Threshold dynamics of a time periodic reaction–diffusion epidemic model with latent period

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Threshold dynamics of a time periodic reaction–diffusion epidemic model with latent period Liang Zhang a,b , Zhi-Cheng Wang a,∗,1 , Xiao-Qiang Zhao b,2 a School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China b Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada

Received 8 December 2014

Abstract In this paper, we first propose a time-periodic reaction–diffusion epidemic model which incorporates simple demographic structure and the latent period of infectious disease. Then we introduce the basic reproduction number R0 for this model and prove that the sign of R0 − 1 determines the local stability of the disease-free periodic solution. By using the comparison arguments and persistence theory, we further show that the disease-free periodic solution is globally attractive if R0 < 1, while there is an endemic periodic solution and the disease is uniformly persistent if R0 > 1. © 2015 Elsevier Inc. All rights reserved. MSC: 35K57; 35B35; 35B40; 92D30 Keywords: Nonlocal model; Spatial diffusion; Basic reproduction number; Periodic solutions; Uniform persistence

1. Introduction Mathematical models have become important tools in analyzing the spread and control of infectious diseases. Understanding the transmission characteristics of infectious diseases in * Corresponding author.

E-mail address: [email protected] (Z.-C. Wang). 1 Research is partially supported by NNSF of China (11371179) and the Scientific Research Foundation for Returned

Overseas Chinese Scholars. 2 Research is partially supported by the NSERC of Canada. http://dx.doi.org/10.1016/j.jde.2014.12.032 0022-0396/© 2015 Elsevier Inc. All rights reserved.

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communities, regions and countries can lead to better approaches to control these diseases (see, e.g., [3,7,16,34]). To describe the transmission of communicable diseases, Kermack and McKendrick [21] proposed a basic SIR model: ⎧  ⎨ S (t) = −βS(t)I (t), I  (t) = βS(t)I (t) − γ I (t), ⎩  R (t) = γ I (t).

(1.1)

Here S(t), I (t) and R(t) denote the sizes of the susceptible, infected and removed individuals, respectively, the constant β is the transmission coefficient, and γ is the recovery rate. Let S0 = S(0) be the density of the population at the beginning of the epidemic with everyone susceptible. It is well known that the basic reproduction number R0 = βS0 /γ completely determines the transmission dynamics (an epidemic occurs if and only if R0 > 1), see also [2,6,16,34]. It should be emphasized that system (1.1) has no vital dynamics (births and deaths) because it was usually used to describe the transmission dynamics of disease within a short outbreak period. However, for an endemic disease, we should incorporate a demographic structure into the epidemic model. The classical endemic model is the following SIR model with vital dynamics: ⎧ βS(t)I (t) ⎪ , S  (t) = μN − μS(t) − ⎪ ⎪ ⎨ N βS(t)I (t) I  (t) = − γ I (t) − μI (t), ⎪ ⎪ ⎪ N ⎩  R (t) = γ I (t) − μR(t),

(1.2)

which is almost the same as the SIR epidemic model (1.1) above, except that it has an inflow of newborns into the susceptible class at rate μN and deaths in the classes at rates μS, μI and μR, where N is a positive constant and denotes the total population size. For this model, the β , which is the contact rate β times the average basic reproduction number is given by R0 = γ +μ

1 death-adjusted infectious period γ +μ (see [3,16]). In reality, many diseases have latency and the length of the latent period differs from disease to disease (see, e.g., [3]). During the latent period, however, the individuals may move from one spatial location at a time to another location at another time, and may disperse from a domain to a larger domain. Therefore, the incorporation of latency and mobility of the individuals in the latent period usually gives rise to nonlocal infection terms. Guo et al. [13] proposed a timedelayed and nonlocal reaction–diffusion epidemic model, and obtained the threshold dynamics in terms of the basic reproduction number. Other deterministic epidemiology models concerning the latency of diseases and the mobility of individuals were developed in quite a few works, see, e.g., [40,27,42]. In particular, it was observed numerically in [40] that the basic reproduction number was a deceasing function of the diffusion rate of susceptible host population. This implies that diffusion rate can impact the disease transmission. For the spatially discrete case, Li and Zou [22] incorporated the mobility of individuals and the latency factor into the classic Kermack–McKendrick SIR model and obtained a two patch SIR model with nonlocal terms. They found that there are multiple outbreaks of the disease before it goes to extinction, which is prominently different from the classic Kermack–McKendrick SIR model. Later, by introducing demographic structure, Li and Zou [23] studied a time-delayed model with nonlocal terms characterizing individuals dispersing among n-patch in a fixed latent period. They showed that nonlocal effects can increase the basic reproduction number, and hence, may cause an otherwise

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dying-out disease to persist. Note that the aforementioned models of infectious diseases are all governed by autonomous systems of differential equations. It is well-known that the population dynamics and the spread of infectious disease are influenced conspicuously by the time varying environments (e.g., due to seasonal variation, see Table 1 in Altizer et al. [1]). Therefore, it is natural and more realistic to incorporate temporal heterogeneity into the disease model, which leads to non-autonomous evolution systems. Bacaër and Guernaoui [5] introduced a general definition of the basic reproduction number R0 in a periodic environment. For further developments, we refer to Bacaër and Dads [4] and references therein. For a large class of periodic compartmental epidemic models, Wang and Zhao [39] characterized the basic reproduction ratio and proved that it is a threshold parameter for the local stability of the disease-free periodic solution. Peng and Zhao [33] introduced the basic reproduction number for a time-periodic reaction–diffusion SIS model, and showed that the combination of spatial heterogeneity and temporal periodicity tends to enhance the persistence of the infectious disease. Lately, Inaba [18] presented the concept of generation evolution operators and gave a new definition of the basic reproduction number in a heterogeneous environment, which unifies the definitions in [11] and [5]. More recently, Zhao [44] established the theory of basic reproduction ratios for periodic and time-delayed compartmental models and applied it to a periodic SEIR model with incubation period. However, there are few investigations on the global dynamics for PDE epidemic models with seasonality and time delay in terms of the basic reproduction number. The purpose of this paper is to propose a new epidemic model by synthesizing disease latency, demographic structure, spatial diffusion and temporal heterogeneity into the SIR model, and to study the spatial dynamics of the derived model. The rest of this paper is organized as follows. In the next section, we derive a new epidemic model, which is a time-periodic reaction–diffusion system with nonlocal and time-delayed nonlinearity, and study its well-posedness. In Section 3, we introduce the basic reproduction number R0 for the model via the next generation operators approach, and show that the sign of R0 − 1 determines the local stability of the disease-free periodic solution. Section 4 is devoted to the threshold dynamics for the model system in terms of R0 . 2. The model In this section, we propose a time-periodic reaction–diffusion epidemic model with latent period and establish the existence of time-global solutions of the model system. In consideration of mobility of individuals and seasonality effect, we assume that a host population lives in a spatially and temporally heterogeneous environment. Let Ω denote the spatial habitat with smooth boundary ∂Ω. We introduce a fixed period of latency into the population, and assume that the disease has full immunity after recovery (regardless of natural recovery or recovery due to treatments). Let S = S(t, x), L = L(t, x), I = I (t, x) and R = R(t, x) be the sub-populations of susceptible, latent, infectious and recovered classes, respectively. Due to the mobility of the host population during the latent period, we introduce the notion of infection age denoted by the variable a. Let E(t, a, x) be the density (with respect to the infection age a) of infected population at time t and location x with infection age a. We assume that all populations remain confined to the region Ω for all time, and subject to no flux boundary condition for E(t, a, x): 

 D(t, a, x)∇E(t, a, x) · n = 0,

t > 0, x ∈ ∂Ω,

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where ∇E(t, a, x) is the gradient of E(t, a, x) with respect to the spatial variable x, n is the outward normal to ∂Ω. By a standard argument on structured population and spatial diffusion (see e.g. [31]), we get   ∂E(t, a, x) ∂E(t, a, x) + = ∇ · D(t, a, x)∇E(t, a, x) ∂t ∂a  − σ (t, a, x) + γ (t, a, x) + d(t, x) E(t, a, x),

(2.1)

where ∇ · [D(t, a, x)∇E(t, a, x)] denote the divergence of D(t, a, x)∇E(t, a, x), D(t, a, x) is the diffusion rate at time t , age a and location x, σ (t, a, x) and γ (t, a, x) are the disease-induced mortality rate and the recovery rate at time t and location x with age a, respectively, d(t, x) is the natural death rate which is independent of the infection age. Suppose that τ is the average latency period, namely, infected individuals do not infect other susceptible individuals until τ time later, we then have

τ L(t, x) =

∞ E(t, a, x)da,

I (t, x) =

E(t, a, x)da.

(2.2)

τ

0

We make some assumptions for functions D(t, a, x), σ (t, a, x) and γ (t, a, x) as follows:  D(t, a, x) =  σ (t, a, x) =  γ (t, a, x) =

DL (t, x), DI (t, x),

for t ≥ 0, a ∈ [0, τ ] and x ∈ Ω, for t ≥ 0, a ∈ [τ, ∞) and x ∈ Ω,

σL (t, x), σI (t, x),

for t ≥ 0, a ∈ [0, τ ] and x ∈ Ω, for t ≥ 0, a ∈ [τ, ∞) and x ∈ Ω,

γL (t, x), γI (t, x),

for t ≥ 0, a ∈ [0, τ ] and x ∈ Ω, for t ≥ 0, a ∈ [τ, ∞) and x ∈ Ω.

Differentiating (2.2) with respect to t and making use of (2.1), we obtain    ∂I (t, x) = ∇ · DI (t, x)∇I (t, x) − σI (t, x) + γI (t, x) + d(t, x) I (t, x) ∂t + E(t, τ, x) − E(t, ∞, x) and    ∂L(t, x) = ∇ · DL (t, x)∇I (t, x) − σL (t, x) + γL (t, x) + d(t, x) I (t, x) ∂t − E(t, τ, x) + E(t, 0, x), respectively. Since the death rate function d(t, x) is positive for t ∈ [0, ∞) and x ∈ Ω, biologically, we can assume that E(t, ∞, x) = 0. As the new infections arise from the contact of infectious and susceptible individuals, we adopt the mass action infection mechanism that the lost of susceptible individuals by infection is at a rate proportional to the number of infectious, leading to the following condition:

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E(t, 0, x) = h(t, x)I (t, x)S(t, x), where h(t, x) > 0 is called infection rate. We use the following simple demographic equation for a population N (t, x) that admits a dynamics of global convergence to a positive periodic solution:   ∂N(t, x) = ∇ · DN (t, x)∇N (t, x) + Λ(t, x) − d(t, x)N(t, x), ∂t where Λ(t, x) is the recruiting rate, DN (t, x) is the diffusion rate and d(t, x) is the natural death rate. We also assume that the disease under consideration does not transmit vertically. On the basis of above assumptions, the disease dynamics is governed by the following system of partial differential equations: ⎧ ∂S(t, x)   ⎪ = ∇ · DS (t, x)∇S(t, x) + Λ(t, x) − d(t, x)S(t, x) ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ − h(t, x)S(t, x)I (t, x), ⎪ ⎪ ⎪ ⎪     ∂L(t, x) ⎪ ⎪ ⎪ = ∇ · DL (t, x)∇L(t, x) − σL (t, x) + γL (t, x) + d(t, x) L(t, x) ⎪ ⎪ ∂t ⎪ ⎪ ⎨ + h(t, x)S(t, x)I (t, x) − E(t, τ, x),     ∂I (t, x) ⎪ ⎪ = ∇ · DI (t, x)∇I (t, x) − σI (t, x) + γI (t, x) + d(t, x) I (t, x) ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ + E(t, τ, x), ⎪ ⎪ ⎪   ∂R(t, x) ⎪ ⎪ ⎪ = ∇ · DR (t, x)∇R(t, x) + γL (t, x)L(t, x) ⎪ ⎪ ∂t ⎪ ⎪ ⎩ + γI (t, x)I (t, x) − d(t, x)R(t, x).

(2.3)

We make the following basic assumption: (H) Functions Λ(t, x), σL (t, x), γL (t, x), σI (t, x), γI (t, x) and h(t, x) are Hölder continuous and nonnegative nontrivial on R × Ω, and periodic in time with the same period ω > 0; the function d(t, x) is Hölder continuous and positive on R × Ω, and periodic in time with the same period ω > 0; the diffusion coefficients DS (t, x), DL (t, x), DI (t, x) and DR (t, x) are Hölder continuous on R ×Ω, and periodic in time with the same period ω > 0. Moreover, we assume Di (t, x) ≥ Di > 0, for all on R × Ω, where i = S, L, I, R. It is then necessary for us to determine E(t, τ, x) by the integration along characteristics. For any ξ ≥ 0, consider solutions of (2.1) along the characteristic line t = a + ξ by letting v(ξ, a, x) = E(a + ξ, a, x). Then for a ∈ (0, τ ], we have   ⎧ ∂a v(ξ, a, x) = ∂t E(t, a, x) + ∂a E(t, a, x) t=a+ξ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ = ∇ · D(a + ξ, a, x)∇E(a + ξ, a, x) ⎪ ⎪ ⎪  ⎨ − σ (a + ξ, a, x) + γ (a + ξ, a, x) + d(a + ξ, x) E(a + ξ, a, x)   ⎪ ⎪ = ∇ · DL (a + ξ, x)∇v(ξ, a, x) ⎪ ⎪  ⎪ ⎪ ⎪ − σL (a + ξ, x) + γL (a + ξ, x) + d(a + ξ, x) v(ξ, a, x), ⎪ ⎪ ⎩ v(ξ, 0, x) = E(ξ, 0, x) = h(ξ, x)I (ξ, x)S(ξ, x).

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For the above last equation, we can regard ξ as a parameter and make an integration to it. We then have

  v(ξ, a, x) = Γ (ξ + a, ξ, x, y) h(ξ, y)S(ξ, y)I (ξ, y) dy, Ω

where Γ (t, s, x, y) with t > s ≥ 0 and x, y ∈ Ω is the fundamental solution associated with the partial differential operator ∂t − ∇ · [DL (t, ·)∇] − βL (t, ·) (see Friedman [12, Chapter 1]) and βL (t, ·) = σL (t, ·) + γL (t, ·) + d(t, ·). Note that Γ (t, s, x, y) = Γ (t + ω, s + ω, x, y) for all t > s ≥ 0 and x, y ∈ Ω due to DL (t + ω, ·) = DL (t, ·) and βL (t + ω, ·) = βL (t, ·) for any t ≥ 0. Since E(t, a, x) = v(t − a, a, x), it follows that

  E(t, a, x) = Γ (t, t − a, x, y) h(t − a, y)S(t − a, y)I (t − a, y) dy. Ω

Set a = τ , then

E(t, τ, x) =

  Γ (t, t − τ, x, y) h(t − τ, y)S(t − τ, y)I (t − τ, y) dy.

(2.4)

Ω

Substituting (2.4) into the second and third equations of (2.3) respectively, and dropping the L(t, x) and R(t, x) equations from (2.3) (since they are decoupled from the S(t, x) and I (t, x) equations), we obtain the following system: ⎧ ∂S(t, x)   ⎪ = ∇ · DS (t, x)∇S(t, x) + Λ(t, x) − d(t, x)S(t, x) − h(t, x)S(t, x)I (t, x), ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂I (t, x)   = ∇ · DI (t, x)∇I (t, x) − βI (t, x)I (t, x) ∂t

⎪ ⎪ ⎪ ⎪ ⎪ + Γ (t, t − τ, x, y)h(t − τ, y)S(t − τ, y)I (t − τ, y)dy, ⎪ ⎩ Ω

where βI (t, x) = σI (t, x) + γI (t, x) + d(t, x) is time periodic with period ω for all (t, x) ∈ [0, ∞) × Ω. For simplicity, letting (u1 , u2 ) = (S, I ), (D1 (·,·), D2 (·,·)) = (DS (·,·), DI (·,·)) and β(·,·) = βI (·,·), we investigate the following time-periodic nonlocal and time-delayed reaction– diffusion system with no flux boundary condition ⎧ ∂u (t, x)   1 ⎪ = ∇ · D1 (t, x)∇u1 (t, x) ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ + Λ(t, x) − d(t, x)u1 (t, x) − h(t, x)u1 (t, x)u2 (t, x), t > 0, x ∈ Ω, ⎪ ⎪ ⎨   ∂u2 (t, x) (2.5) = ∇ · D2 (t, x)∇u2 (t, x) − β(t, x)u2 (t, x) ⎪ ∂t

⎪ ⎪ ⎪ ⎪ + Γ (t, t − τ, x, y)h(t − τ, y)u1 (t − τ, y)u2 (t − τ, y)dy, t > 0, x ∈ Ω, ⎪ ⎪ ⎪ ⎪ ⎪ Ω  ⎩   D1 (t, x)∇u1 (t, x) · n = D2 (t, x)∇u2 (t, x) · n = 0,

t > 0, x ∈ ∂Ω.

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In the rest of this section, we study the existence and uniqueness of time-global solutions of model (2.5). Let X := C(Ω, R2 ) be the Banach space with the supermum norm  · X . For τ ≥ 0, define Cτ = C([−τ, 0], X) with the norm  ·  := maxθ∈[−τ,0] φ(θ)X , ∀φ ∈ Cτ . Then Cτ is a Banach space. Define X+ := C(Ω, R2+ ) and Cτ+ := C([−τ, 0], X+ ), then both (X, X+ ) and (Cτ , Cτ+ ) are strongly order spaces. For a function u : [−τ, σ ) → X for σ > 0, define ut ∈ Cτ by ut (θ ) = u(t + θ ), ∀θ ∈ [−τ, 0]. Consider the following general equation ⎧   ∂u(t, x) ⎪ ˜ x)∇u(t, x) − β(t, ˜ x)u(t, x), ⎪ = ∇ · d(t, ⎨ ∂t   ˜ x)∇u(t, x) · n = 0, d(t, ⎪ ⎪ ⎩ u(0, x) = ϕ(x),

t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω,

(2.6)

x ∈ Ω, ϕ ∈ Y+ ,

˜ x) is Hölder continuous on R × Ω and ω-periodic in t , and there is a positive constant where d(t, ˜ x) ≥ d˜  , β(t, ˜ x) is Hölder continuous and nonnegative nontrivial on R × Ω and d˜ such that d(t, ω-periodic in t , Y := C(Ω, R) and Y+ := C(Ω, R+ ). According to [15, Chapter II], (2.6) admits an evolution operator W (t, s) : Y → Y, 0 ≤ s ≤ t , which satisfies W (t, t) = I, W (t, s)W (s, ρ) = W (t, ρ) for all 0 ≤ ρ ≤ s < t, and W (t, 0)(ϕ)(x) = u(t, x; ϕ) for t ≥ 0, x ∈ Ω and ϕ ∈ Y, ˜ ·) and β(t, ˜ ·) on t , where u(t, x; ϕ) is the solution of (2.6). Due to the time periodicity of d(t, it follows from [9, Lemma 6.1] that W (t + ω, s + ω) = W (t, s) holds for (t, s) ∈ R2 with t ≥ s. Furthermore, for any s, t ∈ R with s < t , W (t, s) is a compact, analytic and strongly positive operator on Y. In particular, W (t, s)(ϕ)(x) > 0 for s < t and x ∈ Ω, provided that ϕ ∈ Y+ and ϕ ≡ 0. In view of [9, Theorem 6.6] with α = 0, there exist positive constants K ≥ 1 and c0 ∈ R such that W (t, s) ≤ Ke−c0 (t−s) ,

∀t ≥ s, t, s ∈ R.

Consider the following periodic reaction–diffusion equation ⎧ ⎨ ∂w(t, x) = ∇ · D(t, x)∇w(t, x) + g(t, x) − μ(t, x)w(t, x), ∂t  ⎩ D(t, x)∇w(t, x) · n = 0,

t > 0, x ∈ Ω,

(2.7)

t > 0, x ∈ ∂Ω,

where D(t, x) ≥ D  > 0 for t > 0 and x ∈ Ω, g(t, x) ≡ 0 is Hölder continuous and nonnegative function for t > 0 and x ∈ Ω, μ(t, x) is Hölder continuous and positive for t > 0 and x ∈ Ω. Furthermore, D(t, ·), g(t, ·) and μ(t, ·) are periodic in t with the same period ω > 0. Recall that a family of operators {Qt }t≥0 is an ω-periodic semiflow on a metric space (Z, ρ) with the metric ρ, provided that {Qt }t≥0 satisfies: (i) Q0 (v) = v, ∀v ∈ Z; (ii) Qt (Qω (v)) = Qt+ω (v), ∀t ≥ 0, v ∈ Z; (iii) Qt (v) is continuous in (t, v) on [0, ∞) × Z. Lemma 2.1. System (2.7) admits a unique positive ω-periodic solution w∗ (t, ·) which is globally attractive in Y+ . Proof. Denote g¯ :=

max t∈[0,ω],x∈Ω

g(t, x),

μ˜ :=

max t∈[0,ω],x∈Ω

μ(t, x),

μ¯ :=

min t∈[0,ω],x∈Ω

μ(t, x).

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For any ϕ ∈ Y+ , (2.7) has a unique solution w(t, x; ϕ) on [0, ∞) with w(0, x; ϕ) = ϕ (see, e.g., [15,32]). Define a family of operators {Qt }t≥0 on Y+ by Qt (ϕ)(x) = w(t, x; ϕ) for any ϕ ∈ Y+ , t ≥ 0 and x ∈ Ω. For any (t, ϕ) ∈ R+ × Y+ and (t0 , ϕ0 ) ∈ R+ × Y+ , we have Qt (ϕ) − Qt (ϕ0 ) ≤ Qt (ϕ) − Qt (ϕ0 ) + Qt (ϕ0 ) − Qt (ϕ0 ) . 0 0 Note that U˜ (t, 0)ϕ is continuous in (t, ϕ) ∈ [0, ∞) × Y, where the evolution operator U˜ (t, s) ˜ x) = D(t, x) and β(t, ˜ x) = μ(t, x). By a similar argument to is defined by Eq. (2.6) with d(t, that in [29, Theorem 8.5.2], it follows that Qt (ϕ) is continuous in (t, ϕ) ∈ [0, ∞) × Y. It is easy to see that Qt satisfies (i) and (ii), thus, {Qt }t≥0 is an ω-periodic semiflow on Y+ . It follows from the maximal principle (see, e.g., [15, Proposition 13.1]) that for any ϕ ∈ Y+ with ϕ ≡ 0, Qt (ϕ) ∈ Y+ and its omega limit set  (ϕ) satisfies  (ϕ) ⊂ {ψ ∈ Y+ : 0 ≤ ψ ≤ μg¯¯ } := M0 , which implies that Qt is point dissipative in Y+ . Applying a similar argument to the proof of [15, Proposition 21.2], we also have that Qt : Y+ → Y+ is continuous and compact for any t > 0. Define the Poincaré operator (map) S : Y+ → Y+ by S(ϕ) = Qω (ϕ), where ϕ ∈ Y+ . With the above argument, we have that S : Y+ → Y+ is continuous, point dissipative and compact. Hence, [14, Theorem 2.4.7] asserts that S : Y+ → Y+ has a global compact attractor ¯ ⊂ M0 . Note that f (t, x, w) := g(t, x) − μ(t, x)w is strictly subhomogeneous in the sense M that f (t, x, αw) > αf (t, x, w) for α ∈ (0, 1) and w > 0. It is not difficult to see that Qt (ϕ) is strictly subhomogeneous as well as Qω (ϕ), i.e., Qω (αϕ) > αQω (ϕ) for any α ∈ (0, 1) and ϕ  0. In addition, Qt is strongly monotone for t > 0. It then follows from [43, Theorem 2.3.2] ¯ that Qω has a fixed point ϕ ∗  0 such that M ¯ = {ϕ ∗ }. This implies that ϕ ∗ is with K = M ∗ globally attractive for Qω in M0 . Consequently, w (t, x) := w(t, x; ϕ ∗ ) is a unique positive ω-periodic solution of (2.7), which attracts every solution w(t, x; ϕ) with ϕ ∈ Y+ and ϕ ≡ 0. 2 In the following, we prove the existence of time-global solution of (2.5). Let U (t, s) be the evolution operators determined by the following reaction–diffusion equation ⎧ ⎨ ∂u1 (t, x) = ∇ · D (t, x)∇u (t, x) − d(t, x)u (t, x), 1 1 1 ∂t  ⎩ ∇ · D1 (t, x)∇u1 (t, x) · n = 0,

t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω

and V (t, s) be the evolution operators determined by the following reaction–diffusion equation ⎧ ⎨ ∂u2 (t, x) = ∇ · D (t, x)∇u (t, x) − β(t, x)u (t, x), 2 2 2  ∂t ⎩ ∇ · D2 (t, x)∇u2 (t, x) · n = 0,

t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω.

Since Di (t, ·), i = 1, 2, d(t, ·) and β(t, ·) are time ω-periodic, [9, Lemma 6.1] implies that U (t + ω, s + ω) = U (t, s) and V (t + ω, s + ω) = V (t, s) holds for (t, s) ∈ R2 with t ≥ s. Moreover, for (t, s) ∈ R2 with t > s, U (t, s) : Y → Y and V (t, s) : Y → Y are compact and strongly  0 positive. By the above arguments, we define U(t, s) := U (t,s) 0 V (t,s) . Then U(t, s) : X → X is an evolution operator for (t, s) ∈ R2 with t ≥ s.

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Define F = (F1 , F2 ) : [0, +∞) × Cτ+ → X by F1 (t, φ) := Λ(t, ·) − r(t, ·)φ1 (0, ·)φ2 (0, ·),

F2 (t, φ) := Γ (t, t − τ, ·, y)h(t − τ, y)φ1 (−τ, y)φ2 (−τ, y)dy Ω

for t ≥ 0, x ∈ Ω and φ = (φ1 , φ2 ) ∈ Cτ+ . Then (2.5) becomes 

∂t u(t, x) = A(t)u(t, x) + F(t, ut ), t > 0, x ∈ Ω, u(ς, x) = φ(ς, x), ς ∈ [−τ, 0], x ∈ Ω,

(2.8)

which can be written as an integral equation

t u(t, φ) = U(t, 0)φ(0) +

U(t, σ )F(σ, uσ )dσ,

t ≥ 0, φ ∈ Cτ+ ,

(2.9)

0

solution is called a mild solution of (2.8), where u(t, x) := (u1 (t, x), u2 (t, x)), A(t) := whose A1 (t) 0 0 A (t) , A1 (t) is defined by 2

    D A1 (t) = ϕ ∈ C 2 (Ω)  D1 (t, ·)∇ϕ · n = 0 on ∂Ω ,    A1 (t)ϕ = ∇ · D1 (t, ·)∇ϕ − d(t, ·)ϕ, ∀ϕ ∈ D A1 (t) , and A2 (t) is defined by     D A2 (t) = ϕ ∈ C 2 (Ω)  D2 (t, ·)∇ϕ · n = 0 on ∂Ω ,    A2 (t)ϕ = ∇ · D2 (t, ·)∇ϕ − β(t, ·)ϕ, ∀ϕ ∈ D A2 (t) . The following result shows that solutions of system (2.5) exist globally on [0, ∞). Theorem 2.2. For any φ ∈ Cτ+ , system (2.5) has a unique solution u(t, φ) on [0, ∞) with u0 = φ. Furthermore, system (2.5) generates an ω-periodic semiflow Φt := ut (·) : Cτ+ → Cτ+ , i.e., Φt (φ)(s, x) = u(t + s, x; φ), ∀φ ∈ Cτ+ , t ≥ 0, s ∈ [−τ, 0], x ∈ Ω, and Φω : Cτ+ → Cτ+ has a global compact attractor in Cτ+ . Proof. Firstly, we show the local existence of the unique mild solution. Clearly, F is locally Lipschitz continuous. It is necessary to show  lim dist φ(0, ·) + θ F(t, φ), X+ = 0,

θ→0+

∀(t, φ) ∈ [0, +∞) × Cτ+ .

(2.10)

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For any (t, φ) ∈ [0, +∞) × Cτ+ and θ ≥ 0, we have 

φ1 (0,  x) + θ [Λ(t, x) − h(t, x)φ1 (0, x)φ2 (0, x)] φ(0, x) + θ F(t, φ)(x) = φ2 (0, x) + θ Ω Γ (t, t − τ, ·, y)h(t − τ, y)φ1 (−τ, y)φ2 (−τ, y)dy   ¯ 2 (0, x)] φ1 (0, x)[1 − θ hφ ≥ φ2 (0, x)



for t ≥ 0 and x ∈ Ω, where h¯ = maxt∈[0,ω], x∈Ω h(t, x). The above inequality implies that φ(0) + θF(t, φ) ∈ X+ if θ is sufficiently small, which yields (2.10). Consequently, by [30, Corollary 4] with K = X+ and S(t, s) = U(t, s), system (2.5) has a unique mild solution u(t, x; φ) with u0 (·, ·; φ) = φ on its maximal interval of existence t ∈ [0, t˜φ ), where t˜φ ≤ ∞, and u(t, ·; φ) ∈ X+ , t ∈ [0, t˜φ ). Moreover, by the analyticity of U(t, s), s, t ∈ R, s < t, u(t, x; φ) is a classical solution when t > τ . Consider the following time-periodic reaction–diffusion equation ⎧   ∂w(t, x) ⎪ ⎪ = ∇ · D1 (t, x)∇w(t, x) + Λ(t, x) − d(t, x)w(t, x), ⎨ ∂t t > 0, x ∈ Ω, ⎪ ⎪  ⎩ D1 (t, x)∇w(t, x) · n = 0, t > 0, x ∈ ∂Ω.

(2.11)

By Lemma 2.1, (2.11) admits a unique positive ω-periodic solution u∗1 (t, x) which is globally asymptotically stable in Y+ . Since the first equation in (2.5) is dominated by (2.11), there is a constant B1 > 0 such that for any φ ∈ Cτ+ , there exists a positive integer l1 = l1 (φ) > 0 satisfying u1 (t, x; φ) ≤ B1 for all t ≥ l1 ω and x ∈ Ω. Next, we use similar arguments to those in [37, Theorem 2.1] to prove the ultimate boundedness of solutions. Set

Λ0 = max

Λ(t, x)dx

t∈[0,ω]

and

d0 =

min

t∈[0,ω],x∈Ω

d(t, x).

Ω

For any given φ ∈ Cτ+ , let (u1 (t, x), u2 (t, x)) := (u1 (t, φ)(x), u2 (t, φ)(x)), t ≥ 0, x ∈ Ω and u¯ i (t) = Ω ui (t, x)dx, i = 1, 2. Integrating the first equation in (2.5), by the Green’s formula, we obtain d u¯ 1 (t) = dt



Λ(t, x)dx − Ω

d(t, x)u1 (t, x)dx −

Ω



≤ Λ0 − d0 u¯ 1 (t) −

h(t, x)u1 (t, x)u2 (t, x)dx Ω

h(t, x)u1 (t, x)u2 (t, x)dx, Ω

that is,

h(t, x)u1 (t, x)u2 (t, x)dx ≤ Λ0 − d0 u¯ 1 (t) − Ω

d u¯ 1 (t) , dt

t > 0.

(2.12)

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11

Let β0 := mint∈[0,ω],x∈Ω β(t, x). By (2.12), the Green’s formula and the property of the fundamental solutions (see [12]), integrating the second equation of (2.5) yields d u¯ 2 (t) d u¯ 1 (t − τ ) ≤ −β0 u¯ 2 (t) − k1 u¯ 1 (t − τ ) − k2 + k3 , dt dt

∀t ≥ l1 ω + τ,

(2.13)

where k1 , k2 and k3 are some positive numbers independent of φ. We can choose k1 ≤ β0 k2 in (2.13), so that  d u¯ 2 (t) + k2 u¯ 1 (t − τ ) ≤ −β0 u¯ 2 (t) − k1 u¯ 1 (t − τ ) + k3 dt k1 ≤ − u¯ 2 (t) − k1 u¯ 1 (t − τ ) + k3 k2  k1  u¯ 2 (t) + k2 u¯ 1 (t − τ ) + k3 , ≤− k2

∀t ≥ l1 ω + τ,

(2.14)

which yields u¯ 2 (t) + k2 u¯ 1 (t − τ ) ≤ kk2 k1 3 + 1 for t ≥ l1 ω + τ , where l1 > l1 is some integer. Since Γ (t, t − τ, x, y) and u1 are bounded, it follows from the second equation in (2.5) that   ∂u2 (t, x) ≤ ∇ · D1 (t, x)∇u2 (t, x) − β0 u2 (t, x) + cu¯ 2 (t) ∂t for some constant c > 0. By the standard parabolic maximum principle, there exist a positive number B2 independent of the initial value φ, and a positive integer l2 = l2 (φ) > l1 (φ) such that u2 (t, x; φ) ≤ B2 for any t ≥ l2 ω + τ and x ∈ Ω. Therefore, we have t˜φ = ∞ for each φ ∈ Cτ+ . Define a family of operators {Φt }t≥0 on Cτ+ by Φt (φ)(s, x) = ut (s, x; φ) = u(t + s, x; φ) for t ≥ 0, s ∈ [−τ, 0], x ∈ Ω and φ ∈ Cτ+ . Similar to the proof of Lemma 2.1, we can show that {Φt }t≥0 is an ω-periodic semiflow on Cτ+ . From the above proofs, we conclude that Φt : Cτ+ → Cτ+ is point dissipative. Moreover, Φω : Cτ+ → Cτ+ is κ-contraction (see Lemma 4.1), and hence, Φω is asymptotically smooth. By [14, Theorem 2.4.7] (see also [43, Theorem 1.1.2]), it follows that Φω : Cτ+ → Cτ+ has a global compact attractor. 2 3. The basic reproduction number Let Cω (R, Y) be the ordered Banach space consisting of all ω-periodic and continuous functions from R to Y, where ψCω (R,Y) := maxθ∈[0,ω] ψ(θ )Y for any ψ ∈ Cω (R, Y). Denote Cω+ (R, Y) as the positive cone of Cω (R, Y), namely,

 Cω+ (R, Y) := φ ∈ Cω (R, Y) : φ(t)(x) ≥ 0, ∀t ∈ R, x ∈ Ω . For τ ≥ 0, define E = C([−τ, 0], Y) with the norm ϕ := maxθ∈[−τ,0] ϕ(θ )Y , ∀ϕ ∈ E , and E + := C([−τ, 0], Y + ). Then (E, E + ) is a strongly ordered Banach space. Setting u2 = 0 in (2.5), we obtain the following equation for the density u1 (t, x) of susceptible host population:

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⎧   ∂u1 (t, x) ⎪ ⎪ = ∇ · D1 (t, x)∇u1 (t, x) + Λ(t, x) − d(t, x)u1 (t, x), ⎨ ∂t t > 0, x ∈ Ω, ⎪ ⎪  ⎩ D1 (t, x)∇u1 (t, x) · n = 0, t > 0, x ∈ ∂Ω.

(3.1)

According to Lemma 2.1, it is easy to see that Eq. (3.1) admits a positive solution u∗1 (t, x) which is unique, globally asymptotically stable and ω-periodic in t ∈ R. Generally, the function (u∗1 , 0) is called the disease-free periodic solution of (2.5). Linearizing system (2.5) at (u∗1 , 0), we obtain the following periodic time-delayed nonlocal equation for the infectious component: ⎧   ∂w(t, x) ⎪ ⎪ = ∇ · D2 (t, x)∇w(t, x) − β(t, x)w(t, x) ⎪ ⎪ ∂t ⎪

⎪ ⎪ ⎪ ⎪ + Γ (t, t − τ, x, y)h(t − τ, y)u∗1 (t − τ, y)w(t − τ, y)dy, ⎨ Ω ⎪ ⎪ ⎪ t > 0, x ∈ Ω,  ⎪  ⎪ ⎪ ⎪ D2 (t, x)∇w(t, x) · n = 0, t > 0, x ∈ ∂Ω, ⎪ ⎪ ⎩ w(s, x) = ϕ(s, x), ϕ ∈ E, s ∈ [−τ, 0], x ∈ Ω.

(3.2)

As discussed in Section 2, there exist positive constant M ≥ 1 and c ∈ R such that V (t, s) ≤ Mec(t−s) ,

∀t ≥ s, t, s ∈ R.

(3.3)

It then follows that c∗ := ω(V ¯ ) ≤ c, where ω(V ¯ ) is the exponential growth bound of the evolution family V and defined by 

 ˜ . ω(V ¯ ) = inf ω˜  ∃M ≥ 1 : ∀s ∈ R, t ≥ 0 : V (t + s, s) ≤ Meωt Note that the evolution operator V (t, s) is compact and strongly positive operator on Y for t > s. The Krein–Rutman theorem [15, Theorem 7.2] implies that r(V (ω, 0)) > 0. It then follows from [15, Lemma 14.2] that  r V (ω, 0) < 1.

(3.4)

Thus, [36, Proposition 5.6] with s = 0 indicates that c∗ = ω(V ¯ ) < 0. Furthermore, similar to Section 2 or [19, Section 3], we know that (3.2) has a unique mild solution w(t, x; ϕ) with w0 (·, ·; ϕ) = ϕ and wt (·, ·; ϕ) ∈ E + := C([−τ, 0], Y+ ) for all t ≥ 0. Moreover, w(t, x; ϕ) is a classic solution when t > τ . Define P : E → E by P (ϕ) = wω (ϕ) for all ϕ ∈ E , where wω (ϕ)(s, x) = w(ω + s, x; ϕ) for all (s, x) ∈ [−τ, 0] × Ω, and wt is the solution map of (3.2). Let r0 = r(P ) be the spectral radius of P . By similar arguments to [19, Section 3] (see also [35, Section 5.3]), we can show that w(t, x; ϕ) > 0 for t > τ , x ∈ Ω, ϕ ∈ E + with ϕ ≡ 0, and wt (·, ·; ϕ) is strongly positive for t > 2τ . Moreover, wt is compact on E + for all t > 2τ . Thus, there is an integer n0 ≥ 2τ ω , such that P n0 = wn0 ω is compact and strongly positive. It then follows from [24, Lemma 3.1] that r0 ¯ and the modulus of any is a simple eigenvalue of P having a strongly positive eigenvector φ, ¯ ¯ = φ(s, ¯ x) other eigenvalue is less than r0 . Let w(t, x; φ) be the solution of (3.2) with w(s, x; φ) ¯ ¯ for all s ∈ [−τ, 0], x ∈ Ω. We can conclude from the strong positivity of φ that w(·, ·; φ)  0.

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¯ for all t > −τ, x ∈ Ω. By arguments similar to those Let μ = lnωr0 and v∗ (t, x) = e−μt w(t, x; φ) in [41, Theorem 2.1] (see also [19, Lemma 3.2]), we have the following observation. Lemma 3.1. Let μ = lnωr0 . Then there exists a positive ω-periodic function v∗ (t, x) such that eμt v∗ (t, x) is a solution of (3.2). Suppose that φ(s, x) = φ(s)(x) ∈ Cω (R, Y+ ) is the initial distribution of infectious individuals at time s ∈ R and the spatial location x ∈ Ω. Define an operator C(t) : Y → Y as 





C(t)ψ (x) :=

Γ (t, t − τ, x, y)h(t − τ, y)u∗1 (t − τ, y)ψ(y)dy,

ψ ∈ Y.

Ω

Give t ∈ R. Due to the synthetical influences of mobility, mortality and recovery, V (t − τ, s)φ(s)(x) represents the density distribution at location x of those infective individuals who were infective at time s (s < t − τ ) and remain infective at time t − τ when  t−τ time evolved from s to t − τ . Consequently, −∞ V (t − τ, s)φ(s)(x)ds denotes the density distribution of the accumulative infective individuals at location x and time t − τ for all previous time s < t − τ when  t−τ time evolved from the previous time s to t − τ . Thus, the term h(t − τ, x)u∗1 (t − τ, x) −∞ V (t − τ, s)φ(s)(x)ds represents new infected individuals due to the  t−τ infective distribution −∞ V (t − τ, s)φ(s)(x)ds at location x and time t − τ . In view of the latent period τ , the term

Γ (t, t − τ, x, y)h(t

− τ, y)u∗1 (t

 t−τ  − τ, y) V (t − τ, s)φ(s)(y)ds dy −∞

Ω

=

Γ (t, t − τ, x, y)h(t

− τ, y)u∗1 (t

 ∞  − τ, y) V (t − τ, t − s)φ(t − s)(y)ds dy τ

Ω

∞

=

Γ (t, t − τ, x, y)h(t − τ, y)u∗1 (t − τ, y)V (t − τ, t − s)φ(t − s)(y)dyds

τ Ω

=

∞ 

  C(t) V (t − τ, t − s)φ(t − s) (x)ds

τ

=

∞ 

 K(t, s)φ(t − s) (x)ds

0

denotes the distribution of new infected individuals at location x and time t , where K(t, s), t ∈ R, s ≥ 0, is defined by  K(t, s) :=

C(t)V (t − τ, t − s), 0,

s > τ, s ∈ [0, τ ].

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[m1+; v1.201; Prn:19/01/2015; 11:14] P.14 (1-26)

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Therefore, we can define the next generation infection operator L as

∞ L(φ)(t) :=

K(t, s)φ(t − s)ds,

∀t ∈ R, φ ∈ Cω (R, Y).

(3.5)

0

It is easy to see that L is a positive and bounded linear operator on Cω (R, Y). Motivated by [11,38,5,39,36,44], we define the spectral radius of L to be the basic reproduction number for the model (2.5), that is, R0 := r(L).

(3.6)

Next, we introduce an operator Lˆ : Cω (R, Y) → Cω (R, Y), ˆ L(φ)(t) =

∞

ˆ s)φ(t − s)ds, K(t,

0

ˆ s) is given by where K(t,  V (t, t − s + τ )C(t − s + τ ), ˆ K(t, s) := 0,

s > τ, s ∈ [0, τ ].

ˆ s), t ∈ R, s ≥ 0, is a compact, positive and bounded linear operator. Let A and B Clearly, K(t, be two bounded linear operators on Cω (R, Y) defined by

∞ A(φ)(t) = C(t)φ(t − τ ),

B(φ)(t) =

V (t, t − s + τ )φ(t − s + τ )ds. τ

ˆ where r(L) ˆ is the spectral radius Since L = AB and Lˆ = BA, it follows that R0 = r(L) = r(L), ˆ of the operator L. In the rest of this section, we adopt the general procedure presented in [44, Section 2] to prove that R0 − 1 has the same sign as r0 − 1. For any given λ ∈ (c∗ , ∞), we introduce an operator Lˆ λ on Cω (R, Y): Lˆ λ (φ)(t) =

∞

ˆ s)φ(t − s)ds, e−λs K(t,

∀t ∈ R, φ ∈ Cω (R, Y).

0

ˆ In view of (3.3), it follows that operator Lˆ λ is bounded for λ ∈ (c∗ , ∞). MoreClearly, Lˆ 0 = L. over, the compactness of V (t, s), t > s, implies that Lˆ λ is compact. Let ρ(λ) be the spectral ˆ = ρ(0). In what follows, radius of Lˆ λ for λ ∈ (c∗ , ∞). It is easy to see that R0 = r(L) = r(L) we prove some properties of the function ρ(λ). Lemma 3.2. For λ ∈ (c∗ , +∞), the following statements are valid for ρ(λ).

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(1) ρ(λ) is continuous and non-increasing; (2) ρ(∞) = 0; (3) ρ(λ) = 1 has at most one solution; ρ(λ) is either strictly decreasing in λ ∈ (c∗ , ∞), or strictly decreasing in λ ∈ (c∗ , b) for some b > c∗ , and ρ(λ) = 0 in λ ∈ [b, ∞). ˆ λ. Proof. The statement (1) easily follows from the positivity and compactness of L Let VL (t, s) be the evolution operator determined by the following reaction–diffusion equation ⎧ ⎨ ∂z(t, x) = ∇ · D (t, x)∇z(t, x) − β (t, x)z(t, x), L L ∂t  ⎩ DL (t, x)∇z(t, x) · n = 0,

t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω

and c1∗ := ω(V ¯ L ) be the exponential growth bound of VL . In fact, in terms of the definition of V (t, s) and C(t), ρ(λ) ≤ Lˆ λ Cω (R,Y) ≤ M  ∗

1 , λ − c∗

∗

where M  = M · e(c1 −c )τ · Hˆ and Hˆ := maxt∈[0,ω], x∈Ω [h(t, x)u∗1 (t, x)]. Letting λ → ∞, it follows that ρ(∞) = 0. By arguments similar to those in the proof of [4, Lemma 1(iv)], it follows that ρ(λ) is logconvex. Therefore, by the monotonicity of ρ(λ) and the conclusion that ρ(∞) = 0, we see that statement (3) is valid. 2 Lemma 3.3. Let μ =

ln r0 ω .

If r0 > r(V (ω, 0)), then ρ(μ) = 1.

Proof. According to Lemma 3.1, there is a positive periodic function v∗ (t, x) such that w ∗ (t, x) := eμt v∗ (t, x) is a solution of (3.2). Thus, w ∗ (t, x) satisfies  w (t, ·) = V (t, s)w (s) + ∗



∗

t



 V (t, η)C(η)w ∗ (η − τ ) dη,

∀t ≥ s, s ∈ R,

s

and hence,   e v∗ (t, ·) = V (t, s) eμs v∗ (s) + μt



t



  V (t, η)C(η) eμ(η−τ ) v∗ (η − τ ) dη,

s

∀t ≥ s, s ∈ R.

(3.7)

In view of [36, Proposition A.2], we have c∗ = ω(V ¯ ) = ln r(Vω(ω,0)) . Since r0 > r(V (ω, 0)), it fol∗ lows from (3.4) that μ > c , and hence, [V (t, s)(eμs v∗ (s))] → 0, as s → −∞. Letting s → −∞ in (3.7), we get

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t

  e−μ(t−η+τ ) V (t, η)C(η)v∗ (η − τ ) dη

v∗ (t, ·) = −∞

∞ =

  e−μs V (t, t − η + τ )C(t − η + τ )v∗ (t − η) dη

τ

∞ =

ˆ η)v∗ (t − η)dη, e−μs K(t,

0

which implies that Lˆ μ (v∗ )(t)(·) = v∗ (t, ·). Since h(t, x) is nonnegative on (t, x) ∈ [0, ω] × Ω, it follows that Lˆ μ may not be strongly positive. In order to show that the spectral radius of Lˆ μ is 1, we use a perturbation argument similar to that in [44, Proposition 2.2]. For any given  > 0, we first consider the following periodic time-delayed nonlocal equation ⎧   ∂w(t, x) ⎪ ⎪ = ∇ · D2 (t, x)∇w(t, x) − β(t, x)w(t, x) ⎪ ⎪ ∂t ⎪

⎪ ⎪  ⎪ ⎪ + Γ (t, t − τ, x, y) h(t − τ, y) +  u∗1 (t − τ, y)w(t − τ, y)dy, ⎨ Ω ⎪ ⎪ ⎪ t > 0, x ∈ Ω, ⎪ ⎪ ⎪ D (t, x)∇w(t, x) · n = 0, t > 0, x ∈ ∂Ω, ⎪ ⎪ 2 ⎪ ⎩ w(s, x) = ϕ(s, x), ϕ ∈ E, s ∈ [−τ, 0], x ∈ Ω.

(3.8)

By analysis similar to (3.2), we define the Poincaré map of (3.8) P  : E → E by P  (ϕ) = wω (ϕ) for all ϕ ∈ E, where wω (ϕ)(s, x) = w  (ω + s, x; ϕ) for all (s, x) ∈ [−τ, 0] × Ω, and wt is the solution map of (3.8). By the continuity of solution with respect to parameter , we have lim→0 P  = P . Let r0 = r(P  ) be the spectral radius of P  . By the upper semicontinuity of the spectrum [20, Section IV.3.1] and the continuity of a finite system of eigenvalues [20, ln r  Section IV.3.5], it follows that lim→0+ r0 = r0 . Let μ = ω 0 . By Lemma 3.1, there is a pos itive periodic function v∗ (t, x) such that eμ t v∗ (t, x) is a solution of (3.8), and lim→0+ μ = ln r 

lim→0+ ω 0 = μ. Let n = n1 , n ≥ 1. We define 

C n (t)ψ(t − τ ) (x) :=



 Γ (t, t − τ, x, y) h(t − τ, y) + n u∗1 (t − τ, y)ψ(t − τ, y)dy,

Ω

Lˆ n (φ)(t) =

∞ 0

and

Kˆ n (t, s)φ(t − s)(·)ds

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Lˆ λn (φ)(t) =

∞

17

e−λs Kˆ n (t, s)φ(t − s)(·)ds,

0

where Kˆ n (t, s) is given by  V (t, t − s + τ )C n (t − s + τ ), n ˆ K (t, s) := 0,

s > τ, s ∈ [0, τ ]. ln r n

Applying the same method as above, it follows that there are μn = ω0 and a positive periodic function v∗n (t, x) such that Lˆ μn v∗n (t)(·) = v∗n (t, ·). Denote ρ n (λ) as the spectral radius of Lˆ λn for λ ∈ (c∗ , ∞). Since h(t, x) + n > 0 on R × Ω, it follows that Lˆ λn : Cω (R, Y) → Cω (R, Y) is continuous, compact and strongly positive. Thus, the positivity of v∗n and the Krein– Rutman theorem (see, e.g., [15, Theorem 7.2]) implies that ρ n (μn ) = 1. It is easy to see that  Lˆ λn ψ ≥ Lˆ λn+1 ψ for all ψ ∈ Cω+ . Let gn (λ) = ρ n (λ). It then follows from [8, Theorem 1.1] that the sequence {gn }n≥1 is nonincreasing. By the upper semicontinuity of the spectrum [20, Section IV.3.1] and the continuity of a finite system of eigenvalues [20, Section IV.3.5], we see that for any fixed λ ∈ [a, b] ⊂ (c∗ , ∞), limn→∞ ρ n (λ) = ρ(λ). Hence, Dini’s theorem implies that limn→∞ ρ n (λ) = ρ(λ) uniformly for λ ∈ [a, b]. Choose a sufficient small δ > 0 such that μ − δ > c∗ . By the above analysis, it follows that there exists an N1 = N1 (δ) ≥ 1 such that for any n ≥ N1 , μ − δ ≤ μ n =

ln r0n ≤ μ + δ. ω

On one hand, we see from the continuity of ρ(λ) for λ ∈ (c∗ , ∞) that for any η > 0, when n ≥ N1 ,     ρ μ n − ρ(μ) < η/2. On the other hand, for any η > 0, there is an N2 ≥ 1 such that when n ≥ N := max{N1 , N2 },       ρ n μ n − ρ μn  < η/2. Thus, for any η > 0, we have             ρ n μ n − ρ(μ) ≤ ρ n μn − ρ μn  + ρ μn − ρ(μ) ≤ η/2 + η/2 ≤ η, when n ≥ N . Letting n → ∞, we have ρ n (μn ) → ρ(μ). Therefore, ρ(μ) = 1. We are now in a position to prove the main result of this section. Theorem 3.4. The following statements are valid: (i) R0 > 1 if and only if r0 > 1.

2

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(ii) R0 = 1 if and only if r0 = 1. (iii) R0 < 1 if and only if r0 < 1. Proof. (i) (a) If R0 > 1, then ρ(0) > 1. According to Lemma 3.2(1) and (2), there is a λ0 > 0 such that ρ(λ0 ) = 1. Since Lˆ λ0 is compact and positive on Cω (R, Y), it follows from the Krein– Rutman theorem (see, e.g., [15, Theorem 7.1]) that ρ(λ0 ) is an eigenvalue of Lˆ λ0 , with a positive eigenfunction ϕ ∗ ∈ Cω (R, Y), that is, Lˆ λ0 ϕ ∗ = ϕ ∗ . Since V (t, s) = V (t, r)V (r, s), ∀t ≥ r ≥ s, we have  Lˆ λ0 ϕ ∗ (t) =

∞

e−λ0 s V (t, t − s + τ )C(t − s + τ )ϕ ∗ (t − s)ds

τ

t =

e−λ0 (t−s+τ ) V (t, s)C(s)ϕ ∗ (s − τ )ds

−∞

r =

e−λ0 (t−s+τ ) V (t, s)C(s)ϕ ∗ (s − τ )ds

−∞

t +

e−λ0 (t−s+τ ) V (t, s)C(s)ϕ ∗ (s − τ )ds

r

=e

−λ0 (t−r)

r V (t, r)

e−λ0 (r−s+τ ) V (r, s)C(s)ϕ ∗ (s − τ )ds

−∞

+ e−λ0 t

t

eλ0 (s−τ ) V (t, s)C(s)ϕ ∗ (s − τ )ds,

∀t ≥ r, r ∈ R.

r

It then follows that ∗

ϕ (t) = e

−λ0 t

 V (t, r) eλ0 r ϕ ∗ (r) + e−λ0 t

t

 V (t, s)C(s) eλ0 (s−τ ) ϕ ∗ (s − τ ) ds,

r

that is,

e

 ϕ (t) = V (t, r) eλ0 r ϕ ∗ (r) +

λ0 t ∗

t

 V (t, s)C(s) eλ0 (s−τ ) ϕ ∗ (s − τ ) ds,

r

∀t ≥ r, r ∈ R.

(3.9)

Let ϕt∗ (θ, ·) = ϕ ∗ (t + θ, ·), ∀θ ∈ [−τ, 0]. It is easy to see from (3.9) that w(t, x) := eλ0 t ϕ ∗ (t, x) is a solution of (3.2) with w0 = eλ0 · ϕ0∗ . Note that wt (θ, ·) = w(t + θ, ·) = eλ0 (t+θ) ϕ ∗ (t + θ, ·) =

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eλ0 t (eλ0 θ ϕt∗ (θ, ·)) for θ ∈ [−τ, 0]. Since w(t, ·) ≡ 0 on [0, ∞), we have eλ0 · ϕ0∗ ∈ E + \ {0}. Due to the ω-periodicity of ϕ ∗ (t, ·), we conclude that    P eλ0 · ϕ0∗ = wω = eλ0 ω eλ0 · ϕω∗ = eλ0 ω eλ0 · ϕ0∗ , and hence, eλ0 ω is an eigenvalue of P . Thus, r0 ≥ eλ0 ω > 1. (b) If r0 > 1, then μ > 0. In view of (3.4), it follows from Lemma 3.3 that ρ(μ) = 1. By the strict monotonicity of ρ(λ) (see Lemma 3.2(3)), we have 1 = ρ(μ) < ρ(0) = R0 . ˆ = ρ(0) = 1 > 0. By similar arguments to the proof of conclusion (ii) (a) If R0 = 1, then r(L) (i) with λ0 = 0, we can prove that r0 ≥ 1. In view of (3.4), we see that r0 ≥ 1 > r(V (0, ω)). It then follow from Lemma 3.3 that ρ(μ) = 1. As the solution of ρ(λ) = 1 is unique (see Lemma 3.2(3)), we obtain that μ = 0, and hence r0 = 1. (b) If r0 = 1, then we can conclude from (3.4) that r0 > r(V (ω, 0)). Thus, by Lemma 3.3, we have ρ(μ) = ρ(0) = R0 = 1. (iii) is a straightforward consequence of conclusions (i) and (ii). 2 4. Threshold dynamics In this section, we establish the threshold dynamics of system (2.5) in terms of the basic reproduction number R0 . For any (σ, φ) ∈ R × Cτ+ , system (2.8) admits a unique solution u(t, σ, φ)(x) on [σ − τ, ∞) with u(s, σ, φ)(x) = φ(s, x), ∀s ∈ [σ − τ, σ ), x ∈ Ω. Define [T(t, σ )φ](θ, x) = u(t + σ + θ, σ, φ)(x), ∀−τ ≤ θ ≤ 0, 0 ≤ t < ∞, that is, the solution in Cτ+ through (σ, φ) at time t + σ . Thus, system (2.8) define a process on R+ × R × Cτ+ , for all t ≥ 0, σ ∈ R and φ ∈ Cτ+ (see [14, Section 3.6]). Due to the ω-periodicity of F(t, φ), it follows that T(t, σ ), t ≥ 0, σ ∈ R is ω-periodic process, that is, T(t, σ + ω) = T(t, σ ) for t ≥ 0, σ ∈ R. Recall that the Kuratowski measure of noncompactness on the Banach space Cτ+ (see [10]), κ, is defined by κ(B) := inf{d : B has a finite cover of diameter < d},

(4.1)

for any bounded set B of Cτ+ . We set κ(B) = ∞ whenever B is unbounded. It is easy to see that B is precompact if and only if κ(B) = 0. By [14, Theorems 4.1.1 and 4.1.11] and [25, Section 4], we have the following result. Lemma 4.1. Suppose T(t, 0) be defined above for (2.8). Then T(t, 0) = V(t) + U(t, 0), where the operator V(t) : Cτ+ → Cτ+ is defined  V(t)[φ](θ, x) =

φ(t + θ, x) − φ(0, x), 0,

t + θ < 0, t +θ ≥0

and the operator U(t, 0) : Cτ+ → Cτ+ is defined  U(t, 0)[φ](θ, x) =

φ(0, x),  t+θ U(t + θ, 0)φ(0, x) + 0 U(ι, 0)F(ι, T(ι, 0)φ)dι,

t + θ < 0, t + θ ≥ 0.

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Furthermore, for any α > 0, there is an equivalent norm | · |∗ so that |V(t)|∗ ≤ e−αt , t ≥ 0 and {T (t, 0)}t≥0 is a κ-contraction in this norm. Before proving the main result on the disease persistence, we need the following lemma. Lemma 4.2. Suppose (u1 (t, x; φ), u2 (t, x; φ)) is the solution of system (2.5) with initial value φ = (φ1 , φ2 ) ∈ Cτ+ . (i) If there exists some t0 ≥ 0 such that u2 (t0 , ·; φ) ≡ 0, then u2 (t, x; φ) > 0 for all t > t0 and x ∈ Ω. (ii) For any φ ∈ Cτ+ , we have u1 (t, ·; φ) > 0, ∀t > 0 and lim inf u1 (t, x; φ) ≥ η t→∞

uniformly for x ∈ Ω, where η is a positive constant. Proof. From Theorem 2.2 and the second equation of (2.5), it is easy to see that u2 (t, x; φ) satisfies ⎧ ⎨ ∂u2 (t, x) ≥ ∇ · D (t, x)∇u (t, x) − β(t, x)u (t, x), 2 2 2 ∂t  ⎩ D2 (t, x)∇u2 (t, x) · n = 0,

t > 0, x ∈ Ω,

(4.2)

t > 0, x ∈ ∂Ω.

If u2 (t0 , ·; φ) ≡ 0 for some t0 ≥ 0, it then follows from the maximum principle (Hess [15, Proposition 13.1]) that u2 (t, x; φ) > 0 for all t > t0 , x ∈ Ω. By Theorem 2.2, there is a constant B > 0, such that u2 (t, x) ≤ B for t > 0 and x ∈ Ω. Let v(t, x; φ) be the solution of ⎧ ∂v(t, x)     ⎪ = ∇ · D1 (t, x)∇v(t, x) + Λ(t, x) − d(t, x) + Bh(t, x) v(t, x), ⎪ ⎪ ⎪ ⎨ ∂t t > 0, x ∈ Ω,   ⎪ ⎪ D1 (t, x)∇v(t, x) · n = 0, t > 0, x ∈ ∂Ω, ⎪ ⎪ ⎩ v(0, x) = φ1 (0, x), x ∈ Ω.

(4.3)

In view of the comparison principle, it follows that u1 (t, x; φ) ≥ v(t, x; φ1 ) > 0,

∀t > 0, x ∈ Ω.

Furthermore, by Lemma 2.1 and the maximum principle, we have lim inf u1 (t, x; φ) ≥ t→∞

inf

v ∗ (t, x)

t∈[0,ω], x∈Ω

uniformly for x ∈ Ω, where v ∗ (t, x) is the unique positive ω-periodic solution of (4.3). Thus, part (ii) is proved. 2

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The following result indicates that R0 is a threshold index for disease extinction or persistence. Theorem 4.3. Let u(t, x; φ) be the solution of (2.5) with u0 = φ ∈ Cτ+ . Then the following statements hold: (i) If R0 < 1, then the disease free ω-periodic solution (u∗1 (t, x), 0) is globally attractive in Cτ+ ; (ii) If R0 > 1, then system (2.5) admits at least one positive ω-periodic solution (u˜ 1 (t, x), u˜ 2 (t, x)), and there exists a δ > 0 such that for any φ ∈ Cτ+ with φ2 (0, ·) ≡ 0, we have lim inf ui (t, x; φ) ≥ δ, t→∞

∀i = 1, 2

uniformly for all x ∈ Ω. Proof. (i) In the case where R0 < 1, it follows from Theorem 3.4(iii) that r0 < 1, and hence, μ = lnωr0 < 0. Consider the following equation with parameter ε > 0: ⎧ ε   ∂u (t, x) ⎪ ⎪ = ∇ · D2 (t, x)∇uε (t, x) − β(t, x)uε (t, x) ⎪ ⎪ ∂t ⎪ ⎪

⎪ ⎪  ⎪ ⎪ + Γ (t, t − τ, x, y)h(t − τ, y) u∗1 (t − τ, y) + ε uε (t − τ, y)dy, ⎨ (4.4)

Ω ⎪ ⎪ ⎪ t ≥ kω, x ∈ Ω, ⎪ ⎪   ⎪ ⎪ ε ⎪ ⎪ ⎪ D2 (t, x)∇u (t, x) · n = 0, t ≥ kω, x ∈ ∂Ω, ⎩ uε (s, x) = ϕ(s, x), ϕ ∈ E, s ∈ [−τ, 0], x ∈ Ω.

Define the Poincaré map of (4.4) Pε : E → E by Pε (ϕ) = uεω (ϕ)

∀ϕ ∈ E,

where uεω (ϕ)(s, x) = uε (ω + s, x; ϕ) ∀(s, x) ∈ [−τ, 0] × Ω and uε (t, x; ϕ) is the solution of (4.4) with uε (s, x) = ϕ(s, x) for all s ∈ [−τ, 0], x ∈ Ω. Let rε = r(Pε ) be the spectral radius of Pε . Thus, we can conclude from r0 < 1 that there exists a sufficiently small positive number ε1 such that rε < 1 for all ε ∈ [0, ε1 ). We fix an ε ∈ (0, ε1 ). Then, we have με = lnωrε < 0. According to Lemma 3.1, there is an ω-periodic function v∗ε (t, x) such that uε (t, x) = eμε t v∗ε (t, x) is a solution of (4.4). In particular, v∗ε (t, x) > 0 for any t ∈ R and x ∈ Ω. On the other hand, it is easy to see that u1 (t, x) satisfies ⎧ ⎨ ∂u1 (t, x) ≤ ∇ · D (t, x)∇u (t, x) + Λ(t, x) − d(t, x)u (t, x), 1 1 1 ∂t  ⎩ D1 (t, x)∇u1 (t, x) · n = 0,

t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω.

(4.5)

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According to Lemma 2.1 and the comparison principle, it follows that there is an integer k > 0 such that u1 (t, x; φ) ≤ u∗1 (t, x) + ε, ∀t ≥ kω, x ∈ Ω. Therefore, for all t ≥ kω, we have ⎧   ∂u2 (t, x) ⎪ ⎪ ≤ ∇ · D2 (t, x)∇u2 (t, x) − β(t, x)u2 (t, x) ⎪ ⎪ ⎪ ∂t ⎪

⎪ ⎪  ⎨ + Γ (t, t − τ, x, y)h(t − τ, y) u∗1 (t − τ, y) + ε u2 (t − τ, y)dy, ⎪ ⎪ Ω ⎪ ⎪ ⎪ t ≥ kω, x ∈ Ω, ⎪ ⎪ ⎪  ⎩ D2 (t, x)∇u2 (t, x) · n = 0, t ≥ kω, x ∈ ∂Ω.

(4.6)

For any given φ ∈ Cτ+ , since u2 (t, x; φ) is globally bounded, there exists some α > 0 such that u2 (t, x; φ) ≤ αeμε t v∗ε (t, x), ∀t ∈ [kω, kω + τ ], x ∈ Ω. Similar to [19, Section 2], using (4.6), (4.4) and the comparison theorem for abstract functional differential equation [30, Proposition 3], we have u2 (t, x; φ) ≤ αeμε t v∗ε (t, x), ∀t ≥ kω + τ , which further implies limt→∞ u2 (t, x; φ) = 0 uniformly for x ∈ Ω. Then, the equation for u1 in system (2.5) is asymptotic to system (2.11). Lemma 2.1 implies that u∗1 (t, x) is a global attractive solution of (2.11). Next, we use the theory of chain transitive sets (see, e.g., [17] or [43]) to prove that limt→∞ u1 (t, x; φ) = u∗1 (t, x) uniformly for x ∈ Ω. Let P = Φω and J =  (φ) be the omega limit set of φ ∈ Cτ+ for P, that is, J=



  φ1∗ , φ2∗ ∈ Cτ+ : ∃{ki } → ∞ such that lim P ki (φ1 , φ2 ) = φ1∗ , φ2∗ . i→∞

It then follows from [17, Lemma 2.1] (see also [43, Lemma 1.2.1]) that J is an internally chain ˆ transitive set for P. Since limt→∞ u2 (t, x; φ) = 0 uniformly for x ∈ Ω, we have J = J1 × {0}. ˆ By Lemma 4.2, we know 0 ∈ / J1 . For any ϕ ∈ E + , let w(t, x; ϕ(0, ·)) be the solution of (2.11) with initial value w(0, x) = ϕ(0, x). Define  wt (θ, x; ϕ) =

w(t + θ, x; ϕ(0)) t + θ > 0, t > 0, θ ∈ [−τ, 0], ϕ(t + θ, x) t + θ ≤ 0, t > 0, θ ∈ [−τ, 0].

Then wt defines a solution semiflow of (2.11) on E + . Let P¯ = wω (ϕ). It then follows from Lemma 2.1 that  (ϕ) = {u∗1,0 }, where  (ϕ) denotes the omega limit set for P¯ , and u∗1,0 ∈ E + ˆ ≡ 0, is defined by u∗ (θ, ·) = u∗ (θ, ·) for θ ∈ [−τ, 0]. Since P(J ) = J and u2 (t, x; (φ1 , 0)) 1,0

1

ˆ = J1 × {0}, ˆ and hence, P¯ (J1 ) = J1 . Consequently, J1 is an we have P(J ) = P¯ (J1 ) × {0} ¯ internally chain transitive set for P . According to Lemma 2.1 and above discussion, we know that {u∗1,0 } is globally attractive in E + . It then follows from [17, Theorem 3.1] (see also [43, ˆ By the definition of J , we have Theorem 1.2.1]) that J1 = {u∗ }. Thus, J = {(u∗ , 0)}. 1,0

1,0

  lim u1 (t, ·; φ), u2 (t, ·; φ) − u∗1 (t, ·), 0 = 0.

t→∞

(ii) In the case where R0 > 1, by Theorem 3.4(ii), we have r0 > 1, and hence, μ = Let

ln r0 ω

> 0.

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 W0 = φ ∈ Cτ+ : φ2 (0, ·) ≡ 0 , and

 ∂W0 = Cτ+ \ W0 = φ ∈ Cτ+ : φ2 (0, ·) ≡ 0 . By Lemma 4.2, we know that u2 (t, x; φ) > 0 for any φ ∈ W0 , t > 0 and x ∈ Ω. It follows that Φωk (W0 ) ⊆ W0 , ∀k ∈ N. Let

 M∂ := φ ∈ ∂W0 : Φωk (φ) ∈ ∂W0 , ∀k ∈ N ˆ and J (φ) be the omega limit set of the orbit γ + (φ) := {Φωk (φ) : ∀k ∈ N}. Set M = (u∗1,0 , 0). k For any given ψ ∈ M∂ , we have Φω (ψ) ∈ ∂W0 , ∀k ∈ N. Thus u2 (t, ·; ψ) ≡ 0, ∀t ≥ 0. Therefore, it follows from Lemma 2.1 that limt→∞ u1 (t, ·; ψ) − u∗1 (t, ·) = 0. Thus, we have J (ψ) = {M} for any ψ ∈ M∂ . Consider the following time-periodic parabolic system: ⎧ ρ   ∂u (t, x) ⎪ ⎪ = ∇ · D2 (t, x)∇uρ (t, x) − β(t, x)uρ (t, x) ⎪ ⎪ ∂t ⎪ ⎪

⎪  ⎪ ⎪ ⎪ + Γ (t, t − τ, x, y)h(t − τ, y) u∗1 (t − τ, y) − ρ uρ (t − τ, y)dy, ⎨ (4.7)

Ω ⎪ ⎪ ⎪ t > 0, x ∈ Ω, ⎪ ⎪   ⎪ ⎪ ⎪ D2 (t, x)∇uρ (t, x) · n = 0, t > 0, x ∈ ∂Ω, ⎪ ⎪ ⎩ ρ u (s, x) = ϕ(s, x), s ∈ [−τ, 0], x ∈ Ω, ϕ ∈ E. ρ

ρ

Define the Poincaré map of (4.7) Pρ : E → E, Pρ (ϕ) = uω (ϕ), where uω (ϕ)(s, x) = uρ (ω + s, x; ϕ) for (s, x) ∈ [−τ, 0] × Ω, and uρ (t, x; ϕ) is the solution of (4.7) with uρ (s, x) = ϕ(s, x) for all s ∈ [−τ, 0], x ∈ Ω. Since r0 > 1, there exists a sufficiently small positive number ρ1 such that rρ = r(Pρ ) > 1 for all ρ ∈ [0, ρ1 ), where r(Pρ ) is the spectral radius of Pρ . Fix a ρ¯ ∈ (0, ρ1 ). By the continuous dependence of solutions on the initial value, there exists ρ0 ∈ (0, ρ1 ) such that     u1 (t, x; φ), u2 (t, x; φ) − u∗ (t, x), 0  < ρ, ¯ 1

∀t ∈ [0, ω], x ∈ Ω,

if |φ(s, x) − (u∗1 (s, x), 0)| < ρ0 , ∀s ∈ [−τ, 0], x ∈ Ω. We now prove the following claim. Claim. M is a uniform weak repeller for W0 in the sense that lim sup Φωk (φ) − M ≥ ρ0

for all φ ∈ W0 .

k→∞

Suppose, by contradiction, there exists φ0 ∈ W0 such that lim sup Φωk (φ0 ) − M < ρ0 . k→∞

(4.8)

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Then there exists k0 ∈ N such that |u1 (kω + s, x; φ0 ) − u∗1 (kω + s, x)| < ρ0 and |u2 (kω + s, x; φ0 )| < ρ0 for all k ≥ k0 , s ∈ [−τ, 0] and x ∈ Ω. In view of (4.8), it follows that u1 (t, x; φ0 ) > u∗1 (t, x) − ρ¯ and 0 < u2 (t, x; φ0 ) < ρ¯

(4.9)

for any t > k0 ω and x ∈ Ω. In particular, u2 (t, x; φ0 ) satisfies ⎧   ∂u2 (t, x) ⎪ ⎪ ≥ ∇ · D2 (t, x)∇u2 (t, x) − β(t, x)u2 (t, x) ⎪ ⎪ ∂t ⎪ ⎪

⎪ ⎪  ⎨ + Γ (t, t − τ, x, y)h(t − τ, y) u∗1 (t − τ, y) − ρ¯ u2 (t − τ, y)dy, (4.10) ⎪ Ω ⎪ ⎪ ⎪ ⎪ t ≥ (k0 + 1)ω, x ∈ Ω, ⎪ ⎪ ⎪  ⎩ D2 (t, x)∇u2 (t, x) · n = 0, t ≥ (k0 + 1)ω, x ∈ ∂Ω. Let ψ¯ ∈ E be the positive eigenfunction of Pρ¯ associated with rρ¯ . Since u2 (t, x; φ0 ) > 0 for all t > τ , x ∈ Ω, there exists a ξ > 0 such that  ¯ u2 (k0 + 1)ω + s, x; φ0 ≥ ξ ψ,

∀s ∈ [−τ, 0], x ∈ Ω.

By (4.10) and the comparison principle, we have  u2 (t, x; φ0 ) ≥ ξ uρ¯ t − (k0 + 1)ω, x; ψ¯ ,

∀t ≥ (k0 + 1)ω, x ∈ Ω.

¯ = ξ(rρ¯ )(k−k0 −1) ψ(0, ¯ Therefore, we have u2 (kω, x; φ) = ξ uρ¯ ((k − k0 − 1)ω, x; ψ) x) → +∞ as k → +∞, which contradicts to (4.9). It follows from the above claim that M is an isolated invariant set for Φω in W0 , and W s (M) ∩ W0 = ∅, where W s (M) is the stable set of M. By appealing to the acyclicity theorem on uniform persistence for maps (see, e.g., [43, Theorem 1.3.1 and Remark 1.3.1]), we have that Φω : Cτ+ → Cτ+ is uniformly persistent with respect to (W0 , ∂W0 ), that is, there exists a δ˜ > 0 such that  ˜ lim inf d Φωk (φ), ∂W0 ≥ δ, k→∞

∀φ ∈ W0 .

It then follows from [43, Theorem 3.1.1] that the periodic semiflow Φt : Cτ+ → Cτ+ is also uniformly persistent with respect to (W0 , ∂W0 ). According to Lemma 4.1, we know that T(ω, 0) is a κ-contraction, and hence, P = T(ω, 0) is κ-condensing. Therefore, it follows from [28, Theorem 4.5] with ρ(x) = d(x, ∂W0 ) that P : W0 → W0 has a global attractor A0 and system (2.5) has an ω-periodic solution (u˜ 1 (t, ·), u˜ 2 (t, ·)) with (u˜ 1,t (·)(·), u˜ 2,t (·)(·)) ∈ W0 . In order to prove the practice uniform persistence in conclusion (ii), we use the arguments similar to [26, Theorem 4.1]. Define a continuous function p : Cτ+ → [0, ∞) by p(φ) := min φ2 (0, x), x∈Ω

∀φ = (φ1 , φ2 ) ∈ Cτ+ .

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Since A0 = P(A0 ) = Φω (A0 ), we have that φ2 (0, ·) > 0 for all φ ∈ A0 . Let B0 :=  t∈[0,ω] Φt (A0 ). It then follows that B0 ⊂ W0 and limt→∞ d(Φt (φ), B0 ) = 0 for all φ ∈ W0 . Since B0 is a compact subset of W0 , we have minφ∈B0 p(φ) > 0. Thus, there exists a δ ∗ > 0 such that lim inft→∞ u2 (t, ·; φ) ≥ δ ∗ . Furthermore, in view of Lemma 4.2, there exists a 0 < δ ≤ δ ∗ such that lim inf ui (t, ·; φ) ≥ δ, t→∞

∀φ ∈ W0 , i = 1, 2,

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