Abstract
In this study, we construct a convergent algorithm for generating an approximate analytic solution for the fractional HIV infection of T cells with Atangana–Baleanu fractional derivatives in the Caputo sense. We compute the solution by utilizing the fractional homotopy analysis transform method (FHATM) and achieved a convergence region of the solution by employing an auxiliary parameter. Moreover, we apply a numerical scheme proposed by Toufik and Atangana for solving this kind of problem and compared with our results. A good agreement between the new algorithm and the numerical scheme is remarkable. The solution via the present algorithm can be obtained without any linearization or discretization which makes it reliable and easy to apply.
1. Introduction
Fractional calculus has played a significant role within the field of science and engineering, and many mathematicians and scientists have been working in this field lately. In recent decades, fractional calculus has been used in several areas of physics, biology, engineering, and others. Further details about fractional calculus and its applications can be found in the literature [1–9].
Because most nonlinear fractional differential equations cannot be solved exactly, it is necessary to use approximate and numerical methods. Various powerful mathematical techniques such as the Adomian decomposition method (ADM) [10, 11], homotopy analysis method (HAM) [12–15], optimal homotopy asymptotic method (OHAM) [16], homotopy perturbation method (HPM) [17], and variational iterative method (VIM) [18, 19] have been used to obtain an exact and approximate analytical solution.
The HAM was first introduced and employed in 1992 by Liao [20], after which many researchers successfully applied this method to solve linear and nonlinear differential equations. In recent years, many researchers have devoted their attention to obtaining a solution of linear and nonlinear differential equations using a variety of methods based on Laplace transform such as the Laplace decomposition method (LDM) [21] and the homotopy perturbation transform method (HPTM) [22]. Khan et al. [23] and Kumar et al. [24–26] coupled the HAM with Laplace transform to solve a nonlinear differential equation and Volterra integral equation. The homotopy analysis transform method (HATM) is a combination of HAM with Laplace transformation. The main advantage of this method is its ability to combine two powerful methods to obtain a rapid convergent series for fractional differential equations. This method provides us with a convenient way to control the convergence of the series solution.
Recently, Toufik and Atangana [27] developed a numerical scheme to solve a nonlinear fractional differential equation considering the Atangana–Baleanu fractional derivative. This method is a combination of the fundamental fractional calculus theorem with two-step Lagrange polynomial which is successfully used to solve many real-world problems [28–30].
Since the early 1980s, researchers have made an enormous effort to mathematically model the human immunodeficiency virus (HIV), the virus responsible for causing acquired immune deficiency syndrome (AIDS). In 1989, Perelson [31] considered the interaction of uninfected (T) and infected (I) T cells, and free virus molecules (F) in his model, following which Perelson et al. [32] extended the original model [31]. Culshaw and Ruan [33] reduced the model discussed in [32] as follows:where , , and represent the concentration of healthy T cells at time , infected T cells, and the free HI virus at time , respectively. Table 1 summarizes the meanings of functions and parameters. Equation (1) describes the rate of change in the uninfected population of T cells. The first term is the constant rate at which the body produces T cells from precursors in the bone marrow. Because the virus can infect both thymocytes and T cells, as for all cells in the body, these cells have a finite lifetime; thus, the second term describes the decreasing source. The third term describes the logistic growth of the healthy T cells, and the proliferation of infected T cells is neglected. The last term models the rate at which the free virus infects a T cell. Once a T cell has been infected, it becomes an infected cell; therefore, FT is subtracted from equations (1) and (3) and added to equation (2). Hence, and decrease concurrently.
Equation (2) describes the rate of change in the infected population of actively infected T cells. The first term represents the rate of infection of T cells by the virus. The second term represents the rate of disappearance of infected cells.
The three terms in equation (3) refer to the rate of production and destruction of the free infection virus. An actively infected T cell produces M virus particles; thus, the rate at which the virus is produced is set equal to M times the lytic death rate for the infected cell. A free virus is lost as a result of binding to an uninfected T cell at FT. The third term accounts for the loss of viral infectivity, viral death, and/or clearance from the body.
In this study, our approach to solving the fractional HIV model is to determine the order in which the fractional derivative changes by extending the classical HIV model (1)–(3) to the following set of fractional ordinary differential equations of the order , , and :with initial conditionswhere , , and are the Atangana–Baleanu fractional derivative in the Caputo sense (ABC). To the best of our knowledge, this is the first work that solves fractional HIV infection of the CD4+ T cells model in ABC sense analytically and numerically. To indicate the strength of our proposed method, we compare our findings with the algorithm of Toufik and Atangana [27].
2. Preliminaries and Notations
The Atangana–Baleanu fractional derivative in the Caputo sense (ABC) is defined as [6, 34]where , is a normalization function satisfyingwith , denotes the Mittag-Leffler function, defined byand denotes Euler’s gamma function defined as
The fractional integral for the ABC, which is newly defined with a nonlocal kernel and does not have singularities at , is defined as follows [6]:Here, when equals zero, the initial function is recovered, and when equals unity, the classical ordinary integral is obtained.
The Laplace transform of the fractional definitions with ABC is given as follows [6]:
3. Homotopy and Laplace Transform for FHATM
Applying the Laplace transform to equations (4)–(6) and using the formula for the Laplace transform of the ABC and then simplifying these equations, we find that
Next, defining the nonlinear operators aswhere , , and are the nonlinear operators. Let be a nonzero auxiliary parameter. Using the embedding parameter , we construct the so-called zeroth-order deformation equation:where is the Laplace operator, subject to the initial conditionsClearly if and we obtainwhen varies from zero to unity, the solution of the model (4)–(6) will vary from the initial guesses , , and to the exact solution , , and of the model (4)–(6). Expanding , by the Taylor series with respect to the embedding parameter yieldswhere
The convergence of equations (21)–(23) depends on the nonzero auxiliary parameters [20]. Moreover, if the initial values guessed for , , and and the auxiliary parameter are appropriately selected, then at , series (21)–(23) convergeswhich must be one of the solution of model (4)–(6), as proved by [20]. The equations governing the unknown functions can be deduced from the zeroth-deformation equations (16)–(18). Define the vectors
Differentiating the zeroth-deformation equations (16)–(18) -times with respect to the embedding parameter , then setting , and finally dividing them by , enables the -order deformation equations to be obtained:with initial conditions
It should be emphasized that the -order deformation equations (29)–(31) are linear; hence, they can be solved by Mathematica or MATLAB. For simplicity, we can specify the auxiliary functions to be equal to unity. Applying the inverse Laplace transform to equations (29)–(31), we obtainwhere
Next, the solution of the -order deformation equations (29)–(31) is given as
Taking initial conditions and equations (35)–(37), we obtainwhere , are given by
In a similar way, , , and , for can be obtained. Finally, the solution of model (2) is given byand by choosing a suitable value for for the convergence of the series according to [20]. The analysis of the convergence of the HATM can found in the literature [35].
4. Numerical Illustration
According to [36], it should be noted that the solution of the series contain the auxiliary parameter , which offers an easy way to control the convergence of the solution of the series. Because it is essential to assure that the series equations (25)–(27) is convergent, we plotted the curve of 6 terms of the FHATM solution for the fractional-time ABC equations in Figures 1–3. Using these curves, we note that the straight line that parallels the axis provides the region of convergence. These valid regions are listed in Table 2.
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Furthermore, if is appropriately chosen, equations (25)–(27) may converge fast. To this end, we have to compute the optimal values of the convergence control parameters from the minimum of the averaged residual errors.
Niu and Chun [37] introduced several methods to obtain the optimal value of . The optimal value of the convergence control parameter is defined by using the concept of the square residual error. An error analysis is presented to determine the optimal values of . We substitute equations (47)–(49) into equations (4)–(6) and obtain the residual functions as follows:Then, the square residual error for the sixth-order approximation is defined asThe use of the first derivative test enables us to determine the values of the auxiliary parameters , , and for which , , and are minimized. It should be emphasized that the approximation procedures that are used to select the optimal value of in FHATM are similar to those of HAM [38].
In Table 3, the minimum values of the square residual error are given for the optimal values of , , and when .
The absolute residual errors that were calculated for various are listed in Table 2. These results show that the FHATM obtains an accurate approximate solution for the fractional HIV model (4)–(6). The residual errors are plotted in Figure 4 for and various values of . The square residual errors are plotted in Figure 5, and Figure 6 shows the absolute residual functions for the optimal . As these figures show, the solution obtained by using FHATM provides us with a sufficiently accurate analytical solution that only requires a few iterative steps. Mathematica software was used to calculate the six-term approximations for , , and , respectively.
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Note that, if we set , then the FHATM solution is the same as that obtained with the HAM in [13]. The numerical results are plotted in Figure 7.
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5. Numerical Scheme
In this section, we solve fractional a HIV model numerically using the numerical scheme introduced by Toufik and Atangana [27]. The numerical solution and FHATM solution are compared in Figure 8.
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6. Conclusion and Further Work
In this study, we successfully solved the fractional HIV infection by using the T cells model numerically and analytically, which includes an operator of the type of the Atangana–Baleanu fractional derivative in the Caputo sense (ABC). Analytically approximate solution was obtained for this derivative by incorporating the FHATM in the model of the fractional HIV infection of T cells. The solution includes the auxiliary parameter , which provides an easy way to control the convergence region of the resulting infinite series. The results we obtained show that the FHATM is a successful technique for obtaining an approximate solution of the fractional HIV infection of T cells. Moreover, our result agrees strongly with the computation of Toufik and Atangana [27]. Studying the dynamics and stability of the system based on the ABC fractional definition and the FHATM algorithm is an interesting idea for the researchers.
Data Availability
No data were used.
Conflicts of Interest
The authors declare that they have no conflicts of interest.