Abstract and Applied Analysis

Volume 2014 (2014), Article ID 291614, 13 pages

http://dx.doi.org/10.1155/2014/291614

Research Article

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

Yiliang Liu,¹ Peifen Lu,¹ and Ivan Szanto²

¹College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, China

²Departamento de Matematica, Universidad Tecnica Federico Santa Maria, Casilla 110-V, Valparaiso, Chile

Received 2 October 2013; Accepted 11 December 2013; Published 12 February 2014

Academic Editor: Abdon Atangana

Copyright © 2014 Yiliang Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study a fractional differential model of HIV infection of T-cell, in which the 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 . 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 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.

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 life-threatening opportunistic infections. HIV infects primarily vital cells in the human immune system such as helper T-cell (to be specific, T-cell), macrophages, and dendritic cells. When 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 T-cell. In 1992, Perelson et al. [3] examined a model for the interaction of HIV with 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 T-cell, infected 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 [5–12]. 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 T-cell 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 T-cell:

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 fractional-order derivatives into a model of HIV infection of T-cell with time delay:

with the initial conditions:

Motivated by the works mentioned above, we will consider this model where the 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 . 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 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 T-cell at time , infected 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.

Table 1: Parameters and values of model (4).

Parameter
Description
Value

Uninfected CD T-cell population size
1000 mm⁻³

Infected CD T-cell density
0

Free infectious virus particles
 mm⁻³

Noninfectious virus particles
 mm⁻³

CD T-cell population for HIV-negative persons
 mm⁻³

Natural death rate of CD T-cell
day⁻¹

Blanket death rate of infected CD T-cell
day⁻¹

Death rate of free virus
day⁻¹

Rate of CD T-cell becoming infected with virus
 mm³ day⁻¹

Rate of infected cells becoming active
 mm³ day⁻¹

Source term for uninfected CD T-cell
day⁻¹ mm⁻³

Number of virions produced by infected CD T-cell
Varies

The maximal proliferation rate ()
Varies

The half saturation constant of the proliferation process
Varies

The effectiveness of RTIs ( = 0 means the therapy is totally ineffective, while = 1 indicates the therapy is 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

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].

Figure 1: Stability region of system (4) with order .

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 two-dimensional stable manifold and a one-dimensional 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 two-dimensional stable manifold and a one-dimensional 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).

Table 2: We take ,  ,  ,  , and , and we get the following.

(a)  

Line

1
0.1
0.05
0.00002
0.5454
1.6016
(194.9317, 30.9602, 1911.7950)
not exist

2
0.5
0.25
0.00001
0.5340
1.0607
(444.4444, 21.3511, 1040.8668)
not exist

3
0.5425
0.27125
0.00001
0.5326
1.0001
(499.8953, 19.2137, 910.1286)
not exist

4
0.5426
0.2713
0.00001
0.5326
1.0000
(500.0390, 0.000003, 0.0002)
(500.0389, 19.2082, 909.8039)

5
0.55
0.275
0.00001
0.5323
0.9893
(510.8556, 0.0009, 0.0444)
(510.8557, 18.7911, 885.5296)

6
0.6
0.3
0.000009
0.5305
0.9165
(595.2381, 0.0104, 0.4716)
(595.2381, 15.5354, 706.8618)

7
0.7
0.35
0.000007
0.5265
0.7649
(854.7009, 0.1186, 5.0099)
(854.7009, 5.4462, 230.1012)

8
0.7252
0.3626
0.000007
0.5254
0.7249
(951.5244, 0.8287, 34.3354)
(951.5245, 1.0116, 41.9104)

9
0.7253
0.3627
0.000007
0.5253
0.7247
Not exist
Not exist

10
0.8
0.4
0.000005
0.5217
0.6000
Not exist
Not exist

11
0.82
0.41
0.000005
0.5206
0.5644
Not exist
Not exist

12
0.8446
0.4223
0.000004
0.5191
0.5190
Not exist
Not exist

13
0.85
0.425
0.000004
0.5188
0.5087
Not exist
Not exist

(b)  

Line

()()
Stability

1
−1.7882
2.7113
0.1317
0.0258
0.3570
(ii)

2
−0.5611
2.6825
0.0550
0.0078
0.1477
(ii)

3
−0.4497
2.6800
0.0484
0.0062
0.1297
(ii)

4
0.0165, −0.4495
2.6800, 2.6800
0.0484, 0.0484
−0.000001, 0.0062
0.1297, 0.1297
Unsuited (ii)

5
0.0346, −0.4305
2.6796, 2.6796
0.0473, 0.0473
−0.0003, 0.0060
0.1267, 0.1267
Unsuited (ii)

6
0.1125, −0.3054
2.6768, 2.6768
0.0399, 0.0399
−0.0014, 0.0042
0.1068, 0.1068
Unsuited (ii)

7
0.0681, −0.0707
2.6717, 2.6717
0.0263, 0.0263
−0.0008, 0.0010
0.0703, 0.0703
Unsuited (ii)

8
0.0060, 0.0015
2.6705, 2.6705
0.0232, 0.0232
−0.00003, 0.00003
0.0618, 0.0618
Unsuited (i)

Table 3: We take ,  ,  ,  ,  , and , and we have the following.

(a)  

Line

14
10
0.0001
0.6035
1.0607
(444.4444, 21.2037, 1033.6821)
Not exist

15
100
0.0012
0.7864
1.0607
(444.4444, 19.7599, 963.2939)
Not exist

16
300
0.0036
0.9200
1.0607
(444.4444, 16.7808, 818.0663)
Not exist

17
500
0.0060
0.9746
1.0607
(444.4444, 14.1982, 692.1614)
Not exist

18
700
0.0084
0.9965
1.0607
(444.4444, 12.0858, 589.1841)
Not exist

19
830
0.0100
1.0000
1.0607
(444.4444, 10.9728, 534.9223)
Not exist

20
900
0.0108
0.9992
1.0607
(444.4444, 10.4534, 509.6013)
Not exist

(b)  

Line

Stability

14
−0.5531
2.6825
0.0550
0.0077
0.1477
(ii)

15
−0.4782
2.6825
0.0551
0.0067
0.1477
(ii)

16
−0.3449
2.6825
0.0550
0.0049
0.1477
(ii)

17
−0.2523
2.6825
0.0550
0.0037
0.1477
(ii)

18
−0.1926
2.6825
0.0551
0.0029
0.1477
(ii)

19
−0.1668
2.6825
0.0550
0.0025
0.1477
(ii)

20
−0.1561
2.6825
0.0550
0.0024
0.1477
(ii)

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 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.

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 integer-order ODE.

(a)

(b)

(c)

(d)

Figure 2: In (a)–(d), , , and .

(a)

(b)

(c)

(d)

Figure 3: In (a)–(d), , , , and .

(a)

(b)

(c)

(d)

Figure 4: In (a)–(d), , , , , , and .

(a)

(b)

(c)

(d)

Figure 5: In (a)–(d), , , , , , and .

(a)

(b)

(c)

(d)

Figure 6: In (a)–(d), , , , , , and .

(a)

(b)

(c)

(d)

Figure 7: In (a)–(d), , , , , , and .

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 integer-order 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 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 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 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.

Table 4: We take ,  ,  ,  ,  , and , we give the following.

(a)  

Line

21
0
0.0001
0.9940
1.0607
(444.4444, 4.2735, 208.3333)
Not exist

22
0.004
0.0001
0.8632
1.0607
(444.4444, 10.9858, 535.5567)
Not exist

23
0.008
0.0001
0.6920
1.0607
(444.4444, 17.7929, 867.4018)
Not exist

24
0.015
0.0001
0.3782
1.0607
(444.4444, 29.7389, 1449.7703)
Not exist

25
0.019
0.0001
0.1950
1.0607
(444.4444, 36.5710, 1782.8352)
Not exist

(b)  

Line

Stability

21
−0.0952
2.6825
0.0550
0.0016
0.1477
(ii)

22
−0.2749
2.6825
0.0551
0.0040
0.1477
(ii)

23
−0.4600
2.6825
0.0550
0.0064
0.1477
(ii)

24
−0.7866
2.6825
0.0550
0.0108
0.1477
(ii)

25
−0.9740
2.6825
0.0550
0.0550
0.1477
(ii)

Table 5: We take ,  ,  ,  ,  , and , and we get the following.

(a)  

Line

26
10
0.00001
0.5340
0.3354
Not exist
Not exist

27
30
0.00001
0.5340
0.5809
Not exist
Not exist

28
40
0.00001
0.5340
0.6708
Not exist
Not exist

29
47
0.00001
0.5340
0.7271
Not exist
Not exist

30
48
0.00001
0.5340
0.7348
(925.9259, 0.6491, 15.1879)
(925.9259, 2.1572, 50.4788)

31
60
0.00001
0.5340
0.8216
(740.7407, 0.0641, 1.8756)
(740.7407, 9.8732, 288.7910)

32
88
0.00001
0.5340
0.9950
(505.0505, 0.0005, 0.0204)
(505.0505, 19.0127, 815.6462)

33
89
0.00001
0.5340
1.0006
(499.3758, 19.2318, 834.4192)
Not exist

34
100
0.00001
0.5340
1.0607
(444.4444, 21.3511, 1040.8668)
Not exist

35
200
0.00001
0.5340
1.5000
(222.2222, 29.9116, 2916.3810)
Not exist

36
600
0.00001
0.5340
2.5981
(74.0741, 35.6123, 10416.5867)
Not exist

37
1000
0.00001
0.5340
3.3541
(44.4444, 36.7520, 17916.6202)
Not exist

38
10000
0.00001
0.5340
10.6066
(4.4444, 38.2906, 186666.6622)
Not exist

39
20000
0.00001
0.5340
15.0000
(2.2222, 38.3761, 374166.6644)
Not exist

(b)  

Line

Stability

30
0.1109, 0.0417
2.6742, 2.6723
0.0328, 0.0278
−0.0014, −0.0005
0.0878, 0.0743
Unsuited, unsuited

31
0.0747, −0.1399
2.6784, 2.6738
0.0443, 0.0319
−0.0008, 0.0020
0.1185, 0.0853
Unsuited (i)

32
0.0173, −0.4352
2.6800, 2.6799
0.0483, 0.0482
−0.00001, 0.0060
0.1296, 0.1291
Unsuited (i)

33
−0.4454
2.6801
0.0487
0.0062
0.1307
(ii)

34
−0.5570
2.6826
0.0553
0.0077
0.1483
(ii)

35
−1.5273
2.7050
0.1150
0.0218
0.3110
(ii)

36
−4.7830
2.7950
0.3543
0.0780
0.9903
(ii)

37
−7.1387
2.8850
0.5937
0.1342
1.7129
(ii)

38
30.8901
4.9100
5.9802
1.3978
29.3628
(i)

39
482.3545
7.1600
11.9652
11.9652
85.6708
(i)

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.

References

R. A. Weiss, “How does HIV cause AIDS?” Science, vol. 260, no. 5112, pp. 1273–1279, 1993.

Y. Yan and C. Kou, “Stability analysis for a fractional differential model of HIV infection of T-cells with time delay,” Mathematics and Computers in Simulation, vol. 82, no. 9, pp. 1572–1585, 2012.

A. S. Perelson, D. E. Kirschner, and R. De Boer, “Dynamics of HIV infection of T cells,” Mathematical Biosciences, vol. 114, no. 1, pp. 81–125, 1993.

R. V. Culshaw and S. Ruan, “A delay-differential equation model of HIV infection of T-cells,” Mathematical Biosciences, vol. 165, no. 1, pp. 27–39, 2000.

J. Arino, C. C. McCluskey, and P. van den Driessche, “Global results for an epidemic model with vaccination that exhibits backward bifurcation,” SIAM Journal on Applied Mathematics, vol. 64, no. 1, pp. 260–276, 2003.

A. Atangana and N. Bildik, “Approximate solution of tuberculosis disease population dynamics model,” Abstract and Applied Analysis, vol. 2013, Article ID 759801, 8 pages, 2013.

J. Dushoff, W. Huang, and C. Castillo-Chavez, “Backwards bifurcations and catastrophe in simple models of fatal diseases,” Journal of Mathematical Biology, vol. 36, no. 3, pp. 227–248, 1998.

H. Gómez-Acevedo and M. Y. Li, “Backward bifurcation in a model for HTLV-I infection of T cells,” Bulletin of Mathematical Biology, vol. 67, no. 1, pp. 101–114, 2005.

R. Qesmi, J. Wu, J. Wu, and J. M. Heffernan, “Influence of backward bifurcation in a model of hepatitis B and C viruses,” Mathematical Biosciences, vol. 224, no. 2, pp. 118–125, 2010.

O. Sharomi, C. N. Podder, A. B. Gumel, E. H. Elbasha, and J. Watmough, “Role of incidence function in vaccine-induced backward bifurcation in some HIV models,” Mathematical Biosciences, vol. 210, no. 2, pp. 436–463, 2007.

P. van den Driessche and J. Watmough, “A simple SIS epidemic model with a backward bifurcation,” Journal of Mathematical Biology, vol. 40, no. 6, pp. 525–540, 2000.

H. Shu and L. Wang, “Role of T-cell proliferation in HIV infection under antiretroviral therapy,” Journal of Mathematical Analysis and Applications, vol. 394, no. 2, pp. 529–544, 2012.

B. Buonomo and C. Vargas-De-León, “Global stability for an HIV-1 infection model including an eclipse stage of infected cells,” Journal of Mathematical Analysis and Applications, vol. 385, no. 2, pp. 709–720, 2012.

M. Y. Li and H. Shu, “Global dynamics of a mathematical model for HTLV-I infection of T cells with delayed CTL response,” Nonlinear Analysis: Real World Applications, vol. 13, no. 3, pp. 1080–1092, 2012.

S. Liu and L. Wang, “Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,” Mathematical Biosciences and Engineering, vol. 7, no. 3, pp. 675–685, 2010.

X. Liu, H. Wang, Z. Hu, and W. Ma, “Global stability of an HIV pathogenesis model with cure rate,” Nonlinear Analysis: Real World Applications, vol. 12, no. 6, pp. 2947–2961, 2011.

G. P. Samanta, “Permanence and extinction of a nonautonomous HIV/AIDS epidemic model with distributed time delay,” Nonlinear Analysis: Real World Applications, vol. 12, no. 2, pp. 1163–1177, 2011.

R. Xu, “Global stability of an HIV-1 infection model with saturation infection and intracellular delay,” Journal of Mathematical Analysis and Applications, vol. 375, no. 1, pp. 75–81, 2011.

M. Février, K. Dorgham, and A. Rebollo, “T-cell depletion in human immunodeficiency virus (HIV) infection: role of apoptosis,” Viruses, vol. 3, no. 5, pp. 586–612, 2011.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.

Z. Liu and X. Li, “Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 6, pp. 1362–1373, 2013.

Z. Liu and X. Li, “On the controllability of impulsive fractional evolution inclusions in banach spaces,” Journal of Optimization Theory and Applications, vol. 156, no. 1, pp. 167–182, 2013.

Z. Liu and J. Sun, “Nonlinear boundary value problems of fractional functional integro-differential equations,” Computers and Mathematics with Applications, vol. 64, no. 10, pp. 3228–3234, 2012.

Z. Liu and J. Sun, “Nonlinear boundary value problems of fractional differential systems,” Computers and Mathematics with Applications, vol. 64, no. 4, pp. 463–475, 2012.

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

B. Ross, Fractional Calculus and Its Applications, vol. 457 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.

Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods and Applications, vol. 72, no. 2, pp. 916–924, 2010.

M. Benchohra, J. R. Graef, and S. Hamani, “Existence results for boundary value problems with non-linear fractional differential equations,” Applicable Analysis, vol. 87, no. 7, pp. 851–863, 2008.

Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.

V. D. Gejji, “Positive solutions of a system of non-autonomous fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 302, no. 1, pp. 56–64, 2005.

D. Jiang and C. Yuan, “The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,” Nonlinear Analysis: Theory, Methods and Applications, vol. 72, no. 2, pp. 710–719, 2010.

E. R. Kaufmann and E. Mboumi, “Positive solutions of a boundary value problem for a nonlinear fractional differential equation,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1–11, 2008.

C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1363–1375, 2010.

Z. Liu and D. Motreanu, “A class of variational-hemivariational inequalities of elliptic type,” Nonlinearity, vol. 23, no. 7, pp. 1741–1752, 2010.

Z. Liu, “Anti-periodic solutions to nonlinear evolution equations,” Journal of Functional Analysis, vol. 258, no. 6, pp. 2026–2033, 2010.

Z. Liu, “Existence results for quasilinear parabolic hemivariational inequalities,” Journal of Differential Equations, vol. 244, no. 6, pp. 1395–1409, 2008.

Z. Liu, “Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities,” Inverse Problems, vol. 21, no. 1, pp. 13–20, 2005.

Z. Peng, Z. Liu, and X. Liu, “Boundary hemivariational inequality problems with doubly nonlinear operators,” Mathematische Annalen, vol. 356, no. 4, pp. 1339–1358, 2013.

S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 2006, no. 36, pp. 1–12, 2006.

Z.-Y. Zhang, Z.-H. Liu, X.-J. Miao, and Y.-Z. Chen, “Stability analysis of heat flow with boundary time-varying delay effect,” Nonlinear Analysis: Theory, Methods and Applications, vol. 73, no. 6, pp. 1878–1889, 2010.

Z.-Y. Zhang, Z.-H. Liu, X.-J. Miao, and Y.-Z. Chen, “New exact solutions to the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 3064–3072, 2010.

A. S. Perelson and P. W. Nelson, “Mathematical analysis of HIV-1 dynamics in vivo,” SIAM Review, vol. 41, no. 1, pp. 3–44, 1999.

D. Kirschner, “Using mathematics to understand HIV immune dynamics,” Notices of the American Mathematical Society, vol. 43, no. 2, pp. 191–202, 1996.

C. Kou, Y. Yan, and J. Liu, “Stability analysis for fractional differential equations and their applications in the models of HIV-1 infection,” Computer Modeling in Engineering and Sciences, vol. 39, no. 3, pp. 301–317, 2009.

D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Computational Engineering in Systems Applications, pp. 963–968, 1996.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience, New York, NY, USA, 1993.

Y.-K. Chang, M. M. Arjunan, G. M. N'Guérékata, and V. Kavitha, “On global solutions to fractional functional differential equations with infinite delay in Fréchet spaces,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1228–1237, 2011.

Z. Ouyang, Y. Chen, and S. Zou, “Existence of positive solutions to a boundary value problem for a delayed nonlinear fractional differential system,” Boundary Value Problems, vol. 2011, Article ID 475126, 17 pages, 2011.

E. Ahmed and A. S. Elgazzar, “On fractional order differential equations model for nonlocal epidemics,” Physica A, vol. 379, no. 2, pp. 607–614, 2007.