Mathematical modeling and the dynamical analysis of HIV/TB interaction

The thesis is based on the design and analysis of a series of realistic deterministic models for the transmission dynamics of Human Immunodeficiency Virus (HIV) and Tuberculosis (TB) in a population. In addition to the enormous public health and socio-economic burden each of these diseases inflicts globally, the synergistic relationship between the diseases (where one disease accelerates progression of the other) has equally devastating effects. First of all, a basic model for the transmission dynamics of the two diseases, incorporating the essential epidemiological and biological features of the two diseases, is designed. Appropriate dynamical systems theories and methodologies, such as stability/bifurcation theory, center manifold theory, comparison theory, Lyapunov function theory and Lasalle Invariance principle, are used to analyze the qualitative dynamics of the model. The TB component of the model was shown to undergo backward bifurcation, where a stable disease-free equilibrium co-exists with a endemic equilibrium when the associated reproduction threshold is less than unity, due to the exogenous re-infection property of TB disease. The HIV component of the basic model has a globally stable disease-free equilibrium when its reproduction number is less than unity.; The model is then extended to include an imperfect HIV vaccine and the directly observed treatment, short-course (DOTS) for TB. Some of the main public health contributions of the thesis include the following. (i) an imperfect HIV vaccine, with modest efficacy, can lead to the elimination of HIV from the community if the coverage rate is high enough. (ii) TB transmission by latently-infected individuals play a vital role in the spread of TB. (iii) TB can be effectively controlled using DOTS with large enough coverage if latent transmission and duration in the latent class are minimized. (iv) a backward bifurcation in TB modelling was caused by exogenous re-infection, whilst that in HIV in the presence of a vaccine arises due to the imperfect nature of the vaccine. (v) the treatment of people with active TB in the mixed HIV/TB class, with large enough coverage, can eliminate the mixed HIV/TB infection in the community.