GLOBAL STABILITY OF A VIRAL INFECTION MODEL WITH TWO DELAYS AND TWO TYPES OF TARGET CELLS∗

In this paper, incorporating the delay of viral cytopathicity within target cells, we first presented a basic model of viral infection with delay, and then extended it into a model with two delays and two types of target cells. For the models proposed here, both their basic reproduction numbers are found. By constructing Lyapunov functionals, necessary and sufficient conditions ensuring the global stability of the models with delays are given. The obtained results show that, when the basic reproduction number is not greater than one, the infection-free equilibrium is globally stable in the feasible region, which implies that the viral infection goes extinct eventually; when it is greater than one, the infection equilibrium is globally stable in the feasible region, which implies that the viral infection persists in the body of host.


Introduction
In order to understand the action of in-host free virus on target cells, a number of mathematical models have been used to describe in-host virus dynamics.Nowak et al. [6,7] proposed one of the earliest of these models where x = x(t), y = y(t) and v = v(t) are the numbers of uninfected cells, infected cells and viral particles (virions) at time t, respectively.In model (1.1), uninfected target cells are assumed to be produced at a constant rate λ and die at a rate dx.Infection of target cells by in-host free virus is assumed to occur at a bilinear rate βxv.Infected cells are lost at a rate ay.Free virus are produced by infected cells at a rate kay in which k is the average number of viral particles produced over the lifetime of a single infected cell, and die at a rate γv.Model (1.1) is a basic model, which has been used widely to investigate infection of some viruses (such as, HIV, HBV, HCV, HLMV, etc.) In 1997, Perelson et al. [9] observed that the HIV attack two types of target cells, CD4 + T cells and macrophages.On the other hand, it was also detected that, except for liver tissue, HCV may be produced in some extrahepatic tissues, such as bone marrow [12], peripheral blood mononuclear cells (PBMC) [4], brain [5] and lymph nodes [8].Then, according to these virological findings, based on model (1.1) some viral dynamical models with two types of target cells were proposed [1,10,11], which are expressed by ordinary differential equations.
In [14], Wodarz and Levy pointed out that the term ay in model (1.1) should consist of two parts: one represents the natural death of infected cells, the other is that infected cells are lost due to viral cytopathicity.In this paper, we assume 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 τ , that is, the time period of viral cytopathicity within target cells is τ .So the objective of our work is to investigate the basic virus dynamical model with delay of viral cytopathicity within target cells and further consider the model with two types of target cells and the associated delays.
The global analysis of viral infection models is an important issue for understanding the pathogenesis of in-host free virus.Usually, it is difficult for models of delay differential equations to obtain the global properties.For the models with delays established in this paper, the necessary and sufficient conditions ensuring their global stability are obtained by constructing the Lyapunov functionals.
The paper is organized as follows: In Section 2, we first present a basic model of viral infection with delay of viral cytopathicity, and then extend it into a model with two delays and two types of target cells.In Sections 3 and 4, the global properties of the two models established here are analyzed.The last section is the conclusion.

Models
In this section, we present two models of viral infection with delay of viral cytopathicity, in which the infected target cells are the same type and two types, respectively.
When the delay of viral cytopathicity within target cells is τ , and the natural death rate of per target cell is d, the number of infected cells at time t (t > τ ) can be represented by where e −d(t−θ) is the probability that target cells survive from time θ to time t, and βx(θ)v(θ)e −d(t−θ) is the number of target cells being infected at time θ and still surviving at time t.Differentiating y(t) of (2.1) yields where the term βe (2.2) Since the variable y does not appear in the first and the third equations of (2.2), we only focus on the following subsystem of (2.2) which has the same dynamics with system (2.2).
In [2], Elaiw studied the global properties of a viral dynamical model with two types of target cells (CD4+ T cells and macrophages) where x i and y i (i = 1, 2) are the numbers of uninfected and infected cells for type i, respectively, and all the parameters in (2.4) have the same biological meanings as given in model (1.1).According to the idea of establishing delay differential equations (2.3), corresponding to model (2.4) we can give the following model with two delays and two types of target cells where τ i (i = 1, 2) is the delay of viral cytopathicity within target cells of type i.
respectively.In the following, we will investigate global dynamics of systems (2.6) and (2.7).For models (2.6) and (2.7) or (2.3) and (2.5), all parameters are assumed to be positive.

Analysis for system (2.6)
To investigate the dynamics of (2.6), we set a suitable phase space.Denote by From the biological meaning, the initial conditions for system (2.6) are given as follows: where The following theorem establishes the non-negativity and boundedness of solutions of (2.6).
Proof.Assume that there is t 1 (t 1 > 0) such that x(t 1 ) = 0, then it follows from x(0) > 0 and the continuity of solution of (2.6) that there is From the last equation of (2.6) we have where For a positive integer k, when kτ ≤ t < (k + 1)τ , from (3.2) we have Assume that v(t) > 0 for 0 ≤ t < kτ , then the similar inference can show that v(t) > 0 for kτ ≤ t < (k +1)τ .It follows from mathematical induction that v(t) > 0 for t > 0.
The positivity of solution of (2.6) is proved completely.To prove the ultimate boundedness of solution of (2.6), we define a functional L 10 = bx(t) + v(t + τ ), then the derivative of L 10 along solutions of (2.6) } is positively invariant with respect to system (2.6).We will analyze the dynamics of system (2.6) on the region Ω 1 .
Denote R 01 = (βbλ)/(dγ), then direct calculation shows that system (2.6) always has the infection-free equilibrium E 01 (λ/d, 0), and that, besides E 01 , system (2.6) also has a unique infection equilibrium With respect to the global stability of system (2.6), we have Theorem 3.2.For system (2.6), the infection-free equilibrium E 01 is globally stable on the region Ω 1 as R 01 ≤ 1; the infection equilibrium E * 1 is globally stable in the region Ω 1 as R 01 > 1.
To simplify the proof of the global stability of the infection equilibrium E * 1 , we first introduce an inequality as lemma.

and the equality holds if and only
It is easy to see that function f (x) = 1 − x + ln x ≤ 0 for x > 0 and the equality holds if and only if x = 1.Thus Lemma 3.1 holds.Proof of Theorem 3.2.To prove the global stability of the infection-free equilibrium E 01 of (2.6), we define a Lyapunov functional then the derivative of L 11 along solutions of (2.6) is given by When R 01 ≤ 1, dL 11 /dt ≤ 0. And it is easy to see that, when R 01 ≤ 1, the largest invariant set of system (2.6) on the region { (x(t), v(t)) T ∈ Ω 1 : dL 11 /dt = 0} is the singleton {E 01 }.Then it follows by the LaSalle's Invariance Principle [3] that the infection-free equilibrium E 01 is globally stable on the region Ω 1 .
To prove the global stability of the infection equilibrium E * 1 , define a Lyapunov functional where m, r and p are positive and left unspecified, then the derivative of L 12 along solutions of system (2.6) is given by
By the relationship between the arithmetical and geometrical means and Lemma 3.1, we have dL 12 /dt ≤ 0, and the equality holds if and only if Obviously, the largest invariant set of system (2.6) on the region Therefore, it follows by the LaSalle's Invariance Principle [3] that E * 1 is globally stable in Ω 1 when R 01 > 1.

Analysis for system (2.7)
To investigate the dynamics of (2.7), we set a suitable phase space.For From the biological meanings, the initial conditions for system (2.7) are given as follows: where ϕ i ∈ C + and ϕ i (0) > 0 for i = 1, 2, 3. Applying the mathematical induction similar to the proof of Theorem 3.1, we can know that, for any solution (x 1 (t), x 2 (t), v(t)) T of system (2.7) under the initial conditions (4.1), x 1 (t), x 2 (t) and v(t) are positive for t > 0, The proof is omitted here.
We initially consider the ultimate boundedness of solutions of system (2.7).

Proof. Define a functional
, then, the derivative of L 20 along solutions of (2.7) is given by where ρ = min {d 1 , d 2 , γ}.Thus, It implies that all solutions of (2.7) are ultimately bounded.The proof is complete.
Therefore, the region is positively invariant with respect to model (2.7).We will analyze the dynamics of model (2.7) in the region Ω 2 .
Obviously, (2.7) always has the infection-free equilibrium E 02 (x 10 , x 20 , 0), where From the first two equations of (4.2) we have , and Substituting them into the third equation of (4.2) yields Since the function of v at the left hand side of (4.3) is strictly decreasing, it is easy to see that (4.3) has a positive root if and only if and that the positive root is unique, denoted by v * .Therefore, with respect to the existence of equilibria of (2.7), we have Then, when R 02 ≤ 1, system (2.7)only has the infection-free equilibrium E 02 ; when R 02 > 1, besides E 02 system (2.7) also has a unique infection equilibrium and v * is determined by (4.3).
In the following, we consider the global stability of equilibria of (2.7).Theorem 4.3.When R 02 ≤ 1, the infection-free equilibrium E 02 of system (2.7) is globally stable on Ω 2 ; when R 02 > 1, the infection equilibrium E * 2 of (2.7) is globally stable in the region Ω 2 .
Proof.We first prove the global stability of the infection-free equilibrium E 02 .
Since x 10 = λ 1 /d 1 and x 20 = λ 2 /d 2 , system (2.7) can be rewritten as Define a Lyapunov functional Direct calculation shows that the derivative of L11 along solutions of (4.4) is given by the derivative of L 21 along solutions of (4.4) is According to the property that the arithmetical mean is greater than or equal to the geometrical mean, it follows from R 02 ≤ 1 that dL 21 /dt ≤ 0. When R 02 < 1, dL 21 /dt = 0 if and only if x 1 = x 10 , x 2 = x 20 and v = 0. Thus E 02 is globally stable in Ω 2 by the Lyapunov Stability Theorem [13].When R 02 = 1, dL 21 /dt = 0 if and only if x 1 = x 10 and x 2 = x 20 .From the first two equations of (2.7) the largest invariant set of system (2.7) on the set {(x 1 , x 2 , v) ∈ Ω 2 : dL 21 /dt = 0} is the singleton {E 02 }.Then, it follows by the LaSalle's Invariance Principle [3] that E 02 is globally stable on Ω 2 when R 02 = 1.
Summarizing the inference above, the infection-free equilibrium E 02 is globally stable on Ω 2 when R 02 ≤ 1.The proof is complete.
Next, we prove the global stability of the infection equilibrium E * 2 .For the infection equilibrium then (2.7) can be rewritten as ]} , (4.5) which has the same dynamics as system (2.7) in the interior of the region Ω 2 .
Define a Lyapunov functional where m i , r i and p i (i = 1, 2) are positive and left unspecified, then the derivative of L 22 along solutions of (4.5) is given by } .
In order to eliminate the terms x i (t)v(t) and . Thus, we have ] .
Notice that L21 can be reexpressed by ] .
Similarly, when ] .This completes the proof of Theorem 4.3.

Conclusion
In this paper, assuming that the time period of viral cytopathicity within target cells is a constant number, we incorporated a constant delay into the basic viral dynamical model proposed in [6,7], established a basic viral dynamical model (2.3) with viral cytopathicity delay, and then extended model (2.3) into the case with two types of target cells.The modeling idea may be applied into the case with n(n ≥ 2) types of target cells.
For the two viral infection models with delay proposed here, we found their thresholds determining their dynamics, respectively.By the definition of the basic reproduction number of viral infection, we can know that the obtained thresholds are the basic reproduction numbers of the associated viral infection models, respectively.By constructing Lyapunov functionals, we obtained the main results on the two models: when the basic reproduction number is not greater than one, the infectionfree equilibrium is globally stable in the feasible region, which implies that the viral infection goes extinct eventually; when it is greater than one, the infection equilibrium is globally stable in the feasible region, which implies that the viral infection persists in the body of host.Mathematically, the method of constructing Lyapunov functions here is suitable for some delay differential equations of higher order (i.e., the system with n(n ≥ 2) types of target cells), and the introduction of Lemma 3.1 may simplify the proof of the global stability.