Numerical Analysis for a Fractional Differential Time-Delay Model of HIV Infection of CD4 + T-Cell Proliferation under Antiretroviral Therapy

We study a fractional differential model of HIV infection of CD4 + T-cell, in which the CD4 + T-cell proliferation plays an important role in HIV infection under antiretroviral therapy. An appropriate method is given to ensure that both the equilibria are asymptotically stable for 𝜏 ≥ 0 . We calculate the basic reproduction number 𝑅 0 , the IFE 𝐸 0 , two IPEs 𝐸 ∗1 and 𝐸 ∗2 , and so on, and judge the stability of the equilibrium. In addition, we describe the dynamic behaviors of the fractional HIV model by using the Adams-type predictor-corrector method algorithm. At last, we extend the model to incorporate the term which we consider the loss of virion and a bilinear term during attacking the target cells.


Introduction
Human immunodeficiency virus (HIV) is a lentivirus (a member of the retrovirus family) that causes acquired immunodeficiency syndrome (AIDS) [1], a condition in humans in which the immune system begins to fail, leading to life-threatening opportunistic infections. HIV infects primarily vital cells in the human immune system such as helper Tcell (to be specific, CD4 + T-cell), macrophages, and dendritic cells. When CD4 + T-cell numbers decline below a critical level, cell-mediated immunity is lost and the body becomes progressively more susceptible to opportunistic infections (see [2]).
There are only a few results for dynamics of HIV infection of CD4 + T-cell. In 1992, Perelson et al. [3] examined a model for the interaction of HIV with CD4 + T-cell who considered four populations: uninfected T-cell, latently infected T-cell, actively infected T cells, and free virus, and they also considered effects of AZT on viral growth and T-cell population dynamics. In 2000, Culshaw and Ruan [4] firstly simplified their model into one consisting of only three components: the healthy CD4 + T-cell, infected CD4 + T-cell, and free virus and discussed the existence and stability of the infected steady state, and they studied the effect of the time delay on the stability of the endemically infected equilibrium; criteria were given to ensure that the infected equilibrium is asymptotically stable for all delay.
For backward bifurcations in other disease models, we refer the reader to [5][6][7][8][9][10][11][12]. In [12], this paper analyzed the backward bifurcation sources and application in infectious disease model. HIV/AIDS infection model is a special case of infectious disease model. For recent work on global analysis and persistence of HIV models, we refer the reader to [13][14][15][16][17][18] and references therein. A discussion on HIV infection and CD4 + T-cell depletion is given in the review paper [19].
Many mathematicians and researchers in the field of application are trying to model fractional order differential equations. In biology, the researchers found that biological membranes with fractional order have the nature of electronic conductivity, so it can be classified as a model of the fractional order. Because of the memory property of fractional calculus, we introduce the fractional calculus into HIV model. Both in mathematics and biology, fractional calculus will correspond with objective reality more than ODE. It is particularly important for us to study fractional HIV model. Furthermore, delay plays an important role in the process of spreading infectious diseases; it can be used to simulate the incubation period of infectious diseases, the period of patients infected with disease, period of patients immune to disease, and so on. The basic fact reflected by the specific mathematical model with time delay is that the change of trajectory about time not only depends on the moment itself but also is affected by some certain conditions before, even the reflection of some certain factors before. This kind of circumstance is abundant in the objective world.
Recently, Yan and Kou [2] have introduced fractionalorder derivatives into a model of HIV infection of CD4 + Tcell with time delay: with the initial conditions: Motivated by the works mentioned above, we will consider this model where the CD4 + T-cell proliferation does play an important role in HIV infection under antiretroviral therapy; a more appropriate method is given to ensure that both equilibria are asymptotically stable for ≥ 0. We calculate the basic reproduction number 0 , the IFE 0 , two IPEs * 1 and * 2 , and so on under certain conditions and judge the stability of the equilibrium. In addition, we describe the dynamic behaviors of the fractional HIV model by using the Adams-type predictor-corrector method algorithm. At last, we extend the model to incorporate the term which we consider the loss of virion and a bilinear term during attacking the target cells. In this paper, we establish mathematical model as follows: with the initial conditions: where denotes Caputo's fractional derivative of order with the lower limit zero. ( ), ( ), and ( ) represent the concentration of healthy CD4 + T-cell at time , infected CD4 + T-cell at time , and free HIV virus particles in the blood at time , respectively. The positive constant represents the length of the delay in days. A complete list of the parameter values for the model is given in Table 1.
Furthermore, we assume that ( ) > 0, ( ) ≥ 0, and ( ) ≥ 0 for all ≥ − . This paper is organized in the following way. In the next section, some necessary definitions and lemmas are presented. In Section 3, the stability of the equilibria is given. In Section 4, we calculate some of the data and judge the stability of the equilibrium. In Section 5, we will give the numerical simulation for the fractional HIV model. Finally, the conclusions are given.

Preliminaries
In this section, we introduce definitions and lemmas which will be used later.

Number of virions produced by infected CD4 + T-cell Varies
The maximal proliferation rate ( < 1 ) Varies The half saturation constant of the proliferation process Varies rt The effectiveness of RTIs ( rt = 0 means the therapy is totally ineffective, while rt = 1 indicates the therapy is 100% effective and the cell-to-cell infection is completely stopped) Varies The effectiveness of PIs ( = 1 meaning the therapy with PIs is 100% effective and no newly infectious virus particles will be produced [42]) Varies +

The stimulation of T-cell to proliferate in the presence of virus [43] Varies
is locally asymptotically stable if all the eigenvalues of the Jacobian matrix evaluated at the equilibrium point satisfying the following condition: The stable and unstable regions for 0 < ≤ 1 are shown in Figure 1 [45,46]. Proposition 4 (see [12]). Consider model (4).
(a) If 0 < 1 − , then the IFE 0 is the only equilibrium. (b) If 1 − < 0 < 1, then there are three equilibria: the IFE 0 and two IPEs, denoted by  [47,48] to prove the existence of solution of the fractional delay equations, and the one without no delay time is also parallel to [29,30]. By the next generation matrix method [11], we easily get we obtain the basic reproduction number of model (4)

The Stability of the Equilibria
In this section, we investigate the existence of equilibria of system (4). In order to find the equilibria of system (4), we put Following the analysis in [12], we get that system (13) has always the infection free equilibrium (IFE) 0 = (( / 1 ), 0, 0) and the infection persistent equilibrium (IPE) Next, we will discuss the stability for the local asymptotic stability of the viral free equilibrium 0 and the infected equilibrium * .
For the local asymptotic stability of the viral free equilibrium 0 , we have the following result. Lemma 6. If 0 < 1, then 0 is locally asymptotically stable for ≥ 0. If 0 = 1, then 0 is locally stable for ≥ 0. If Proof. The associated transcendental characteristic equation at 0 = ( 0 , 0, 0) is given by Obviously, the above equation has the characteristic root Next, we consider the transcendental polynomial For = 0, we get We have If 2 0 > 1, the characteristic equation has a positive eigenvalue and two negative eigenvalues. 0 is thus unstable with a two-dimensional stable manifold and a one-dimensional unstable manifold. If 2 0 = 1, 2,3 = 0, then 0 is locally stable. If 2 0 < 1, the other two eigenvalues have negative real parts if and only if 2 3 − 2 0 2 3 > 0, then 0 is locally asymptotically stable.

Comparison with Some of the Data
In this section, we calculate the basic reproduction number 0 , the IFE 0 , two IPEs * 1 and * 2 , ( ), + , + , + , and ( + )( + ). On the basis of these data, we apply all the conditions in Lemma 7 to judge the stability of the equilibrium * 1 and * 2 (

Numerical Simulations
In this section, we use the Adams-type predictor-corrector method for the numerical solution of nonlinear system (4) with time delay.
Firstly, we will replace system (4) by the following equivalent fractional integral equations: Next, we apply the predict, evaluate, correct, evaluate (PECE) method.

Extending the Model
In this section, we add the term − 3 in the third equation of model (4) which we consider the loss of virions due to all causes, and we also add the bilinear term − (1 − rt ) ( − ) ( − ) which we consider free infectious virions when they enter the target cells. We extend model (4) to the following system of differential equations: Following the analysis in [12], we get that system (30) has always the uninfected equilibrium 0 = (( / 1 ), 0, 0), and the infected equilibrium * * = ( * * , * * , * * ), where * * = By the next generation matrix method [11], we obtain the basic reproduction number of model (30)

Conclusion
In this paper, we modified the ODE model proposed by Shu and Wang [12] and the fractional model proposed by Yan and Kou [2] into a system of fractional order. We study a fractional differential model of HIV infection of CD4 + T cell. We will consider this model where the CD4 + T-cell proliferation does play an important role in HIV infection under antiretroviral therapy. The more appropriate method is given to ensure that both the equilibria are asymptotically stable for ≥ 0 under some conditions. We calculate the basic reproduction number 0 , the IFE 0 , two IPEs * 1 and * 2 , and so on, under certain conditions and judge the stability of the equilibrium. According to Tables 1 and 4   term which we consider the loss of virion and a bilinear term during attacking the target cells.