Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay

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Original Articles

Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay Ye Yan a , Chunhai Kou b,∗ a

College of Information Sciences and Technology, Donghua University, Shanghai 201620, PR China b Department of Applied Mathematics, Donghua University, Shanghai 201620, PR China Received 15 August 2010; accepted 18 January 2012

Abstract In this paper, we introduce fractional-order derivatives into a model of HIV infection of CD4+ T-cells with time delay. We deal with the stability of both the viral free equilibrium and the infected equilibrium. Criteria are given to ensure that both the equilibria are asymptotically stable for all delay under some conditions. Numerical simulations are carried out to illustrate the results. © 2012 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: HIV infection; Fractional-order; Time delay; Asymptotic stability

1. Introduction Fractional calculus is a generalization of classical differentiation and integration to arbitrary (non-integer) order. In recent years fractional calculus has been a fruitful field of research in science and engineering [12,23,25,26,30]. In fact, many researchers are currently paying attention to the fractional calculus concepts and we can refer its adoption in chemistry, electromagnetic waves [8], quantitative finance [16], quantum evolution of complex system [21], chaos and fractals [7], robotics [31], control systems [20], etc. Thus, as mentioned in [23], there is no field that has remained untouched by fractional derivatives. In particular, fractional differential equations as an important research branch of fractional calculus gain much attention. Many results on local existence, uniqueness and structural stability of solutions of specific fractional differential equations are successively established [10,17,32,37]. Also varieties of schemes for numerical solutions of fractional differential equations are proposed [15,19]. Meanwhile, the applications of fractional differential equations to physics, biology and engineering are a recent focus of interest [9,12]. Many systems are known to display fractional-order dynamics, such as viscoelastic systems [4,13], electrode–electrolyte polarization [11] and complex adaptive systems in biology [2]. Human immunodeficiency virus (HIV) is a lentivirus (a member of the retrovirus family) that causes acquired immunodeficiency syndrome (AIDS) [34], 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-cells (to be specific, CD4+ T-cells), macrophages, and dendritic cells. When CD4+ T-cell numbers decline below a ∗

Corresponding author. E-mail addresses: [email protected] (Y. Yan), [email protected], [email protected] (C. Kou).

0378-4754/$36.00 © 2012 IMACS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.matcom.2012.01.004

Please cite this article in press as: Y. Yan, C. Kou. Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay, Math. Comput. Simul. (2012), http://dx.doi.org/10.1016/j.matcom.2012.01.004

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critical level, cell-mediated immunity is lost, and the body becomes progressively more susceptible to opportunistic infections. Mathematical modeling has proven to be valuable in understanding the dynamics of HIV infection [27–29]. Simple HIV models have played a significant role in the development of a better understanding of the disease and the various drug therapy strategies used against it. Time delays of one type or another have been introduced to describe the time between viral entry into a target cell and the production of new virus particles by many authors [5,24,35,38]. Nelson and Perelson [24] developed and analyzed a set of models that included intracellular delays, combination antiretrovial therapy, and the dynamics of both infected and uninfected T-cells. Culshaw and Ruan [5] first simplified their model into one consisting of only three components (the healthy CD4+ T-cells, infected CD4+ T-cells, and free virus) and discussed the existence and stability of the infected steady state. They obtained a restriction on the number of viral particles released per infectious cell in order for infection to be sustained. Under this restriction, the system had a positive equilibrium—the infected steady state. By using stability analysis they obtained sufficient conditions on the parameters for the stability of the infected steady state. Then, they introduced a discrete time delay to the model to describe the time between infection of a CD4+ T-cells and the emission of viral particles on a cellular level. The same restriction on the number of viral particles released per infectious cell is required. By analyzing the transcendental characteristic equation, they analytically derived stability conditions for the infected steady state in terms of the parameters and independent of the delay. For more references, see also Zhou et al. [38], Yang et al. [35] and Li and Ma [18] for related work. Recently, more and more investigators begin to study the qualitative properties and numerical solutions of fractionalorder biological models [1,3,6,36]. The main reason is that fractional-order equations are naturally related to systems with memory which exists in most biological systems. Also they are closely related to fractals which are abundant in biological systems. In [3], the fractional-order predator–prey model and the fractional-order rabies model were investigated; the existence and uniqueness of solutions were proved; the stability of equilibrium points were studied; numerical solutions of these models were given. In [1], a fractional-order model for nonlocal epidemics was given; stability of fractional-order equations was studied; the results were expected to be relevant to foot-and-mouth disease, SARS and avian flu. In addition, Ding and Ye have also introduced some kinds of models of HIV infection and considered the stability properties of the equilibria of the corresponding systems [6,36]. In our previous work, we have studied stability properties for fractional order differential equations and applied the results obtained to analyze the stability of the equilibria for the model of HIV-1 infection [14]. To our knowledge, no works are contributed to the analysis for fractional delay-differential equations of HIV infection of CD4+ T-cells. Motivated by this situation, we introduce fractional-order derivatives into a HIV infection model with time delay proposed by Culshaw and Ruan [5]. The new model is described as follows: ⎧   ⎪ α T (t) = s − μ T (t) − rT (t) 1 − T (t) + I(t) − k T (t)V (t), ⎪ D α ∈ (0, 1] ⎪ T 1 ⎨ Tmax (1) Dα I(t) = k1 T (t − τ)V (t − τ) − μI I(t), ⎪ ⎪ ⎪ ⎩ α D V (t) = Nμb I(t) − k1 T (t)V (t) − μV V (t), with the initial conditions T (θ) = T0 ,

I(0) = 0,

V (θ) = V0 ,

θ ∈ [−τ, 0],

(2)

where Dα is in the sense of the Caputo fractional derivative, T(t), I(t) represent the concentration of healthy CD4+ T-cells and infected CD4+ T-cells at time t, respectively, V(t) represents the concentration of free HIV at time t, and the positive constant τ represents the length of the delay in days. Parameters s, r, N, Tmax , k1 , k1 , μT , μI , μb , μV are positive constants. s is the source of CD4+ T-cells from precursors, μT is the natural death rate of CD4+ T-cells, r is their growth rate (thus, r > μT in general), and Tmax is their carrying capacity. The parameter k1 represents the rate of infection of T-cells with free virus, k1 is the rate at which infected cells become actively infected (the ratio k1 /k1 is the proportion of T-cells, which ever become actively infected). μI is a blanket death term for infected cells, to reflect the assumption that we do not initially know whether the cells die naturally or by bursting. In addition, μb is the lytic death rate for infected cells. N is the number of virus produced by infected CD4+ T-cells during its lifetime. Finally, μV is the loss rate of virus. A complete list of the parameter values for the model is given in Table 1 [5]. For biological meaning, T0 ≥ 0, V0 ≥ 0 and the solution (T(t), I(t), V(t)) of (1) with Please cite this article in press as: Y. Yan, C. Kou. Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay, Math. Comput. Simul. (2012), http://dx.doi.org/10.1016/j.matcom.2012.01.004

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3

Table 1 Parameters and values of model (1). Parameter

Description

Value CD4+

T I V T0 μT μI μb μV k1 k1 r N Tmax s

Uninfected 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-cells Blanket death rate of infected CD4+ T-cells Lytic death rate for infected cells Death rate of free virus Rate CD4+ T-cells become infected with virus Rate infected cells become active Growth rate of CD4+ T-cells population Number of virions produced by infected CD4+ T-cells Maximal population level of CD4+ T-cells Source term for uninfected CD4+ T-cells

1000 mm−3 0 10−3 mm−3 1000 mm−3 0.02 day−1 0.26 day−1 0.24 day−1 2.4 day−1 2.4 × 10−5 mm3 day−1 2×10−5 mm3 day−1 0.03 day−1 Varies 1500 mm−3 10 (day)−1 (mm)−3

the initial condition (2) exists for all t ≥ 0 and is unique. Furthermore, we assume that T(t) > 0, I(t) ≥ 0 and V(t) ≥ 0 for all t > 0. This paper is organized as follows. In Section 2, a fractional order model of HIV infection of CD4+ T-cells is deduced and some necessary definitions and notations are presented. A detailed analysis on local stability of the equilibria is carried out in Section 3. Simulations are given in Section 4. And the conclusions are given in the last section. 2. Preliminaries We first give the definition of fractional-order integration and fractional-order differentiation [30]. There are several forms of definitions of fractional integral and derivative, such as, Riemann–Liouville fractional integral and derivative, Caputo’s fractional derivative, Gr¨unwald–Letnikov fractional derivative, and so on. As is well known, in fractional differential equations, the initial conditions are specified in terms of fractional derivatives of the unknown function in the Riemann–Liouville approach. But, in the Caputo approach, the initial conditions could be specified in terms of integer derivatives. Another advantage of this definition is that the Caputo derivative for a constant equals to zero while in the Riemann–Liouville sense, the fractional derivative of a constant is nonzero. Therefore, in this article, we deal with the systems of fractional-order differential equations involving the Caputo derivative. Definition 1. [12] The fractional integral of order α(R(α) > 0) of a function f : R+ → R is given by  x 1 α (x − t)α−1 f (t) dt. I f (x) = (α) 0 Here (α) is the Gamma function. Definition 2. [12] The Caputo fractional derivative of order α ∈ (n − 1, n) of a function f : R+ → R is given by Dα f (x) = I n−α Dn f (x),

D=

d . dt

In particular, when 0 < α < 1, we have  x f  (t) 1 Dα f (x) = dt. (1 − α) 0 (x − t)α The purpose of this paper is to study the stability properties of the equilibria of a fractional differential system, so the following lemma is useful to our arguments. Please cite this article in press as: Y. Yan, C. Kou. Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay, Math. Comput. Simul. (2012), http://dx.doi.org/10.1016/j.matcom.2012.01.004

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Y. Yan, C. Kou / Mathematics and Computers in Simulation xxx (2012) xxx–xxx Im(eig(A))

Stable Unstable

Stable απ/2 απ/2

Re(eig(A))

Unstable Stable

Stable

Fig. 1. Stability region of system (3) with order 0 < α ≤ 1.

Lemma 3. [14] The equilibrium point (xeq , yeq ) of the fractional differential system Dα x(t) = f1 (x, y), Dα y(t) = f2 (x, y), α ∈ (0, 1], x(0) = x0 ,

y(0) = y0

(3)

is locally asymptotically stable if all the eigenvalues of the Jacobian matrix

 ∂f1 /∂x ∂f1 /∂y A= ∂f2 /∂x ∂f2 /∂y evaluated at the equilibrium point satisfies the following condition: |arg(eig(A))| >

απ . 2

The stable and unstable regions for 0 < α ≤ 1 are shown in Fig. 1 [22,33]. 3. Main results In this section, we investigate the stability of the fractional-order model of HIV infection of CD4+ T-cells with time delay, i.e., system (1). In order to find the equilibria of system (1), we put ⎧   T (t) + I(t) ⎪ ⎪ − k1 T (t)V (t) = 0, ⎪ ⎨ s − μT T (t) − rT (t) 1 − Tmax (4) k1 T (t − τ)V (t − τ) − μI I(t) = 0, ⎪ ⎪ ⎪ ⎩ Nμb I(t) − k1 T (t)V (t) − μV V (t) = 0. For the existence of the nonnegative equilibria of (1), it is not difficult to see that the algebraic system (4) has two equilibria: the uninfected equilibrium E0 = (T0 , 0, 0), where Tmax 4sr T0 = [r − μT + (r − μT )2 + ], 2r Tmax and the infected equilibrium E* = (T* , I* , V* ), where T∗ =

μ V μI , k1 Nμb − k1 μI

I∗ =

k1 T ∗ V ∗ , μI

V∗ =

μI [(s + (r − μT )T ∗ )Tmax − rT ∗ 2 ] . T ∗ [k1 rT ∗ + k1 μI Tmax ]

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5

Following the analysis in [5], we define a parameter Ncrit as Ncrit =

μI (μV + k1 T0 ) . k1 μb T0

This section is to give a detailed analysis for the local asymptotic stability of the viral free equilibrium E0 and the infected equilibrium E* . To discuss the local asymptotic stability of system (1), let us consider the following coordinate transformation x(t) = T (t) − T ,

y(t) = I(t) − I,

z(t) = V (t) − V ,

where (T , I, V ) denotes any equilibrium of (1). Hence, we have that the corresponding linearized system of (1) is of the form ⎧   ⎪ α x(t) = − μ + 2rT + rI + k V − r x(t) − rT y(t) − k T z(t), ⎪ D ⎪ T 1 1 ⎪ ⎪ Tmax Tmax ⎨ (5) Dα y(t) = k1 V x(t − τ) − μI y(t) + k1 T z(t − τ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ α D z(t) = −k1 V x(t) + Nμb y(t) − (k1 T + μV )z(t). The Jacobian matrix at (T , I, V ) is given by ⎛   r(2T + I) + k1 V − r ⎜ − μT + T max ⎜ J(M) = ⎜  −λτ ⎝ k1 V e −k1 V

⎞

rT − Tmax −μI

−k1 T

Nμb

−(k1 T + μV )

k1 T e−λτ

⎟ ⎟ ⎟. ⎠

Then the associated transcendental characteristic equation of (5) is given by |λE − J(M)| = 0, where E is the identity matrix. First, for the local asymptotic stability of the viral free equilibrium E0 , we have the following result. Theorem 4. Let N < Ncrit , then the uninfected equilibrium E0 is locally asymptotically stable for any time delay τ ≥ 0. Proof. It is clear that, at E0 = (T0 , 0, 0) = (T , I, V ), the associated transcendental characteristic equation of (5) becomes     2rT 0 − r) · (λ + μI )(λ + (k1 T0 + μV )) − Nk1 T0 μb e−λτ = 0. (6) λ + (μT + Tmax Obviously, (6) has the characteristic root λ1 = r − μT − (2rT0 /Tmax ). Because of   2rT 0 2r Tmax 4sr r − μT − = r − μT − · r − μT + (r − μT )2 + Tmax Tmax 2r Tmax  4sr < 0, = − (r − μT )2 + Tmax we know that λ1 < 0. Now, we consider the transcendental polynomial (λ + μI )(λ + (k1 T0 + μV )) − Nk1 T0 μb e−λτ = 0.

(7)

We rewrite (7) in the following form: λ2 + (k1 T0 + μI + μV )λ + (k1 T0 + μV )μI − Nk1 T0 μb e−λτ = 0.

(8)

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For τ = 0, we have had (k1 T0 + μV )μI − Nk1 T0 μb > 0 by N < Ncrit and k1 T0 + μI + μV > 0. Then, we have λ=

−(k1 T0 + μI + μV ) ±



(k1 T0 + μI + μV )2 − 4(Ncrit − N)k1 T0 μb 2

.

This shows that the roots of (8) have negative real parts for τ = 0. For τ = / 0, assume that (8) has pure imaginary roots λ = ± iω for some ω > 0 and τ > 0, then we have from (8) that −ω2 + (k1 T0 + μI + μV )ωi + (k1 T0 + μV )μI − Nk1 T0 μb (cos ωτ − i sin ωτ) = 0. Separating the real and imaginary parts, we have −ω2 + (k1 T0 + μV )μI = Nk1 T0 μb cos ωτ, −(k1 T0 + μI + μV )ω = Nk1 T0 μb sin ωτ,

(9)

which implies that   ω4 + (k1 T0 + μV )2 + μ2I ω2 + (k1 T0 + μV )2 μ2I − (Nk1 T0 μb )2 = 0. Because of N < Ncrit , we get ω2

=

−((k1 T0 + μV )2 + μI ) ±



2 − N 2 ) · (k  μ T )2 ((k1 T0 + μV )2 + μI )2 − 4(Ncrit 1 b 0

2

< 0. The contradiction shows that any root of (8) must have negative real part. Hence, according to Lemma 3, the uninfected equilibrium E0 is local asymptotic stable for any time delay τ ≥ 0. This proves the conclusion. 

Remark 1. If τ = 0, system (1) is just the model which Ding and Ye [6] have proposed. From Fig. 1, we see that the stability region of a system with fractional-order α ∈ (0, 1) is always larger than that of a corresponding ordinary differential system. This means that an unstable equilibrium of an ordinary differential system may be likely to be stable in a fractional differential system. Because of this reason, we have to point out that the results obtained by Ding and Ye are not proper enough. In [6], it was proven that if N > Ncrit , then at least one eigenvalue would be positive real root and the uninfected state E0 was unstable. From Lemma 3 and Fig. 1, we get that even if an eigenvalue has positive real part, the system can also be stable. We can show this through the simulations (Fig. 2).

For the parameter values given in Table 1, Ncrit = 131.3. Then we take N = 132 > Ncrit , so from Fig. 2, we see that the equilibrium E0 of the model with α = 0.51 is still stable. However, in the case of the ordinary differential model, we have no such a result. Next, for the sake of convenience, we give the following symbols: r(T ∗ + I ∗ ) + k1 V ∗ − r, A = μI + μV + k1 T ∗ + M, Tmax B = M(k1 T ∗ + μI + μV ) + μI (μV + k1 T ∗ ) − k12 T ∗ V ∗ ,   rμV V ∗  ∗ ∗ C = k1 T k1 Nμb V + − MNμb , T  ∗  max rV D = k1 T ∗ − Nμb , E = MμI (μV + k1 T ∗ ) − μI k12 T ∗ V ∗ . Tmax

M = μT +

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T

1000 α=1

999.9999 999.9999 999.9999 999.9999 999.9999 999.9998

0

100

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t−time (days) P2

−5

1.4

x 10

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0.4

0.2

0

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−3

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x 10

0.9 0.8 0.7

V

0.6 0.5 0.4 α=1 0.3

α=0.51

0.2 0.1 0

0

100

200

300

400

500

t−time (days)

Fig. 2. In P1–P3, τ = 0, N = 132.

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Denote

D(λ)

=

=

 1   0  −  3  0  0

B+D

C+E

1 A 2A B + D 3 2A

B+D

0

2A

A

3

0 B+D

    C+E   0   0   B + D 0

18A(B + D)(C + E) + [A(B + D)]2 − 4A3 (C + E) − 4(B + D)3 − 27(C + E)2 .

For the local asymptotic stability of the infected equilibrium E* of (1), we have the following result. Theorem 5. Let E2 − C2 ≥ 0, B2 − 2AE − D2 > 0, then the infected equilibrium E* of (1) is asymptotically stable for any time delay τ ≥ 0 if either (i) D(λ) > 0, A > 0, (C + E) > 0, A(B + D) − (C + E) > 0 or (ii) D(λ) < 0, A > 0, B + D > 0, A(B + D) = C + E, α ∈ (2/3, 1) is satisfied. Proof. It is clear that the associated transcendental characteristic equation of (5) at E∗ = (T ∗ , I ∗ , V ∗ ) = (T , I, V ) becomes λ3 + Aλ2 + Bλ + (C + Dλ)e−λτ + E = 0,

(10)

For τ = 0, we have from (10) that λ3 + Aλ2 + (B + D)λ + C + E = 0. Using the results of [1], we get that the infected equilibrium E* is asymptotically stable if either (i) D(λ) > 0, A > 0, (C + E) > 0, A(B + D) − (C + E) > 0 or (ii) D(λ) < 0, A > 0, B + D > 0, A(B + D) = C + E, α ∈ (2/3, 1) is satisfied. For τ = / 0, similar to the proofs of Theorem 4, we assume that (10) has pure imaginary roots λ = ± iω for some ω > 0 and τ > 0, then we will finally obtain that if E2 − C 2 ≥ 0

and

B2 − 2AE − D2 > 0

(11)

are satisfied, then the real parts of all the eigenvalues of (10) are negative for all delay τ ≥ 0. Summarizing the above analysis, we reach the results.  4. Simulations In this section, we are to illustrate our results on stability by some numerical simulations. The number of infectious viruses released N varies in the literature. It has been suggested to be hundreds and even thousands. We first take N = 800, τ = 2, then E* = (162.7645, 35.2022, 2811.6) and M = 0.0647,

A = 2.7286,

C = −0.04,

D = −0.6248,

B = 0.7971, E = 0.0404,

and E2 − C2 = 3.216 × 10−5 > 0,

B2 − 2AE − D2 = 0.0245 > 0,

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T

600 500 α=1

400

α=0.8

α=0.9

300 200 100 0

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α=0.8 150 α=0.9

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α=1

50

0

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t−time (days) 4

2.5

(A6)

x 10

2

α=0.8

V

1.5

1

α=0.9 α=1

0.5

0

0

50

100

150

200

250

300

t−time (days)

Fig. 3. In (A4)–(A6), τ = 2, N = 800.

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T

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200 100 0

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I

250 200 150

α=0.8 α=1

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α=0.9

50 0

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t−time (days) (A9)

4

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x 10

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V

4

3

α=0.8

2

α=0.9

α=1

1

0

0

50

100

150

200

250

300

t−time (days)

Fig. 4. In (A7)–(A9), τ = 2, N = 1400.

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D(λ) = 25.1438 > 0,

C + E = 0.0004 > 0,

11

A(B + D) − (C + E) = 0.8297 > 0.

Thus, all the conditions in Theorem 5 are satisfied and the infected equilibrium E* is asymptotically stable. Numerical simulations show that trajectories of system (1) approach to the steady state (Fig. 3). Moreover, when N is larger (N = 1400), the effect of the delay is weaker (Fig. 4). Remark 2. In the simulations above, we can clearly see that as the order α is far away from 1, the results are different from those of the model with the integer-order 1. That means, compared with the case of the model with the order α = 0.8, the trajectory of the model with order α = 0.9 is closer to the trajectory of the model with the integer-order 1. Why do such differences happen and what influence does the order α have on the HIV model? They are all the problems which we need to solve in our future studies.

Remark 3. In Theorem 5, in the case that E2 − C2 ≥ 0, B2 − 2AE − D2 > 0, and   2 D(λ) < 0, A ≥ 0, B + D ≥ 0, C + E > 0, α ∈ 0, , 3 the infected equilibrium E* is still asymptotically stable. But, according to Remark 2, we know that the trajectory of the model with the order α which is closer to zero can not describe the situation properly, so we remove this case in Theorem 5. (A10)

(A11)

320

80

310 75 300 70

290

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Fig. 5. In (A10)–(13), α = 0.9, τ = 0.0001, N = 10, the figure depicts the infected equilibrium E* bifurcating into a periodic solution.

Please cite this article in press as: Y. Yan, C. Kou. Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay, Math. Comput. Simul. (2012), http://dx.doi.org/10.1016/j.matcom.2012.01.004

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(A15)

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Fig. 6. In (A14)–(17), α = 0.9, τ = 20, N = 10, the figure depicts the infected equilibrium E* unstably.

Remark 4. In the models with integer-order, Hopf bifurcations might occur under some conditions. However, for the parameter values given in Table 1, no Hopf bifurcation occurs. So, we use another list of parameter values to show that in fractional-order models Hopf bifurcation can still occur. Let μT = 0.2, μI = 2.4, μb = 1, μV = 2.4, k1 = k1 = 0.0027, s = 0.1, r = 0.95. Moreover, we take N = 10 and α = 0.9. For these parameters, from Figs. 5 and 6, we see that Hop bifurcation occurs. 5. Conclusions Incorporating a time delay into HIV infection models has been done by some researchers. It is still a hot topic to determine how the intercellular delay affects overall disease progression and, mathematically, how the delay affects the dynamics of systems. In the present paper, we propose a fractional-order model of HIV infection of CD4+ T-cells with time delay. The time delay is introduced to describe the time between infection of a CD4+ T-cell and the emission of viral particles on a cellular level. In our analysis, there are two equilibria: the uninfected equilibrium and the infected equilibrium. By using the results in [14], we analyze the stability properties of the above two equilibria. First, we consider the stability properties of the uninfected equilibrium. We get a sufficient condition on the parameters for the stability of the uninfected steady point. Moreover, we point out an improper result in [6] and prove our conclusion by simulations. Next, we consider the stability properties of the infected equilibrium. By analyzing the transcendental characteristic equation, we analytically derive the stability conditions for the infected steady state. We know that the infected steady point is stable despite of the size of the delay under certain conditions, though the time delay does affect all the components. Biologically, it implies that the intercellular delay can make the cell and virus populations fluctuate in the early stage of infection, and they will converge to the infected steady state after a period of Please cite this article in press as: Y. Yan, C. Kou. Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay, Math. Comput. Simul. (2012), http://dx.doi.org/10.1016/j.matcom.2012.01.004

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time. We notice that the models with fractional derivatives are more complicated than the ones with classical derivatives. Though, by simulations, we show that there really exists Hopf bifurcation in fractional-order models under certain conditions, we have not been able to prove this theoretically. Further analysis and proofs on this issue are needed and we will concern about this problem in our further studies. Acknowledgements This work is supported by the National Natural Science Foundation of PR China (No. 10971221) and Shanghai Natural Science Foundation (No. 10ZR1400100). References [1] E. Ahmed, A.S. Elgazzar, On fractional order differential equations model for nonlocal epidemics, Physica A 379 (2007) 607–614. [2] E. Ahmed, A.S. Elgazzar, A.S. Hegazi, Modeling the avian flu, lessons form complex adaptive systems in biology, Applied Mathematics and Computation 195 (2008) 351–354. [3] E. Ahmed, A.M. Elsayed, H.A.A. 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Please cite this article in press as: Y. Yan, C. Kou. Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay, Math. Comput. Simul. (2012), http://dx.doi.org/10.1016/j.matcom.2012.01.004