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.


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][3][4][5][6][7]. Moreover, optimal control methods have been applied to the derivation of optimal therapies for this HIV infection [8][9][10][11][12][13][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. Section 3 proposes the delayed optimal control model of HIV infection. In Section 4, we utilize the Fourier series approximation to solve our model and results are demonstrated by some figures. Finally, the conclusions are summarized in Section 5.

Properties of Fourier Series
A measurable function defined over the interval 0 to may be expanded into a Fourier series as follows: Journal of Nonlinear Dynamics where the Fourier coefficients and * are given by The series in (1) has an infinite number of terms. To obtain an approximate expression for ( ), one can truncate the series up to the (2 + 1)th term as follows: where the Fourier series coefficient vector and the Fourier series vector ( ) are defined as where ( ) = cos ( 2 ) , = 0, 1, 2, 3, . . . , , (6a) * ( ) = sin ( 2 ) , = 1, 2, 3, . . . , .
The elements of Φ( ) are orthogonal in the interval [0, ]. By integrating both sides of (6a) and (6b) with respect to , we obtain Now, we approximate the function in the (7) by a truncated Fourier series. Consequently, the forward integral of the Fourier series vector Φ( ) can be represented by where (2 +1)×(2 +1) is the Fourier series operational matrix of forward integration and is given as ] .
Journal of Nonlinear Dynamics 3 As a result, we obtain Moreover, ] .
The delay function Φ( − ) is the shift of the function Φ( ) defined in (5) along the time axis by . The general expression is given by where is the delay operational matrix of Fourier series, which is as follows: ] .
Using (3) and (10) leads to We can get wherẽis the product operational matrix for the vector (see [22]). As a result,
In some systems of this type, it is desirable to select the optimal pair ( * (⋅), * ( )) which satisfies (22a), (22b), and (22c) and minimizes performance criterion modeled by a cost function of the form The following model is a delayed control system of HIV infection of CD4 + T-cells (for more details, see [15]): where control function ( ) is showing the effect of drugs on HIV virus production. For the delayed control model (24a), (24b), (24c), (24d), (24e), (24f), (24g), and (24h), we consider the objective functional to be defined as where , the weight of current costs of treatment, is assumed to be 100. Our goal is to maximize the objective functional (25) subject to the delayed control system (24a), (24b), (24c), (24d), (24e), (24f), (24g), and (24h), that is, to maximize the total count of CD4 + T-cells and to minimize the costs of treatment. The treatment interval is assumed to be = [ 0 , ] = [0, 300], which shows the duration on treatment in terms of days.
Finally, the optimal control problem now is reduced to Now, one can use various toolboxes like Matlab software to solve the above problem. The parameters of viral spread in Journal of Nonlinear Dynamics 7 Growth rate of CD4 T-cell population 0.03  Table 1.
The results of this optimization problem for = 45 and = 65 are depicted in Figure 1. As can be seen, utilizing the treatment increases the CD4 + T-cells ( ) (Figure 1(a)) and decreases the infected CD4 + T-cells ( ) (Figure 1(b)) and viral load ( ) (Figure 1(c)).

Conclusion
In this paper, we consider a delayed mathematical model describing HIV infection of CD4 + T-cells during therapy.
Our objective is to minimize the cost of treatment and to maximize the uninfected CD4 + T-cells. We use Fourier series approximation in order to solve the so-called mathematical model. By applying operational matrices for integration, product, and delay, the nonlinear optimal control model is reduced to a set of nonlinear algebraic equations. Since the set of sines and cosines is orthogonal, the operational matrices contain many zero elements. Hence, it makes the method computationally easy and attractive. The results show the efficiency of the method.