Threshold dynamics of a time periodic and two--group epidemic model with distributed delay.

In this paper, a time periodic and two--group reaction--diffusion epidemic model with distributed delay is proposed and investigated. We firstly introduce the basic reproduction number R0 for the model via the next generation operator method. We then establish the threshold dynamics of the model in terms of R0, that is, the disease is uniformly persistent if R0>1, while the disease goes to extinction if R0< 1. Finally, we study the global dynamics for the model in a special case when all the coefficients are independent of spatio--temporal variables.


1.
Introduction. Mathematical modeling is a basic but efficient tool to study the spread mechanism of diseases, by which the future course of an outbreak can be predicted and then be controlled. In order to establish a theoretical framework for mathematical analysis of transmission of malaria, Ross [44] firstly proposed a system of ordinary differential equations which is the origin of the modern susceptibleinfected-recovered (SIR) compartmental model. Since then the SIR compartmental model and many of its extensions, which are independent of the spatial variables, have been well investigated by many scholars [2,8,20,40,36]. At the same time, the heterogeneity of living environment and mobility of the host individuals play a crucial role in the geographic spread of infectious disease. In fact, there have been many articles which have analyzed mathematically the spatial dynamics of epidemic models, see [3,16,42,45,43,46,61,59,64,65,66] and the references therein. As reported by [29], multi-group epidemic models have been proposed to describe the spread of various infectious diseases in heterogeneous populations, such as measles, mumps, gonorrhea, and HIV/AIDS. In such models, a heterogeneous host population can be divided into several homogeneous groups according to the modes of transmission, contact patterns, or geographic distributions, so that withingroup and inter-group interactions could be modeled separately. The works involved with multi-group models with or without spread diffusion can be found in [7,13,14,15,17,21,23,32,34,50,60,70,68].
Many infectious diseases, such as measles, chicken pox, cholera, influenza, HIV, SARS, etc., exhibit a latent period, namely, the infected individuals do not infect other susceptible individuals until some time later. Meanwhile, the infected individuals may move from one spatial location to another spatial location with time, which give rise to spatial nonlocal effect. Generally, such nonlocal infection may effect the outbreak and transmission of the diseases (see, e.g. Li and Zou [27], Lou and Zhao [32], Wang and Zhao [58]). Li and Zou [28] proposed a time-delayed SIR epidemic model with nonlocal terms among n-patches in a fixed latent period, where a demographic structure is incorporated by adding recruitment (including births) and natural deaths. They found that nonlocal terms can enhance the basic reproduction number R 0 , and thus, may leads to an otherwise dying-out disease to persist. When the habitat is a continuous domian, Guo et al. [18] derived a reaction-diffusion epidemic model with time-delay and non-locality in a fixed latent period and investigated the threshold dynamics of the epidemic model by means of the basic reproduction number R 0 . In addition, there have been other papers studying diffusion-reaction epidemic models with fixed latent period, see [27,32,58,63,67] and the references therein.
However, it is common that the length of the latent period differs from disease to disease; even for the same disease, the length of the latent period is also different from individuals to individuals. Based on this point, instead of using the discrete (fixed) delay, we employ distributed delay to characterize the variable latency (see, e.g., van den Driessche et al. [53]). The distributed delay allows infectivity to be a function of the duration since infection, up to some maximum duration (see [38]). To characterize the distributed delay, a distribution function p(u) : [0, ∞) → [0, ∞) which accounts for the variance that the infected individuals become infectious and is assumed to have compact support, p(u) ≥ 0 and ∞ 0 p(u)du = 1 can be used. Epidemic models with distributed delay independent of the spatial variables have been studied, see [6,10,22,29,49,56] and the references therein.
It is well known that seasonality can impact host-pathogen interactions, including seasonal changes in host social behaviour and contact rates, variation in encounters with infective stages in the environment, annual pulses of host births and deaths and changes in host immune defences (see [1]). For an infectious disease, it is crucial and more realistic to take into account temporal heterogeneity, which gives rise to non-autonomous evolution equations. Bacaër and Guernaoui [5] defined the basic reproduction number R 0 in a periodic environment. For further developments, we refer to Bacaër et al. [4] and Inaba [24] and the references therein. Wang and Zhao [57] developed the basic reproduction number R 0 of a large class of compartmental epidemic models in periodic environments and studied the impact of periodic contacts or periodic migrations on the disease transmission by analyzing the global dynamics of a periodic epidemic model with patch structure. Peng and Zhao [41] studied the threshold dynamics of a time-periodic reaction-diffusion SIS model and showed that the persistence of the infectious disease can be enhanced by incorporating the spatial heterogeneity and temporal periodicity into the model. Recently, the theory of the basic reproduction number on the periodic and timedelayed compartmental models is established by Zhao [71] and can be applied to periodic SEIR models with incubation period. Zhang et al. [67] proposed a timeperiodic reaction-diffusion epidemic model which incorporates simple demographic structure and a fixed latent period of the infectious disease, introduced the basic reproduction number R 0 via a next generation operator, and investigated the threshold dynamics of the epidemic model in terms of R 0 . Some other studies on the dynamics of time heterogeneous epidemic models can be found in [33,54,55,65,68] and the references therein. However, for such non-autonomous (even autonomous) diffusion-reaction epidemic models with distributed delays, much less is done. The purpose of this paper is to incorporate spatial diffusion, distributed latency of the disease and temporal heterogeneity into a multi-group SIR disease model and to investigate the threshold dynamics of the derived model.
The rest of this paper is organized as follows. In the next section, we derive a two-group reaction-diffusion epidemic model with seasonality and distributed delay. In section 3, we introduce the basic reproduction number R 0 for the system via the next generation operator method and then establish the threshold dynamics for the system in term of R 0 , namely, the disease is uniformly persistent if R 0 > 1, while the disease goes to extinction if R 0 < 1. Section 4 is devoted to the global dynamics for the model in a special case where all the coefficients are independent of spatio-temporal variables.
2. Model formulation. Assume that an infectious disease spreads in two populations or sub-populations living in a bounded domain Ω ∈ R n with smooth boundary ∂Ω. We always define two populations or sub-populations by the subscript 1 and 2. Without loss of generality, we divide each population/sub-population into four compartments: the susceptible compartment, the latent compartment, the infectious compartment and the removed compartment. Then we denote the densities of four compartments at time t and location x by Let E 1 (t, a, x) and E 2 (t, a, x) be the densities of two exposed populations or sub-populations at time t ≥ 0, infection age variable a ≥ 0 and location x ∈Ω, repectively. Then E i (i = 1, 2) satisfy the following model where n is the outward normal to ∂Ω, D i represents the diffusion rate of the i-th population,D i (t, a, x) and M i (t, a, x) mean the disease-induced mortality rate and the recovery rate of the i-th population which are dependent upon the infection age a, time t and location x, respectively and d i (t, x) denotes the natural death rate of the i-th population at time t and location x for i = 1, 2.
Suppose that an infectious disease has a period of latency which is not fixed. Namely, for each population, we assume that infectious individuals must have capable of infecting others after the infection age τ i ∈ [0, ∞). But, between the infection age 0 and τ i , the infected individual may or may not have an infection ability. Assume that f i (r)dr denotes the probability of becoming into the individuals who are capable of infecting others between the infection ages r and r + dr and F i (a) := a 0 f i (r)dr represents the probability of turning into the individuals with infecting others before the infection age a for i = 1, 2. Then we have For the sake of simplicity, we assume that the functionsD i (t, a, x) and M i (t, a, x) are independent of the infection age a, namely, For a convenience, we assume We now aim to find partial differential equations satisfied by L i (t, x) and I i (t, x). Integrating (1) with respect to a and using the expressions of L i (t, x) and and where i = 1, 2. Let E i (t, ∞, x) = 0(i = 1, 2), then we can obtain As the new infection individuals come from the contact of the infectious and susceptible individuals, we adopt the following form: where β ij (t, x) ≥ 0 is called the infection rate for i, j = 1, 2. In this paper, we assume that the contacts between susceptible individuals and infectious individuals are defined by incidence functions g ij (u, v)(i, j = 1, 2), which satisfy the following conditions: (ii) g ij (u, 0) = 0 and g ij (0, v) = 0 for all u, v ≥ 0 and i, j = 1, 2; (iii) ∂ ∂u g ij (u, v) ≥ 0 and ∂ ∂v g ij (u, v) ≥ 0 for all u, v ≥ 0 and i, j = 1, 2. In particular, ∂ u g ij (u, 0) = 0 and ∂ v g ij (u, 0) > 0 for all u > 0; (iv) there exist η i > 0(i = 1, 2) such that g ij (u, v) ≤ η i u for all u, v ≥ 0; Note that the class of g ij (u, v)(i, j = 1, 2) satisfying (H1) include many common incidence functions such as where a ij , b ij , c ij > 0 for i, j = 1, 2, see [39]. We use the following simple demographic equation for a population Q(t, x) that admits a dynamics of global convergence to a positive periodic solution where µ(t, x) is the recruiting rate, D Q 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 the above assumptions, the disease dynamics is expressed by the following system (2) We make the following basic assumptions: The reminder is to derive functions E i (t, a, x)(i = 1, 2) by integration along characteristics. For a convenience, let r i (t, ·) =D i (t, ·) + M i (t, ·) + d i (t, ·). For any ξ ≥ 0, we consider the solutions of (1) along the characteristic line t = a + ξ by letting v i (ξ, a, x) = E i (a + ξ, a, x)(i = 1, 2). Then for a ∈ (0, τ i ], we have ∂vi(ξ,a,x) ∂a

LIN ZHAO, ZHI-CHENG WANG AND LIANG ZHANG
where i = 1, 2. For the above system, we can regard ξ as a parameter. Then we have v i (ξ, a, x) = Ω Γ i (ξ + a, ξ, x, y) β i1 (ξ, y)g i1 (S i (ξ, y), I 1 (ξ, y)) where Γ i (t, s, x, y) with t > s and x, y ∈ Ω is the fundamental solution associated with the partial differential operator ∂ t − D i ∆ − r i (t, ·) and Neumann boundary condition for i = 1, 2. Note that Γ i (t, s, x, y) = Γ i (t + T, s + T, x, y) for all t > s ≥ 0 and x, y ∈ Ω because of r i (t + T, x) = r i (t, x) for any t ≥ 0. It then follows from Substituting (3) into the second equation and the third equation of (2) respectively, and ignoring the L i (t, x) and R i (t, x) equations from (2) because they are decoupled from the S i (t, x) and I i (t, x) equations, we obtain the following system: for i, j = 1, 2. For simplicity, letting (u S1 , u S2 , u 1 , u 2 ) = (S 1 , S 2 , I 1 , I 2 ), we focus on the following reaction-diffusion system with Neumann boundary condition: 3. Threshold dynamics. In this section, we explore the threshold dynamics of system (6).

Global existence of solution.
In this subsection, we investigate the existence and uniqueness of time-global solutions of system (6). Set τ = max{τ 1 , τ 2 }.
, then (X, X + ) and (C τ , C + τ ) are strongly ordered spaces. For σ > 0 and a given function Set Y := C(Ω, R) and Y + := C(Ω, R + ). Furthermore, we consider the following system: where D Si > 0(i = 1, 2) and d i (t, x)(i = 1, 2) are Hölder continuous and nonnegative nontrivial functions on R ×Ω and T -periodic in t. It follows from [19, Chapter II] that (7) admits an evolution operator V Si (t, s) : (7) for i = 1, 2. Similarly, we take into account the following system: where D i > 0(i = 1, 2) and r i (t, x)(i = 1, 2) are Hölder continuous and nonnegative nontrivial functions on R ×Ω and T -periodic in t. Let V i (t, s)(i = 1, 2) be the evolution operators determined by the above system and have the similar properties as V Si (t, s). Due to the periodicity of coefficients, it follows from [11, Lemma 6.1] that V Si (t, s) = V Si (t + T, s + T ) and V i (t, s) = V i (t + T, s + T ) hold for (t, s) ∈ R 2 , t ≥ s and i = 1, 2. In addition, for any t, s ∈ R and s < t, V Si (t, s) and V i (t, s) are compact, analytic and strongly positive operators on Y + for i = 1, 2. Together [11, Theorem 6.6] with α = 0, we get that there exist constants Q ≥ 1 and and U (t, s) : X → X be an evolution operator for (t, s) ∈ R 2 with t ≥ s. Let and respectively. Then (6) can be written as the following Cauchy problem: where . Moreover, it can be rewritten as the following integral equation A solution of (9) is called a mild solution of (8).
Lemma 3.1. For every initial value function φ ∈ C + τ , system (6) has a unique mild solution u(t, φ) on [0, +∞) with u 0 = φ. Furthermore, system (6) generates a T-periodic semiflow Φ t (·) := u t (·) : For The above inequality implies that (10) holds when k is small enough. Consequently, by [37,Corollary 4] with K = X + and S(t, s) = U (t, s), system (6) admits a unique mild solution is a classic solution for t > τ by using the analytic of U (t, s) for any s, t ∈ R with s < t. Consider the following time-periodic reaction-diffusion equation: where i = 1, 2. It follows from [67, Lemma 2.1] that system (11) admits a unique positive T -periodic solution ω * i (t, x) which is globally asymptotically stable in Y + for i = 1, 2. Since the u Si (i = 1, 2) equations of system (6) are dominated by (11), respectively, there exists a positive constant B s such that for any In view of (iv) of (H1), we have for t > 0 and x ∈ Ω, It follows from the comparison principle that there exists a constantB > 0 such that for any From the above discussion, we have that Φ t is point dissipative. Let n 0 := min{n ∈ N : nT > 2τ }. Then by the standard parabolic estimates, we conclude that Φ n0 T = u n0T is compact. Following from [35, Theorem 2.9], one has that Φ T : C + τ → C + τ has a global compact attractor. The proof is completed.
3.2. Basic reproduction number. Let C T (R ×Ω, R) be the ordered Banach space consisting of all T -periodic and continuous functions from R ×Ω to R, where φ C T = max t∈[0,T ],x∈Ω |φ(t, x)| for any φ ∈ C T . Denote C + T as the positive cone of C T , that is, Similarly, we define C + T as the positive cone of C T , namely, Setting u 1 ≡ 0 and u 2 ≡ 0, we have the following equations for the densities of the susceptible population u Si (t, x)(i = 1, 2)  (12) admit positive solutions u * Si (i = 1, 2) which are unique, globally asymptotically stable and T -periodic in t ∈ R, respectively. As a consequence, the function (u * S1 , u * S2 , 0, 0) is called the disease-free periodic solution of (6). Linearizing the third and the forth equations of system (6) at (u * S1 , u * S2 , 0, 0) and according to (iii) of (H1), we have the following system: is the initial distribution of infectious individuals at time s ∈ R and the spatial location x ∈Ω. Given t ∈ R. Due to the synthetical influence of mobility, mortality and recovery, (V i (t − a, s)φ i (s)) (x), where s < t − a represents the density distribution at location x of those infective individuals who were infected at time s and remained infective at time t − a when time evolved from s to t−a for a ∈ [0, τ ]. Furthermore, represents the distribution of new infected individuals of the i-th group at location x and time t for i = 1, 2. As a consequence, we can define the next generation infection operator L as It is obvious that L is a positive and bounded linear operator on C T . Let r(L) be the spectral radius of L. Similar to [5,12,57,71,67], denote the spectral radius of L as the basic reproduction number R 0 of model (6), that is, Next, we define an operatorL(φ)(t, x) : Clearly,L is a compact, positive and bounded linear operator on C T . Let Then one has L = AB andL = BA. It follows that R 0 = r(L) = r(L), where r(L) is the spectral radius of the operatorL. As the previous discussion, there exist constants Q > 1 and c i ∈ R such that V i (t, s) ≤ Qe ci(t−s) , ∀t ≥ s, t, s ∈ R, i = 1, 2.
Obviously, e λ0T is an eigenvalue of P. Thus, one has r 0 ≥ e λ0T > 1.
The conclusions of (iii) is immediately followed from conclusion (i) and (ii). This completes the proof.
Proof. The proof of the lemma is similar to those of [67,Lemma 4.2], so we omit the details.
Secondly, we present the main theorem of this paper.
Next, similar to the proof of [67, Theorem 4.3 (i)], we get uniformly for x ∈Ω.
. The claim is proved. It follows from the above claim that M is an isolated invariant set for Φ T in W 0 , and W s (M ) ∩ W 0 = ∅, where W s (M ) is the stable set of M . According to the acyclicity theorem on uniform persistence for maps (see [71, Theorem 1.3.1 and Remark 1.3.1]), one has that Φ T : C + τ → C + τ is uniformly persistence with respect to (W 0 , ∂W 0 ), namely, there exists aδ > 0 such that It then follows from [71, Theorem 3.1.1] that the periodic semiflow Φ t : C + τ → C + τ is also uniformly persistent with respect to (W 0 , ∂W 0 ). It is easy to see that Φ n0 T is compact and point dissipative on W 0 . Therefore, according to [35,Theorem 2.9], one obtains that Φ n0 T : W 0 → W 0 has a global attractor Z 0 .
≤ 0 for all v > 0 and j = 1, 2. In short, we consider the following spatio-temporally homogeneous reaction -diffusion epidemic model with Neumann boundary condition: By a straightforward computation, one has Next, we give the explicit expression with the basic reproduction number R 0 . Let φ = (φ 1 , φ 2 ) T be the initial distribution of infective individuals such that Ω φ i (x)dx = 1 for i = 1, 2. Then T i (t)φ i represents the solution of the following system whereΓ i (t, x, z) for t > 0 and x, z ∈ Ω is the fundamental solution associated with the partial differential operator ∂ t − D i ∆ − r i and Neumann boundary condition for i = 1, 2 and ΩΓ i (t, x, z)dz = ΩΓ i (t, x, z)dx = e −rit for i = 1, 2. Let T (t)φ = (T 1 (t)φ 1 , T 2 (t)φ 2 ) T be the remaining distribution of infective individuals at time t. Also in this case, V is the positive linear operator on C(Ω, R × R) defined by is the distribution of newly infected individuals at time t. Thus, the next generate operator can be represented by Furthermore, the total number of infectious individuals is given by Let ϑ 1 := τ1 0 f 1 (a) Ω Γ 1 (a, x, y)dxda. According to (v) of (A1), we can obtain By the same methods, one has As a consequence, it follows that

LIN ZHAO, ZHI-CHENG WANG AND LIANG ZHANG
Let r(L) be the spectral radius of L. Finally, we can define the spectral radius of L as the basic reproduction number R 0 , that is, We are in a position to show the main results in this section.
In the following, we prove the conclusion (2) by using a Volterra like Lyapunov functional. Similarly to the proof of [48, Theorem 2.1], we can show that the corresponding ordinary differential equations of (21): admits at least one endemic equilibrium u * = (S * 1 , S * 2 , I * 1 , I * 2 )(S * i , I * i > 0, i = 1, 2), which is also a positive constant steady state of (21).
On the basis of (22) and (23), we have dW dt ≤ 0. By using the similar arguments introduced in [52, Theorem 12.1], we can prove that the attractor Z 0 in Theorem 3.7 is a singleton set which is formed by the endemic equilibrium u * . This completes the proof.