Stability analysis for HIV infection of CD4+ T-cells by a fractional differential time-delay model with cure rate
© Liu and Lu; licensee Springer. 2014
Received: 26 June 2014
Accepted: 13 November 2014
Published: 28 November 2014
In this paper, a fractional differential model of HIV infection of CD4+ T-cells is investigated. We shall consider this model, which includes full logistic growth terms of both healthy and infected CD4+ T-cells, time delay items, and cure rate items. A more appropriate method is given to ensure that both equilibria are asymptotically stable for under some conditions. Furthermore, the dynamic behaviors of the fractional HIV models are described by applying an Amads-type predictor-corrector method algorithm.
Mathematical models have played an important role in understanding the dynamics of HIV infection; there are several papers introducing the Human Immunodeficiency Virus (HIV) [1, 2]. When HIV infects the body, its target is the CD4+ T-cell. In these years, mathematical models have been proven valuable in the dynamics of HIV infection. Meanwhile, there are only some works for the dynamics of HIV infections of CD4+ T-cells [3, 4].
The consideration of the cure (or recovery) rate of infected cells is significant in the modeling for viral dynamics. The covalently closed circular (ccc) DNA of Hepatitis B viral has been shown to be eliminated from the nucleus of infected cells in the absence of hepatocyte injury during transient infections . In 2010, Wang et al.  built and studied an improved HBV model with a standard incidence function and ‘cure’ rate. Inspired by the HBV dynamic model with cure rate, Zhou et al.  firstly introduced the cure rate into the HIV infection model. In recent years, the HIV model with cure rate has received a great deal of attention (see e.g. [8–11]).
Fractional differential equations have been widely used in various fields, such as physics, chemical technology, biotechnology, and economics in recent years (see e.g. [13–16]). As is well known, the boundary value problem is an important topic, there is a great deal of attention for this (see [17–26]).
We introduce the fractional calculus into the HIV model for the memory property of fractional calculus. Both in mathematics and biology, fractional calculus will be more in line with the actual situation. It is particularly of significance for us to study the fractional HIV model.
Parameters and values of model ( 1.4 )
Uninfected CD4+ T-cell population size
Infected CD4+ T-cell density
Initial density of HIV RNA
CD4+ T-cell population for HIV-negative persons
Natural death rate of CD4+ T-cell
Blanket death rate of infected CD4+ T-cell
Death rate of free virus
Lytic death rate for infected cells
Rate CD4+ T-cell become infected with virus
Rate infected cells become active
Rate of each infected cells reverting to the uninfected state
Growth rate of CD4+ T-cell population
Number of virions produced by infected CD4+ T-cell
Maximal population level of CD4+ T-cell
Source term for uninfected CD4+ T-cell
Furthermore, we assume that , and for all .
This article 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 will give the numerical simulation for the fractional HIV model. Finally, the conclusions are given.
In this section, we introduce some definitions and lemmas, which will be used later.
Definition 2.2 ()
3 The stability of the equilibria
In this section, we investigate the existence of equilibria of system (1.4).
Next, we shall 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.
Theorem 3.1 If , the uninfected state is locally asymptotically stable for .
if , the characteristic roots have negative real parts for .
that is , .
For the parameter values given in Table 1, we take any , the infected equilibrium , and we find that the above equation is unequal for . Therefore, .
According to Lemma 2.1, the uninfected equilibrium is locally asymptotically stable. The proof is completed. □
Remark 3.1 ()
The stability region of a system with fractional order is always larger than that of a corresponding ordinary differential system. This means that a unstable equilibrium of an ordinary differential system may be stable in a fractional differential system.
Proof According to (3.5).
that is, , .
For the parameter values given in Table 1, we take any ; then we get the specific value on the infected equilibrium and we can see that the above equation is unequal for .
Due to , we have . Hence, neither nor is positive. Thus, (3.10) does not have positive roots. Since , , it follows that (3.9) has no positive roots.
Because of , the roots of (3.7) are positive, that is, .
The proof is completed. □
4 Numerical simulations
In this section, we use the Adams-type predictor-corrector method for the numerical solution of the nonlinear system (1.4) and (1.5) with time delay.
Next, we apply the PECE (Predict, Evaluate, Correct, Evaluate) method.
For the parameter values given in Table 1, we take , then .
Hence, all the conditions in Theorem 3.2 are satisfied and the infection case is asymptotically stable. In addition, when we take , , all the conditions in Theorem 3.2 are also satisfied and the infection case is asymptotically stable.
Remark 4.2 Figure 4 shows that, as τ increases, the fluctuation of the trajectory of the system is smaller during the previous period of the time.
Remark 4.3 If , Figure 5 shows that, as α closes in to 1, the number of steady states of T approaches the initial value, the numbers of steady states of I and V approach zero.
Remark 4.4 Figures 6, 7, and 8 show that, as p increases, the number of infected T-cells is decreased, the level of the steady state of T is higher, the fluctuations of the trajectories of I and V are smaller. For , the trajectory of the system is fluctuating during the previous period of the time. As ρ (>0.6) is increasing, the fluctuation of the trajectory of the system is stronger. It is noticeable that, for ρ in a certain range, drugs can resist the virus. For , the trajectory of the system is fluctuating during the previous period of the time, and it will tend to the steady state later. For , the trajectory of the system is unstable.
Remark 4.5 Figure 9 shows that, as N decreases, the number of steady states of T increases, the numbers of steady states of I and V are decreased and the trajectories of the system of I and V are also close to stable.
In this paper, we modified the ODE model proposed by Liu et al.  and the fractional model proposed by Yan and Kou  into a system of fractional order. We study a fractional differential model of HIV infection of the CD4+ T-cells. We shall consider this model, which includes full logistic growth terms of both healthy and infected CD4+ T-cells, time delay items, and cure rate items. Moreover, we study α, τ, N, and ρ, and we obtain some significant conclusions. For example, if the cure rate gets large in a certain range, it will control the HIV infection efficiently. In our analysis, the more appropriate method is given to ensure that both equilibria are asymptotically stable for . Both in mathematics and biology, it is particularly important to show stability of the infected and uninfected equilibrium point. In addition, we describe the dynamic behaviors of the fractional HIV model by using the Amads-type predictor-corrector method algorithm.
This project supported by NNSF of China Grant Nos. 11271087, 61263006 and NSF of Guangxi Grant No. 2014GXNSFDA118002.
- Weiss RA: How does HIV cause AIDS? Science 1993, 260: 1273-1279. 10.1126/science.8493571View ArticleGoogle Scholar
- Yan Y, Kou CH: Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay. Math. Comput. Simul. 2012, 82: 1572-1585. 10.1016/j.matcom.2012.01.004MathSciNetView ArticleGoogle Scholar
- Culshaw RV, Ruan S: A delay-differential equation model of HIV infection of CD4+ T-cells. Math. Biosci. 2000, 165: 27-39. 10.1016/S0025-5564(00)00006-7View ArticleGoogle Scholar
- Perelson AS, Kirschner DE, De Boer R: Dynamics of HIV infection of CD4+ T-cells. Math. Biosci. 1993, 114: 81-125. 10.1016/0025-5564(93)90043-AView ArticleGoogle Scholar
- Guidotti LG, Rochford R, Chung J, Shapiro M, Purcell R, Chisari FV: Viral clearance without destruction of infected cells during acute HBV infection. Science 1999, 284: 825-829. 10.1126/science.284.5415.825View ArticleGoogle Scholar
- Wang KF, Fan AJ, Torres A: Global properties of an improved hepatitis B virus model. Nonlinear Anal., Real World Appl. 2010, 11: 3131-3138. 10.1016/j.nonrwa.2009.11.008MathSciNetView ArticleGoogle Scholar
- Zhou XY, Song XY, Shi XY: A differential equation model of HIV infection of CD4+ T-cells with cure rate. J. Math. Anal. Appl. 2008, 342: 1342-1355. 10.1016/j.jmaa.2008.01.008MathSciNetView ArticleGoogle Scholar
- Zack JA, Arrigo SJ, Weitsman SR, Go AS, Haislip A, Chen IS: HIV-1 entry into quiescent primary lymphocytes: molecular analysis reveals a labile latent viral structure. Cell 1990, 61: 213-222. 10.1016/0092-8674(90)90802-LView ArticleGoogle Scholar
- Zack JA, Haislip AM, Krogstad P, Chen IS: Incompletely reverse-transcribed human immunodeficiency virus type 1 genomes in quiescent cells can function as intermediates in the retroviral cycle. J. Virol. 1992, 66: 1717-1725.Google Scholar
- Essunger P, Perelson AS: Modeling HIV infection of CD4+ T-cell subpopulations. J. Theor. Biol. 1994, 170: 367-391. 10.1006/jtbi.1994.1199View ArticleGoogle Scholar
- Srivastava PK, Chandra P: Modeling the dynamics of HIV and CD4+ T-cells during primary infection. Nonlinear Anal., Real World Appl. 2010, 11: 612-618. 10.1016/j.nonrwa.2008.10.037MathSciNetView ArticleGoogle Scholar
- Liu XD, Wang H, Hu ZX, Ma WB: Global stability of an HIV pathogenesis model with cure rate. Nonlinear Anal., Real World Appl. 2011, 12: 2947-2961.MathSciNetGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Ross B 475. In The Fraction Calculus and Its Applications. Springer, Berlin; 1975.View ArticleGoogle Scholar
- Liu ZH, Li X, Sun J: Controllability of nonlinear fractional impulsive evolution systems. J. Integral Equ. Appl. 2013, 25(3):395-405. 10.1216/JIE-2013-25-3-395MathSciNetView ArticleGoogle Scholar
- Bai Z: On positive solutions a nonlinear fractional boundary value problem. Nonlinear Anal. TMA 2010, 72: 916-927. 10.1016/j.na.2009.07.033View ArticleGoogle Scholar
- Benchohra M, Graef JR, Hamani S: Existence results for boundary value problems with nonlinear fractional differential equations. Appl. Anal. 2008, 87: 851-863. 10.1080/00036810802307579MathSciNetView ArticleGoogle Scholar
- Bai Z, Lv H: Positive solution for boundary value problem of nonlinear differential equation. J. Math. Anal. Appl. 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleGoogle Scholar
- Geiji VD: Positive solution of a system of non-autonomous fractional differential equations. J. Math. Anal. Appl. 2005, 302: 56-64. 10.1016/j.jmaa.2004.08.007MathSciNetView ArticleGoogle Scholar
- Jiang D, Yuan C: The positive properties of the Green function for Dirichlet-type boundary value problem of nonlinear fractional differential equations and its application. Nonlinear Anal. TMA 2010, 72: 710-719. 10.1016/j.na.2009.07.012MathSciNetView ArticleGoogle Scholar
- Kaufmann ER, Mboumi E: Positive solution of a boundary value problem for a nonlinear fractional differential equations. Electron. J. Qual. Theory Differ. Equ. 2008., 2008: Article ID 3Google Scholar
- Li CF, Luo XN, Zhou Y: Existence of positive solution for boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 2010, 59: 1363-1375. 10.1016/j.camwa.2009.06.029MathSciNetView ArticleGoogle Scholar
- Liu ZH: Anti-periodic solutions to nonlinear evolution equations. J. Funct. Anal. 2010, 258: 2026-2033. 10.1016/j.jfa.2009.11.018MathSciNetView ArticleGoogle Scholar
- Liu ZH, Migorski S: Analysis and control of differential inclusions with anti-periodic conditions. Proc. R. Soc. Edinb. A 2014, 144(3):591-602. 10.1017/S030821051200090XMathSciNetView ArticleGoogle Scholar
- Zhang S: Positive solution for boundary value problem of nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 2006., 2006: Article ID 36Google Scholar
- Ahmed E, El-Sayed AMA, El-Saka HAA: Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J. Math. Anal. Appl. 2007, 325: 542-553. 10.1016/j.jmaa.2006.01.087MathSciNetView ArticleGoogle Scholar
- Kou CH, Yan Y, Liu J: Stability analysis for fractional differential equations and their applications in the models of HIV-1 infection. Comput. Model. Eng. Sci. 2009, 39: 301-317.MathSciNetGoogle Scholar
- Matignon D: Stability results for fractional differential equations with applications to control processing. Computational Engineering in Systems Applications 1996, 963-968.Google Scholar
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York; 1993.Google Scholar
- Ahmed E, Elgazzar AS: On fractional order differential equations model for nonlocal epidemics. Physica A 2007, 379: 607-614. 10.1016/j.physa.2007.01.010MathSciNetView ArticleGoogle Scholar
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