Author(s)

Leontine Nkague Nkamba^1,2, Thomas Timothee Manga³, Noboru Sakamoto^2,4

¹Department of Mathematics, Higher Teacher Training College, University of Yaoundé I, Yaoundé, Cameroon.
²Deustotech Laboratory Chair of Computational Mathematics, University of Deusto, Bilbao, Spain.
³AIDEPY Association des Ingénieurs Diplomés de l’Ecole Polytechnique de Yaoundé, Yaoundé, Cameroon.
⁴Faculty of Science and Engineering, Department of Mechatronics, Nanzan University, Nagoya, Japan.

This paper focuses on the study and control of a non-linear mathematical epidemic model ( SS_vihVELI ) 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 R_{vac < 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.

Tuberculosis, Basic Reproduction Number, Global Stability, Prevalence, CD4 Cells, Immune Deficiency, Optimal Control, Optimality, HIV

Nkamba, L.N., Manga, T.T. and Sakamoto, N. (2019) Stability and Optimal Control of Tuberculosis Spread with an Imperfect Vaccine in the Case of Co-Infection with HIV. Open Journal of Modelling and Simulation, 7, 97-114. doi: 10.4236/ojmsi.2019.72005.