Dynamics of spatially heterogeneous viral model with time delay

A delayed reaction-diffusion virus model with a general incidence function and spatially dependent parameters is investigated. The basic reproduction number for the model is derived, and the uniform persistence of solutions and global attractively of the equilibria are proved. We also show the global attractivity of the positive equilibria via constructing Lyapunov functional, in case that all the parameters are spatially independent. Numerical simulations are finally conducted to illustrate these analytical results.


1.
Introduction. In recent years, mathematical modeling of in vivo dynamics of viral infections has been employed to understand the mechanisms of disease transmission. For instance, the dynamics of a general in-host model with intracellular delay and spatial homogeneity were studied in [5]; Shu et al. in [15] considered a viral model with nonlinear incidence functions, state dependent removal functions, infinitely distributed intracellular delays, and the cytotoxic T lymphocyte response (CTL). In the framework of mathematical modeling by ordinary differential equations, Perelson and Ribeiro [12] reviewed some developments in HIV modeling, emphasizing quantitative findings about HIV biology uncovered by studying acute infection, the response to drug therapy and the rate of generation of HIV variants that escape immune responses.
When spatial heterogeneity is taken into account in the modelling, many partial differential equations for describing virus dynamics are also proposed. In [4], Lewis et al. studied a spatially-independent model for West Nile virus, proving the existence of traveling waves and calculating the spatial spreading speed of infection. Zhao and Lou in [6] put forward a nonlocal, time-delayed reaction-diffusion malaria model, where some parameters in the model is assumed to be spatially heterogeneous for designing the spatial allocation of resources. In [17], the author considered the dynamics of steady states of an in-host virus dynamics model with spatial heterogeneity on the general bounded domain. Xu and Ma in [22] investigated a hepatitis B virus (HBV) model with spatial diffusion and saturation response of the infection rate, where the intracellular incubation period was also considered by adding a discrete time delay to the model. Another diffusive HBV model with delayed Beddington-DeAngelis response is studied in [27]. The authors proved the global stability of the two steady states for the model, by constructing proper Lyapunov functionals. Other than that, they also showed the existence of traveling wave solutions connecting the these two steady states when R 0 > 1, while there is no traveling wave solution connecting the uninfected steady state to itself when R 0 < 1. For more references, we refer the readers to [1,9,18,23,24,25].
A basic in-host model of viral dynamics was initially proposed in [11]: (1) The model includes three compartments: uninfected target cells (S), infected target cells that produce virus (I), and free virus particles (v). Uninfected target cells are assumed to be produced at a constant rate λ, and die at the rate of dS(t); infection of target cells by free virus is assumed to occur at the rate of βS(t)v(t); the removal rates of infected cells and virus are a and u, respectively, new viruses are produced from infected cells at the rate of kI(t). The basic reproductive number of (1) is R 0 = λkβ dau . It is argued in [11] that, (1) may be adopted to describe in vivo dynamics of HIV-1, HBV, and other viruses. By additionally considering spatial movement of virus, Wang and Wang in [18] introduced the following the viral infection model with spatial heterogeneity: where ∆ is the Laplacian operator and D the diffusion coefficient. There are no diffusion terms in the first two compartments, by assuming that the uninfected and infected cells move extremely slow in the host environment. In [18], the authors proved the existence of traveling waves via the geometric singular perturbation approach. For (2) with periodic boundary conditions on a bounded square domain, when the recruitment parameter λ is spatially-dependent, Brauner et al. in [1] studied the dynamics of the system by analyzing the principle eigenvalue. Later, by assuming that all parameters are spatially dependent except for the diffusion coefficient D, Wang et al. in [17] further extended the results in [1] to (2) on a general bounded domain in a finite (not necessarily two) dimensional space, with zero-flux boundary condition.
Motivated by the work of [17], in the paper we consider Compared with (2), a time delay τ is involved in the system, for describing the period between the entry of a virus into a target cell and the production of a new virus. Due to this time delay, the term e −nτ should be included, representing the probability of surviving rate for infected cells from time t − τ to t. Here, the parameter n is the constant death rate for infected cells that can produce virus. In addition, the parameters λ and k are assumed to be spatially dependent. Again, both uninfected and infected cells are assume to be spatially immobile. For (3), we assign the following initial and boundary conditions and where Ω ⊂ R N is bounded with smooth boundary ∂Ω. Throughout this paper, we assume that k :Ω → (0, ∞) and λ :Ω → [0, ∞) are both continuous, and the rest parameters a, d, D, n, τ and u are positive constants. The initial functions ϕ = (ϕ 1 , ϕ 2 , ϕ 3 ) are also assumed to be uniformly continuous, and f (·, S) ∈ C([0, ∞)) ∩ C 1 ((0, ∞)) satisfies the following properties: f (·, 0) = 0, f (·, S) > 0, f (·, S) ≤ 0 for all S ∈ (0, ∞).
The rest of this paper is organized as follows. In section 2, we first prove some basic results of spatially heterogeneous systems, including existence, uniqueness, positivity, as well as boundedness of solutions. Then, we discusses the compactness and persistence properties of the solution semiflow generated by (3). The basic reproduction number R 0 of (3) is also derived. Section 3 devotes to the global attractivity of the equilibria for (3) without spatial homogeneity (on k and λ). In section 4, some numerical simulations are carried out to illustrate the analytic results.
2. Spatial heterogeneity. In this section, we shall investigate the global attractivety of the disease-free equilibrium and the uniform persistence of solutions for (3).
Then, system (3)-(5) can be rewritten as the integral equation: It is easy to obtain that By [8], we can obtain that there exists a unique noncontinuable solution z(x, t; ϕ) System (3) becomes Since G is quasipositive, the solutions of system (3) remain nonnegative for all t ≥ 0 (see [8, It is easy to see that D is a positively invariant set with respect to model (3). Now, we prove the boundedness of solutions.
whereλ and c are defined in (6). Consider By the comparison principle, it follows that From (12) we know v satisfies the following system wherek = max x∈Ω k(x). Let v 1 (t) be the solution to the following equation Then, we obtain and therefore, S(t), I(t) and v(t) are ultimately bounded in Y + .

2.2.
Compactness. In this part, we shall discuss the compactness property of the semiflow generated by system (3)- (5). Define the Ψ t : where z t (·, θ; ϕ) is the solution of system (3)-(5) with z 0 (·, θ; ϕ) = ϕ ∈ Y + . Since the first two equations in system (3) have no diffusion terms, the semiflow associated with system (3) is not compact. However, we can show the asymptotic compactness of forward orbits, with the aid of the Kuratowski measure κ of noncompactness (see [2]). By the similar arguments as in [26, Lemma 2.1], we have Lemma 2.2. For every ϕ ∈ Y + , the forward orbit γ + (ϕ) := z t (·, θ; ϕ), t ≥ 0 for system (3) is asymptotically compact in the sense that for any sequences t n → ∞, there exists a subsequence t n k such that z tn k (·, θ; ϕ) converges in Y + as k → ∞.
Proof. For a given ϕ = (ϕ 1 , ϕ 2 , ϕ 3 ) ∈ Y + , there exists ϑ > 0 such that In view of the Arezla-Ascoli theorem, it suffices to prove that {z tn (x, θ; ϕ n )} n≥1 is equicontinuous in x ∈Ω for all n ≥ 1. We first show that this is the case for {S tn (x, θ; ϕ n )} n≥1 . It is easy to see that is uniformly continuous in x ∈Ω, uniformly for t ≥ 0, that is, for any given > 0, there exists δ > 0 such that Letting t = 0 and s = −t n in the above inequality, we further obtain that is, This proves the equicontinuity of {S tn (x, θ; ϕ n )} n≥1 . Similarly, we can verify that {I tn (x, θ; ϕ)} n≥1 and {v tn (x, θ; ϕ)} n≥1 are also equicontinuous in x ∈Ω for all n ≥ 1. Consequently, Ψ t is asymptotically compact.

Basic reproduction number and global attractivity.
Obviously, E 0 (x) = (S 0 (x), 0, 0) = λ(x) d , 0, 0 is the disease-free steady state for system (3)-(5). Linearizing the system (3) at E 0 (x), we get the following cooperative system for the infected host cells and free virus Suppose that the initial population distribution for the system is given by ϕ = (ϕ 1 , ϕ 2 , ϕ 3 ) ∈ Y + . As time evolves, the spatial distribution of the infective individuals under the synthetical influences of mortality, mobility, and transfer of individuals among the infected compartments is described by Let (I(t, ϕ), v(t, ϕ)) be the distribution of the infective individuals and virus for t > 0. Since there are no infective agents for t < 0, it is easy to see that and Moreover, the distribution of the new infection rate of cells induced by viruses at time t is and the distribution of the new fission rate of produced free virion from infected cells at time t is

Consequently, the distribution of total new infections of cell is
and the distribution of total new produced free virions is Obviously,F = (F 1 ,F 2 ) is a continuous and positive operator, which maps the initial infection distribution ϕ to the distribution of the total infective members produced during the infection period. Now, we define the spectral radius ofF , r(F ), as the basic reproduction number R 0 of model (3) (see [20,Theorem 3.1]). ForF 2 , we haveF Define the operator B 2 by with the Neumann boundary condition. By [16,Theorem 3.12], we have Hence, it follows from (19) . We are now in a position to show that R 0 is a threshold value for disease invasion. Consider the following auxiliary system of equation (16) ∂I For ϕ = (ϕ 1 , ϕ 2 , ϕ 3 ) ∈ Y + , we define the operator C = (C 1 , C 2 ) by . Let s(Â) be the spectral bound of an operatorÂ, that is, s(Â) = sup{Reζ : ζ ∈ σ(Â)}, where σ(Â) is the spectral set of the operatorÂ. It is easy to see that s(B) is negative. From s(B) < 0, it follows that As a consequence, we have (22) and This proves r(−CB −1 ) = r(F ) = R 0 .
Our next goal is to show that s(A H ) is a principal eigenvalue of A H on C(Ω, R 2 ).
For ζ > −a, we obtain from the first equation of (25) that Substituting it into the second equation of (25), we get Set Then it is easy to see that ζ * > −a. If we choose ϕ 3 = 1, then we have
Proof. By the properties of f (·, S), it follows from the mean value theorem that for each x ∈ Ω, t > 0. Consider We know that the system (28) admits a unique positive equilibrium λ(x) d+M which is globally asymptotically stable in C(Ω, R). Thus, by the comparison theorem, we can obtain that there exists h(x) = λ(x) d+M > 0, such that lim t→∞ inf S(x, t; ϕ) ≥ h(x), ∀x ∈Ω.
For (iii), suppose that there exist x 0 ∈Ω andt >t such that I(x 0 ,t; ϕ) = 0. From the second equation of system (3), it follows that Hence, which implies v(x,t − τ ) = 0. This contradicts to (ii). Now, we state the main theorem in this section.

Proof. By Theorem 2.4, there exists a positive eigenfunction φ such that
We first consider the case that R 0 < 1 (or equivalently, s(A H ) < 0). It is easy to see that s(A H ) is the principal eigenvalue of the problem (30). By Theorem 2.4 and the continuity, there exists a ρ 0 > 0 such that s(A f (x,S0(x)+ρ0) ) is still the principal eigenvalue of the eigenvalue problem (30) and s(A f (x,S0(x)+ρ0) ) < 0.
From the comparison principle, it follows that there is a t 0 = t 0 (ϕ) such that By Theorem 2.4, there is a strongly positive eigenfunctionφ corresponding to for any ϕ ∈ C([−τ, 0], C(Ω, R 2 )). The comparison principle implies that Choose a sufficiently large number K > 0 such that 0 ≤ ϕ(x, θ) ≤ Kφ(x) for all (x, θ) ∈Ω × [−τ, 0]. By the comparison principle again, we obtain Since s(A f (x,S0(x)+ρ0) ) < 0, it is easy to see that w + (x, t) is an upper solution of system (32). Then, Let w t (ϕ)(θ) = w(·, t + θ; ϕ) for all t ≥ 0, θ ∈ [−τ, 0]. It then follows from (33) that w t (φ) ≤φ for all t ≥ 0. By the comparison principle, we obtain w t+s (φ) = w t (w s (φ)) ≤ w s (φ) for all t, s ≥ 0. This implies that w t (φ) is nonincreasing in t ∈ [0, ∞). In view of [29, Lemma 3.1], w t is a κ-contraction for each t > 0, and it then follows that w t is asymptotically smooth. Therefore, the omega limit set ω(φ) of the bounded orbit γ + (φ) = {w t (φ) : t ≥ 0} is nonempty, compact, and invariant. Since w t (φ) is nonincreasing for t ≥ 0, it then follows that ω(φ) = e(x), where e(x) is a nonnegative steady state of system (32). Clearly, e(x) is also an equilibrium of the auxiliary system of linear system (32) as the auxiliary system of (16). Due to s(A H1 ) < 0, e s(A H 1 )tφ (x) is a solution of the auxiliary system of system (32). It implies that e(x) = 0, and it follows from the comparison arguments that every solution of the last two equations of (3) converges to zero. Note that the asymptotic equation of the first equation (3) of is It is easy to see lim t→∞ S(x, t; ϕ) = S 0 (x) uniformly for x ∈Ω. This proves part (i). Next, we consider the case R 0 > 1 or s(A H ) > 0. Let By Lemma 2.5, it follows that for any ϕ ∈ W 0 , and ω(ϕ) be the omega limit set of the orbit O + (ϕ) := {Ψ t ϕ : t ≥ 0}.
3. Spatially homogeneous model. In this section, we study the global attractivity of system (3) in case that all the coefficients are positive constants.
3.1. Existence of equilibria. When all the coefficients are positive constants, system (3) becomes in (x, t) ∈ Ω × [0, ∞). The equilibria of the model (38) satisfies We know that system (38) always admits an infection-free equilibrium E 0 = (S 0 , I 0 , v 0 ) = ( λ d , 0, 0). Aside from E 0 , the system may have a positive equilibrium E * = (S * , I * , v * ). The basic reproduction number for ODE model corre- . By the similar arguments as in [19] and [23], one can show that Lemma 3.1. The basic reproduction number for system (38) is also given by R 0 . Proposition 1. System (38) always has a disease-free equilibrium E 0 . Moreover, if R 0 > 1, then system (38) has a unique positive equilibrium E * satisfying (39).

Global attractivity.
In this subsection, we show the global attractivity of equilibria for system (38) via Lyapunov functionals.
Theorem 3.2. For the system (38), the following statements are valid (i) If R 0 < 1, then the disease-free equilibrium E 0 is globally attractive in Y + .
(ii) If R 0 > 1, then the positive equilibrium E * is globally attractive in Proof. The statement (i) has already been shown in Theorem 2.6, when λ and k are spatially dependent. Of course, it is also can be proved by constructing a Lyapunov function, which is omitted here. For the part (ii), we choose where g(q) = q − 1 − ln q for q ∈ R + . The function V is non-negative with respect to positive solutions of system (38). Denote From the properties of f (S), it follows that Thus,V (ϕ) ≤ 0 holds. Applying LaSalle Invariance Principle, we know that the By Lemma 2.2, the orbit γ + (ϕ) = z t (·, θ; ϕ) is precompact in Y + . According to Theorem 4.3.4 in [3], we know E * is globally attractive.

5.
Summary and discussion. In this paper, a diffusive virus model (3) with time delay and a general incidence function in a bounded domain, is proposed and analyzed. Heterogeneity in the production rates of uninfected target cells as well as that of new viruses, together with zero-flux boundary condition, are assumed. It is shown that model (3) admits two equilibria: a disease-free equilibrium and an endemic equilibrium. The dynamics of this model are shown to be determined by the basic reproduction number R 0 . Precisely, it is proved that the virus dies out if R 0 < 1, while uniformly persists if R 0 > 1 (in terms of the persistence theory). In particular, we also prove global attractivity of equilibria for system (38), provided that all parameters are constants.
Observe that the basic reproduction number R 0 is independent of the diffusion coefficient D, the results reveal that the spatial diffusion of the virus has no impact on the global dynamics of (3), subject to the Neumann boundary condition.