Numerical Treatment of the Model for HIV Infection of CD 4 T Cells by Using Multistep Laplace Adomian Decomposition Method

A new method for approximate analytic series solution called multistep Laplace Adomian Decomposition Method MLADM has been proposed for solving the model for HIV infection of CD4 T cells. The proposed method is modification of the classical Laplace Adomian Decomposition Method LADM with multistep approach. Fourth-order Runge-Kutta method RK4 is used to evaluate the effectiveness of the proposed algorithm. When we do not know the exact solution of a given problem, generally we use the RK4 method for comparison since it is widely used and accepted. Comparison of the results with RK4method is confirmed that MLADM performs with very high accuracy. Results show that MLADM is a very promising method for obtaining approximate solutions to the model for HIV infection of CD4 T cells. Some plots and tables are presented to show the reliability and simplicity of the methods. All computations have been made with the aid of a computer code written in Mathematica 7.


Introduction
In this study, we consider that the HIV infection model of CD4 T cells is examined 1 .This model is characterized by a system of the nonlinear differential equations

Laplace Adomian Decomposition Method
Application of the LADM to the HIV infection model of CD4 T cells is introduced in this section.In this model initial conditions were given as T 0 0.1, I 0 0, V 0 0.1.To solve this model by using the LADM, the Laplace transform is recalled.As known, the Laplace transform of x t is defined as We consider the following HIV infection model of CD4 T cells:

2.2
If we apply the Laplace transform to both sides of 2.2 we obtain the following equations:

2.3
To address the nonlinear terms, 3 , the Adomian decomposition method and the Adomian polynomials can be used.Solutions in this method are represented by infinite series such as where the components T k , I k , and V k are recursively computed.However, the nonlinear terms F T 2 t , G T t • I t , and H V t T t at the right side of 2.3 will be represented by an infinite series of Adomian polynomials: where A k , B k , and C k , k ≥ 0 are defined by

2.6
Substitution of 2.4 and 2.5 into 2.3 leads to

2.7
An iterative approximation algorithm by means of both sides of 2.7 could be obtained as follows:

2.8
The inverse Laplace transform of the first part of 2.8 gives the first terms of solutions T 0 , I 0 and V 0 which will be used to calculate, A 0 , B 0 , and C 0 .Consequently, the first term of Adomian polynomials, A 0 , B 0 , and C 0 is used to evaluate T 1 , I 1 , and V 1 .Subsequently, the determination of T 1 , I 1 , and V 1 leads to the determination of A 1 , B 1 , and C 1 , which are used to determine T 2 , I 2 , and V 2 and so on.Finally, the components of T k , I k , and V k , k ≥ 0, are determined by the second part of 2.8 and the series solutions of the 2.5 are obtained.

Multistep Laplace Adomian Decomposition Method
The multistep approach is used by many authors for different methods to find the solutions of various problems 11, 20-23 .The multistep approach for LADM proposed in this section is as a new idea for constructing the approximate solutions for the given HIV infection model of CD4 T cells.Let 0, T be the interval over which we want to find the solution of the initial value problem 1.1 .The solution interval, 0, T , is divided into M subintervals t m−1 , t m , m 1, 2, . . ., M of equal step size, h T/M by using the nodes, t m mh.The solution algorithm of the MLADM consists of the following steps.Initially, the LADM is applied to obtain the approximate solutions of T 1 , I 1 , and V 1 on the interval 0, t 1 by using the initial conditions, T 0 0.1, I 0 0 and V 0 0.1, respectively.For obtaining the 3.1

Application
To demonstrate the effectiveness of the proposed algorithm, the MLADM and RK4 are applied to the HIV infection model of CD4 T cells.Firstly for comparison purpose we implement the present method on small interval t ∈ 0, 1 as given in 7 .Tables 1, 2, and 3 show the comparison between the results of MLADM solution and results of classical LADM solution.
As could be seen in Figures 1-3 we obtain better results than Classical LADM solutions given in 7 for the same interval t ∈ 0, 1 .Now we implement the MLADM for larger time interval t ∈ 0, 520 .We obtain MLADM results for M 2000, T 520, and n 10.These results, obtained by MLADM and the RK4 method for T t , I t and V t are presented as figures.Figures 1-3 show the graphical outputs for MLADM and RK4 for t 0 to t 520.Figures 1-3 show that the multistep LADM solutions are very close to the Runge-Kutta solutions.Additionally, Table 4 shows the absolute errors between MLADM solutions and RK4 solutions.According to the Table 4 the amount of the absolute errors is small according to the values of variables.Figures 1-3 and Table 4 show that there is a good agreement between MLADM and RK4 for given time interval.It is observed that the MLADM gives a much better performance in approximate solutions compared to other mentioned methods in the literature for larger time interval.Table 4 shows the absolute errors between MALDM solutions and RK4 solutions.As could be seen from Figures 1-4, large oscillations have occurred between t 0 and t 100.Due to large oscillations big absolute errors have occurred from t 0 to t 100.But absolute errors become smaller after t 100.Initial oscillations effectively disappear after t 200.Damped oscillations are clearly visible after t 200.
As could be seen in Figure 1, the concentration of susceptible CD4 T cells approaches around 90 by oscillating with time while CD4 T cells infected by the HIV viruses converges to around 520 by oscillating as shown in Figure 2 and free HIV virus particles in the blood converges to around 650 by oscillating as shown in Figure 3.The main aim of this study is to find mathematical solution to given model for HIV infection of CD4 T cells.Besides Figures 5, 6

Conclusions
In this study, a new method called multistep LADM for solution of the HIV infection model of CD4 T cells is introduced.Figures 1, 2, 3 and Table 4 shows that the MLADM approximate solutions for the HIV infection model of CD4 T cells are very close to the Runge-Kutta approximate solutions.As can be seen clearly from the graphics, MLADM gives considerably good results on a longer time interval of t ∈ 0, 520 .This confirms that this new algorithm of the LADM increases the interval of convergence for the series solution.We have shown that the proposed algorithm is a very accurate and efficient method compared with RK4 method for the HIV infection model of CD4 T cells and it can be applied to other nonlinear systems.

Figure 3 :
Figure 3: Graphical comparison of V t .

Figure 5 :Figure 6 :
Figure 5: Phase portrait for HIV infection model of CD4 T cells 1.1 by using MLADM.

Figure 7 :Figure 8 :
Figure 7: T t − V t : phase portrait for HIV infection model of CD4 T cells 1.1 by using MLADM.

Table 1 :
Numerical comparison for T t .

Table 2 :
Numerical comparison for I t .

Table 3 :
Numerical comparison for V t .
approximate solutions of 1.1 over the interval t m−1 , t m , the LADM for m > 2 is used with the initial conditionsT 1 t m−1 , I 1 t m−1 , V 1 t m−1 .The similar process is repeated to generate a sequence of approximate solutions of T m t , I m t , V m t , m 1, 2, . . ., M. Consequently, final approximate MLADM solutions are obtained as follows:

Table 4 :
Absolute errors obtained by using Runge Kutta fourth-order method and MLADM for M 2000, T 520, n 10.