A deterministic model for the transmission dynamics of tuberculosis (TB) with optimal control

Abstract

In this paper, we employed a deterministic model in the analysis of the dynamics of Tuberculosis with a keen interest in vaccination and drug resistance as the first line of treatment. It was assumed that some of the susceptible population were vaccinated but with temporal immunity. This is due to the fact that vaccines do not confer permanent immunity. Moreover, Part of the infected individual after treatment grow resistance to the drug. Infective immigrants were also considered to be part of the population. The basic reproductive number for the model is estimated using the Next Generation Matrix method. The equilibrium points of the TB model and their local and global stability were determined. It was established that if the basic reproductive number was less than unity (R₀ < 1), then the disease free equilibrium is stable and unstable if R₀ > 1. Furthermore, we investigated the optimal prevention, treatment and vaccination as control measures for the disease. It was established that the best control measure in combating Tuberculosis infections is prevention and vaccination of susceptible population.