected individuals . 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 . 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 rate (iii)All populations die naturally at the rate (iv)Cholera infected individuals die at the rate (v)HIV/AIDS infected individuals die at the rate (vi)Co-infected individuals die at the rate (vii)Transmission rate of cholera is (viii)Transmission rate of HIV/AIDS is (ix)Recovery rate of only cholera infected with temporary immunity is (x)Recovery rate of co-infected person with temporary immunity is (xi) is the immunity loss rate of cholera recovered (xii) is the immunity loss rate of cholera recovered

The total size of population under study is denoted by and defined as , where is the size of susceptible population at time , is the size of cholera population at time , is the size of only cholera recovered population at time , is the size of HIV/AIDS population at time , is the size of co-infected population at time , and is the size of cholera recovered HIV population at time .

Based on Figure 1 and stated assumptions, the subsequent model is stated.

with conditions .

3. Mathematical Analysis of the Model

3.1. Invariant Region

Theorem 1. The solutions of the model (1) are positively invariant in the region .

Proof. Using [1], and adding all expressions on the left and right of equality in the model (1), we obtain

Logically ignoring the expressions and in the preceding equality, we get

Solving the preceding inequality over time interval , we obtain

In the preceding inequality as time , the total population size for all possible initial population size. Therefore, by the works presented in [13–20], the total population size is bounded for all time .

3.2. Positivity Property

Theorem 2. All solutions of the model are positive for all time in the solution region .

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

Ignoring the non-negative terms of the right expression in the preceding equality, we obtain

The solution of the foregoing inequality over time interval is computed as

Since the exponential expression and are positive, the state variable is positive for all time . Similarly, all state variables are non-negative for all time . 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 .

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 so that model (1) reduces to the HIV only model as

with non-negative initial conditions.

3.4.2. Invariant Region

Theorem 4. For HIV only model (2), the solution region is defined as

Proof. To prove Theorem 4, we add expressions on the left and right sides of the HIV only model so that

Solving the preceding inequality, we get

In the foregoing inequality, as . Therefore, the HIV only model is invariant in the region such that

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 in the HIV only model by solving the following equation:

Solving for the state variable from the preceding equation, we get

Therefore, the disease-free equilibrium of the HIV only model is given by

(2) Endemic Equilibrium of HIV Only Model. The endemic equilibrium of the HIV only model is obtained by solving the equations:

Solving the preceding equation, we obtain

By taking this, and the endemic equilibrium of the HIV only model exist if and given by

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 . The Jacobian matrix evaluated at disease-free equilibrium is given by . Hence, is defined by and given by

3.4.5. Bifurcation Analysis

Theorem 5. The HIV only model exhibits forward bifurcation at .

Proof. The prove of this theorem follows from the forward bifurcation diagram plotted in Figure 2.

3.4.6. Stability of Equilibriums of HIV Only Model

Theorem 6. The disease-free equilibrium of the HIV only model is locally asymptotically stable if and unstable if .

Proof. To determine the local stability, we computed the Jacobian matrix from the HIV only model at disease free-equilibrium as

The eigenvalues of the computed preceding matrix are

Since both foregoing eigenvalues are negative, by stability theory, the disease-free equilibrium of the HIV only model is locally asymptotically stable if and unstable if .

Theorem 7. The disease-free equilibrium of the HIV only model is globally asymptotically stable if and unstable if .

Proof. We follow from Figure 2.

Theorem 8. The endemic equilibrium of the HIV only model is locally asymptotically stable if and unstable if .

Proof. The Jacobian matrix at endemic equilibrium is given by

Trace

Again, .

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 if .

Theorem 9. The endemic equilibrium of HIV only model is globally asymptotically stable if and unstable if .

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 has both locally and globally endemic equilibrium.

3.5. Cholera Only Model

The cholera only model is obtained from model (1) by setting so that and given by

with non-negative initial conditions.

3.5.1. Invariant Region

Theorem 10. For cholera only model (2), the solution region is defined as

Proof. To prove Theorem 4, we add expressions on the left and right sides of the HIV only model so that

Solving the preceding inequality, we get

In the foregoing inequality, as . Thus, the cholera only model is invariant in the region such that

3.5.2. Equilibria of Cholera Only Model

In this section, we compute the disease-free equilibrium and endemic equilibrium of the cholera only model.

(1) Disease Free Equilibrium. The disease-free equilibrium of the cholera only model is computed by setting in the cholera only model at the equilibrium and given by

(2) Endemic Equilibrium. The endemic equilibrium of the cholera only model is computed by setting the rate of change state variables equal to zero. That is,

Adding first and second equation and using fourth equation such that

, we get the following mathematical equations:

Letting and using , the preceding equation reduces to the form:

Furthermore, letting , the preceding equations reduce to

This gives

Furthermore, solving for and , we get

And

Hence,

Thus, the endemic equilibrium is given by

3.5.3. Reproduction Number

The basic reproduction number is computed using the next-generation method. Following [3], the next-generation matrix , computed from the cholera only model is given by

Therefore, the reproduction number is given by

3.5.4. Bifurcation Analysis

Theorem 11. The cholera only model exhibits forward bifurcation at .

Proof. The prove of this theorem follows from forward bifurcation diagram of the cholera only model plotted in Figure 3.

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 if and unstable if .

Proof. It is corollary of forward bifurcation [Ref. 7, 18] Ded: , this condition is satisfied if, and is satisfied as where

Therefore, by stability theory, the disease-free equilibrium is locally asymptotically stable if .

Theorem 13. The disease-free equilibrium of the cholera only model is globally asymptotically stable if and unstable if .

Proof. We follow from Figure 3.

Theorem 14. The endemic equilibrium of cholera only model is locally asymptotically stable if and unstable if .

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 point is a point where there is no disease in the population. The disease-free equilibrium computed from model (1) is given by

3.6.2. Endemic Equilibrium (EE)

The endemic equilibrium point of model (1) is a state where the disease persists in the population with the zero rate of change of population. Hence, we obtain

with positive initial conditions.

Moreover, logically taking the parameter values and initial population size, the endemic equilibrium of model (1) can be written as

with initial conditions .

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

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 be a vector consists of newly infected individuals arriving at compartments, and be a vector consists of the remaining terms in the infected compartments, where . That is,

Let be the Jacobian matrix computed from vector and be the Jacobian matrix computed from vector and evaluated at disease-free equilibrium . The computed matrices are

Let .

The next-generation matrix is computed from matrices and as

The eigenvalues computed from the preceding next-generation matrix are

Since reproduction number is the spectral radius of next-generation matrix, we have

Hence, in this study,

3.8. Stability of Equilibria

Theorem 15. The disease-free equilibrium of model (1) is locally asymptotically stable if .

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 of model (1) evaluated at is given by

The computed eigenvalues of the foregoing matrix J arewhere .

Since all computed eigenvalues are negative if and , by stability theory, the disease-free equilibrium of model (1) is locally asymptotically stable if .

Theorem 16. The disease-free equilibrium of co-infection model (1) is globally asymptotically stability if .

Proof. Following from Figure 2, forward bifurcation of model (1) at , (see [7, 18]).

Theorem 17. The endemic equilibrium of the constructed mode (1) is both locally and globally asymptotically stable if .

Proof. The prove of this theorem follows from forward bifurcation behavior illustrated in Figure 2 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, is the common parameter of both and . Hence, we compute partial derivative . Taking the computed results, and , we have

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 (), HIV prevention control , and immunity control . In this model, we assume that cholera vaccination is administered for cholera recovered population. Furthermore, we normalized the state variables such that

Therefore, incorporating the above suggested assumptions, model (1) can be written as

The objective function:

The Hamiltonian function:

Similar to the work done in [1, 14, 16], to solve the optimal problem, we defined Hamiltonian function aswhere are adjoint variables corresponding to state variables , respectively.

Optimal controls:

Let us consider the next Hamiltonian function as defined earlier:

The optimal controls are obtained by solving the following equations:

The first equation of (61) gives

second equation of (61) gives

Also, from the second equation, we get

Also, solving for , we get

Thus,

Solving for from the preceding equation, we get

Therefore, the maximum controls and are given by

5. Numerical Simulations

The parameters and initial conditions used in this study are taken logically and given by

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 at . 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 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 and . 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.

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 Figures 2–8, HIV and cholera that can be controlled for basic reproduction number is less than unity. The co-infected model exhibits forward bifurcation at , and the optimal control reduces the cholera-infected individuals. Since the model exhibits forward bifurcation at , the disease extinct and persists in the population if . 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.

Data Availability

No collected data are used in this manuscript.

Conflicts of Interest

Authors declare that there are no conflicts of interest in the publication of this manuscript.

Acknowledgments

The first author would like to thank Wollega University and Hawasaa College of Teacher Education and Ministry of Higher Education for the support to pursue Ph.D program and continue in research work. This manuscript has no specific fund but part of Ph.D thesis at Wollega University.

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Copyright © 2022 Kumama Regassa Cheneke et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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