Analytical and Numerical Approaches for Complicated Nonlinear Equations
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Numerical Analysis for a Fractional Differential TimeDelay Model of HIV Infection of CD4^{+} TCell Proliferation under Antiretroviral Therapy
Abstract
We study a fractional differential model of HIV infection of Tcell, in which the Tcell 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 . We calculate the basic reproduction number , the IFE , two IPEs and , 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 Adamstype predictorcorrector 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.
1. 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 lifethreatening opportunistic infections. HIV infects primarily vital cells in the human immune system such as helper Tcell (to be specific, Tcell), macrophages, and dendritic cells. When Tcell numbers decline below a critical level, cellmediated 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 Tcell. In 1992, Perelson et al. [3] examined a model for the interaction of HIV with Tcell who considered four populations: uninfected Tcell, latently infected Tcell, actively infected T cells, and free virus, and they also considered effects of AZT on viral growth and Tcell population dynamics. In 2000, Culshaw and Ruan [4] firstly simplified their model into one consisting of only three components: the healthy Tcell, infected Tcell, 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 [5–12]. 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 [13–18] and references therein. A discussion on HIV infection and Tcell depletion is given in the review paper [19].
In 2012, Shu and Wang [12] considered a new model frame that included full logistic growth terms of both healthy and infected Tcell:
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in various fields, such as physics, mechanics, chemical technology, population dynamics, biotechnology, and economics (see e.g., [20–26]). As one of the important topics in the research differential equations, the boundary value problem has attained a great deal of attention (see [27–41] and the references therein).
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 Tcell with time delay: with the initial conditions:
Motivated by the works mentioned above, we will consider this model where the Tcell 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 . We calculate the basic reproduction number , the IFE , two IPEs and , 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 Adamstype predictorcorrector 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 Tcell at time , infected Tcell 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 , and 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.
2. Preliminaries
In this section, we introduce definitions and lemmas which will be used later.
Definition 1 (see [20, 25]). The fractional (arbitrary) order integral of the function of order is defined by
Definition 2 (see [20]). Let , , where denotes the integer part of number . If , the Caputo fractional derivative of order of is defined by
Lemma 3 (see [44]). The equilibrium point of the fractional differential system: 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 are shown in Figure 1 [45, 46].
Proposition 4 (see [12]). Consider model (4).(i)Assume that (H1): is satisfied.(a)If , then the IFE is the only equilibrium (Table 3).(b)If , then there are two equilibria: the IFE and a unique IPE .(ii)Assume that (H2): is satisfied. Let ; then ( follows from the assumption that ).(a)If , then the IFE is the only equilibrium.(b)If , then there are three equilibria: the IFE and two IPEs, denoted by and such that .(c)If or , then there are two equilibria: the IFE and a unique IPE.
Remark 5. It is similar to [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)
3. 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) and the infection persistent equilibrium (IPE) , where
Next, we will discuss the stability for the local asymptotic stability of the viral free equilibrium and the infected equilibrium .
For the local asymptotic stability of the viral free equilibrium , we have the following result.
Lemma 6. If , then is locally asymptotically stable for . If , then is locally stable for . If , then is a saddle point with a twodimensional stable manifold and a onedimensional unstable manifold.
Proof. The associated transcendental characteristic equation at is given by
Obviously, the above equation has the characteristic root
Next, we consider the transcendental polynomial
For , we get
We have
If , the characteristic equation has a positive eigenvalue and two negative eigenvalues. is thus unstable with a twodimensional stable manifold and a onedimensional unstable manifold. If , , then is locally stable. If , the other two eigenvalues have negative real parts if and only if , then is locally asymptotically stable.
For , we get
Assume that the above equation has roots for and ; we get
Separating the real and imaginary parts gives
From the second equation of (22), we have that is , .
For , , substituting into the first equation of (22), we have
For the parameter values given in Table 1, we take any , then the infected equilibrium , and we get that the above equation is unequal for . Therefore, .
According to Lemma 3, the uninfected equilibrium is locally asymptotically stable. The proof is completed.
Next, for the sake of convenience, at , we give the following symbols:
Then the characteristic equation of the linear system is Using the results in [49], we get Denote
Lemma 7 (see [49]). The infected equilibrium is asymptotically stable for any time delay if either(i) if ;(ii)if , , , , ;(iii)if , , , , then all roots of satisfy ;(iv)if for all .
4. Comparison with Some of the Data
In this section, we calculate the basic reproduction number , the IFE , two IPEs and , , , , , and . On the basis of these data, we apply all the conditions in Lemma 7 to judge the stability of the equilibrium and (Table 2).
(a)  
 
(b)  

(a)  
 
(b)  

Remark 8. (1) If , there are two equilibria: the IFE and a unique IPE . In addition, the system at satisfies the second condition in Lemma 7, then the IPE is locally asymptotically stable (Table 5).
(2) If , there are three equilibria: the IFE and two IPEs. The system at doesn't satisfy all the conditions in Lemma 7, then the IPE is unstable. The system at satisfies the second condition in Lemma 7, then the IPE is locally asymptotically stable.
(3) If , the IFE is the only equilibrium.
Remark 9. If , there are two equilibria: the IFE and a unique IPE . In addition, the system at satisfies the second condition in Lemma 7, then the IPE is locally asymptotically stable.
Remark 10. If , there are two equilibria: the IFE and a unique IPE . In addition, the system at satisfies the second condition in Lemma 7, then the IPE is locally asymptotically stable.
Remark 11. (1) If , the IFE is the only equilibrium.
(2) If , there are three equilibria: the IFE and two IPEs. The system at and does not satisfy all the conditions in Lemma 7, then the IPEs and are unstable.
(3) If , there are two equilibria: the IFE and a unique IPE . In addition, the system at does not satisfy all the conditions in Lemma 7, then the IPE is unstable. The system at satisfies the second condition in Lemma 7, then the IPE is locally asymptotically stable.
(4) If , there are two equilibria: the IFE and a unique IPE . The system at satisfies the second condition in Lemma 7, then the IPE is locally asymptotically stable.
(5) If , there are two equilibria: the IFE and a unique IPE . The system at satisfies the first condition in Lemma 7, then the IPE is locally asymptotically stable.
5. Numerical Simulations
In this section, we use the Adamstype predictorcorrector 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.
The approximate solution is displayed in (Figures 2(a)–2(d), 3(a)–3(d), 4(a)–4(d), 5(a)–5(d), 6(a)–6(d), and 7(a)–7(d)). When , system (4) is the classical integerorder ODE.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
Remark 12. Figures 2 and 3 show that, if and , the system at is locally stable.
Remark 13. Figures 4 and 5 show that, if and , the system at is locally stable.
Remark 14. Figures 6 and 7 show that, if and , the system at is locally stable. As increases, the trajectory of the system approaches the steady state faster and gets close to the integerorder ODE.
6. Extending the Model
In this section, we add the term 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 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 , and the infected equilibrium , where
By the next generation matrix method [11], we obtain the basic reproduction number of model (30)
7. 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 T cell. We will consider this model where the Tcell 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 under some conditions. We calculate the basic reproduction number , the IFE , two IPEs and , and so on, under certain conditions and judge the stability of the equilibrium. According to Tables 1 and 4, we get that, if and , there is only the IFE for . In addition, if under some conditions, there is only the IFE for . We describe the dynamic behaviors of the fractional HIV model by using the Adamstype predictorcorrector 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.
(a)  
 
(b)  

(a)  
 
(b)  

Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This project was supported by NNSF of China Grant nos. 11271087 and 61263006.
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