Development of a mathematical model for optimal treatment strategies to control acute HIV infection

Abstract

In this study, we seek to apply a system of nonlinear ordinary differential equations to analyze how the dynamics of primary infection affect the proliferation of Hu- man Immunodeficiency Virus (HIV). We prove existence, uniqueness, positivity, and boundedness of the solution. Also investigate the qualitative behavior of the models, and find a threshold parameter that guarantee the asymptotic stability of the equilibrium points, this parameter is known as basic reproduction number. The terms in the equations introduce parameters which are determined by fitting the model to matching clinical data sets using nonlinear least-squares method. The aim of this work is to determine the optimal drug administration schemes useful in improving patient’s health especially in poor resourced settings. The optimal treatments represent the efficacy of drug inhabiting viral production and preventing new infections with an objective functional which maximizes the T-cell (the white cells that coordinate activities of the immune system) and minimizes the systematic cost based on the percentage effect of the drug. The existence and the uniqueness of the optimal pair are discussed. A characterization of the optimal drug doses via adjoint variables is established. We obtain an optimality system that we solve numerically by a competitive Forward-Backward Sweep method.