A mathematical study of co-infection dynamics of tuberculosis and hepatitis B under vaccination and therapeutic control measures.

Abstract

Mathematical model describing the co-infection dynamics of tuberculosis and hepatitis B virus diseases incorporate optimal control were developed with the aim of investigating the effect of vaccination against hepatitis B virus and treatment for both diseases in a population. The disease-free equilibrium of the co-infection model were studied. The effective Basic Reproduction Number () for the co-infection model were obtained using the next generation matrix method. Stability of the disease-free equilibrium of the co-infection model were analyzed and it shown to be Locally and Asymptotically Stable if the  and Unstable if the . We performed a sensitivity analysis to illustrate the influence of different parameters on the effective Basic Reproduction Numbers of TB-HBV Co-infection model, . The co-infection model was extended to optimal control by incorporating four control interventions. The optimality system of the model was obtained using Pontryagin’s maximum principle. Simulation of the optimal control system was done and three strategies was proposed to check the effect of the controls. We implored the used of maple 18 (Runge-Kutta-Fehlberg 4-5^th order) in numerical solutions of system.  Firstly, prevention only for both diseases was considered. Secondly, treatment only for both diseases was considered and thirdly by using all control interventions (i.e both prevention and treatment to the diseases) were considered. We obtained that the three strategies were effective in minimizing the spread of TB-HBV Co-infection in the population in the specified period of time.