Global Stability of a Virus Infection Model with Time Delay and Absorption

In this paper, a virus infection model with time delay and absorption is studied. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the model is established. By using comparison arguments, it is shown that the infection free equilibrium is globally asymptotically stable when the basic reproduction ratio is less than unity. When the basic reproduction ratio is greater than unity, sufficient conditions are derived for the global stability of the virus-infected equilibrium. Numerical simulations are carried out to illustrate the theoretical results.


Introduction
Mathematical modelling has been proven to be valuable in studying the dynamics of virus infection, such as HIV/AIDS, HBV, and HCV.In recent years, great attention has been paid by many researchers to the pathogen infectious agent, or germ, which can cause disease or illness to its host see, e.g., 1-3 .Based on the clinical experiment of chronic HBV carriers treated with various doses of lamivudine, Nowak and Bangham 4 and Bonhoeffer et al. 5 proposed a classical mathematical model describing the interaction between the susceptible host cells x , infected host cells y , and free virus particles v , which is formulated by the following differential equations: where hepatocytes are produced at a rate λ, die at rate dx, and become infected at rate βxv; infected hepatocytes are produced at rate βxv and die at rate ay; free viruses are produced from infected cells at rate ky and are removed at rate uv.It is assumed that parameters a, d, k, u, λ, and β are positive constants.The basic reproductive ratio R 0 was found in 4 .Furthermore, in 6 , by constructing suitable Lyapunov function, Korobeinikov showed that the infection-free equilibrium is globally asymptotically stable if R 0 < 1, and the virusinfected equilibrium is globally asymptotically stable if R 0 > 1.
It is assumed in model 1.1 that the infection process follows the principle of mass action 7 , namely, for each uninfected cell and free virus particle is assumed to be constant β between the rate of infection.However, studies of parasitic infections have shown that the relationship between the dose and rate of infection is clearly nonlinear.In 8 , a more general saturated infection rate βxv p / 1 αv q was suggested, where p, q, and α are positive constants.
The virus life cycle plays a crucial role in disease progression.The binding of a viral particle to a receptor on a target cell initiates a cascade of events that can ultimately lead to the target cell becoming productively infected, that is, producing new virus.In model 1.1 , it is assumed that as soon as virus contacts a target cell, the cell begins producing virus.This is not biologically sensible.In reality, there is a time delay between initial viral entry into a cell and subsequent viral production 9 .There have been some works on virus infection model in the literature see, e.g., 9-14 .In 10 , Herz et al. examined the effect of including a constant delay in the source term for productively infected T-cells.In 11 , Li and Ma considered the following more general HIV-1 infection model with saturation incidence rate and time delay:

1.2
By analyzing the transcendental characteristic equations, sufficient conditions for the local asymptotic stability of the equilibria were studied, and by using Lyapunov-LaSalle invariance principal, the global asymptotic stability of the viral-free equilibrium was given.We note that in 1.1 and 1.2 , the loss of pathogens due to the absorption into uninfected cells are ignored.In reality, when a pathogen enters an uninfected cell, the number of pathogens in the blood decreases by one.This is called the absorption effect 7 .In 15 , considering the erythrocytic cycle in the absence of an immunological response by the host,

1.4
The initial conditions for system 1.4 take the form where It is easy to show that all solutions of system 1.4 with initial condition 1.5 is defined on 0, ∞ and remain positive for all t ≥ 0.
The organization of this paper is as follows.In the next section, we introduce some notations and state several lemmas which will be essential to our proofs.In Section 3, by analyzing the corresponding characteristic equations, the local stability of each of nonnegative equilibria of system 1.4 is discussed.In Section 4, by using an iteration technique, we study the global stability of the uninfected equilibrium of system 1.4 .By comparison arguments, we discuss the global stability of the virus-infected equilibrium of system 1.4 .Numerical simulations are carried out in Section 5 to illustrate the main theoretical results.The paper ends with conclusions in Section 6.

Preliminaries
In this section, based on the work developed by Xu and Ma 16 , we introduce some notations and state several results which will be useful in next section.
Let R n be the cone of nonnegative vectors in R n .If x, y ∈ R n , we write x ≤ y x < y if x i ≤ y i x i < y i for 1 ≤ i ≤ n.Let {e 1 , e 2 , . . ., e n } denote the standard basis in R n .Suppose, r ≥ 0 and let C C −r, 0 , R n be the Banach space of continuous functions mapping the interval −r, 0 into R n with supremum norm.If φ, ψ ∈ C, we write φ ≤ ψ φ < ψ when the indicated inequality holds at each point of −r, 0 .Let C {φ ∈ C : φ ≥ 0}, and let denote the inclusion R n → C −r, 0 , R n by x → x, x θ x, θ ∈ −r, 0 .Denote the space of functions of bounded variation on −r, 0 by BV −r, 0 .If t 0 ∈ R, A ≥ 0 and x ∈ C −t 0 − r, t 0 A , R n , then for any t ∈ t 0 , t 0 A , we let x t ∈ C be defined by x t θ x t θ , −r ≤ θ ≤ 0. We now consider ẋ t f t, x t .

2.1
We assume throughout this section that f : R × C → R n is continuous, f t, φ is continuously differentiable in φ,f t T, φ f t, φ for all t, φ ∈ R × C , and some T > 0.Then, by 17 , there exists a unique solution of 2.1 through t 0 , φ for t 0 ∈ R, φ ∈ C .This solution will be denoted by x t, t 0 , φ if we consider the solution in R n , or by x t t 0 , φ if we work in the space C. Again, by 17 , x t, t 0 , φ x t t 0 , φ is continuously differentiable in φ.In the following, the notation x t 0 φ will be used as the condition of the initial data of 2.1 , by which we mean that we consider the solution x t of 2.1 which satisfies x t 0 θ φ θ , θ ∈ −r, 0 .
To proceed further, we need the following results.Let r r 1 , r 2 , . . ., r n ∈ R n , |r| max i {r i }, and define We write φ φ 1 , φ 2 , . . ., φ n for a generic point of C r .Let C r {φ ∈ C r : φ ≥ 0}.Due to the ecological applications, we choose C r as the state space of 2.1 in the following discussions.
Fix φ 0 ∈ C r arbitrarily.Then, we set L t, • D φ 0 f t, φ 0 , D φ 0 f t, φ 0 denotes the Frechet derivation of f with respect to φ 0 .It is convenient to have the standard representation of L L 1 , L 2 , . . ., L n as where η ij •, t is continuous from the left in −r j , 0 .We make the following assumptions for 2.1 .
h2 The matrix A t defined by h3 For each j, for which r j > 0, there exists i such that for all t ∈ R and for positive constant ε sufficiently small, η ij −r j ε, t > 0.
The following result was established by Wang et al. 18 .

Lemma 2.1. Let (h1)-(h4) hold. Then, the hypothesis (h0) is valid
i If φ and ψ are distinct elements of C r with φ ≤ ψ and t 0 , t 0 σ with n|r| < σ ≤ ∞ is the intersection of the maximal intervals of existence of x t, t 0 , φ and x t, t 0 , ψ , then This lemma shows that if h1 -h4 hold, then the positivity of solutions of 2.1 follows.
The following definition and results are useful in proving our main result in this section.

Definition 2.2. Let A
a ij n×n be an n × n matrix, and let P 1 , . . ., P n be distinct points of the complex plane.For each nonzero element a ij of A, connect P i to P j with a directed line P i P j .The resulting figure in the complex plane is a directed graph for A. We say that a directed graph is strongly connected if, for each pair of nodes P i , P j with i / j, there is a directed path We now consider the following delay differential system: with initial conditions

2.10
System 2.9 always has a trivial equilibrium A 0 0, 0 .If ka 1 − aa 2 > au, then system 2.9 has a unique positive equilibrium A * u * 1 , u * 2 , where The characteristic equation of system 2.9 at the positive equilibrium A 0 takes the form λ 2 p 1 λ p 0 q 0 e −λτ 0, 2.12 where p 0 a a 2 u , p 1 a a 2 u, q 0 −ka 1 .

2.13
Noting that Kuang 21 , we see that the equilibrium A 0 is locally asymptotically stable for all τ > 0. If ka 1 − aa 2 > au, then A 0 is unstable for all τ > 0.
The characteristic equation of system 2.9 at the positive equilibrium A * is of the form where

2.16
Note that 2.17 Hence, if ka 1 − aa 2 > au, then the positive equilibrium A * is locally stable when τ 0. If ka 1 − aa 2 < au, then A * is unstable when τ 0.
It is easy to see that Discrete Dynamics in Nature and Society 7 If ka 1 − aa 2 > au, then g 2 0 − h 2 0 > 0. By Kuang 21 , we see that the positive equilibrium A * is locally asymptotically stable for all τ > 0. If ka 1 − aa 2 < au, then A * is unstable for all τ > 0.
i If ka 1 − aa 2 > au, then the positive equilibrium A * u * 1 , u * 2 is globally stable.ii If ka 1 − aa 2 < au, then the equilibrium A 0 0, 0 is globally stable.
Proof.We represent the right-hand side of 2.9 by f t, x t f 1 t, x t , f 2 t, x t , and set By direct calculation, we have

2.20
We now claim that the hypotheses h1 -h4 hold for system 2.9 .It is easily seen that h1 and h4 hold for system 2.9 .We need only to verify that h2 and h3 hold.
The matrix A t takes the form

2.21
Clearly, the matrix A t is irreducible for each t ∈ R.
Thus, the conditions of Lemma 2.1 are satisfied.Therefore, the positivity of solutions of system 2.9 follows.It is easy to see that system 2.9 is cooperative.By Lemma 2.3, we see that any solution starting from D C τ converges to some single equilibrium.However, system 2.9 has only two equilibria: A 0 and A * .Note that if ka 1 − aa 2 > au, then the positive equilibrium A * is locally stable, and the equilibrium A 0 is unstable.Hence, any solution starting from D converges to Using a similar argument one can show the global stability of the equilibrium A 0 when ka 1 − aa 2 < au.This completes the proof.

Local Stability
In this section, we discuss the local stability of equilibria of system 1.4 by analyzing the corresponding characteristic equations.System 1.4 always has an infection free equilibrium E 0 λ/d, 0, 0 .Let R 0 is called the basic reproduction ratio of system 1.4 .It is easy to show that if R 0 > 1, system 1.4 has a virus infected equilibrium E * x * , y * , v * , where The characteristic equation of system 1.4 at the infection free equilibrium E 0 is of the form s d s 2 p 1 s p 0 q 0 e −sτ 0, 3.3 where Obviously, 3.3 always has a negative real root s −d.All other roots of 3.3 are determined by s 2 p 1 s p 0 q 0 e −sτ 0.

3.5
It is easy to show that p 1 > 0, p 0 q 0 > 0, then the infection free equilibrium E 0 of system 1.4 is locally asymptotically stable when τ 0. If iω ω > 0 is a solution of 3.3 , by calculating, we have the following: Note that
The characteristic equation of system 1.4 at the virus infected equilibrium E * x * , y * , v * takes the form s 3 g 2 s 2 g 1 s g 0 h 1 s h 0 e −sτ 0, 3.8 where

3.9
When τ 0, 3.8 becomes By direct calculation, we have

3.11
Clearly, all roots of 3.10 have only negative real parts.If iω ω > 0 is a solution of 3.8 , separating real and imaginary parts, it follows that

3.15
Hence, 3.13 has no positive roots.Accordingly, if R 0 > 1, the virus-infected equilibrium E * of system 1.4 exists and is locally asymptotically stable for all τ > 0.
Based on the discussions above, we have the following result.
Theorem 3.1.For system 1.4 , one has the following.
ii If R 0 > 1, the virus infected equilibrium E * x * , y * , v * is locally asymptotically stable.

Global Stability
In this section, we discuss the global stability of the uninfected equilibrium and the virus infected equilibrium of system 1.4 , respectively.The technique of proofs is to use a comparison argument and an iteration scheme 22 .
Theorem 4.1.Let R 0 > 1.The virus infected equilibrium E * x * , y * , v * of system 1.4 is globally asymptotically stable provided that Proof.Let x t , y t , v t be any positive solution of system 1.4 with initial condition 1.5 .Let It follows from the first equation of system 1.4 that ẋ t ≤ λ − dx t .

4.2
By comparison, we derive that Hence, for ε > 0 sufficiently small, there exists a T 1 > 0 such that if t > T 1 , x t ≤ M x 1 ε.We therefore, derive from the second and the third equations of system 1.4 that for t > T 1 τ,

4.4
Consider the following auxiliary equations:

4.6
By comparison, we obtain that

4.7
Since these inequalities are true for arbitrary ε > 0, it follows that where Hence, for ε > 0 sufficiently small, there is a For ε > 0 sufficiently small, we derive from the first equation of system 1.4 that for t > T 2 , Since these inequalities are true for arbitrary ε > 0, it follows that V 1 ≥ N x 1 , where

4.11
Hence, for ε > 0 sufficiently small, there is a For ε > 0 sufficiently small, it follows from the second and the third equations of system 1.4 that for t > T 3 τ,

4.12
Consider the following auxiliary equations:

4.13
Since H1 holds, by Lemma 2.5, it follows from 4.13 that lim

4.14
By comparison, we derive that

4.15
Since these two inequalities hold for arbitrary ε > 0 sufficiently small, we conclude that where Therefore, for ε > 0 sufficiently small, there exists a For ε > 0 sufficiently small, it follows from the first equation of system 1.4 that for t > T 4 , By comparison, we derive that Since this is true for arbitrary ε > 0, it follows that U 1 ≤ M x 2 , where

4.19
Hence, for ε > 0 sufficiently small, there is a T 5 ≥ T 4 such that if t > T 5 , x t ≤ M x 2 ε.It therefore, follows from the second and the third equations of system 1.4 that for t > T 5 τ,

4.20
By Lemma 2.5 and a comparison argument, we derive from 4.20 that

4.21
Since these inequalities are true for arbitrary ε > 0, it follows that where Hence, for ε > 0 sufficiently small, there is a Again, for ε > 0 sufficiently small, we derive from the first equation of system 1.4 that for t > T 6 , Since this is true for arbitrary ε > 0, it follows that V 1 ≥ N x 2 , where

4.25
Hence, for ε > 0 sufficiently small, there is a T 7 ≥ T 6 such that if t > T 7 , x t ≥ N x 2 − ε.For ε > 0 sufficiently small, it follows from the second and the third equations of system 1.4 that for t > T 7 τ,

4.26
Since H1 holds, by Lemma 2.5 and a comparison argument, it follows from 4.26 that

4.27
Since these two inequalities hold for arbitrary ε > 0 sufficiently small, we conclude that

4.29
Clearly, we have

4.30
It is easy to show that the sequences M x n , M

4.31
We, therefore, obtain from

Numerical Examples
In this section, we give two examples to illustrate the main theoretical results above.By calculation, we have R 0 16/9 > 1, and system 1.4 has a virus-infected equilibrium E * 11/5, 7/10, 7/30 .Clearly, H1 and H2 hold.By Theorem 4.1, we see that E * is globally asymptotically stable.Numerical simulation illustrates the above result see Figure 1 .

Conclusions
In this paper, we have discussed a virus infection model with time delay, absorption, and saturation incidence.The basic reproduction ratio R 0 was found.We investigated the global asymptotic stability of the infection-free equilibrium and the virus-infected equilibrium of system 1.4 , respectively.When the basic reproduction ratio is greater than unity, by using the iteration scheme, we have established sufficient conditions for the global stability of the virus-infected equilibrium of system 1.4 .By Theorem 4.1, we see that when R 0 > 1 and H1 and H2 hold, the virus-infected equilibrium is globally stable.Biologically, these indicate that when the death rate of infected cells and the production rate of free viruses from infected cells are sufficiently large, then the solutions of system 1.4 tend to the virus infected equilibrium which means that the virus persists in the host.On the other hand, by Theorem 4.2, we see that if the basic reproduction ratio is less than unity, the infection free equilibrium is globally asymptotically stable.Biologically, if the rate at which new uninfected cells are generated and the average number of effective contacts of one infective individual per unit time are small enough and the death rates of uninfected cells, infected cells and pathogens are large enough such that R 0 < 1, then the virus is cleared.We would like to point out here that Theorem 4.1 has room for improvement, we leave this for future work.

yn
, and M v n are nonincreasing, and the sequences N x n , N y n , N v n are nondecreasing.Hence, the limit of each sequence in M x n ,
Anderson et al. presented the following model for malaria infection: Motivated by the works of Anderson et al. 15 , Li and Ma 11 , and Song and Neumann 8 , in this paper, we study the following virus infection model with a time delay and absorption: where the term −βxv in the third equation of 1.3 represents the absorption effect.
Note that if H1 and H2 hold, by Theorem 3.1, the virus-infected equilibrium E * is locally stable, we conclude that E * is globally stable.The proof is complete.Hence, for ε > 0 sufficiently small satisfying 4.39 , there exists a T 1 > 0 such that if t > T 1 , x t ≤ λ/d ε.We, therefore, derive from the second and the third equations of system 1.4 that for t > T 1 τ,