# A Modified Iterative Algorithm for Numerical Investigation of HIV Infection Dynamics
###### Abstract  
The human immunodeficiency virus (HIV) mainly attacks CD4+ T cells in the host. Chronic HIV infection gradually depletes the CD4+ T cell pool, compromising the host's immunological reaction to invasive infections and ultimately leading to acquired immunodeficiency syndrome (AIDS). The goal of this study is not to provide a qualitative description of the rich dynamic characteristics of the HIV infection model of CD4+ T cells, but to produce accurate analytical solutions to the model using the modified iterative approach. In this research, a new efficient method using the new iterative method (NIM), the coupling of the standard NIM and Laplace transform, called the modified new iterative method (MNIM), has been introduced to resolve the HIV infection model as a class of system of ordinary differential equations (ODEs). A nonlinear HIV infection dynamics model is adopted as an instance to elucidate the identification process and the solution process of MNIM, only two iterations lead to ideal results. In addition, the model has also been solved using NIM and the fourth order Runge-Kutta (RK4) method. The results indicate that the solutions by MNIM match with those of RK4 method to a minimum of eight decimal places, whereas NIM solutions are not accurate enough. Numerical comparisons between the MNIM, NIM, the classical RK4 and other methods reveal that the modified technique has potential as a tool for the nonlinear systems of ODEs. Footnote †: journal: _algorithms_  
0000-0001-9991-9999-2  
numerous standard books [3; 4; 5; 6; 7] on the solution of ordinary differential equations (ODEs). A first attempt at classifying ODE systems is with respect to the initial or boundary conditions connected with them, both mathematically and computationally [8]. It should be mentioned that the numerical solution of ordinary differential equations is a very active and ongoing topic of research [9]. Infectious diseases frequently affect a huge number of people spread across vast geographical regions. Mathematical models of ordinary differential equations have significance in studying the dynamic behavior of infectious diseases. In recent times, many mathematical models including mumps virus [10], ebola virus disease [11], dengue fever disease [12], rubella disease [13], influenza transmission [14], zika virus transmission [15], COVID-19 pandemic [16; 17; 18] and many others have been formulated using differential equations. In the present study, the HIV infection model of CD4\({}^{+}\) T cells has been investigated mathematically. HIV infection targets CD4\({}^{+}\) T cells, the immune system's biggest white blood cells. HIV infection has a severe devastating impact on CD4\({}^{+}\) T cells, killing them and weakening the immune system. When the number of CD4\({}^{+}\) T cells falls below a particular threshold, the cell-mediated immune system vanishes, the immune system weakens, and the body is more likely to be infected [19]. The HIV infection model is represented by the standard three-compartmental system [20], such as susceptible CD4\({}^{+}\) T cell concentration, CD4\({}^{+}\) T cell infection and free HIV viral substance in the blood. They are denoted by time dependent variables such as \(X(t),Y(t)\) and \(Z(t)\). A system of nonlinear differential equations characterizes this model are as follows:  
\[\frac{d}{dt}\;X(t)=p-\alpha X(t)+rX(t)\bigg{(}1-\frac{X(t)+Y(t)}{T_{\text{max} }}\bigg{)}-kZ(t)X(t) \tag{1}\]  
\[\frac{d}{dt}\;Y(t)=kZ(t)X(t)-\beta Y(t) \tag{2}\]  
\[\frac{d}{dt}\;Z(t)=N\beta Y(t)-qZ(t) \tag{3}\]  
Let us assume the initial conditions are as follows:  
\[X(0)=x_{0},\;Y(0)=y_{0},\;Z(0)=z_{0} \tag{4}\]  
Several typical approaches to numerically address the HIV infection of the CD4\({}^{+}\) T cells model have recently been introduced in the literature. For instance, [PERSON] [21] used the Laplace-Adomian decomposition method (LADM) to solve the HIV infection model. [PERSON] et al. [22] had developed the multi-stage variational iteration approach (MSVIM).