An almost periodic epidemic model with age structure in a patchy environment

An almost periodic epidemic model with age structure in a patchy 
environment is considered. The existence of the almost periodic 
disease-free solution and the definition of the basic reproduction 
ratio $R_{0}$ are given. Based on those, it is shown that a 
disease dies out if the basic reproduction number $R_{0}$ is less 
than unity and persists in the population if it is greater than 
unity.


1.
Introduction. Recently, the study of epidemic models with population dispersal has become increasingly important. Population movements between countries, regions or cities are one of main reasons for the spread of many diseases, such as influenza, measles, tuberculosis (TB) and sexually transmitted diseases. Mathematical models and field observations show that population dispersal has a positive impact for the control of a disease sometimes (see [4]), while it may lead to a disease outbreaks (see [28,29]). Originally, the epidemic model involving population dispersals is between two patches (see [7,12]). Hereafter, a transmission epidemic model among n patches is considered (see [28,32]).
Since the work of McKendrick [16], more and more researchers have been aware that the age-structure of a population affects the dynamics of disease transmission and should be taken into account in the realistic model of some infectious diseases. For example, TB is highly age-dependent (see [5]), and the population who is likely to be infected is the adult individuals since newborns are vaccinated in many countries, but the immunity has decreased with increasing age. As noted in [15], infectious diseases like meals admit the same situation because the babies from their birth to semiprecession have a strong immune system which was passed on from their mothers and older children are immunized during their school time under the supervision of the school and government health organizations. Consequently, when we study some diseases, it is realistic to assume that the adult individuals may be only infected and the childhood has a immunity against the disease. Simultaneously, the adults are responsible for the reproduction of the population.
It is well known that seasonal variations in temperature, rainfall, resource availability, contact rates, the birth and death rates of populations and immune defences are ubiquitous and can exert strong pressures on population dynamics. Consequently, many infectious diseases are significantly affected by seasonal factors. As 292 BIN-GUO WANG, WAN-TONG LI AND LIANG ZHANG noted in [1], Watts et al. [31] showed that dengue haemorrhagic fever has a timing of outbreak during hot-dry and rainy season because rainfall and temperature affect mosquito vector abundance, temperature influences parasite replication in vectors. Fine and Clarkson [9] implied that respiratory-aerosol and contact-borne pathogens Measles increases in fall or spring due to host aggregation during school terms increases transmission. From an applied perspective, a periodic epidemic model where the role of seasonality has been relatively well explored offers important insights into understanding the mechanisms of a disease outbreak. Empirical evidence points to several biologically distinct mechanisms by which the birth rate, the death rate, the recovery rate and the emigration rates are not necessary to share a common period. Especially, if the periods of these periodic coefficients have no common integer multiple, then we can treat such a model as an almost periodic system. In this paper, we consider the dynamics of an almost periodic epidemic model with age-structure in a patchy environment.
We use M i and G i to denote the number of juvenile individuals and adult individuals in path i, respectively. Refer to [15], we assume that M i satisfies where F i (t, G i ) and ν i (t) are the per capita birth rate and the per capita death rate of juveniles, respectively, in patch i at time t.
Consider the following epidemic model with age-structure: where S i , I i are the numbers of susceptible and infectious individuals in patch i, respectively. Naturally, G i = S i + I i . D i (t, G i ) is the transition rate of juvenile individuals from juvenile stage to adult stage in patch i at time t, d i (t) is the death rate of the population in the ith patch at time t, i (t) is the contact rate of susceptible individuals with infectious individuals at time t, and γ i (t) is the recovery rate of infectious individuals in the ith patch at time t. r ij (t), h ij (t) and o ij (t), i = j, represent the immigration rates of susceptible individuals, infectious individuals and juvenile individuals, respectively, from the jth patch to the ith patch. −r ii (t) ≥ 0, −h ii (t) ≥ 0 and −o ii (t) ≥ 0 are the emigration rates of susceptible individuals, infectious individuals and juvenile individuals, respectively, in the ith patch. During the dispersal process, we assume that the death and birth of individuals are neglected, then we have   Furthermore, we assume that (A1) Two n × n matrices (r ij (t)) and (h ij (t)) are strongly irreducible. (A2) F i (t, G i ) > 0 for all G i > 0, t > 0 and i = 1, . . . , n. (A3) The continuously differentiable function F i (t, G i ) satisfies ∂Fi(t,Gi) ∂Gi < 0, ∀G i > 0, t > 0, i = 1, . . . , n.
In the following, applying the similar arguments to those in [15], we simplify model (1) without including the item of juvenile individuals by the way of seeking the expression of D i (t, G i ). Let M (t, a) = (M 1 (t, a), . . . , M n (t, a)) T , where M i (t, a) is the number of the juveniles in the ith patch at time t with age a. Hence, the birth law can be denoted by Assume that τ is the length of the juvenile period. Define D(t) := (D 1 (t), . . . , D n (t)) := M (t, τ ). As noted in [15] (see also [25]), M (t, a) satisfies the following differential equation: where Let Φ(t, a), t ≥ a ≥ 0, be the fundamental solution matrix of (3), that is, ∂Φ(t,a) ∂t = E M (t)Φ(t, a) and Φ(a, a) is an identity matrix. Hence, we conclude that It then follows that . . , n and t > 0. Obviously, D(t) is independent of the variables of juveniles, and hence, we can rewrite model (1) as follows: where C + := C([−τ, 0], R n + ). Throughout this paper, we assume that F i (t, G i ) and D i (t, G i ) are uniformly almost periodic, and ν i (t), d i (t), i (t), γ i (t), r ij (t), h ij (t) and o ij (t) are almost periodic in t. Hence, a similar argument to system (17) implies that p ij (t) is also almost periodic. Thus, model (4) is an almost periodic time-delayed differential system. Here, the Poincaré map is invalid, we draw support from the tool of skew-product semiflows and the theory developed in [27].
The remaining parts of this paper are organized as follows. In Section 2, we present the theory of almost periodic delay differential equations. In Section 3, we prove the existence of an almost periodic disease-free solution and define the basic reproduction ratio R 0 . In Section 4, we establish threshold results on uniform persistence and global extinction.
2. Almost periodic delay differential equations. A function f ∈ C(R, R k ) is said to be almost periodic if for any * > 0, there exists l = l( * ) > 0 such that every interval of R of length l contains at least one point of the set where | · | is the usual Euclidean norm in R k . A matrix is said to be almost periodic in the sense that every entry is almost periodic. Let AP (R, R k ) := {f ∈ C(R, R k ) : f is an almost periodic function}. Then AP (R, R k ) is a Banach space equipped with the supremum norm · . Let D ⊂ R k . A function f ∈ C(R × D, R k ) is said to be unif ormly almost periodic in t if f (·, x) is almost periodic for each x ∈ D, and for any compact set E ⊂ D, f is uniformly continuous on R × E (see [6,10] A q e iλqt associated with the function f , i.e., We call λ q , q = 1, 2, . . ., the Fourier exponent of f (t), and A q the Fourier coefficient of f (t). For a function f ∈ AP (R, R k ), the module of f , mod(f ), is defined as the smallest additive group of real numbers that contains the Fourier exponent of f (t). A square matrix M is said to be cooperative if all off-diagonal entries of M are nonnegative. If all entries of M are nonnegative, then we say M is nonnegative.
A nonnegative square matrix M is called positive if M is not the zero matrix. A square matrix M = (m ij ) is called quasi-positive if it is not the zero matrix and all off-diagonal entries of M are nonnegative, i.e., m ij ≥ 0 for i = j.
A k×k square matrix (m ij ) is said to be strongly irreducible if there exists a θ 0 > 0 such that if two nonempty subsets K 1 , K 2 form a partition of K = {1, 2, . . . , k}, then there exist i ∈ K 1 and j ∈ K 2 = K \ K 1 with |m ij | ≥ θ 0 .
Proof. For any β ∈ (0, 1), let z g (t) = βu(t, g, x). We assume that all solutions of (6) are defined for t ≥ 0. Hence, we have Since x ≥0 and f is sublinear, we conclude that It then follows from comparison theorems for cooperative system that z , and hence, the skew-product semiflow is sublinear. In the following, if x 0 , the strict sublinearity implies that The strong irreducibility and comparison theorems imply that z f (t) < u(t, f, βx) Hence, from z f (1) < u(1, f, βx) and comparison theorems yield to z f (t) < u(t, f, βx) Remark 1. Indeed, for any give x 0 and β ∈ (0, 1), the strict sublinearity implies that there exists Since π t1 • π t2 = π t1+t2 , we conclude that v(t, f, βx) βv(t, f, x), ∀t ≥ t 1 + t 2 , that is, the skew-product semiflow is eventually strongly sublinear.
is called the upper Lyapunov exponent over K.
If λ K ≤ 0, then K is said to be linearly stable.
The next result shows that the Lyapunov exponent of a point of K can be obtained at discrete times. n .
In order to obtain the main result of this section, we have the following preparation.
Proof. Since Lemma 2.1 implies that the skew-product is sublinear, a simple calcu- It then follows from the compactness and positive invariance of Hence, by Definition 2.2, we conclude that λ Og ≥ 0. Thus, it is only sufficient to prove λ Og = 0.
On the contrary, we assume that λ Og = 0. Since f (t, 0, 0) = 0 implies that g(t, 0, 0) = 0 for all g ∈ H(f ), a completely similar argument to one in [18, Proposition 6.2] for the concave case implies that there exists the subset By the structure theorem of skew-product semiflows (see [20,Theorem 1]), as applied to the flow π : O * → O * , we conclude that . For e 0, we define e-norm by Let e = b(g 0 ) 0 in (8) and define Since card(O * ∩p −1 (g)) = 1, from the monotonicity and the eventually strongly sublinearity of the skew-product semiflow at f , there exists 0 < ϑ 0 < ϑ and t 0 > τ (depending on g 0 , x(g 0 ) and x * ) such that Hence, we can find (g 0 ,y) ∈ O * such thať On the contrary, assume that v(t, g,0) =0, and hence u(t+ς, g,0) = 0, ∀ς ∈ [−τ, 0]. Thus, 0 = u (t + ς, g,0) = g(t + ς, 0, 0) > 0, a contradiction. In addition, by [19,Theorem 3.2], for any (g, . It then follows from the strong monotonicity and the invariant of the omega limit set that ω(g, there exists a positive almost periodic solution of (6) which is globally attractive in (ii) Similar to the above, for any (g, ). It then follows from the strong monotonicity of the skew-product semiflow and the compactness and invariance of the omega limit set that either ω(g, In the following, we claim that ω(g, On the contrary, if for some (g 0 , (g 0 , x 0 ). Thus, we can prove that for any sequence t n → ∞ as n → ∞, lim n→∞ u(t n , g 0 , x 0 ) = 0 holds. Otherwise, there exists a sequence t k → ∞ as k → ∞, the strong monotonicity implies that ω(g 0 , x 0 ) ⊂ H(f ) × Int(C + ), which contradicts to O g = ω(g 0 , x 0 ). Hence, for any > 0 there exists a n 0 such that for n ≥ n 0 , Since u(t, g,0) = 0, a simple computation and Lemma 2.3 imply that λ Og ≤ 0, a contradiction. We have proved the claim. It then follows from the similar arguments to the above that there exists a positive almost periodic solution of (6) which is globally attractive in H(f ) × (C + \ {0}).
(iii) On the contrary, if O g is not globally attractive in H(f ) × C + , then the similar arguments to the above imply that there exists a compact and minimal invariant subset K * with K * ⊂ H(f ) × Int(C + ), that is, O g C K * . It then follows from Proposition 1 that λ Og > 0, which is a contradiction.
3. The disease-free solution and basic reproduction ratio. From now on, we embed system (4) into a skew-product semiflow, and draw support from the conclusions in Section 2 to consider the existence of the almost periodic disease-free solution. Furthermore, we give the definitions of the exponential growth bound and the basic reproduction ratio of system (4).
(13) In order to assure the global attractivity of the positive almost periodic solution of system (12), we assume that the trivial solution is unstable as J(t, 0, 0) = 0. According to the conclusion of the above section, we have the following assumption.
where λ Oη is the upper Lyapunov exponent associated with system (12).
Proof. We can clarify that J(t, S(t), S(t − τ )) = (J 1 (t, S(t), S(t − τ )), . . . , J n (t, S(t), S(t − τ ))) satisfies the conditions (B1)-(B4). Then the results can be obtained from Theorem 2.4 straightforwardly. In the following, we introduce the basic reproduction ratio of (4). In epidemiology, the basic reproduction number R 0 (sometimes called basic reproductive rate, basic reproductive ratio) of an infection is defined as the expected number of secondary cases produced by a single (typical) infection in a completely susceptible population (see [8]). It is used to measure the transmission potential of a disease, i.e., it helps determine whether or not an infectious disease can spread through a population. The basic reproduction ratio for periodic cases can be found in [3]. A generalized case for periodic cases is showed in [30]. Almost periodic functions are a generalization of periodic functions. Here, we will apply the theory developed in [27].
Linearizing system (4) at A 0 (t) = (S * (t), 0), a simple computation implies that the infectious classes variable z = (I 1 , . . . , I n ) admits the following linear almost periodic system: where Let Ψ −Z (t, s) (t ≥ s, s ∈ R) be the evolution operator of the linear almost periodic system that is, the n × n matrix Ψ −Z (t, s) satisfies where i d is the n×n identity matrix. Then the fundamental solution matrix Φ −Z (t) of (17) equals Ψ −Z (t, 0), t ≥ 0. We define the exponential growth bound of the evolution operator Ψ −Z (t, s) as Note that ω(Ψ −Z ) is finite because Z(t), as an almost periodic function, is bounded on R.
Note that the internal evolution of individuals in the infectious patches due to deaths, recoveries and movements among the patches is dissipative, and exponentially decays in many cases because of the loss of infective members from natural mortalities and disease-induced mortalities. According to (2) and applying mathematical inductions, we can verify that ω(Ψ −Z ) < 0. It then follows from (18) that there exist K 1 > 0 and σ 1 > 0 such that Furthermore, the existence of a unique almost periodic solution (see [ which induces a linear map L by It then follows from [27, Lemma 3.1], L is a positive and continuous linear operator from AP (Y,Z) to AP (Y,Z) . Using the ideas in [30], in an epidemic model, assume that φ(s) ∈ AP (Y,Z) is the initial distribution of infectious individuals. Then, Y (s)φ(s) is the distribution of new infections produced by the infected individuals who were introduced at time s. Given t ≥ s, then Ψ −Z (t, s)Y (s)φ(s) denotes the distribution of those infected individuals who were newly infected at time s and remain in the infected compartments at time t. V(t) is the distribution of accumulative new infections at time t produced by all those infected individuals φ(s) introduced at previous time to t. Following the appellation in [8,26,30], L is called the next infection operator, and define the spectral radius of L R 0 := ρ(L) as the basic reproduction ratio of model (4).
Let Ψ −Z+Y (t, s) (t ≥ s, s ∈ R) be the evolution operator of (14). Similarly, we can define the exponential growth bound ω(Ψ −Z+Y ) of Ψ −Z+Y (t, s). 4. Threshold dynamics. In this section, we consider the uniform persistence and the global extinction of system (4) in terms of its basic reproduction ratio. Proof. We use the skew-product semiflows approach to prove the desired uniform persistence and practical uniform persistence. (see, e.g., [15,Theorem 3.3
Since O is isolated for Π t in ∂P 0 , the claim above implies that O is isolated for Π t in P . The claim above also implies that W s (O) is the stable set of O for the semiflow Π t : P → P .
In the case of J(t, 0, 0) > 0, then O 1 is the unique compact invariant set for Π t in M ∂ . A similar result can be obtained to refer to the case of k * = 1.
By the continuous-time version of [34, Theorem 1.3.1 and Remark 1.3.1], the skew-product semiflow Π t : P → P is uniformly persistent with respect to (P 0 , ∂P 0 ).
This completes the proof.
In the case of R 0 < 1, we show the global extinction whenever the initial value of infective individuals is sufficiently small or susceptible and infective individuals have the same dispersal rate.

Remark 2.
Here, we consider the threshold dynamics of an almost periodic epidemic model with age structure in a patchy environment in terms of the basic reproduction ratio R 0 . The arguments above implies that R 0 plays a crucial role in the control of a disease. Refer to [27,Theorem 3.4], we can create a method how to compute R 0 .