Mathematical model of HIV/AIDS, tuberculosis and their coevolution with optimal control: A case study in Ethiopia

Abstract

The communicable disease tuberculosis (TB), human immunodeficiency virus/acquired immune deficiency syndrome (HIV/AIDS) disease, and their co-infection are the most serious public health issues in the world. In this thesis, three population level mathematical models of the three infections min Ethiopia are developed and analyzed. The first model considers the dynamics of HIV/AIDS, which comprise the following exclusive classes of individuals, the aware and unaware susceptibles, undiagnosed HIV infectious, diagnosed HIV infectious with and without AIDS symptoms, and those under HIV treatment. This model considers the rate of becoming aware and unaware as a function of media campaigns, whereas screening and treatment rates are constant. The effective reproduction number, equilibria, and nature of stability were formulated. The bifurcation occurs when the effective reproduction number is equal to unity. This model is extended to a new model which incorporates interventions such as preventive, screening, and treatment strategies. In this model, the optimal control problem is formulated and solved analytically. In addition to this, the optimality system is derived and solved numerically using the forward-backward sweep method (FBSM). Finally, the cost-effectiveness of some combined control strategies is derived. The second model reflects the TB transmission dynamics with drug resistance TB (DR-TB). The two infectious TB stages, namely drug-sensitive TB and drug-resistant TB, are considered in the model. Assuming that drug-sensitive TB can be cured by first-line anti-TB drugs. In fact, once the Tubercle Bacilli become resistant to one or more anti-TB drugs, the drug-resistance TB occurs. The model is analyzed analytically and extended to an optimal control problem via incorporating preventive efforts, case finding, and case holding. In the study, four different strategies are introduced based on different combination of measures. The optimal control problem is examined both analytically and numerically. The third model describes a new mathematical model of human immunodeficiency virus (HIV) associated with tuberculosis (TB). This full TB-HIV co-infection model is analyzed analytically. Which is extended to an optimal control problem by using controlling variables such as preventive efforts, case finding effort for TB, and HIV treatment. We proposed four strategies, which are combinations of two or more control measures at a time. The model with controls is analyzed both analytically and numerically. The numerical results are derived using the classical Runge-Kutta method of order four (RK4-method). The finding suggests that optimal combination strategies are used to reduce both the disease burden and the cost of intervention. Further, the cost- effectiveness of each strategy is assessed to identify the best cost-effective approach the fight against TB-HIV co-infection in Ethiopia.