A mathematical simulation and optimal control of a VIH model with different infectious level

Abstract

In this paper, we consider a mathematical model of propagation HIV disease. We propose a case with three different levels of infection. The model was analyzed using the stability theory of a nonlinear differential equation. We describe the equilibrium point of the model and the basic reproduction number. This equilibrium point is both locally and globally stable under certain conditions. A control problem is formulated, we use an optimal control strategies to reduce the number of deaths and to reduce the spread of HIV. Some results concerning the existence and the characterization of the optimal control will be given. The Pontryagin’s maximum principle is used to characterize the optimal control. We obtained an optimality system that we sought to solve numerically by an iterative discrete schema that converges following an appropriate test similar the one related to the forward-backward sweep method. Numerical simulations are given to illustrate the obtained results.