Research Article  Open Access
Linli Zhang, Gang Huang, Anping Liu, Ruili Fan, "Stability Analysis for a Fractional HIV Infection Model with Nonlinear Incidence", Discrete Dynamics in Nature and Society, vol. 2015, Article ID 563127, 11 pages, 2015. https://doi.org/10.1155/2015/563127
Stability Analysis for a Fractional HIV Infection Model with Nonlinear Incidence
Abstract
We introduce the fractionalorder derivatives into an HIV infection model with nonlinear incidence and show that the established model in this paper possesses nonnegative solution, as desired in any population dynamics. We also deal with the stability of the infectionfree equilibrium, the immuneabsence equilibrium, and the immunepresence equilibrium. Numerical simulations are carried out to illustrate the results.
1. Introduction
1.1. The History of Fractional Calculus
Fractional calculus which is a branch of mathematical analysis extends derivatives and integrals to an arbitrary order (real, even, or complex order). The study on fractional calculus started at the end of seventeenth century and the first reference was proposed by Leibniz and L’Hospital in 1695, in which halforder derivative was mentioned. The emergence of fractional calculus aroused interest of some famous mathematicians such as Lacroix, Lagrange, Fourier, and Laplace. J. L. Lagrange developed the law of exponents for differential operators in 1772, which promoted indirectly the development of fractional calculus. P. S. Laplace defined a fractional derivative by means of integral in 1812 and S. F. Lacroix and J. B. J. Fourier in succession mentioned the arbitrary order derivative, respectively, in 1819 and in 1822. However, the basic theory of fractional calculus was established with the studies of Liouville, Grunwald, Letnikov, and Riemann until the end of the nineteenth century.
At the initial stage, the development of fractional calculus had been restricted to the pure mathematics research. Until the nineteen nineties, the theory of fractional calculus was applied to nature and social sciences. In the field of anomalous diffusion, the researchers applied fractional calculus to describe the diffusion processes which do not conform to the brown motion and proposed the fractional anomalous diffusion model. According to the theoretical analysis and the experimental data or results, they found that the fractional model is more reasonable to describe these processes [1]. Afterwards, many mathematicians and applied researchers also have tried to demonstrate applications of fractional differentials in the areas of nonNewtonian fluids [2], signal processing [3–5], viscoelasticity [6, 7], fluiddynamic traffic model [8], colored noise [9], bioengineering [10], solid mechanics [11], continuum and statistical mechanics [12], and economics [13] and brought new research view for those fields.
1.2. Mathematical Modeling
In biology, it has been deduced that the membranes of cells of biological organism have fractionalorder electrical conductance [14]; hence some mathematical models which describe cells behavior are classified into groups of nonintegerorder models. In the field of rheology, fractionalorder derivatives embody essential features of cell rheological behavior and have enjoyed greatest success [15]. Some researchers also found that fractional ordinary differential equations are naturally related to systems with memory which exists in most biological systems. And fractionalorder equations are also closely related to fractals, which are abundant in biological systems.
Mathematical models have been proven valuable in understanding the dynamics of viral infection. Although a large number of works on modeling the dynamics of viral infection have been done [16–20], it has been restricted to integerorder (delay) differential equations. In recent years, it has turned out that many phenomena in virus infection can be described very successfully by the models using fractionalorder differential equations [21, 22]. Motivated by these references, in this paper, we will consider a fractionalorder HIV model.
Among a large number of virus infection models described with integerorder or delay differential equations, a typical model is given as the following simple threedimensional system:Here represents the concentration of uninfected cells at time ; represents the concentration of infected cells that produce virus at time ; represents the concentration of viruses at time . and are the death rate of uninfected cells and the rate constant characterizing infection of the cells, respectively. is the death rate of infected cells due to either the virus or the immune system. Free virus is produced from infected cells at the rate and removed at rate .
The infection rate in model (1) is usually assumed to be bilinear with respect to virus and uninfected target cells . However, there are some evidences which show that a bilinear infection might not be an effective assumption when the number of target cells is large enough [23–25]. And Huang et al. [17] incorporated Holling type II functional response infection rate into basic virus dynamics (1), which was expressed as where () is a constant. Hence, system (1) can be modified into the following system:
The immune response following viral infection is universal and necessary to eliminate or control the disease. Antibodies, cytokines, natural killer cells, B cells, and T cells are all essential components of a normal immune response to viral infection. Since cytotoxic T lymphocytes (CTLs) play a critical role in antiviral defense by attacking the virusinfected cells, Huang et al. [17] considered the immune response to model (3) with .
System (3) with can be further simplified if we take into consideration the fact that an average life span of viral particles is usually significantly shorter than one of the infected cells. It can be assumed, therefore, that, compared with a “slow” variation of the infected cells level, the virus load relatively quickly reaches a quasiequilibrium level. The equality holds in the quasiequilibrium state and hence . This assumption is referred to as “separation of time scales” and is in common use in the virus dynamics [18]. We have to stress that this assumption does not imply that the virus concentration remains constant; on the contrary, it is assumed to be proportional to the varying concentration of infected cells . Accordingly, system (3) with can now be reformulated as a system of three differential equations:
Further we introduce the fractional order into model (4) and a new system can be described by the following set of FODEs of order :where the immune response is assumed to get stronger at a rate , which is proportional to the number of infected cells and their current concentration; the immune response decays exponentially at a rate , which is proportional to their current concentration; the parameter expresses the efficacy of nonlytic component.
The paper is organized as follows. In the next section, we give two definitions and two lemmas about fractional calculus. In Section 3, we show the nonnegativeness of the solutions of system (5) with initial condition (6). In Section 4, we give a detailed stability analysis for three equilibria. Finally numerical simulations are presented to illustrate the obtained results in Section 5.
2. Fractional Calculus
Definition 1 (see [26]). The RiemannLiouville (RL) fractional integral operator of order of a function is defined as Here is the Euler gamma function which is defined as This function is generalization of a factorial in the following form:
Definition 2 (see [26]). The Caputo (C) fractional derivative of order , , , is defined as where the function has absolutely continuous derivatives up to order . In particular, when , one has
In this paper we use Caputo fractional derivative definition. The main advantage of Caputo’s definition is that the initial conditions for fractional differential equations with Caputo derivatives take the same form as that for integerorder differential equations.
Lemma 3 (see [26]). Consider the following commensurate fractionalorder system:with and . The equilibrium points of system (12) are calculated by solving the following equation: . These points are locally asymptotically stable if all eigenvalues of Jacobian matrix evaluated at the equilibrium points satisfy
Definition 4 (see [27]). The discriminant of a polynomial is defined by , where is the derivative of . If , is the determinant of the corresponding Sylvester matrix. The Sylvester matrix is formed by filling the matrix beginning with the upper left corner with the coefficients of and then shifting down one row and one column to the right and filling in the coefficients starting there until they hit the right side. The process is then repeated for the coefficients of .
Lemma 5 (see [27]). For the polynomial equation,the conditions which make all the roots of (15) satisfy (13) are displayed as follows: (i)for , the condition is ;(ii)for , the conditions are either RouthHurwitz conditions or (iii)for ,(a)if the discriminant of , is positive, then RouthHurwitz conditions are the necessary and sufficient conditions; that is, , , and if ;(b)if , , , and , then (13) for (15) holds when ;(c)if , , and , then (13) for (15) holds when ;(d)if , , , and , then (13) for (15) holds for all .
3. Nonnegative Solutions
Let and . For the proof of the theorem about nonnegative solutions, we would need the following lemma.
Lemma 6 (see [28] generalized mean value theorem). Let and for . Then one has with , , where .
Remark 7. Suppose that and , for . It is clear from Lemma 6 that if , , then is nondecreasing for each . If , , then is nonincreasing for each .
We now prove the main theorem.
Theorem 8. There is a unique solution for the initial value problem (5) with (6) and the solution remains in .
Proof. From Theorem 3.1 and Remark 3.2 of [29], we obtain the solution on solving the initial value problem (5) with (6) which is not only existent but also unique. Next, we will show the nonnegative orthant is a positively invariant region. What is needed for this is to show that, on each hyperplane bounding the nonnegative orthant, the vector field points into . From (5), we findBy Remark 7, the solution will remain in .
4. Equilibrium States and Their Stability
In this section, we investigate the stability of the fractionalorder model of HIV infection of CD4 + T cells, that is, system (5) with (6). Consider the initial value problem (5) with (6) with satisfying .
In order to obtain the equilibria of system (5), we set , , and and we haveSystem (5) has three types of relevant nonnegative equilibrium states. System (5) always has an infectionfree equilibrium , where which means that the infected cells are cleared. The basic reproductive number of the viruses for system (5) is given byThis number describes the average number of newly generated infected cells from one infected cell at the beginning of the infection process. When , in addition to the infectionfree equilibrium , there is an immuneabsence equilibrium , where Note that implies . When the HIV infection is in the immuneabsence equilibrium , the infected cells and virus exist but the immune response is not activated yet. Further, we denoteNote that , which implies that is equivalent to . The latter is seen as the immune reproductive number, which expresses the average number of activated CTLs generated from one CTL during its life time through the stimulation of the infected cells . It is reasonable that immune response is activated in the case where . When , in addition to the infectionfree equilibrium and the immuneabsence equilibrium , there is an interior immunepresence equilibrium , where expresses the state where CTLs immune response is present.
Next, we establish the local asymptotic stability of model (5) by the characteristic equation.
Theorem 9. Consider system (5). (1)The infectionfree equilibrium is locally asymptotically stable if .(2)If , the equilibrium is unstable, and if , it is a critical case.
Proof. The characteristic equation for the infectionfree equilibrium is given as follows: It is reduced toIt is clear that (26) has the characteristic roots which means , which means , and . Since the imaginary part of characteristic root is zero, which means is necessary and sufficient to ensure the local asymptotic stability of . If , ; hence is unstable. If , , which is a critical case.
Theorem 10. Consider system (5). (1)The immunefree equilibrium is locally asymptotically stable if .(2)If , the immunefree equilibrium is unstable; if , it is a critical state.
Proof. The characteristic equation for the immunefree equilibrium is given as follows: Using , it is reduced toThe root of (28) is negative and when , positive and when , and zero when , which is a critical case.
Now, we consider the equationDenoteSince and , (29) has two negative real roots and we denote them by and . It is easy to see and . Hence, when , the immunefree equilibrium is locally asymptotically stable, when , is unstable, and when , it is a critical case.
To discuss the local stability of the immunepresent equilibrium state , we consider the linearized system of (5) at . The Jacobian matrix at is given by The characteristic equation of the linearized system iswhereBased on Definition 4, we obtain the discriminant of (32)Using the result (iii) of Lemmas 5 and 3, we have the following theorem.
Theorem 11. Consider system (5). Under the condition of , (1)if the discriminant of , is positive, namely, , then the immunepresent equilibrium is locally asymptotically stable for .(2)if , then the immunepresent equilibrium is locally asymptotically stable for .
5. Numerical Method
Atanackovic and Stankovic introduced a numerical method to solve the single linear FDE in 2004 [1]. A few years later, they developed again a method to solve the nonlinear FDE [30]. It was shown that the fractional derivative of a function with order satisfying may be expressed aswherewith the following properties:We approximate by using terms in sums appearing in (35) as follows:We can rewrite (38) as follows:where We set for . We can rewrite system (19) as the following form:whereNow we can rewrite (39) and (42) as the following form:with the following initial conditions:Now we consider the numerical solution of system of ordinary differential equations (44) with the initial conditions (45) by using the wellknown RungeKutta method of fourth order.
6. Numerical Simulation and Discussion
Firstly, by using GEM (generalized Euler method) [21], we simulate system (5) with the parameter values as shown in Table 1.

By direct calculation, we have , , and and the simulations display that the immunepresent equilibrium is asymptotically stable for , , and (see Figure 1). From Figure 1, we can clearly see that, compared with the case of order , the trajectory of model with order is closer to the trajectory of the model with the integerorder 1. That is, the farther from to 1, the bigger of the trajectory difference of them.
(a) Time evolution of the state variables
(b) Time evolution of the state variables
(c) Time evolution of the state variables
We choose the parameter values as shown in Table 2.

Based on the parameter values in Table 2, we have , , and . With the same simulated method, it is shown that the immunepresent equilibrium is asymptotically stable for and (see Figure 2).
(a) Time evolution of the state variables
(b) Time evolution of the state variables
(c) Time evolution of the state variables
Next, we use the method that is shown in the previous section to simulate system (5) by transforming system (5) to one order ordinary differential equation. Here we set the parameters values as shown in Table 3 with , . We have , , and . The approximate solutions , , and for , , , and are displayed in Figure 3. It shows that the immunepresent equilibrium is asymptotically stable for and there exists the limit circle for .

We also simulate the situation of system (5) by the method in the previous section when with the parameter values of Table 3 in Figure 4. Both Figures 1 and 4 show that the immunepresent equilibrium is asymptotically stable for and that is nothing to do with or . Hence the results of fractionalorder system when are consistent with the result of integerorder HIV model (4).
(a)
(b)
(c)
(d)
7. Conclusion
Fractional differential equations have garnered a lot of attention and appreciation due to their ability to provide an exact description of different nonlinear phenomena. The advantage of fractionalorder systems is that they allow greater degrees of freedom in the model. Nowadays, more and more investigators begin to study the qualitative properties and numerical solutions of fractionalorder virus infection models. In this paper, we introduced a fractionalorder HIV infection model with nonlinear incidence and dealt with the mathematical behaviors of the model.
We showed that system (5) possesses nonnegative solutions and studied the stability behavior of the infectionfree equilibrium, the immuneabsence equilibrium, and the immunepresence equilibrium. We found that the stability of the infectionfree equilibrium and the immuneabsence equilibrium of system (5) is the same as that of system (4). When the basic reproduction number of viruses () is less than one, the infectionfree equilibrium is stable; however, when is more than one, the infectionfree equilibrium is unstable and when the immune reproduction number () is less than one, the immuneabsence equilibrium is stable; however, when is more than one, the immuneabsence equilibrium is unstable. However, the results for the immunepresence equilibrium of system (5) are different to those of system (4). In system (4), the immunepresence equilibrium is stable when is more than one, while, in system (5), when is more than one, the immunepresence equilibrium is not always stable. In the condition of , when , the immunepresence equilibrium is stable for , while when , the immunepresence equilibrium is stable only for . But using the simulation, we found when , the immunepresence equilibrium is stable for . From the simulation, we also found the farther from to 1, the bigger of their trajectory difference. These results show that the integerorder model can be viewed as a special case from the more general fractionalorder model. Although a large part of results is illustrated by both theory analysis and numerical simulation, the result for the immunepresence equilibrium when and can be just verified by the simulation in this paper.
In this paper, we introduce the fractional calculus into the HIV infection model with nonlinear incidence and, from the theory analysis and numerical simulations, it is illustrated that the integerorder HIV infection model can be viewed as a special case of fractionalorder model. We hope that this work can create interest and further do research effort in this field, since the fractional modeling might provide more insight into understanding the dynamical behaviors of such systems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was partially supported by the Hainan Natural Science Foundation (112006) and Natural Science Foundation of Hainan Provincial Department of Education (Hjkj201347). Gang Huang was supported by the Fundamental Research Funds for the Central Universities (no. CUG130415) and the National Natural Science Foundation of China (no. 11201435).
References
 T. M. Atanackovic and B. Stankovic, “An expansion formula for fractional derivatives and its application,” Fractional Calculus & Applied Analysis, vol. 7, no. 3, pp. 365–378, 2004. View at: Google Scholar  MathSciNet
 G. Bognár, “Similarity solution of boundary layer flows for nonNewtonian fluids,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 1112, pp. 1555–1566, 2009. View at: Google Scholar
 M. Benmalek and A. Charef, “Digital fractional order operators for Rwave detection in electrocardiogram signal,” IET Signal Processing, vol. 3, no. 5, pp. 381–391, 2009. View at: Publisher Site  Google Scholar
 Y. Ferdi, “Some applications of fractional order calculus to design digital filters for biomedical signal processing,” Journal of Mechanics in Medicine and Biology, vol. 12, no. 2, Article ID 12400088, 2012. View at: Publisher Site  Google Scholar
 Y. Ferdi, A. TalebAhmed, and M. R. Lakehal, “Efficient generation of ${1/f}^{\beta}$ noise using signal modeling techniques,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 55, no. 6, pp. 1704–1710, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 R. L. Bagley and R. A. Calico, “Fractional order state equations for the control of viscoelastically damped structures,” Journal of Guidance, Control, and Dynamics, vol. 14, no. 2, pp. 304–311, 1991. View at: Publisher Site  Google Scholar
 G. L. Jia and Y. X. Ming, “Study on the viscoelasticity of cancellous bone based on higherorder fractional models,” in Proceedings of the 2nd International Conference on Bioinformatics and Biomedical Engineering (ICBBE '08), pp. 1733–1736, 2006. View at: Publisher Site  Google Scholar
 J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,” Bulletin of Science Technology & Society, vol. 15, pp. 86–90, 1999. View at: Google Scholar
 B. Mandelbrot, “Some noises with $1/f$ spectrum, a bridge between direct current and white noise,” IEEE Transactions on Information Theory, vol. 13, no. 2, pp. 289–298, 1967. View at: Publisher Site  Google Scholar
 R. L. Magin, “Fractional calculus in bioengineering,” Critical Reviews in Biomedical Engineering, vol. 32, pp. 1–377, 2004. View at: Publisher Site  Google Scholar
 Y. A. Rossikhin and M. V. Shitikova, “Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids,” Applied Mechanics Reviews, vol. 50, no. 1, pp. 15–67, 1997. View at: Publisher Site  Google Scholar
 F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378, pp. 291–348, Springer, Berlin, Germany, 1997. View at: Publisher Site  Google Scholar  MathSciNet
 R. T. Baillie, “Long memory processes and fractional integration in econometrics,” Journal of Econometrics, vol. 73, no. 1, pp. 5–59, 1996. View at: Publisher Site  Google Scholar  MathSciNet
 K. S. Cole, “Electric conductance of biological systems,” in Proceedings of the Cold Spring Harbor Symposia on Quantitative Biology, pp. 107–116, Cold Spring Harbor, NY, USA, 1993. View at: Google Scholar
 V. D. Djordjević, J. Jarić, B. Fabry, J. J. Fredberg, and D. Stamenović, “Fractional derivatives embody essential features of cell rheological behavior,” Annals of Biomedical Engineering, vol. 31, no. 6, pp. 692–699, 2003. View at: Publisher Site  Google Scholar
 R. V. Culshaw and S. Ruan, “A delaydifferential equation model of HIV infection of CD4^{+} Tcells,” Mathematical Biosciences, vol. 165, no. 1, pp. 27–39, 2000. View at: Publisher Site  Google Scholar
 G. Huang, H. Yokoi, Y. Takeuchi, T. Kajiwara, and T. Sasaki, “Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics,” Japan Journal of Industrial and Applied Mathematics, vol. 28, no. 3, pp. 383–411, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 G. Huang, Y. Takeuchi, and A. Korobeinikov, “HIV evolution and progression of the infection to AIDS,” Journal of Theoretical Biology, vol. 307, pp. 149–159, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 A. S. Perelson, D. E. Kirschner, and R. de Boer, “Dynamics of HIV infection of CD4^{+} T cells,” Mathematical Biosciences, vol. 114, no. 1, pp. 81–125, 1993. View at: Publisher Site  Google Scholar
 A. S. Perelson and P. W. Nelson, “Mathematical analysis of HIV1 dynamics in vivo,” SIAM Review, vol. 41, no. 1, pp. 3–44, 1999. View at: Publisher Site  Google Scholar  MathSciNet
 A. A. M. Arafa, S. Z. Rida, and M. Khalil, “Fractional modeling dynamics of HIV and CD4^{+} Tcells during primary infection,” Nonlinear Biomedical Physics, vol. 6, article 1, 2012. View at: Publisher Site  Google Scholar
 Y. Ding and H. Ye, “A fractionalorder differential equation model of HIV infection of CD4^{+} Tcells,” Mathematical and Computer Modelling, vol. 50, no. 34, pp. 386–392, 2009. View at: Publisher Site  Google Scholar
 A. Korobeinikov, “Global properties of infectious disease models with nonlinear incidence,” Bulletin of Mathematical Biology, vol. 69, no. 6, pp. 1871–1886, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 A. Korobeinikov, “Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate,” Mathematical Medicine and Biology, vol. 26, pp. 225–239, 2009. View at: Google Scholar
 A. Korobeinikov, “Stability of ecosystem: global properties of a general predatorprey model,” Mathematical Medicine and Biology, vol. 26, no. 4, pp. 309–321, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 I. Petras, FractionalOrder Nonlinear Systems: Modeling, Analysis and Simulation, Springer, New York, NY, USA, 2011.
 E. Ahmed and A. S. Elgazzar, “On fractional order differential equations model for nonlocal epidemics,” Physica A: Statistical Mechanics and Its Applications, vol. 379, no. 2, pp. 607–614, 2007. View at: Publisher Site  Google Scholar
 Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 W. Lin, “Global existence theory and chaos control of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 709–726, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 T. M. Atanackovic and B. Stankovic, “On a numerical scheme for solving differential equations of fractional order,” Mechanics Research Communications, vol. 35, no. 7, pp. 429–438, 2008. View at: Publisher Site  Google Scholar  MathSciNet
Copyright
Copyright © 2015 Linli Zhang 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.