Stability and Optimal Control of an Mathematical Model of Tuberculosis/AIDS Co-infection with Vaccination

Abstract

This paper focuses on the study and control of a non linear mathematical epidemic model (S _Svih V E LI ) based on a system of ordinary differential equation modeling the spread of tuberculosis infectious with VIH/AIDS coinfection. Existence of both disease free equilibrium and endemic equilibrium are discussed. Reproduction number R₀ is determined. Using Lyapunov-Lasalle methods, we analyze the stability of epidemic system around the equilibriums (Disease free and endemic equilibrium). The global asymptotic stability of the disease free equilibrium whenever R_vac < 1 is proved, where R₀ is the reproduction number. We prove also that when R₀ is less then one, Tuberculosis can be eradicated. Numerical simulations, are conducted to approve analytic results. To achieve control of the disease, seeking to reduce the infectious group by the minimum vaccine coverage we can have. a control problem is formulated. The Pontryagin’s maximum principle is used to characterize the optimal control. The optimality system is derived and solved numerically
using the Runge Kutta fourth procedure.