GLOBAL STABILITY OF THE VIRUS DYNAMICS MODEL WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE

In this paper, a virus dynamics model with Crowley-Martin functional response of the infec- tion rate is investigated. By analyzing the corresponding characteristic equations, the local stability of an infection-free equilibrium point and infection equilibrium point are discussed. By constructing suitable Lyapunov functions and using LaSalles invariance principle, the global stability also are established, it is proved that if the basic reproductive number, R0, is less than or equal to one, the infection-free equilib- rium point is globally asymptotically stable, if R0, is more than one, the infection equilibrium point is globally asymptotically stable.


Introduction
Mathematical models have been used to model the dynamic of viral infections, such as human immunodeficiency virus type I (HIV-I), hepatitis C virus (HCV), hepatitis B virus (HBV), and human T-cell lymphotropic virus I (HTLV-I) (Perelson et al., 1993(Perelson et al., , 1996Bonhoeffer et al., 1997;Perelson and Nelson, 1999;Nowak and May, 2000;Korobeinikov, 2004 ).  and Nowak and May(2000) proposed a basic mathematical model for uninfected susceptible host cells, T , infected host cells, I, and free virus particles, V , as follows: dT dt = λ − dT − βT V, t > 0, dI dt = βT V − pI, t > 0, dV dt = kI − µV, t > 0, (1.1) where susceptible cells are produced at rate λ, die at rate dT and become infected at rate βT V ; infected cells are produced at rate βT V and die at rate pI; free viruses are produced from infected cells at rate kI and are removed at rate µV .
In this paper, we consider the following more general the virus dynamics model with Crowley-Martin functional response (1.2) and with the following initial conditions: (1.3) This work was partially supported by the Longdong University Grant XYZK-1007. EJQTDE, 2012 No. 9, p. 1 Here the state variables T, I and V , and all parameters λ, β, µ, d, k, p have the same biological meanings as in model (1.1), the term βT V has been replaced with Crowley-Martin the functional response term, βT V (1+aT )(1+bV ) , where a, b ≥ 0 are constants. The Crowley-Martin type of functional response was introduced by Growley and Martin(1989).
Note that when a = b = 0, (1.2) reduces to the system (1.1); when a > 0, b = 0, the Crowley-Martin functional response reduces to the Holling type II functional response (e.g., Ma and Li, 2007); when a = 0, b > 0, it expresses a saturation response (see, Song and Neumann, 2007); Huang et al. (2009) considered a mathematical model with Beddington-DeAngelis functional response of the infection rate, and sufficient conditions were derived for the global stability of an infected steady state and uninfected steady state.
Recently, Zhou and Cui (2010) proposed the model (1.2), and showed that the infection equilibrium E * is globally asymptotically stable if the following conditions hold. The proof uses the theory of competitive systems as developed in Smith (1987), with conditions (1.4), (1.5) and (1.6) being used to establish the local stability of E * . In this article, we will study the global dynamics of (1.2) by constructing a suitable Lyapunov function and using LaSalle's invariance principle rather than by using the theory of competitive systems, as has been done in Zhou and Cui (2010). This will enable us to obtain the global asymptotic stability of the infection equilibrium point under weaker hypotheses than those used in Zhou and Cui (2010) and by a simpler method. In our setting, the persistence condition R 0 > 1 used in Zhou and Cui (2010) will appear in a natural way as a monotonicity condition. Finally, we will discuss the biological significance of our results and indicate possible extensions to the study of more comprehensive models in section 4.

2.
Boundedness, Equilibria and their stability 2.1. Boundedness. First, we shows that the solution of system (1.2) is bounded.
Theorem 2.1. Let (T (t), I(t), V (t)) be a solution of (1.2) with initial conditions (1.3), and let [0, T ) be the maximal existence interval of the solution.. Then there is an M > 0 such that EJQTDE, 2012 No. 9, p. 2 Denote m = min{d, p}, it follows that Therefore, there exists a t 1 > 0 and M 1 > 0 such that Z(t) < M 1 for t > t 1 . So that T (t) and I(t) are bounded. On the other hand, by the third of system (1.2) we obtain Similar to the argument on Z(t), we can conclude that V (t) is also ultimately bounded. Therefore, from the extension theorem of solutions, we have T = +∞. This completes the proof.
which is independent of the number of total cells of liver. Now we consider the stability of equilibrium points of (1.2). By the simple calculation, system (1.2) has the following three equilibrium points: (iii) if R 0 > 1, then the system (1.2) has a unique infection equilibrium point E * = (T * , I * , V * ), E * represents persistent, chronic HBV infection. E * is given by if and only if We note that R 0 = βT0 δ(d+aT0) > βT * δ(1+aT * ) , since I * = 1 dp ( λ d − T * ). Hence, the system (1.2) has a unique infection equilibrium point E * = (T * , I * , V * ) if R 0 > 1.
Remark 2.2. Notice that R 0 > R * , therefore, Theorem 2.2.(iii) becomes: (iii) ′ If R 0 > 1 holds, then E * is also locally asymptotically stable. EJQTDE, 2012 No. 9, p. 3 Proof. (i) The linearization matrix of the system (1.2) at E 0 is The characteristic equation at E f is Therefore, by the Routh-Hurwitz criterion, if R 0 < 1 hold, then E f is locally asymptotically stable.
(iii) The linearization matrix of the system (1.2) at E * is The characteristic equation at E * is Furthermore, EJQTDE, 2012 No. 9, p. 4 Then by the Routh-Hurwitz criterion, we have that E * = (T * , I * , V * ) is locally asymptotically stable if R * > 1.
2.4. Analysis at R 0 = 1. To use the center manifold theory, as described in Castillo-Chavez and Song (2004) (Theorem 4.1). To apply this method, the following simplification and change of variables are made first.
. The linearization matrix of system (2.1) around the infection-free equilibrium when β = β * is The matrix D x f has eigenvalues (0, −d, −p − µ) T , which meets the requirement of a simple zero eigenvalue and others having negative real part. A right eigenvector w corresponding to the zero eigenvalue is w = (− p d , 1, k µ ) T , and the left eigenvector satisfying v · w = 1 is v = (0, µ µ+p , p(µ+p) kµ ). For system (2.1) the associated non-zero the second partial derivative of f 2 , f 3 are given by It follows that Thus, a < 0, b > 0, by item (iv) of Theorem 4.1 in Castillo-Chavez and Song (2004), we can give the following result: Theorem 2.3. The infection equilibrium point E * is locally asymptotically stable for R 0 near 1.
Note that the result in Theorem 2.3 holds for R 0 > 1 but close to 1.

Global stability and uniformly persistent
3.1. Global stability. The following two theorems is the global stability results of the infection-free equilibrium point E f and infection equilibrium point E * .
Proof. (i) Let us consider the Lyapunov function It is easily seen that V 1 (T, I, V ) ≥ 0 and V 1 (T, I, V ) = 0 if and only if T = T 0 , I = V = 0. We now compute the time derivative of V 1 along the solutions of (1.2). One then haṡ EJQTDE, 2012 No. 9, p. 6 It follows from R 0 ≤ 1 thatV 1 ≤ 0 for all T, I, V > 0. Hence, the uninfected steady E f is stable. Anḋ V 1 = 0, when T = T 0 and V = 0. Let Σ 0 be the largest invariant set in set We have from the third equation of (1.2) that Σ 0 = {E f }. By the Lyapunov-LaSalle invariance principle (LaSalle, 1976), E f is global asymptotically stable.
(ii) Let us consider the Lyapunov function  EJQTDE, 2012 No. 9, p. 7 From the AM − GM inequality, we get 3.2. Uniformly persistent. We recall that the system (1.2) is said to be uniformly persistent if there is ε 0 such that any solution of (1.2) which starts with T (0), Theorem 3.2. If R 0 > 1, then the system (1.2) is uniformly persistent.

Discussion.
This paper presents a mathematical study on the global dynamics of virus dynamic model with Crowley-Martin functional response. At first, we discussed the existence and local stability of the infection-free and infection equilbria. The results (see, Theorem 2.2.) showed that the basic reproduction number of virus R 0 is a sharp threshold parameter. Next, the global asymptotical stability for infection-free equilibrium E f when R 0 < 1 and infection equilibrium E * when R 0 > 1 are proved (see, Theorem 3.1.). The proof relies on the construction of a global Lyapunov function that are motivated by earlier works (Korobeinikov, 2004;Huang et al., 2009) Now, we discuss the biological significance of our results. Theorem 3.1 (i) implies that a person with R 0 < 1 cannot be infected by virus forever. Theorem 3.1 (ii) implies that a person with R 0 > 1 will be very difficult to prevent to be infected. Consequently, HBV vaccines may be the first line choice for preventing HBV infection. It also implies that if a patient's R 0 > 1, and drug anti-virus therapy cannot activate the patient's immune response, then the anti-virus treatment cannot stop until all virus has been cleared. EJQTDE, 2012 No. 9, p. 8 In another approach, Zhou and Cui (2010) studied the model (1.2), and obtained global asymptotic stability of the infection equilibrium E * result by the theory of competitive systems. However, their results (see, Theorem 5.2 of Zhou and Cui (2010)) need more hypotheses than our results here (Theorem 3.1).
Furthermore, our considerations may be easily extended to systems of the form 1+aT +bV , n(T ) = λ − dT . On the other hand, we can consider with "intracellular" delay (see, Herz et al., 1996)   And, we also consider the system (1.2) with CTL immune response (Nowak and Bangham, 1996):

Acknowledgment
This work was partially supported by the Longdong University Grant XYZK-1007 and XYZK-1010. The author wish to thank the editor and the referees for their useful feedback and comments that greatly improved this paper.