Revista Electrónica Nova Scientia Soluciones aproximadas de la dinámica de infección de VIH-1 con tasa de curación Approximate solutions for HIV-1 infection dynamics with cure rate

In this paper, two approximate solutions of HIV-1 infection dynamics model with cure rate are presented. The proposed solutions are obtained using homotopy perturbation method (HPM) and Boubaker Polynomials expansion scheme (BPES). A comparison of obtained solutions shows that HPM and BPES are powerful tools to solve nonlinear host viral infection models.

It is commonly known that HIV-1 targets mainly CD4 + T-cells and causes their death.It decreases the body's ability to fight infections.The standard infection process starts when HIV-1 enters its target T-cell and elaborates DNA copies of its viral RNA, with the help of the reverse transcriptase enzyme RT.Consequently, the viral DNA is inserted into the DNA of the infected cell; which will produce, from itself, viral particles that can bud off the cell and infect other cells.
Throughout the world, already over 16 million deaths at average age of 43 years have been caused by this virus [2][3][4]; bringing into attention an increasing need to understand and study its action and dynamics.Mathematical models have been proven valuable in understanding the dynamics of HIV infection [4][5][6].
One of the earliest models to primary infection with HIV is the one developed by Perelson [7], which considered a standard four-population model involving uninfected CD4+ T cells, latently infected CD4 + T cells, productively infected CD4 + T cells, and virus population.
The paper is organized as follows.Section 2 provides an idea about the model and its governing equations.We will describe the basic concepts of HPM in Section 3. Section 4 and Section 5 show the solution procedure for HIV using HPM and BPES, respectively.In Section 6, we discuss the obtained results.Finally, Section 7 summarizes the study and provides a global conclusion.

Basic concept of HPM method
The HPM method can be considered as a combination of the classical perturbation technique [37,38] and the homotopy (whose origin is in the topology) [39][40], but not restricted to a small parameter like traditional perturbation methods.For instance, HPM requires neither small parameter nor linearization, but only few iterations to obtain accurate solutions.
To figure out how HPM method works, consider a general nonlinear equation in the form with the following boundary conditions: where A is a general differential operator, B is a boundary operator, ) (r f a known analytical function, and  is the domain boundary for  .A can be divided into two operators L and N , where L is linear and N nonlinear; from this last statement, (2) can be rewritten as Generally, a homotopy can be constructed in the form [16][17][18]37] 0 where p is a homotopy parameter whose values are within the range of 0 and 1; 0 u is the first approximation for the solution of (4) that satisfies boundary conditions.
where operator L possesses trivial solution.
Assuming that solution for (5) can be written as a power series of Substituting ( 8) into (5) and equating identical powers of p terms, there can be found values for the sequence ,... , , (8), it yields in the approximate solution for (4) in the form

Solution by using HPM method
From (1) and ( 5), we establish the homotopy formulation

vt H v v v p p v p v t s dv t av t v t v t v t T H v v v p p v p v t v t v t v t v t H v v v p p v p v t qv t cv t
with boundary conditions : From ( 8), we assume that solution for (10) can be written as a power series of p as follows Where ), are functions yet to be determined.Substituting (11) into (10), and rearranging the coefficients of p powers, we have In addition, in order to fulfil the boundary conditions, we consider In order to obtain the unknown j i v , ( 1, 2,3, and 1, 2,3, ij  ), we must construct and solve the following system of equations, considering initial conditions , (0) We obtained , , , We set the values of parameters and initial conditions ( 0 (0) xx  , 0 (0) yy  , and 0 (0) zz  ) as reported in Table 1 [41].In order to increase the domain of convergence, we apply the Padé [20,35] approximant to (15)
where k B 4 are the 4k-order Boubaker polynomials [29-30], The main advantage of this formulation is the verification of boundary conditions, expressed in (1), in advance to the resolution process.In fact, thanks to the properties expressed in ( 16) and (17), these conditions are reduced to the inherently verified linear equations The final solution is

Results and analysis
In order to provide a reference point, the obtained results were compared to the numerical solution obtained using the Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant (RKF45) [42,43] built-in routine from Maple 17 Software.The routine was configured using an absolute error of 10 -7 and a relative error of 10 -6 .Figure 2  HPM-Padé technique is able to produce easy computable rational expression that exhibit a wide convergence region in comparison to polynomial solutions schemes.Nonetheless, further research is required in order to obtain solutions with even larger domain of convergence that can lead to a better understanding of the dynamics of the HIV infection and its relationship with the parameters of Table 1.    .Further research is required in order to obtain solution with larger domain of convergence that can lead to a better understanding of the dynamics of the HIV infection and the relationship with its parameters.

Figure 1 :
Figure 1: A synopsis of model's dynamics unknown pondering real coefficients.
for (1) is obtained, according to the principles of the BPES, by determining the non-null set of coefficients through 4 show the graphical comparison of the HPM (15), HPM-Padé and BPES (21) solutions.HPM and BPES solutions exhibit similar domains of convergence; the accuracy of both approximations decrease rapidly for t>0.4 as depicted in figures 2-4.Nonetheless, from the same figures, we can observe that the HPM-Padé solution possesses wider domain of convergence than BPES and standard HPM.
In this paper, a comparison of HPM, HPM-Padé and BPES was studied by solving an HIV-1 infection dynamics model with cure rate.The HPM-Padé solution exhibited a wider domain of convergence than HPM and BPES, reaching a good agreement to the exact solution for range

Table 1
[36] parameters values[36] + T-cells is in concordance to the mass action principle under mixing homogeneity.In this case, the concentration of new infected cells is proportional to the product and obtain approximations of order