Global stability of a delayed and diffusive virus model with nonlinear infection function

This paper studies a delayed viral infection model with diffusion and a general incidence rate. A discrete-time model was derived by applying nonstandard finite difference scheme. The positivity and boundedness of solutions are presented. We established the global stability of equilibria in terms of by applying Lyapunov method. The results showed that if is less than 1, then the infection-free equilibrium is globally asymptotically stable. If is greater than 1, then the infection equilibrium is globally asymptotically stable. Numerical experiments are carried out to illustrate the theoretical results.


Introduction
Mathematical models that describe within-host dynamics have been proposed and studied by constructing corresponding differential equations to get a better understanding of viral processes, particularly, their global dynamics behaviour has been investigated [1,2,14,17,21,28,29,32,[35][36][37]. For example, Wang and Zhou [32] studied the following model: where T, I, C and V represent the concentrations of the uninfected cell, shorted-lived infected cells, chronically infected cells and free virus particles, respectively. s is the source term for uninfected cells. β represents the infection rate. is the efficacy of the therapy. d, δ, μ and c are the mortality rates of uninfected cells, short infected cells, chronically infected cells and virus, respectively. The fractions α and (1 − α) are the probabilities that, upon infection, an uninfected cell will become either chronically infected or short-lived infected. k 1 = N 1 (1 − γ 1 ) and k 2 = N 2 (1 − γ 2 ) where N 1 and N 2 are the average numbers of virions produced in the lifetime of short-lived and chronically infected cells, respectively. γ 1 and γ 2 are the efficacy of the therapy. τ 1 , τ 2 , τ 3 and τ 4 are the intracellular delays. The global dynamics of the model have been studied by constructing Lyapunov functionals. For more details, one can refer to [32]. The key assumption in model (1) is that cells and viruses are well mixed, and the mobility of viruses was ignored. So, in order to study the influences of spatial structures of virus dynamics, Wang et al. [30] studied the following model: where T(x, t), I(x, t) and V(x, t) represent the densities of uninfected cells, infected cells and free virus at position x and at time t, respectively. τ is the intracellular delay. D is the diffusion coefficient and is the Laplacian operator. The bilinear incidence rate βTV used in models (1) and (2) is a simple description of the infection. Though the incidence rates βT q V, βTV 1+aV and βTV 1+aT+bV+abTV are improved forms which are more commonly used [22,26,31], the general incidence rates f (T, V)V, f (T, V) and f (T, I, V)V can help us gain the unification theory by the omission of unessential details [7,8,11,13]. So, motivated by [30,32], we consider the following model with a general incidence rate which is similar to the one in [7] ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (3) f (T, V)V is the general incidence rate and satisfies the following hypotheses: It is easy to check that a class of functions f (T, V)V satisfying (4) includes some common used nonlinear incidence functions such as f (T, The initial conditions of model (3) are given as and we have considered model (3) with homogeneous Neumann boundary conditions where τ = max{τ 1 , τ 2 , τ 3 , τ 4 }, is a bounded interval in R and ∂ ∂ − → n denotes the outward normal derivative on ∂ .
As we know, it is difficult or even impossible to find the exact analytical solutions for most nonlinear models, such as (3). In order to perform numerical simulations, we need to seek an efficient discrete method to discretize such nonlinear continuous models. However, some classical numerical discrete methods are unsuccessful in preserving the quantitative properties of corresponding continuous model. For example, for classical forward Euler's method, if the step size is selected small enough and the positivity conditions can be satisfied, then it can be shown that local asymptotic stability for an equilibrium is preserved while in some special cases numerical instability or Hopf bifurcation may appear. Thus how to design a feasible discrete scheme so that the same quantitative behaviours of solutions to the corresponding continuous models can be efficiently preserved is a challenging and interesting task. Recently, an interesting method which is called non-standard finite difference (NSFD) scheme has been proposed by Mickens [18,19]. NSFD has been applied to obtain discrete-time epidemic models [4,6,9,10,20,24,25,34] and references therein. Hence, motivated by the work of [18,19], we apply the NSFD scheme on model (3), then we obtain The initial conditions of model (7) are and the discrete boundary conditions are The main purpose of this paper is to demonstrate the discretized model (7) derived by applying NSFD scheme can efficiently preserves the global dynamical properties to the original model (3). The rest of this paper is organized as follows. In Section 2, we study the dynamical behaviour of the continuous model (3). In Section 3, we investigate the global dynamics of discrete model (7). An example, along with numerical simulations is presented in Section 4 to validate the theoretical results. A brief conclusion ends the paper.

Preliminaries
Let X = C(¯ , R 4 ) be the space of continuous functions from the topological space¯ into the space R 4 . Denote C = C([−τ , 0], X) be the Banach space of continuous functions from [−τ , 0] into X with the usual supremum normal, and C + = C([−τ , 0], X + ). When convenient, we identify an element φ ∈ C as a function from¯ × [−τ , 0] into R 4 defined by φ(x, s) = φ(s)(x). We adopt the notation that for σ > 0, a function u(·) : T . It follows from [3] that the X-realization of D generates an analytic semi-group T (t) on X.

Proof: For any
We now reformulate (3)-(6) as the abstract functional differential equation where = (T, I, C, V) T and A = (0, 0, 0, D V) T . It is clear that F is locally Lipschitz in X. It follows from [5,15,16,27,33] that system (9) admits a unique local solution on t ∈ [0, T max ), where T max is the maximal existence time for the solution of system (9). In order to demonstrate the boundedness of solutions. Define U(x, t) = T(x, t) + I(x, t + τ 1 ) + C(x, t + τ 2 ) and d 0 = min{d, δ, μ}, it then follows from model (3) that Then we have LetṼ(t) be a solution to the ordinary differential equation follows from the comparison principle [23]. Therefore, Based on the above analysis, we have demonstrated that T(x, t), . It then follows from the standard theory for semilinear parabolic systems [12] that T max = +∞.
Thus we can demonstrate that for small enough ρ, We are now prepared to refer to key results from the literature.
is analytic [3]. Thus it follows from [33] that the mild solution is classic for t ≥ τ . This completes the proof.
The dynamical outcomes of model (3) will be determined by the basic reproduction number R 0 , which is given by It is clear that model (3) always has an infection-free equilibrium E 0 = (T 0 , 0, 0, 0) and any positive equilibrium, denoted by E * = (T * , Simple calculation shows that and T * satisfies It is easy to show that According to (4), calculating shows H (T) > 0. Thus there exists a unique (3) is globally asymptotically stable.

Stabilities of equilibria
Proof: Define a Lyapunov functional For convenience, we let u = u(x, t) and Recall that V dx = 0 and (4), we obtain Recall the condition (4), it is easy to show that dL 1 dt ≤ 0 whenever R 0 ≤ 1. Moreover, it can be shown that the largest invariant set { dL 1 dt = 0} is the singleton {E 0 }. By LaSalle's Invariance Principle, the infection-free equilibrium E 0 of model (3) is globally asymptotically stable when R 0 ≤ 1. This completes the proof.

Theorem 2.4:
If R 0 > 1, then the infection equilibrium E * is globally asymptotically stable.
Proof: Constructing a Lyapunov functional L 2 as follows: where h(x) = x − 1 − ln x ≥ 0 for all x > 0 and with a global minimum h(1) = 0. In the calculation that follows we will use the equilibrium equations and note that V(x, t) dx = 0, and Then, calculating the time derivative of L 2 along a solution of model (3), we obtain Recall the conditions (4), we then obtain that dL 2 dt ≤ 0 whenever R 0 > 1. Moreover, it can be shown that the largest invariant set { dL 2 dt = 0} is the singleton {E * }. By LaSalle's Invariance Principle, the infection equilibrium E * of model (3) is globally asymptotically stable when R 0 > 1. This completes the proof.

Preliminary results
In this section, we dedicate to the investigation of the discrete model (7). It is easy to see that the discrete model (7) has the same equilibria as model (3): the infection-free equilibrium E 0 = (T 0 , 0, 0, 0) and the infection equilibrium E * = (T * , I * , C * , V * ). In the following, we first show that the solution of model (7) is non-negative and bounded. To this end, rewriting the discrete model (7) yields where matrix A of dimension (N + 1) × (N + 1) is given by It is easy to show that A is a strictly diagonally dominant matrix. Hence, A is non-singular.
We then obtain that Proof: We can claim that T n > 0 for all n ∈ N. In fact, assuming the contrary and letting n 1 > 0 be the first time such that T n 1 ≤ 0 and T n > 0, I n > 0, C n > 0, V n > 0 for all n < n 1 . Since, Note that the conditions of (4), we then obtain T m n 1 −1 ≤ 0, which contradicts our assumption and so T n > 0 for all n ∈ N. Moreover, it is easy to prove that the sequences {I n }, {C n } and {V n } are non-negative by using mathematical induction.
Next, we establish the boundedness of solutions. To this end, we define a sequence {G n } as follows: By using induction, we easily obtain implying {V n } is bounded. This completes the proof.
Proof: Consider the following discrete Lyapunov functional: Then we have The last inequality is followed by the condition (4). Thus if R 0 ≤ 1, then we have G n+1 − G n ≤ 0, for all n ∈ N, which implies that G n is a monotone decreasing sequence. Since G n ≥ 0, there is a limit lim n→∞ G n ≥ 0 which implies that lim n→∞ (G n+1 − G n ) = 0, from which we get lim n→∞ T m n = T 0 and lim n→∞ V m n (R 0 − 1) = 0. We discuss two cases: (i) if R 0 < 1, from model (7), we obtain lim n→∞ I m n = 0, lim n→∞ C m n = 0, for all m ∈ {0, 1, . . . , N}; (ii) if R 0 = 1, by lim n→∞ T m n = T 0 and from model (7), we have lim n→∞ I m n = 0, lim n→∞ C m n = 0, lim n→∞ V m n = 0. Thus concluding the above discussion implies that E 0 is globally asymptotically stable. This completes the proof.

Conclusion
In this paper, we have formulated a delayed and diffusive viral infection model incorporating shorted-lived and chronically infected cells and general nonlinear incidence function.
Then, by applying NSFD scheme, we presented an efficient numerical method for the corresponding continuous model. Theoretically, we have shown that the stability conditions for the equilibria are identical in case of both the continuous and discrete models. Specifically, if R 0 ≤ 1, then the infection-free equilibrium E 0 is globally asymptotically stable; if R 0 > 1, then the infection equilibrium E * is globally asymptotically stable. The results show that the NSFD scheme has the advantage that the positivity, boundedness and global properties of solutions for original continuous model are efficiently preserved.
As far as we know, there are few delayed and diffusive virus models considering both the shorted-lived and chronically infected cells, and no theoretical analysis has been made on this kind of models. Here, our main contribution is to construct suitable Lyapunov functional for both the continuous-time and discrete-time virus models, and present a general method to analyse this kind of models. Based on the above obtained results, one can extend this method to more complicated models.