Mathematical modeling of the population dynamics of tuberculosis

Abstract

Tuberculosis (TB) is currently one of the major public health challenges in South Africa, and in many countries. Mycobacterium tuberculosis is among the leading causes of morbidity and mortality. It is known that tuberculosis is a curable infectious disease. In the case of incomplete treatment, however, the remains of Mycobacterium tuberculosis in the human system often results in the bacterium developing resistance to antibiotics. This leads to relapse and treatment against the resistant bacterium is extremely expensive and difficult. The aim of this work is to present and analyse mathematical models of the population dynamics of tuberculosis for the purpose of studying the effects of efficient treatment versus incomplete treatment. We analyse the spread, asymptotic behavior and possible eradication of the disease, versus persistence of tuberculosis. In particular, we consider inflow of infectives into the population, and we study the effects of screening. A sub-model will be studied to analyse the transmission dynamics of TB in an isolated population. The full model will take care of the inflow of susceptibles as well as inflow of TB infectives into the population. This dissertation enriches the existing literature with contributions in the form of optimal control and stochastic perturbation. We also show how stochastic perturbation can improve the stability of an equilibrium point. Our methods include Lyapunov functions, optimal control and stochastic differential equations. In the stability analysis of the DFE we show how backward bifurcation appears. Various phenomena are illustrated by way of simulations.