AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 295060 10.1155/2012/295060 295060 Research Article The Global Behavior of a Periodic Epidemic Model with Travel between Patches Wen Luosheng 1, 2 Long Bin 1 Liang Xin 1 Zeng Fengling 1 Cheung Wing-Sum 1 School of Mathematics and Statistics Chongqing University Chongqing 400044 China cqu.edu.cn 2 Key Laboratory of Dependable Service Computing in Cyber Physical Society, Ministry of Education Chongqing 400044 China 2012 12 7 2012 2012 03 03 2012 14 05 2012 2012 Copyright © 2012 Luosheng Wen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We establish an SIS (susceptible-infected-susceptible) epidemic model, in which the travel between patches and the periodic transmission rate are considered. As an example, the global behavior of the model with two patches is investigated. We present the expression of basic reproduction ratio R0 and two theorems on the global behavior: if R0 < 1 the disease-free periodic solution is globally asymptotically stable and if R0 > 1, then it is unstable; if R0 > 1, the disease is uniform persistence. Finally, two numerical examples are given to clarify the theoretical results.

1. Introduction

Epidemic models have been paid intensive attention for recent decades. In the models, population is divided into several compartments, for example, susceptible (S), infected (I), and recovery (R) by individual state. The classic epidemic models, including SIS model and SIR model, generally aim at the basic reproduction ratio (the epidemic threshold) and the global behavior .

With the development of transportation, the travel becomes more and more easy for people. It has been observed that the travel can affect the spread of infectious disease. In [7, 8], authors showed that international travel is one of the major factors associated with the global spread of infectious disease. Ruan et al. investigated the effect of global travel on the spread of SARS  and pointed out that the basic reproduction ratio is independent upon the travel but the travel increase the number of infected individuals.

On the other hand, many infectious diseases show seasonal behavior, such as measles, chickenpox, rubella, and influenza. Zhang and Zhao  presented a periodic SIS epidemic model with individuals immigration among n patches. By employing the persistence theory, they gave the expression of the epidemic threshold and obtained the conditions under which the positive periodic solution is globally asymptotically stable. In , Wang and Zhao showed that the threshold parameter is the basic reproduction ratio for a wide class of compartmental epidemic model in periodic environments. Applying the method in [10, 11], Nakata and Kuniya  and Bai and Zhou  examined the threshold dynamics of a periodic SEIRS epidemic model.

Combining the mobility and seasonality, we consider an SIS epidemic model, in which people can travel among n patches and the transmission rate is a periodic function. Our SIS epidemic model with mobility and seasonality is as follows: (1.1)dSiidt=Bi(t,Nip)Nip+k=1,kinρik(t)Sik-(σi(t)+dii(t))Sii-k=1nβiik(t)SiiIki+γii(t)Iii,dIiidt=k=1,kinρik(t)Iik+k=1nβiik(t)SiiIki-(σi(t)+dii(t)+γii(t))Iii,dSijdt=σi(t)Vij(t)Sii-(ρij(t)+dij(t))Sij-k=1nβijk(t)SijIki+γij(t)Iij,ijdIijdt=σi(t)Vij(t)Iii+k=1nβijk(t)SijIki-(ρij(t)+dij(t)+γij(t))Iij,ij, where Sij(t) and Iij(t) are the number of susceptible and infected individuals whose current location is the jth patch and home location is the ith patch at time t, respectively. Denote Nij=Sij(t)+Iij(t). Nip=j=1nNji,Nip, is the number of individuals who are physically present in the ith patch at time t. N=i,j=1n(Sij+Iij). B(t,Nip) is the birth rate of the population in the ith patch. dij(t) is the death rate of the individuals whose current location is the jth patch and home location is the ith patch at time t. Individuals are assumed to leave a patch i at a certain constant rate, σi(t). The probability that a person travels from patch i to any other patch j is given by Vij(t). So σi(t)Vij(t) is the travel rate of individuals from the ith patch to the jth patch at time t. A person from patch i who travels to patch j returns home at a rate ρij(t). βikj(t) is the disease transmission coefficient in patch k that a susceptible individual from patch i contacts with an infectious individual from patch j. The recovery rate of infectious individuals from ith patch who are present in region j is γij(t). In , the birth rate Bi(t,Nip) satisfies the following basic assumptions for Nip(0,):

Bi(t,Nip)>0,  i=1,2,,n;

Bi(t,Nip) is continuously differentiable with dBi(t,Nip)/dNip<0,  i=1,2,,n;

Bi(t,)<dii(t),  i=1,2,,n;

and the birth function Bi(t,Nip)=B(t)/Nip+C(t) can be found in the biological literature.

We assume that these coefficients are functions being continuous, positive ω-periodic in t and we can obtain a periodic SIS epidemic model, in which individuals can travel among n patches. For simplicity, we consider an SIS model with travel among two patches, that is, n=2. In this paper, we assume that Bi(t,Nip)=B(t)/(Nip)+C(t),dij(t)=d(t),C(t)<d(t).  βikj(t)=β(t),γij(t)=γ(t),i,j=1,2. Hence j=1,ji2Vij(t)=1, we have Vij(t)=1. σ1(t) and σ2(t) are the travel rate from the 1st patch to the 2nd patch and from the 2nd patch to the 1st patch, respectively. B(t), C(t), d(t), β(t), γ(t), and ρij(t) are continuous, positive ω-periodic functions of t. We have the following system: (1.2)dS11dt=(B(t)N1p+C(t))N1p+ρ12(t)S12-(σ1(t)+d(t))S11-β(t)S11(I11+I21)+γ(t)I11,dS12dt=σ1(t)S11-(ρ12(t)+d(t))S12-β(t)S12(I12+I22)+γ(t)I12,dS21dt=σ2(t)S22-(ρ21(t)+d(t))S21-β(t)S21(I11+I21)+γ(t)I21,dS22dt=(B(t)N2p+C(t))N2p+ρ21(t)S21-(σ2(t)+d(t))S22-β(t)S22(I12+I22)+γ(t)I22,dI11dt=ρ12(t)I12+β(t)S11(I11+I21)-(σ1(t)+d(t)+γ(t))I11,dI12dt=σ1(t)I11+β(t)S12(I12+I22)-(ρ12(t)+d(t)+γ(t))I12.dI21dt=σ2(t)I22+β(t)S21(I11+I21)-(ρ21(t)+d(t)+γ(t))I21,dI22dt=ρ21(t)I21+β(t)S22(I12+I22)-(σ2(t)+d(t)+γ(t))I22.

In this paper, we will study the basic reproduction ratio and global behavior of system (1.2). This paper is organized as follows. In Section 2, we show the existence of the disease-free periodic solution of (1.2) and define the basic reproduction ratio. In Section 3, we show the global asymptotical stability of the periodic disease-free solution and the uniform persistence of the disease. In Section 4, two numerical examples are given to clarify the theoretical results.

2. The Basic Reproduction Ratio

Let (Rn,R+n) be the standard ordered n-dimensional Euclidian space with a norm ·. For u,vRn, we write uv if u-vR+n,u>v, if u-vR+n{0}, and uv if u-vInt(R+n). Let A(t) be a continuous, cooperative, irreducible, and ω-periodic n×n matrix function, and ΦA(t) is the fundamental solution matrix of the linear ordinary differential system (2.1)dXdt=A(t)X, and r(ΦA(ω)) be the spectral radius of ΦA(ω). By the Perron-Frobenius theorem, r(ΦA(ω)) is the principal eigenvalue of ΦA(ω) in the sense that it is simple and admits an eigenvector v*0. The following result is useful for our subsequent comparison arguments.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B10">10</xref>]).

Let P=(1/ω)lnr(ΦA(ω)). Then there exists a positive, ω-periodic function V(t) such that ePt  V(t) is a solution of (2.1).

Lemma 2.2.

Every forward solution of (1.2) eventually into (2.2)Γ={(S(t),I(t))R+80i,j=12(Sij+Iij)2bc}, where b=maxt[0,ω](B(t)),c=mint[0,ω](d(t)-C(t)), and for each N(t)2b/c, Γ is a positively invariant set for (1.2).

Proof.

By the method of variation of constant, it is obvious that any solution of (1.2) with nonnegative initial values is nonnegative. From (1.2), we have (2.3)dNdt=2B(t)-(d(t)-C(t))N2b-cN0if  N(t)2bc. This implies that Γ is a forward invariant compact absorbing set of (1.2). Hence, the proof is complete.

Next, we show the existence of the disease-free periodic solution of (1.2). To find the disease-free periodic solution of (1.2), we consider (2.4)dS11dt=B(t)+C(t)(S11+S21)+ρ12(t)S12-(σ1(t)+d(t))S11,dS12dt=σ1(t)S11-(ρ12(t)+d(t))S12,dS21dt=σ2(t)S22-(ρ21(t)+d(t))S21,dS22dt=B(t)+C(t)(S12+S22)+ρ21(t)S21-(σ2(t)+d(t))S22. Denote (2.5)M(t)=(C(t)-(σ1(t)+d(t))ρ12(t)C(t)0σ1(t)-(ρ12(t)+d(t))0000-(ρ21(t)+d(t))σ2(t)0C(t)ρ21(t)C(t)-(σ2(t)+d(t))). Let Ψ:R+1×R+4R4 be defined by the right-hand side of (2.4). Ψi(t,S)0 for every S0 with Si=0,tR+1,1i4.  Ψ(t,S) is strongly subhomogeneous for SR+4 in the sense that Ψ(t,αS)αΨ(t,S) for any t0,SR+4 and α(0,1).  M(t) is a continuous, cooperative, irreducible, and ω-periodic 4 × 4 matrix function. By Lemma 2.2, the solution of (2.4) is ultimately bounded in R+4. By Theorem 2.3.2 of , applying the Poincare map associated with (2.4), it follows that system (2.4) has a unique positive periodic solution (2.6)S*(t)=(S11*(t),S12*(t),S21*(t),S22*(t)).

We need to assume that r(ϕM(ω))>1,r(ΦM(ω)) be the spectral radius of M(ω). By Theorem 2.1.2 of , it then follows that the unique positive periodic solution S*(t) of (2.4) is globally attractive for S0R+4{0}. Hence, (1.2) has a unique disease-free periodic state (S*,0,0,0).

For convenience, we denote (2.7)S(t)=(S11(t),S12(t),S21(t),S22(t)),I(t)=(I11(t),I12(t),I21(t),I22(t)).

Consider the following system: (2.8)dI11dt=ρ12(t)I12+β(t)S11(I11+I21)-(σ1(t)+d(t)+γ(t))I11,dI12dt=σ1(t)I11+β(t)S12(I12+I22)-(ρ12(t)+d(t)+γ(t))I12,dI21dt=σ2(t)I22+β(t)S21(I11+I21)-(ρ21(t)+d(t)+γ(t))I21,dI22dt=ρ21(t)I21+β(t)S22(I12+I22)-(σ2(t)+d(t)+γ(t))I22. Define function matrix (2.9)F(t)=(β(t)S11*(t)0β(t)S11*(t)00β(t)S12*(t)0β(t)S12*(t)β(t)S21*(t)0β(t)S21*(t)00β(t)S22*(t)0β(t)S22*(t)),V(t)=(σ1(t)+d(t)+γ(t)-ρ12(t)00-σ1(t)ρ12(t)+d(t)+γ(t)0000ρ21(t)+d(t)+γ(t)-σ2(t)00-ρ21(t)σ2(t)+d(t)+γ(t)). Then (2.8) can be rewritten as (2.10)dZdt=(F(t)-V(t))Z, where Z=I(t)T.

Assume that Y(t,s),ts is the evolution operator of the linear periodic system (2.11)dydt=-V(t)y. That is, for each sR, the 4×4 matrix Y(t,s) satisfies (2.12)dY(t,s)dt=-V(t)Y(t,s),ts,Y(s,s)=I, where I is a 2×2 identity matrix.

Let Cω be the ordered Banach space of all ω-periodic function RR4, which is equipped with norm · and the positive cone Cω+={ϕCω:ϕ(t)0,tR}.

Consider the following operator L:CωCω by (2.13)(Lϕ)(t)=0+Y(t,t-a)ϕ(t-a)da,tR,ϕCω.

We can define the basic reproduction ratio R0=r(L), the spectral of radius of L.

Theorem 2.3 (see [<xref ref-type="bibr" rid="B11">11</xref>, Theorem 2.2]).

The following statements are valid:

R0=1 if and only if r(ΦF-V(ω))=1.

R0>1 if and only if r(ΦF-V(ω))>1.

R0<1 if and only if r(ΦF-V(ω))<1.

Thus, (S*,0,0,0) of (1.2) is asymptotically stable if R0<1 and it is unstable if R0>1.

3. The Threshold Dynamics

In this section, we show R0 as a threshold parameter between the extinction and the uniform persistence of the disease.

Theorem 3.1.

If R0<1, the disease-free periodic solution (S*,0,0,0) is globally asymptotically stable and if R0>1, it is unstable.

Proof.

By Theorem 2.3, if R0>1, the disease-free periodic solution (S*,0,0,0) is unstable. If R0<1, the disease-free periodic solution (S*,0,0,0) is locally stable. Hence, it is sufficient to show the global attractivity of (S*,0,0,0) when R0<1.

By (1.2), we have (3.1)dN11dt=B(t)+C(t)(N11+N21)+ρ12(t)N12-(σ1(t)+d(t))N11,dN12dt=σ1(t)N11-(ρ12(t)+d(t))N12,dN21dt=σ2(t)N22-(ρ21(t)+d(t))N21,dN22dt=B(t)+C(t)(N12+N22)+ρ21(t)N21-(σ2(t)+d(t))N22.

By the aforementioned conclusion, the above system has a unique positive fixed point S*(t) which is globally attractive in R+4{0}. It then follows that for any ε1>0, there exists T1>1 such that (3.2)Nij(t)=Sij(t)+Iij(t)Sij*(t)+ε1,t>T1. Obviously, Sij(t)Sij*(t)+ε1,  (i,j=1,2). Hence, we have (3.3)dI11dtρ12(t)I12+β(t)(S11*+ε1)(I11+I21)-(σ1(t)+d(t)+γ(t))I11,dI12dtσ1(t)I11+β(t)(S12*+ε1)(I12+I22)-(ρ12(t)+d(t)+γ(t))I12,dI21dtσ2(t)I22+β(t)(S21*+ε1)(I11+I21)-(ρ21(t)+d(t)+γ(t))I21,dI22dtρ21(t)I21+β(t)(S22*+ε1)(I12+I22)-(σ2(t)+d(t)+γ(t))I22. Denote (3.4)M1(t)=(β(t)0β(t)00β(t)0β(t)β(t)0β(t)00β(t)0β(t)). By Theorem 2.3, we have r(ΦF-V(ω))<1. We restrict ε1>0 such that r(ΦF-V+εM1(ω))<1.

Consider the system (3.5)dI~11dt=ρ12(t)I12+β(t)(S11*+ε1)(I11+I21)-(σ1(t)+d(t)+γ(t))I11,dI~12dt=σ1(t)I11+β(t)(S12*+ε1)(I12+I22)-(ρ12(t)+d(t)+γ(t))I12,dI~21dt=σ2(t)I22+β(t)(S21*+ε1)(I11+I21)-(ρ21(t)+d(t)+γ(t))I21,dI~22dt=ρ21(t)I21+β(t)(S22*+ε1)(I12+I22)-(σ2(t)+d(t)+γ(t))I22.

Applying Lemma 2.1 and the standard comparison principle, there exists a positive ω-positive function V1(t) such that I(t)V1(t)ep1t, where p1=lnr(ΦF-V+εM1(ω))/ω<0. Hence, we have that limtIij(t)=0,(i,j=1,2). Consequently, we obtain that (3.6)limt(S(t)-S*(t))=limt(N~(t)-I(t)-S*(t))=0, where N~(t)=(N11(t),N12(t),N21(t),N22(t)).

Hence, the disease free periodic solution (S*,0,0,0) is globally attractive and the proof is complete.

The following result shows that R0 is the threshold parameter for the extinction and the uniform persistence of the disease.

We define (3.7)X=R+8,X0=R+4×Int(R+4),X0=XX0. Let P:R+8R+8 be the Poincare map associated with (1.2), P(x0)=μ(ω,x0),x0R+8, where μ(t,x0) is the solution of (1.2) with μ(0,x0)=x0.

It is obvious that both X and X0 are positively invariant and X0 is relatively closed in X. Set (3.8)M={(S0,I0)X0:Pm(S0,I0)X0,  m0}. We now show that (3.9)M={(S,0):S0}. Obviously, {(S,0):S0}M. To show that M{(S,0):S0}=, we consider for any (S0,I0)X0{(S,0):S0}.

Firstly, if one element of I0=(I110,I120,I210,I220) is 0, say I110, that is, I110=0,  I120>0,  I210>0,  I220>0, then I21(t)>0, I22(t)>0 for any t>0. From (1.2) (3.10)dI11dt|t=0=ρ12(0)I12(0)+β(0)S11(0)I21(0)>0. It is clear that I11(t)>0, I12(t)>0, I21(t)>0, I22(t)>0.

Secondly, if two elements of I0=(I110,I120,I210,I220) is 0, for example, I110=0,I120=0,I210>0,I220>0. From (1.2), using the method of variation of constant, it is clear that S11(t)>0, S12(t)>0, S21(t)>0, S22(t)>0, for any t>0, (3.11)I11=(I110+0t(ρ12(s)I12(s)+β(s)S11(s)I21(s))e0s(σ1(u)+d(u)+γ(u)-β(s)S11(s))duds)×e-0t(σ1(s)+d(s)+γ(s)-β(s)S11(s))ds. Then I11(t)>0 for any t>0. I12(t)>0 can be proven similarly. So that I11(t)>0, I12(t)>0, I21(t)>0, I22(t)>0.

Thirdly, if three elements of I0=(I110,I120,I210,I220) are 0, for example, we chose I110=0,I120=0,I210=0,I220>0. From (1.2), (3.12)dI21dt|t=0=σ2(0)I22(0)>0. So I21(t)>0 for some small t. From (1.2), using the method of variation of constant, it is clear that I11(t)>  0,I12(t)>  0,I21(t)>  0,I22(t)>  0.

It follows that (S(t),I(t))X0, for 0<t1. Thus, the positive invariance of X0 implies (3.9). It is clear that there are two fixed points of P in M, which are M0=(0,0) and M1=(S*(0),0).

Now we see R0 as a threshold parameter between the extinction and the uniform persistence of the disease.

Theorem 3.2.

If R0>1, then there exists some ε>0 such that any solution (S(t),I(t)) of (1.2) with initial value (S(0),I(0))=(S0,I0)R+4×Int(R+4), satisfies limtinfI(t)ε. Furthermore, (1.2) admits at least one positive periodic solution.

Proof.

First we prove that P is uniformly persistent with respect to (X0,X). By Theorem 2.3, we have that R0>1 if and only if r(ΦF-V(ω))>1. Then we choose η>0 small enough such that r(ΦF-V-ηM1(ω))>1. Note that the perturbed system of (2.4), (3.13)dS^11dt=B(t)+C(t)(S^11+S^21)+ρ12(t)S^12-(σ1(t)+d(t)+β(t)δ)S^11,dS^12dt=σ1(t)S^11-(ρ12(t)+d(t)+β(t)δ)S^12,dS^21dt=σ2(t)S^22-(ρ21(t)+d(t)+β(t)δ)S^21,dS^22dt=B(t)+C(t)(S^12+S^22)+ρ21(t)S^21-(σ2(t)+d(t)+β(t)δ)S^22.

As in our previous analysis of system (2.4), we can choose δ>0 small enough such that the Poincare map associated with (3.13) admits a unique positive fixed point S*(0,δ) which is globally attractive in R+4{0}. By the implicit function theorem, it follows that S*(0,δ) is continuous in δ. Thus, we can fix a small number δ>0 such that S*(t,δ)>S*(t)-η¯, where η¯={η,η,η,η}. By the continuity of solutions with respect to the initial values, there exists δ0*>0 such that for all (S0,I0)X0, with (S0,I0)-Mδ0*. We have μ(t,(S0,I0))-μ(t,Mi)<δ,t[0,ω],i=0,1. We now claim that (3.14)limmsupd(Pm(S0,I0),Mi)δ0*. Suppose, by contradiction, that limmsupd(Pm(S0,I0),Mi)<δ0*, for some (S0,I0)X0, and i=0,1. Without loss of generality, we can assume that d(Pm(S0,I0),Mi)<δ0*,m0. Then, we have μ(t,Pm(S0,I0))-μ(t,Mi)<δ,m0,t[0,ω].

For any t0, let t=mω+t, where t[0,ω) and m=[t/m] is the greatest integer less than or equal to t/m. Then we get (3.15)μ(t,(S0,I0))-μ(t,Mi)=μ(t,Pm(S0,I0))-μ(t,Mi)<δ,t0. Let (S(t),I(t))=μ(t,(S0,I0)). It then follows that 0Iij(t)δ,t0,  i,j=1,2.

We have (3.16)dS11dtB(t)+C(t)(S11+S21)+ρ12(t)S12-(σ1(t)+d(t)+β(t)δ)S11,dS12dtσ1(t)S11-(ρ12(t)+d(t)+β(t)δ)S12,dS21dtσ2(t)S22-(ρ21(t)+d(t)+β(t)δ)S21,dS22dtB(t)+C(t)(S12+S22)+ρ21(t)S21-(σ2(t)+d(t)+β(t)δ)S22. Since the fixed point S*(0,δ) of the Poincare map associated with (3.13) is globally attractive and S*(t,δ)>S*(t)-η-, there is T>0, such that S(t)>S*(t)-η- for t>T, there holds (3.17)dI11dtρ12(t)I12+β(t)(S11*-η)(I11+I21)-(σ1(t)+d(t)+γ(t))I11,dI12dtσ1(t)I11+β(t)(S12*-η)(I12+I22)-(ρ12(t)+d(t)+γ(t))I12,dI21dtσ2(t)I22+β(t)(S21*-η)(I11+I21)-(ρ21(t)+d(t)+γ(t))I21,dI22dtρ21(t)I21+β(t)(S22*-η)(I12+I22)-(σ2(t)+d(t)+γ(t))I22. Since r(ΦF-V-ηM1(ω))>1, by Lemma 2.1, it is obvious that limtIij(t)=,i,j=1,2. This leads to a contradiction. Then (3.14) holds. Note that S*(0) is globally attractive in R+4{0}. By the aforementioned claim, it follows that M0 and M1 are isolated invariance sets in X,Ws(M0)X0=, and Ws(M1)X0=. Clearly, every orbit in M converges to either M0 or M1, M0 and M1 are acyclic in M. By [15, Theorem 1.3.1], P is uniformly persistent with respect to (X0,X). This implies the uniform persistence of the solutions of system (1.2) with respect to (X0,X). By [6, Theorem 1.3.6], P has a fixed point P(S-(0),I-(0))X0. Then, S-(0)R+4,I-(0)Int(R+4). We further claim that S-(0)R+4{0}, suppose that S-(0)=0, by (2.8), we can obtain -4(d(t)+γ(t))(I-11(0)+I-12(0)+I-21(0)+I-22(0))=0. And hence I-ij(0)=0,i,j=1,2, a contradiction. Thus, S-(0)0. Then (S-(0),I-(0)) is a positive ω-periodic solution of (1.2). The proof is complete.

4. Numerical Simulations

In this section, we give the numerical solutions (1.2) to clarify the correctness of our theoretical results. We set B(t)=0.4, C(t)=0.12, ρ21=0.16, ρ12=0.051, d(t)=0.3, γ(t)=0.365, σ1(t)=0.65+0.04cos(πt/6), σ2(t)=0.3+0.04cos(πt/6), β(t)=β+0.06588cos(πt/6). The initial value of the model is S11(0)=0.8, S12(0)=0.02, S21(0)=0.03, S22(0)=0.4, I11(0)=0.1, I12(0)=0.061, I21(0)=0.03, I22(0)=0.6. Figure 1 shows the numerical solutions of (1.2) when β=0.4. Because basic reproduction ratio R0>1, a positive periodic solution exists, and the disease is uniform persistence. In Figure 2, β=0.25, the disease dies out because R0<1.

When R0>1, a positive periodic solution exists and the disease will be uniform persistence.

When R0<1, a periodic disease-free solution exists and the disease dies out.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (CDJZR10100011) and the support of the Natural Science Foundation of Chongqing CSTC (2009BB2184).

Anderson R. M. May R. M. Infectious Diseases of Humans 1991 Oxford, UK Oxford University Press Hethcote H. W. The mathematics of infectious diseases Society for Industrial and Applied Mathematics 2000 42 4 599 653 10.1137/S0036144500371907 1814049 ZBL0993.92033 Daley D. J. Gani J. Epidemic Modeling: An Introduction 2005 New York, NY, USA Cambridge University Press Beretta E. Takeuchi Y. Convergence results in SIR epidemic models with varying population sizes Nonlinear Analysis A 1997 28 12 1909 1921 10.1016/S0362-546X(96)00035-1 1436361 ZBL0879.34054 Li J. Ma Z. Qualitative analyses of SIS epidemic model with vaccination and varying total population size Mathematical and Computer Modelling 2002 35 11-12 1235 1243 10.1016/S0895-7177(02)00082-1 1910450 ZBL1045.92039 Wen L. Zhong J. Global asymptotic stability and a property of the SIS model on bipartite networks Nonlinear Analysis 2012 13 2 967 976 10.1016/j.nonrwa.2011.09.003 2846896 ZBL1238.34110 Ostroff S. M. Kozarsky P. Emerging infectious diseases and travel medicine Infectious Disease Clinics of North America 1998 12 1 231 241 2-s2.0-0031916355 10.1016/S0891-5520(05)70420-7 Wilson M. E. The traveller and emerging infections: Sentinel, courier, transmitter Journal of Applied Microbiology 2003 94 32, supplement 1 1 11 2-s2.0-0038070407 10.1046/j.1365-2672.94.s1.1.x Ruan S. Wang W. Levin S. A. The effect of global travel on the spread of SARS Mathematical Biosciences and Engineering 2006 3 1 205 218 10.3934/mbe.2006.3.205 2192135 ZBL1089.92049 Zhang F. Zhao X.-Q. A periodic epidemic model in a patchy environment Journal of Mathematical Analysis and Applications 2007 325 1 496 516 10.1016/j.jmaa.2006.01.085 2273541 ZBL1101.92046 Wang W. Zhao X.-Q. Threshold dynamics for compartmental epidemic models in periodic environments Journal of Dynamics and Differential Equations 2008 20 3 699 717 10.1007/s10884-008-9111-8 2429442 ZBL1157.34041 Nakata Y. Kuniya T. Global dynamics of a class of SEIRS epidemic models in a periodic environment Journal of Mathematical Analysis and Applications 2010 363 1 230 237 10.1016/j.jmaa.2009.08.027 2559058 ZBL1184.34056 Bai Z. Zhou Y. Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate Nonlinear Analysis 2012 13 3 1060 1068 10.1016/j.nonrwa.2011.02.008 2863936 Cooke K. van den Driessche P. Zou X. Interaction of maturation delay and nonlinear birth in population and epidemic models Journal of Mathematical Biology 1999 39 4 332 352 10.1007/s002850050194 1727839 ZBL0945.92016 Zhao X. Q. Dynamic Systems in Population Biology 2003 New York, NY, USA Springer