THRESHOLD DYNAMICS OF A REACTION-DIFFUSION EPIDEMIC MODEL WITH STAGE STRUCTURE

. A time-delayed reaction-diﬀusion epidemic model with stage structure and spatial heterogeneity is investigated, which describes the dynamics of disease spread only proceeding in the adult population. We establish the basic reproduction number R 0 for the model system, which gives the threshold dynamics in the sense that the disease will die out if R 0 < 1 and the disease will be uniformly persistent if R 0 > 1 . Furthermore, it is shown that there is at least one positive steady state when R 0 > 1 . Finally, in terms of general birth function for adult individuals, through introducing two numbers ˇ R 0 and ˆ R 0 , we establish suﬃcient conditions for the persistence and global extinction of the disease, respectively.


1.
Introduction. As discussed in Thieme [30], when we describe the spread of infectious disease, all the interesting structures that could and should be considered. Spatial heterogeneity of the environment and spatial-temporal movement of individuals play an important role in the dynamics of infectious disease (see, e.g., [27,40]). Assuming some types of host random movement, there are increasing interests to formulate and analyze infectious disease by reaction-diffusion equations (see, e.g., [1,5,10,9]). Sometimes delay or non-local delay effects would be incorporated into reaction-diffusion epidemic models (see, e.g., [11,18,20,26,36] and the references therein). It is necessary to point that another common way to study the spread of disease in a heterogeneous population is to assume the immigration of infective individuals, which is described by patchy models (see, e.g., [17,19,34,35]). In fact, a constant immigration term has a mildly stabilizing effect on the dynamics and tends to increase the minimum number of infective individuals in the models (see, e.g., [2]).
In nature, as usual, the individual members of population undergo life history through two stages, immature and mature. For vector-borne disease, Dengue fever is transmitted to humans by the mature female Aedes aegypti mosquito (see, e.g., [36]). For some disease, such as sexual disease, it is reasonable to consider the disease transmission in adult population and neglect transmission in juveniles(see, e.g., [35]). Sometimes it seems unreasonable for us to assume that all the individuals in a bounded habitat are commonly susceptible for the disease and have the ability to 3798 LIANG ZHANG AND ZHI-CHENG WANG transmit the disease. Therefore, it is important for us to incorporate stage structure of individual into epidemic model to understand the transmission dynamics of the infectious disease. This work intends to take stage structure, spatial heterogeneity and spatial-temporal movement of individuals into consideration of epidemic models. So in the following, we consider that the host population has two stages: juvenile stage and adult stage. For simplicity, we assume that (see, e.g., [35]) (A1): disease transmission occurs only in adult individuals, and juvenile individuals are immune to the disease; (A2): juvenile individuals do not have the ability to reproduce, and adults are responsible for the reproduction of the population. Let u j (t, x) be the density of juvenile individuals at time t and location x. Then where u(t, a, x) is the density of individuals with age a at time t and location x, and τ the length of the juvenile period. Denote A(t, x) as the density of adult individuals at time t and location x. Then u(t, a, x) and A(t, x) satisfy (see, e.g., Metz and Diekmann [23]) where f (x, A(t, x)) and µ(x)A(t, x) is the birth and mortality function of adult individuals, respectively, and µ j (a) denotes the per capita mortality rate of juvenile at age a, ∆ is the Laplacian operator on R N , Ω is a bounded and open subset of R N with a smooth boundary ∂Ω. The term u(t, τ, x) of the third equation in (2) is the adults recruitment term, being those of maturation age τ . For simplicity, we assume d j (a) = d j , µ j (a) = µ j , that is, diffusion rate and mortality rate of juvenile individuals are independent of age a. Let v(r, a, x) = u(a + r, a, x) with r ≥ 0. Then it follows that Regarding r as a parameter and integrating the last equation, we obtain where Γ is the Green function associated with ∆ and the Neumann boundary condition. Since u(t, τ, x) = v(t − τ, τ, x), ∀t ≥ τ, we have Differentiating (1) with respect to t and making use of (2) and (3), it then follows that u j (t, x) and A(t, x) satisfy We consider a disease transmission of SIS type with nonlinear incidence. According to the principle of mass action, bilinear incidence rate which reflects mechanism of disease transmission could be adopted in classical epidemic model. It has been shown that the disease transmission process may have a nonlinear incidence rate (see, e.g. [15,14] and the reference therein). We employ saturating incidence to describe the transmission process of the disease. Let S = S(t, x), I = I(t, x) be sub-population of susceptible, infectious classes, respectively. Then A(t, x) = S(t, x) + I(t, x). Therefore, we obtain the following model: where g(x, I(t, x)) = β(x)I(t,x) 1+α(x)I(t,x) is saturating incidence, both α(x) and β(x) are positive Hölder continuous functions on Ω. Substituting (3) into the second equation of (4), and dropping the u j (t, x) equation from (4) (since u(t, τ, x) does not depend on the variables of juveniles) result in the following system containing S(t, x) and I(t, x) only: For simplicity, letting (u 1 , u 2 ) = (S, I), (d 1 , d 2 ) = (d S , d I ), we investigate the following time-delayed and non-local reaction-diffusion system with Neumann boundary condition: , ∀α ∈ (0, 1), A > 0. The rest of this paper is organized as follows. In the next section, we study the well-posedness of model system (5) and introduce the basic reproduction number for model (5). In section 3, based on the monotonicity of the birth function on the density of adult individuals, we establish threshold dynamics in terms of the basic reproduction number. Section 4 is devoted to establish sufficient conditions for the persistence and global extinction of disease under the general birth function. Furthermore, a spatially homogeneous case of model (5) with the same diffusion rate of susceptible and infectious adult individuals is studied.
For any ϕ = (ϕ 1 , ϕ 2 ) ∈ C + τ , consider the following linear cooperative reactiondiffusion system (6) By [8,Theorem 4.2], we conclude that the unique of the system (6) exists globally on [0, ∞). For any ϕ ∈ C + τ , let v + (t, x, ϕ(0, x)) be the solution of (6) with initial For any ϕ ∈ C + τ , set Then it is easy to see that the functions v Take B(ϕ) = F (ϕ) for any ϕ ∈ C + τ . Then B is Lipschitz continuous on C + τ . For any x) for any φ ∈ C + τ . Thus, the system (5) generates a semiflow Φ(t) = u t (·) : In the following we prove the point dissipativeness of the solution semiflow Φ(t). (5) and Green's formula, it follows that By (F) and the boundedness of Γ(d j τ, x, y), there exists a positive number k 1 independent of φ, such that Consequently, with the aid of [24, Lemma 3.1] (see also, [16, Theorem 1 and Corollary 1]), we conclude that there exists a positive constant K independent of φ such that which implies that the solutions of system (5) are ultimately bounded, and hence, Φ(t) : Consider the following time-delayed reaction-diffusion equation where d > 0, µ(x) is a positive Hölder continuous function on Ω. Let and (C, C + ) are strongly ordered spaces. In view of the proof of [41, Theorem 3.1], we have the following result.
Using arguments similar to those in [28, Theorem 7.6.1] (see also [32,Theorem 2.2]), it is shown that the following nonlocal elliptic eigenvalue problem has a principal eigenvalue denoted by λ 0 (d, τ, ∂ A f (·, 0)). By [32, Theorem 2.2], the following nonlocal elliptic eigenvalue problem For any ψ ∈ Z + , let A(t; ψ)(·) = A(t, ·; ψ) denote the solution of (7). At what follows, we establish the threshold dynamics for system (7). (7) admits at least one positive steady statē A * (·), and there exists a ς > 0 such that for every . then (7) admits a unique positive steady state A * , which satisfies The proofs of Lemma 2.3 (i) and (ii) are completely similar to those in [41, Theorem 3.1] and the proof of Lemma 2.3 (iii) is also similar to that in [41, Theorem 3.2 (1)], so we omit the details of the proofs of Lemma 2.3. In order to find the disease-free equilibrium (infection-free steady state), we set u 2 = 0 in system (5), leading to the following equation for the density of susceptible host population: As in Lemma 2.3, the nonlocal elliptic eigenvalue problem ∂ŵ(x) ∂n = 0, x ∈ ∂Ω has a principal eigenvalue, which is denoted by λ 0 (d 1 , τ, ∂ u1 f (y, 0)). We further make the following assumption:  1 (x) which is globally attractive in Y + \{0}, and hence, system (5) admits a unique disease-free equilibrium (u * 1 (x), 0). Linearizing system (5) at the disease-free equilibrium (u * 1 (x), 0), we get the following system for infectious component u 2 : Substituting u 2 (t, x) = e λt ϕ(x) into (9), we obtain the following eigenvalue problem: It then follows from [28, Theorem 7.6.1] that (10) has a principal eigenvalue denoted by λ(d 2 , u * 1 (·)) with a positive eigenfunction. In the following, we introduce the basic reproduction number for system (5). Suppose that host population is near the disease-free equilibrium. We introduce the distribution of initial infectious individuals ϕ(x) at time t = 0. Under the synthetical influences of mobility and mortality of infected individuals, the distribution of those infective members as time evolves becomes By using the ideas in [37], we define the next generation operator: Motivated by [7,33,31,36,37], we define the spectral radius of L as the basic reproduction number for model (5), that is, By [37, Theorem 3.1] with diffusion rate independent on spatial variable x, we have the following observation.
3. Threshold dynamics. In this section, we establish the threshold dynamics of the system (5) in terms of the basic reproduction number R 0 . Before we show the main results of this section, we will propose the following results which play an important role in establishing persistence of (5).
The following conclusion indicates that R 0 is a threshold index for disease extinction or persistence.
Claim 3. M 2 is a uniform weak repeller for W 0 in the sense that Then there exists t 0 > τ such that Letψ be the strongly positive eigenfunction corresponding to λ d 2 , tψ is a solution of the following linear system : In view of Lemma 3.1, there exists ε 0 > 0 such that for all x ∈ Ω. By the standard comparison principle, we have which implies u 2 (t, x; φ 0 ) is unbounded, a contradiction. Define a continuous function p : It is obvious that p −1 (0, ∞) ⊆ W 0 . By Lemma 3.1, p has the property that if p(φ) > 0 or φ ∈ W 0 with p(φ) = 0, then p(Φ(t)φ) > 0, ∀t > 0. That is, p is a generalized distance function for the semiflow Φ(t) : C + τ → C + τ (see, e.g., [29]). From the above claims, it follows that any forward orbit of Φ(t) in M ∂ converges to M 1 or M 2 . In view of Claim 2 and Claim 3, we conclude that M 1 and M 2 art two isolated invariant sets in C + τ , and that W s (M i ) ∩ W 0 = ∅, i = 1, 2, where W s (M i ) is the stable set of M i . It is clearly that no subset of {M 1 , M 2 } forms a cycle in ∂W 0 . It then follows from [29,Theorem 3] that there exists anη > 0 such that Hence, lim inf t→∞ u 2 (t, ·; φ) ≥η, ∀φ ∈ W 0 . On the other hand, according to Theorem 2.1 and Lemma 3.1, there exists a > 0 such that 0 < u 2 (t, ·; φ) ≤ , ∀t ≥ t 2 = t 2 (φ), x ∈ Ω. Consequently, for large enough t, u 1 (t, x) satisfies that By using similar arguments to [20, Lemma 1], it follows that the following reactiondiffusion equation admits a unique positive steady state w * (·) which is globally attractive in X 1 . With the aid of the standard parabolic comparison principle, we obtain that Therefore, there exists an ι with 0 < ι ≤η such that lim inf t→∞ u i (t, ·; φ) ≥ ι, ∀φ ∈ W 0 , i = 1, 2.
Thus, the global attractivity stated in conclusion holds.

General birth function.
In the previous section, under the condition (F1), we obtained some conclusions on the persistence and extinction of disease. It should be noted that in (F1), the birth function f is monotone on the density of adult individuals A ∈ (0, ∞). In this section, for more general birth function f , we intend to investigate the threshold dynamics for model (5) satisfying d 1 = d 2 = d, that is, Setting u 2 = 0 in system (23), we have the following equation for the density of susceptible host population: Assume that (F) holds. It then follows from Lemma 2.2 that (25) generates a semiflowQ(t) = u 1t (·) : C + → C + , t ≥ 0, (in this case of d 1 = d, we see that Y = X 1 ), which admits a global compact attractor B 0 . As in Lemma 2.3, the following nonlocal elliptic eigenvalue problem ∂ŵ(x) ∂n = 0, x ∈ ∂Ω has a principal eigenvalue, which is denoted by λ 0 (d, τ, ∂ u1 f (y, 0)). We further make the following assumption: (F2): λ 0 (d, τ, ∂ u1 f (y, 0)) > 0. Assume that (F) and (F2) hold. Then by Lemma 2.3(ii), we have that for φ 1 (0, ·) ≡ 0, where ς > 0 and t 0 are determined by Lemma 2.3(ii) with ψ(s, x) = φ 1 (s, x) for all s ∈ [−τ, 0] and x ∈ Ω. Hence, the solution semiflow of (25) defined by Q (t)φ 1 (s, x) := u 1 (t + s, x; φ 1 ) for s ∈ [−τ, 0] and x ∈ Ω admits a positive global compact attractor B 0 ⊂ int(C + ). We mention that under the conditions (F) and (F1), compact attractor B 0 degenerates into a singleton set (see Theorem 2.4). As such, in Section 2, with the aid of the unique disease-free equilibrium (u * 1 , 0), we can define the basic reproduction number R 0 via the next generation operator and get the threshold result, see Theorems 3.2 and 3.3. However, in the present section, due to the non-monotonicity of birth function, the compactor attractor B 0 may not be a singleton set. Therefore, it is impossible to get a threshold dynamics by defining a unique number R 0 as that in Section 2. To establish the similar threshold results, in the foolwing we introduce two R 0 -like numbersŘ 0 andR 0 by virtue of the lower and upper bounds of B 0 respectively and then establish the dynamics of system (23).
Proof. In the following, we shall use the previous analysis of this section and similar arguments to those in [42,Theorem 3.3] to prove the conclusion stated in (1).