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

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 $R_0<1$ and unstable if $R_0>1$ . 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 $R_0<1$ , 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 $\left(\partial R_{0h}/\partial R_{0c}>0\right)$ 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.