APPLICABILITY AND ANALYSIS OF TRIGONOMETRIC – EXPONENTIAL SINGLE-STEP METHOD FOR THE NUMERICAL SOLUTION OF HIV-1 MODEL

. Abstract: This paper proposes the Trigonometric Exponential Single-Step Method (TESSM) for the numerical solution of HIV-1 infection model by using an interpolating function that consists of both trigonometric and exponential functions. The delay argument was approximated using Lagrange interpolation. The analysis of TESSM such as order of accuracy, convergence, consistency and stability was presented. The applicability of TESSM was tested on the HIV-1 infection model. The results generated via TESSM were also presented


INTRODUCTION
Most of the physical models in science and engineering are emanated from Differential Equations (DEs). Some of these DEs are difficult to solve or cannot be solved analytically. An alternative approach is to use numerical integration methods for approximating the solution of DEs using prescribed initial or boundary conditions [1]. There are many methods developed for the numerical solutions of the Initial Value Problems (IVPs) of the form ′ ( ) = ( , ( ), ( − ( , ( )))) , > 0 where Φ( ) is the initial function. In [2], the authors developed a new one-step rational method of order four for solving stiff and non-stiff Delay Differential Equations (DDEs) via interpolating function which consists of rational functions. Niekerk [3] proposed first, second and third order explicit nonlinear methods for singular and stiff IVPs. The algorithms are based on the representation of the solution by finite continued fractions. Fadugba [4] developed an improved numerical integration method via the transcendental function of exponential form for the solution of IVPs in Ordinary Differential Equations (ODEs). Islam [5] compared the numerical solutions of IVPs for ODEs with Euler's method and Runge-Kutta method. Stefanov [6] studied the cases of inverse interpolation of monotone and non-monotone functions. Some applications of inverse interpolation, including approximate solutions of nonlinear equations (root-finding) and analysis of census data, are also considered. Numerical models of nitrogen compound measurements in a stream with a removal mechanism using Saulyev technique with cubic spline interpolation were considered by [7]. Several authors have also studied the solutions of IVPs in ODEs via developed and existing methods, see [8] - [26]. Over the last two decades, there has been extensive research on the area of HIV-1 infection invading the human immune system. Bonhoeffer et al. [27] introduced a population model representing long-term dynamics of HIV infection in response to available drug therapies. According to the Joint United Nations Programme on HIV/AIDS (UNAIDS), 37 million people worldwide are infected with HIV-1 today of whom 24 million are in developing countries, see [28]. Infection with HIV-1, degrading the human immune system and 3 APPLICABILITY AND ANALYSIS OF TESSM FOR HIV-1 MODEL recent advances in drug therapy to arrest HIV-1 infection has generated considerable research interest in the area. Long-term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model were considered, see [28]. In this paper, we propose a new numerical method "Trigonometric -Exponential Single-Step Method (TESSM)" to analyse a mathematical model of HIV-1 infection. The rest of the paper is organized as follows.
Section 2 presents the derivation of TESSM. In Section 3, the properties of TESSM in terms of order of accuracy, consistency, stability and convergence are analyzed and investigated. Section 4 presents the numerical solution of the HIV-1 infection model via TESSM. Section 5 concludes the paper.

ANALYSIS OF THE PROPERTIES OF TESSM
The properties of the new method are analyzed as follows.

]
Thus, Therefore we can say that our derived method is convergent and hence Φ is Lipschitzian.

Order of Accuracy of TESSM
To determine the order of the new scheme, consider the Taylor's series expansion of the form The local truncation error is defined as Substituting (27) and (28) The order of accuracy of the new scheme is 2.
The condition that is the necessary and sufficient condition that TESSM (15) be stable and convergent.

Proof: Let
Then we have, We therefore conclude that TESSM is stable and hence convergent. This completes the proof.

NUMERICAL EXAMPLES
In this section, we analyse a mathematical model of HIV-1 infection to CD4+ T cells including the inhibitor drug via TESSM.

Consider a mathematical model of HIV-1 infection to CD4+ T cells including the inhibitor drug
discussed in the paper [28]. Let x(t) be the number of infected cells and y(t) be the number of virus producing cells and z(t) be the density of the Cytotoxic T-Lymphocyte (CTL) responses against virus-infected cells.

Model 1
In this basic delay HIV-1 infection model, we assume that the virus producing cells are killed by CTL instantaneously. When the delay is small, this model can be represented by the following set of equations

Model 2
In reality, there is a latency period during the process of killing of virus-producing cells by CTL. The variables and parameters used in these three models are given in Table 1. The numerical simulations of these models by TESSM using Table 1 are shown in Figs. 1 -3.