Homotopy Perturbation Method for Solving Human T-cell Lymphotropic Virus I(htlv-i) Infection of Cd4 + T-cells Model

In this article, homotopy perturbation method is implemented to give approximate and analytical solutions of nonlinear ordinary differential equation systems such as human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells model. The proposed scheme is based on homotopy perturbation method (HPM), Laplace transform and Padé approximants. The results to get the homotopy perturbation method (HPM) is applied Padé approximants. Our proposed approach showed results to analytical solutions of nonlinear ordinary differential equation systems. Some plots are presented to show the reliability and simplicity of the methods.


Dynamics of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells is examined
[3] at the study.The components of the basic four-component model are the concentration of healthy CD4+ T-cells at time t, the concentration of latently infected CD4+ T-cells, the concentration of actively infected CD4+ T-cells and the concentration of leukemic cells at time t are denoted respectively by T(t), (t) , ( ) and (t) with the initial conditions: and max 2200 T  A technique for calculating the analytical solutions of nonlinear ordinary differential equation systems is developed in this paper.The developed technique depends only on the fundamental operation properties of Laplace transform and Padé approximants.The calculated results are exactly the same as those obtained by other analytical or approximate methods and demonstrate the reliability and efficiency of the technique.We will use Laplace transform and Pade´ approximant to deal with the truncated series.Pade´ approximant [2] approximates a function by the ratio of two polynomials.The coefficients of the powers occurring in the polynomials are determined by the coefficients in the Taylor series expansion of the function.Generally, the Pade´ approximant can enlarge the convergence domain of the truncated Taylor series and can improve greatly the convergence rate of the truncated Maclaurin series.
The motivation of this paper is to extend the application of the analytic homotopyperturbation method (HPM) [8][9] and Padé approximants [1] to solve of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells (1).The homotopy perturbation method (HPM) and the variational iteration method was first proposed by Chinese mathematician He [8][9][12][13][14][15].The first connection between series solution methods such as an Adomian decomposition method and Padé approximants was established in [6].The transmission and dynamics of HTLV-I feature several biological characteristics that are of interest to epidemiologists, mathematicians, and biologists, see, for example, [10][11]16], etc.Like HIV, HTLV-I targets CD4+ T-cells, the most abundant white cells in the immune system, decreasing the body's ability to fight infection.
And from the difference ( ) ( ) ( ) ( ) : The lower index 1 j N M    in the summation on the right side of ( 6) is chosen because the first NM  derivatives of () fx and , () NM Rx are to agree at 0 x  .When the left side of ( 6) is multiplied out and the coefficients of the powers of i x are set equal to zero for 0,1, 2,..., k N M  , the result is a system of 1 NM  linear equations: ap q a a p q a q a a p q a q a q a a p q a q a a p q a q a q a a q a q a q a a Notice that in each equation the sum of the subscripts on the factors of each product is the same, and this sum increases consecutively from 0 to NM  .The M equations in (8) involve only the unknowns 1 2 3 , , ,..., M q q q q and must be solved first.Then the equations in (7) are used successively to find

HOMOTOPY PERTURBATION METHOD
To illustrate the homotopy perturbation method (HPM) for solving non-linear differential equations, He [8,9] considered the following non-linear differential equation: where A is a general differential operator, B is a boundary operator, f(r) is a known analytic function,  is the boundary of the domain  and n   denotes differentiation along the normal vector drawn outwards from  .The operator A can generally be divided into two parts M and N. Therefore, (9) can be rewritten as follows: He [8,9] constructed a homotopy where is an embedding parameter, and 0 u is an initial approximation of (9).Obviously, we have 0 The changing process of p from zero to unity is just that of H(v,p) from 0 ( ) In topology, this is called deformation and 0 ( ) ( ) and ( ) are called homotopic.According to the homotopy perturbation method, the parameter p is used as a small parameter, and the solution of Eq. ( 12) can be expressed as a series in p in the form 12) corresponds to the original one, Eqs. ( 11) and ( 15) become the approximate solution of Eq. ( 11), i.e., The convergence of the series in Eq. ( 16) is discussed by He in [8,9].

APPLICATIONS
In this section, we will apply the homotopy perturbation method to nonlinear ordinary differential systems (1).

Homotopy perturbation method to a human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells
According to homotopy perturbation method, we derive a correct functional as follows: where "dot" denotes differentiation with respect to t , and the initial approximations are as follows:

t T P v t y t T P v t z t T P v t u t T P
where , , , 1, 2,3,...
are functions yet to be determined.Substituting Eqs.( 18) and (19) into Eq.( 17) and arranging the coefficients of "p" powers, we have In order to obtain the unknowns , ( ), , 1, 2,3, ij v t i j  we must construct and solve the following system which includes nine equations with nine unknowns, considering the initial conditions , (0) 0, , From Eq. ( 16), if the three terms approximations are sufficient, we will obtain:      These results obtained by homotopy perturbation method, three, four, five and six terms approximations for ( ), ( ), ( ) and ( ) T t T t T t T t are calculated and presented follow.

CONCLUSIONS
In this paper, we have presented an after treatment technique for the homotopy perturbation method.Because the Pade´ approximant usually improves greatly the Maclaurin series in the convergence region and the convergence rate, the at leads to a better analytic approximate solution from homotopy perturbation method truncated series The homotopy perturbation method was used for finding the solutions of nonlinear ordinary differential equation systems such as human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ Tcells model.We demonstrated the accuracy and efficiency of these methods by solving some ordinary differential equation systems.We use Laplace transformation and Padé approximant to obtain an analytic solution and to improve the accuracy of homotopy perturbation method.The reliability of the method and reduction in the size of computational domain give this method a wider applicability.It is observed that The results to get the homotopy perturbation method (HPM) applied Padé approximants is an effective and reliable tool for the solution of the nonlinear ordinary differential equation systems considered in the present paper.
The computations associated with the examples in this paper were performed using Maple 7 and Matlab 7
is the quotient of two polynomials NM P x Q x of degrees N and M, respectively.We use the notation , () NM Rx to denote this quotient.The , () NM Rx Padé approximations to a function () fx are given by [1]