Optimal Control of a Delayed HIV Infection Model with Immune Response Using an Efficient Numerical Method

We present a delay-differential equation model with optimal control that describes the interactions between human immunodeficiency virus (HIV), CD4+ T cells, and cell-mediated immune response. Both the treatment and the intracellular delay are incorporated into the model in order to improve therapies to cure HIV infection. The optimal controls represent the efficiency of drug treatment in inhibiting viral production and preventing new infections. Existence for the optimal control pair is established, Pontryagin’s maximum principle is used to characterize these optimal controls, and the optimality system is derived. For the numerical simulation, we propose a new algorithm based on the forward and backward difference approximation. 1. Introduction Human immunodeficiency virus (HIV) is a lentivirus that causes acquired immunodeficiency syndrome (AIDS), a condition in humans in which the immune system begins to fail, leading to life-threatening opportunistic infections. There are several other ways the infection can transfer, for example, open wound, saliva, and ulcers. There are some antiretroviral (ARV) drugs available nowadays which help the immune system in preventing the infection due to HIV even though it is not possible to cure it. Reverse transcriptase inhibitors (RTIs) are one of the chemotherapies which oppose the conversion of RNA of the virus to DNA (reverse transcription), so that the viral population will be minimum and on the other hand the CD count remains higher and the host can survive. Another one is the protease inhibitors (PIs) which prevent the production of viruses from the actively infected CD T cells. In the literature, many mathematical models have been developed in order to understand the dynamics of HIV infection [1–6]. In addition, optimal control methods have been applied to the derivation of optimal therapies for this HIV infection [7–13]. All these methods are based on HIV models which ignore the intracellular delay by assuming that the infectious process is instantaneous; that is, in the very moment that the virus enters an uninfected cell, this one starts to produce virus particles, and we know that this is not biologically reasonable. In this paper, we consider the mathematical model for HIV infection with intracellular delay and cell-mediated immune response presented by Zhu and Zou in [6] and we introduce two controls, one simulating effect of RTIs and the other control simulating effect of PIs, incorporating drug efficacy. The intracellular delay represents the time needed for infected cells to produce virions after