Abstract
This paper focuses on the study and control of a non-linear mathematical epidemic model ( SSvihVELI ) based on a system of ordinary differential equation modeling the spread of tuberculosis infectious with HIV/AIDS coinfection. Existence of both disease free equilibrium and endemic equilibrium is 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 Rvac < 1 is proved, where R0 is the reproduction number. We prove also that when R0 is less than 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, 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.
Highlights
Tuberculosis is one of the top 10 causes of death in the world [1]
This paper focuses on the study and control of a non-linear mathematical epidemic model ( SSvihVELI ) based on a system of ordinary differential equation modeling the spread of tuberculosis infectious with HIV/AIDS coinfection
Stability of Disease Free Equilibrium (DFE) we show that the disease-free equilibrium P0 is globally asymptotically stable with respect to if R0 ≤ 1 ; and P0 is unstable if R0 > 1 : Theorem 1
Summary
As part of the necessary multidisciplinary research approach, mathematical models have been extensively used to provide a framework for understanding tuberculosis transmission dynamics and control strategies of the infection spread in the host population [3] [4] [5] [6] [7]. The papers [17] consider the optimal control of tuberculosis through education, diagnosis campaign and chemoprophylaxis of latently infected. This paper deals with the stability analysis of an SVSvih , ELI transmission model and uses optimal control technique to find and evaluate the impact of a mass vaccination schedule in the spread of TB/VIH coinfection.
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