Global Stability for a Viral Infection Model with Saturated Incidence Rate

and Applied Analysis 3 Multiplying e in both sides of the above equation and integrating it from 0 to t, we have


Introduction
In recent years, study of infectious disease model has been a hot issue; the main cause of infectious disease is the virus invasion.As we know, viral cytopathicity within target cells is very common.A number of mathematical models have been used to study virus dynamics.In 1996, Nowak et al. [1] designed a simple but natural mathematical model based on ordinary differential equation.The model is as follows: where () denotes the number of uninfected cells, () the numbers of infected cells, and V() the numbers of free viral particles at time , respectively.In model (1), uninfected target cells are assumed to be produced at a constant rate  and died at rate .Infection of target cells by in-host free viruses is assumed to occur at a bilinear rate V; infected cells are lost at a rate .Free viruses are produced by infected cells at a rate , in which  is the average number of viral particles produced over the lifetime of a single infected cell.Free viral particles die at a rate V.For model (1), Korobeinikov [2] established the condition of global stability in 2004.Some other viral dynamical models were proposed by later researchers; see for example [3][4][5][6][7][8].
In [7], Wodarz and Levy pointed out that the term () in model (1) should consist of two parts: one is the natural death of infected cells, the other is viral cytopathicity.In 2012, Li et al. [4] assumed that infected cells burst and then release viral particles (i.e., viral cytopathicity occurs) after uninfected cells were infected by a constant period of time ; that is, the time period of viral cytopathicity within target cells is .They incorporated the delay of viral cytopathicity within target cells and built a new model: ( By constructing Lyapunov functionals, necessary and sufficient conditions were obtained ensuring the global stability of the model. In models (1) and ( 2), the researcher studied the viral dynamics with bilinear incidence rate V.As we know, as the viral particles diffuse in the body, the person often takes some actions when V gets large.In order to describe the inhibitory effect from the uninfected cells when the number of viral cytopathicity is large enough, following the idea of [9], we propose an incidence rate V/(1 + V), where V measures the infection force of the viral, and  reflects the level of inhibitory action.
Similar to the discussions in [4], we assume that the viral cytopathicity has time delay.When the delay of viral cytopathicity within target cells is  and the natural death rate of per target cell is , the number of infected cells at time  ( > ) can be represented by where  −(−) is the probability that target cells survive from time  to time , and (()V()/(1 + V())) −(−) is the number of target cells being infected at time  and still surviving at time .
Differentiating () of (3), we get where  − (( − )V( − )/(1 + V( − ))) is the transfer rate of the infected cells being used to produce free viruses at time ; the recruitment rate of free virus at time  is  − (( − )V( − )/(1 + V( − ))), in which  is the average number of viral particles produced by an infected target cell when viral cytopathicity occurs, which implies that the recruitment of virus at time  depends on the number of target cells that were newly infected at time  −  and still alive at time .Therefore following the model (2), we obtain a basic viral dynamical model of delay differential equations: Since the variable  does not appear in the first and the third equations of (5), we only focus on the following equations: which has the same dynamics with system (5).
Let  =  − , by (6), we have where all the parameters are assumed to be positive.The rest of this paper is organized as follows.In the next section we will derive the infection-free equilibrium and the infection equilibrium.In Section 3, we carry out a qualitative analysis of the model, and stability conditions for the infection-free equilibrium and the infection equilibrium are derived, respectively.A brief conclusion will be given in Section 4.
Integrating both sides of inequality above from 0 to , we have It means that   (−) + V  (0) ≤ / for any  ≥ 0 as long as From the first equation of ( 8), we have Similar discussion shows that Then   (0) ≤ / for any  ≥ 0 as long as  1 (0) ≤ /.Moreover, lim sup Thus, the region is an invariant set and an attractor of system (8) with initial condition In what follows, we study the existence of equilibria.We consider algebraic equations It is easy to see that system (24) always has an infectionfree equilibrium  1 (/, 0).To find the other equilibrium, we assume V ̸ = 0.By the second equation of (24), we have Then We put  into the first equation of (24); then Thus the positive root  2 exists if and only if  −  > 0.
For system (8), define the basic reproduction number [10] as follows It is easy to see that (i) If  0 ≤ 1, then system (8) has a unique equilibrium  1 (/, 0), which corresponds to the case that viruses die out, and it is called infection-free equilibrium.

Stability of the Equilibrium
In this section, we consider the stability of the equilibrium.There are two cases,  = 0 and  > 0.

Local Stability of Equilibria.
First we consider the case of  = 0.In this case system ( 8) is reduced to a system of ordinary differential equations.In order to examine local stability of an equilibrium, we should compute the eigenvalues of the linearized operator for system (8) at the equilibrium.By a direct computation, the Jacobian matrix is as follows: Consider infection-free equilibrium  1 (/, 0).The characteristic equation is obtained by the standard method as follows.
Proof.Set (, V) = V/(1 + V).Then system (8) at the equilibrium  2 ( * , V * ) has Jacobian matrix A direct computation shows that the characteristic equation is By Hurwitz criterion, all of the eigenvalues of characteristic equation have negative real parts if and only if Indeed, This implies that all the eigenvalues of characteristic equation have negative real parts.Then the infection equilibrium  2 is locally asymptotically stable.This completes the proof of theorem.
Now we consider the case  > 0. By linearizing system (8) at the infection-free equilibrium  1 (/, 0), we obtain the characteristic equation as follows: It is easy to see that  1 = − < 0; hence we only need to discuss the roots of the following equation: Theorem 5. When  > 0, then (i) If  0 < 1, then the infection-free equilibrium  1 (/, 0) is locally asymptotically stable.
Proof.(i) By implicit function theorem for complex variables, we know that the roots of (35) are continuous on the parameter .
If  0 < 1, then 0 is not a root of (35) for all  > 0. Note that all complex roots of (35) must come in conjugate pairs and the root of (35) is negative for  = 0. Thus, all roots of (35) have negative real parts for small ; that is, 0 <  = 1.Suppose that there exists a positive number  =  0 such that (35) has a pair of purely imaginary roots  = ±; here  is a positive number.We have Then Summing up the square of both equations in (37) we obtain When  0 < 1, then  2 < 0. It is a contradiction with  2 > 0 which leads to the nonexistence of  0 .This contradiction proves the result.
(ii) When  = 0, and  0 > 1, then Therefore equation must have a positive real root for all  > 0.
Proof.By implicit function theorem for complex variables, we know that the root of ( 40) is continuous on the parameter .If  0 > 1, then all roots of (40) have negative real parts as  = 0 and (40) has no zero root for all  > 0. Thus, all roots of (40) have negative real parts for very small ; that is, 0 <  ≪ 1. Assume that there exists a positive  0 such that (40) has a pair of purely imaginary roots ±,  > 0. Then  > 0 must satisfy Separating the real and imaginary parts, we have which implies that Direct computation shows that Let ( By Hurwitz criterion, (44) has no positive roots, which implies the nonexistence of  0 .Thus all roots of (40) have negative real parts for  > 0.

Global Stability of Equilibria.
In the section, we study the global stability of equilibria; we first consider the infectionfree equilibrium  1 .

Proof. (i) Define a Lyapunov function as what follows
It means that (, V)/| (8) is negative semidefinite as  0 ≤ 1.Moreover, the last equality of the above equation shows that the largest invariant set of system (8) on the region {(, V)  ∈ R 2 + : / = 0} is the singleton { 1 }.Therefore, the infection-free equilibrium  1 is global asymptotically stability.
(ii) We rewrite the system ( 8) Choose a Dulac function We have Thus system (49) does not have nontrivial periodic orbits in Ω.The conclusion follows.

Conclusion
The viral infection model addressed in this paper has saturated incidence rate and viral infection with delay.The basic reproductive number  0 is given.When  0 < 1, for the model with or without delay time, the infection-free equilibrium is globally asymptotically stable, which implies that the viral infection goes extinct eventually; when  0 > 1, the infection equilibrium is globally asymptotically stable, which implies that the viral infection persists in the body of the host.