Since the early eighties, tremendous effort has been made in modeling the human immunodeficiency virus (HIV) infection. HIV is the virus that causes AIDS (Acquired Immune Deficiency Syndrome). A number of mathematical models have been developed to describe the interaction of human immune system with the HIV.

Before we present our HIV infection model, we give an introduction of Immunology and HIV infection in Chapter 1. Some other models related to HIV infection are also introduced.

In Chapter 2, seven state variables are used to describe HIV infection in our model. They consist of the concentrations of uninfected T cells, latently infected T cells (cells that are carrying but not producing provirus), actively infected T cells (cells that are producing provirus), uninfected macrophages, infected macrophages, slow/low strain viruses (viruses that are replicating slowly and having low expression in CDT 4⁺cells), and rapid/high strain viruses (viruses that are growing rapidly and causing large number of new viruses produced in T cells). The difference between our model and others is the inclusion of the effect of human immune response to HIV infection. As the HIV infection mechanism is still not fully known yet, we have made different assumptions about that and investigated the subsequent outcomes.

In Chapter 3, the element of control is introduced in the model developed in Chapter 2. We considered the use of chemotherapy for treatment in the early stages of HIV infection of a patient. Our control represents the effect the chemotherapy has on the viral production. In our optimal control problem, the objective function consists of two parts; one part denotes the maximization of T cell counts, and the other part represents the minimization of the systematic cost of chemotherapy. Finally, optimal control theory is employed to obtain optimal chemotherapy treatment strategies for the patient. Numerical computations are carried out using MISER3, a software package for solving optimal control problems developed by Jennings L.S. et al.

In Chapter 4, we present our numerical computation results. For the optimal control problem with objective function linear in control, optimal chemotherapy treatment strategies display the well known pattern of bang-bang control. For the optimal control problem with objective function quadratic in control, optimal chemotherapy treatment strategies are of a continuous nature. It is observed that a patient should be given some form of treatment as early as possible in order to achieve optimal result for all types of objective functions we have considered.