Optimal Control of a Delayed HIV Infection Model via Fourier Series

We present a delayed optimal control which describes the interaction of the immune system with the human immunodeficiency virus (HIV) and CD4+ T-cells. In order to improve the therapies, treatment and the intracellular delays are incorporated into the model. The optimal control in this model represents the efficiency of drug treatment in preventing viral production and new infections. The optimal pair of control and trajectories of this nonlinear delay system with quadratic cost functional is obtained by Fourier series approximation. The method is based on expanding time varying functions in the nonlinear delay system into their Fourier series with unknown coefficients. Using operational matrices for integration, product, and delay, the problem is reduced to a set of nonlinear algebraic equations. 1. Introduction Delays occur frequently in biological, chemical, electronic, and transportation systems [1]. Many mathematical models have been developed in order to understand the dynamics of HIV infection [2–7]. Moreover, optimal control methods have been applied to the derivation of optimal therapies for this HIV infection [8–14]. All these methods are based on HIV models which ignore the intracellular delay by assuming that the infectious process is instantaneous; that is, as soon as the virus enters an uninfected cell, it starts to produce virus particles; however, this is not reasonable biologically. In this paper, we consider the mathematical model of HIV infection with intracellular delay presented in [15] in order to make the model more tangible and closer to what happens in reality. Orthogonal functions (OFs) have received considerable attention in dealing with various problems of dynamic systems. Using operational matrices, the technique is based on reduction of these problems to systems of algebraic equations. Special attention has been given to applications of Walsh functions [16], block-pulse functions [17], Laguerre polynomials [18], Legendre polynomials [19], Chebyshev polynomials [20], and Fourier series [21]. In this paper, we apply Fourier series approximation to find the optimal pair of control and trajectories of the nonlinear delayed optimal control system governed by ordinary delay differential equations which describe the interaction of the human immunodeficiency virus (HIV). Operational matrices of integration, product, and delay have the most important role in our method. The paper is organized as follows. Section 2 consists of an introduction to Fourier series approximation and operational and other matrices, being used in Section 4.