Mathematical Model Of Human Immunode???ciency Virus - Tuberculosis Co-Infection Dynamics With Optimal Control

Abstract

In this thesis, we modified a mathematical model that describes the dynamics of HIV-TB co-infection with optimal control. The model was described using nonlinear ordinary differential equations. The HIV and TB only sub-models were analyzed separately. The basic reproduction numbers for all models are computed using the next-generation matrix and the stability analysis for the equilibrium points are analyzed. The disease free equilibrium points of the HIV and TB sub-models were both locally and globally stable if the respective reproduction numbers are less than one. The finding shows that the endemic equilibrium points of the HIV and TB sub-models were both locally and globally stable if the respective reproduction numbers are greater than one. Likewise, the disease free equilibrium point of the HIV-TB co-infection model is both locally and globally stable provided that the reproduction number is less than one and unstable otherwise. On the other hand, sensitivity analysis is studied and showed that reducing the value of most sensitive parameters could help to lower the basic reproduction number and thereby reducing the rate of infection. Moreover, the study extends co-infection model by incorporating time dependent controls as intervention, using Pontryagin’s minimum principle to derive necessary conditions for the optimal control and optimality system. Finally, numerical simulations of the model equations were carried out using Runge-Kutta 4th order in MATLAB. The simulations result illustrated that applying control strategies can successfully reduce the transmission dynamics of HIV and TB co-infection.