Mathematical Model Of Hiv/Aids Transmission Dynamics With Optimal Control In The Case Of Adama City

Abstract

The human immunodeficiency virus (HIV)/acquired immunodeficiency syndrome (AIDS) continues to infect and kills many people worldwide each year particularly in Sub-Saharan Africa countries despite about four decade of research and attempts to control and minimize this infectious disease. In this thesis, the deterministic compartmental model for HIV/ AIDS is modified using nonlinear ordinary differential equations. We first proved that the model is both mathematically and epidemiologically well-posed by showing that all solutions of the model are positive and bounded with initial conditions in a certain meaningful set. The basic reproductionnumberiscomputedandstabilityconditionsforboththediseasefreeandendemic equilibria are investigated. The analysis indicated that the system has a locally asymptotically stabledisease-freeequilibriumpointwhenthereproductionnumberislessthanoneandlocally asymptotically stable endemic equilibrium point for the reproduction number is greater than one. The simulation result shows the agreement with the analytical results. Moreover, we examined the robustness of the model to some parameter values by examining the sensitivity of thebasicreproductionnumber. Themodelisthencalibratedwithcumulativecasesofinfection by HIV/AIDS data collected from Adama Health Office from 2003 to 2011 E.C. Furthermore, an optimal control model with two control strategies: namely disease prevention through education, and anti-retroviral therapy (ART) treatment is formulated and investigated. We then proved the existence of optimal control, determined the necessary optimality conditions, and then performed numerical simulations. The numerical results are shown that by using ART treatment in combination with health eduction rate of transmission, the spread of the disease can be reduced significantly. Thus, we can conclude that the proposed model well predicts the dynamics of infectious disease under consideration.