Computational and Mathematical Methods in Medicine

Computational and Mathematical Methods in Medicine / 2019 / Article
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Research Article | Open Access

Volume 2019 |Article ID 4059549 | https://doi.org/10.1155/2019/4059549

Bijan Hasani Lichae, Jafar Biazar, Zainab Ayati, "The Fractional Differential Model of HIV-1 Infection of CD4+ T-Cells with Description of the Effect of Antiviral Drug Treatment", Computational and Mathematical Methods in Medicine, vol. 2019, Article ID 4059549, 12 pages, 2019. https://doi.org/10.1155/2019/4059549

The Fractional Differential Model of HIV-1 Infection of CD4+ T-Cells with Description of the Effect of Antiviral Drug Treatment

Academic Editor: Katarzyna Kolasa
Received13 Jun 2018
Revised11 Sep 2018
Accepted17 Dec 2018
Published08 Jan 2019

Abstract

In this paper, the fractional-order differential model of HIV-1 infection of CD4+ T-cells with the effect of drug therapy has been introduced. There are three components: uninfected CD4+ T-cells, , infected CD4+ T-cells, , and density of virions in plasma, . The aim is to gain numerical solution of this fractional-order HIV-1 model by Laplace Adomian decomposition method (LADM). The solution of the proposed model has been achieved in a series form. Moreover, to illustrate the ability and efficiency of the proposed approach, the solution will be compared with the solutions of some other numerical methods. The Caputo sense has been used for fractional derivatives.

1. Introduction

Human immunodeficiency virus (HIV) is a retrovirus that causes acquired immunodeficiency syndrome (AIDS) [1]. HIV infects, damages, and reduces CD4+ T-cells. Therefore, it causes to decrease the resistance of immune system [2]. The body becomes more gradually sensitive to infections and loses its safety. AIDS is one the most important and dangerous diseases in our time. According to UNAIDS 2017 annual report, “36.7 million people globally were living with HIV and 1.8 million people became newly infected with HIV and 1 million people died from AIDS-related illnesses in 2016.” In spite of the great progress in controlling the disease, no vaccine has been yet discovered for HIV. In the last two decades, a lot of efforts have been made to design and solve mathematical models that have essential rule in analyzing to control and prevent the spread of HIV-related diseases [313]. Usually almost all of these mathematical models explain the relation between HIV viruses and uninfected CD4+ cells and the effect of drug therapy to infected cells. Bonhoeffer et al. [4] presented a model for virus dynamics with two components and , where denotes the density of infected cells and shows the density of virus-producing cells.

The proposed model is as follows:where is the rate of production of infected cells, is the natural death rate of infected cells, is the rate of virus-producing cells’ death, and is the rate of infection of uninfected cells. This model and many such models were inspired from Anderson’s model [14, 15]. Anderson’s model is one of the first and the most important models of infectious diseases. Tuckwell and Wan [16] introduced a modified model of equation (1) with three components: uninfected, infected CD4+ T-cells, and density of virions in plasma (, , and , respectively). The presented model with three equations is as follows:with the initial conditions ,, and where , , , , , and are constant coefficients, is the rate of creation or production of CD4+ T-cells, is the natural death rate, is the rate of infected CD4+ cells from uninfected CD4+ cells, is the rate of virus-producing cells’ death, is the rate of creation of virions viruses by infected cells, and is the rate of virus particle death. For the sake of comparison and showing the ability of the proposed approach, we use the parameter values reported in references [6, 16]. The parameter values are as follows: (day/mm3), (day/mm3), (day/virion/mm3), (day/mm3), (virion/CLM/day), and (day). The rate of some coefficients will change if drug therapy is not 100% successful. When the drug treatment begins, infected cells which create virus components are affected. If the drug therapy is not effective, a part of infected cells will improve and remaining cells will begin to produce virus [8].

Mathematical modeling of many problems in biology and other branches of sciences appears as differential equations in fractional order. Because the fractional-order differential equations save memory on themselves and are related to fractals [8, 1719], we prefer to use the fractional-order form of the model (2) as follows:with the same initial conditions, where , . There are some numerical approaches for solving these types of mathematical models. Some of these methods are as follows: homotopy analysis, optimal homotopy asymptotic, homotopy perturbation, Adomian decomposition, and variational iteration [2030]. In [8], system of fractional equation (3) has been solved by homotopy analysis method (HAM) and generalized Euler method (GEM). In [31], equation (3) has also been solved by homotopy perturbation method (HPM). Adomian [32], introduced a decomposition method (ADM) which is a powerful method to get analytic approximate solution of differential equations. Using Laplace transform method with couple of ADM (LADM) to solve systems of differential equations leads to an effective method that finds many applications in applied mathematics. In this paper, we will solve equation (3) by LADM and will compare the results with the results achieved by generalized Euler, homotopy analysis, homotopy perturbation, and Runge–Kutta methods. The structure of the paper is as follows: in Section 2, we will present a brief review of fractional calculus. In Sections 3 and 4, we will solve the fractional-order HIV-1 model by LADM. In Section 5, the convergence of the method will be discussed. In the last section, we present the conclusion.

2. Fractional Calculus

The purpose of this section is to recall a few preliminaries about what appears in this research.

Definition 2.1. The Riemann–Liouville fractional integral of order for a function is defined aswhere (see [33]).

Definition 2.2. The Caputo fractional derivative for a function on the closed interval is defined aswhere is the integer part of . Another presentation of the Caputo fractional derivative can be shown as follows (see [33]):

Lemma 2.1. If and , then the following result holds for fractional calculus:

Proof. See [33, 34].

Definition 2.3. The Laplace transform of Caputo fractional derivative is defined as follows:

3. Solution of Model (3)

In this section, LADM has been implemented to solve system of fractional equation (3) with the initial conditions.

We apply Laplace transform on both sides of each equation of equation (3):which implies that

Substitution of the initial conditions in equation (10) and applying inverse Laplace transform results in

To apply ADM, let us consider , , and as the following series:

To decompose the nonlinear term , let us follow an alternate algorithm [35] to get,where is as the following equation:substituting equations (12)–(14) into (11) reads

We take inverse Laplace transform on both sides of each equation of equation (15):

We have calculated four terms of the infinite series of , , and as an approximate solution. To get any desired accuracy, one is able to proceed the process and obtain more terms. Finally, the solution of mathematical model can be obtained as follows:

4. Numerical Simulation

In this section, constants and initial values are substituted in equation (16) to obtain an approximate solution.

Substituting the following values:  = 0.272 (day/mm3),  = 0.00136 (day/mm3),  = 0.00027 (day/virion/mm3),  = 0.33 (day/mm3),  = 50 (virion/CLM/day), and  = 2.0 (day) and the initial conditions , and in equation (16), we get,

Three terms approximations can be written as the following form:

Let us take , , and equal to , so the solution of fractional-order of model (3) is obtained as follows:

For , the solution of equation (3) will be as follows:

In Tables 13, one can compare the approximate solution of fractional-order of model (3) with the results of GEM, HAM, RK4 in [8], and HPM in [31] using traditional order . The results of LADM are more accurate than the results obtained by other methods.


tLADMGEMHPMHAMRK4

0100100100100100
0.2100.023100.023100.023100.023100.023
0.4100.046100.047100.047100.047100.047
0.6100.070100.071100.071100.071100.071
0.8100.092100.097100.097100.096100.097
1.0100.112100.122100.123100.122100.122


tLADMGEMHPMHAMRK4

000000
0.20.004360.004340.004340.004340.00434
0.40.007500.007150.007210.007140.00715
0.60.010740.009080.009340.009090.00908
0.80.013360.010490.011170.010630.01049
1.00.018660.011610.012760.011940.01161


tLADMGEMHPMHAMRK4

011111
0.20.695810.690300.690710.690590.69070
0.40.538510.511520.512080.512370.51190
0.60.460970.410690.413940.409940.41103
0.80.396070.356560.377490.351480.35684
1.00.276710.330530.424190.328690.33073

Figures 13 show the results for different values of , and the results can be compared.

5. Convergence Analysis of the Method

In this section, convergence of the proposed method, using the idea presented in [36], is studied.

Consider the following functional equation:where is a functional operator and can be decomposed as and is a known function.

is a Caputo fractional derivative operator, is a linear operator, and is a nonlinear analytic operator, respectively. So equation (22) can be written as follows:

The goal is to find a function satisfying equation (22). By applying the Laplace transform on both sides of equation (23) reads

By using definition (2.3), equation (23) can be written as follows:by considering and using inverse of Laplace transform on both sides of equation (25) results inwhich implies that

By implementing ADM and assuming the solution as an infinite series say, , and writing the nonlinear term based on Adomian polynomials such aswhere

Equation (27) can be written as the following form:

So, we have

From which we can define

Theorem 5.1. LADM for equation (21) with the solution (19) is equivalent to

By using the following iterative scheme:where

Proof. For , from equation (33):Then, by assumption of equation (33), we havefor :We know that , so we obtainBy strong induction, let us haveand prove the following for ,Then, from equation (33), we deriveThis entails the statement is true and the theorem is proved.

Theorem 5.2. Let be a Banach space.(i) resulted from equation (31), convergence to , if (ii) satisfies in

Proof. , we haveThis means that , therefore, is a Cauchy sequence in the Banach space of and is convergent. So, .
From equation (33), we deriveFrom equation (35), we haveSo,

Lemma 5.1. Equation (43) is equivalent to equation (22).

Proof. By using Laplace transform on both sides of equation (43) reads toso we can write