# Transmission Dynamics of a Two-City SIR Epidemic Model with Transport-Related Infections

Authors: Yao Chen; Mei Yan; Zhongyi Xiang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

A two-city SIR epidemic model with transport-related infections is proposed. Some good analytical results are given for this model. If the basic reproduction numberℜ0γ≤1, there exists a disease-free equilibrium which is globally asymptotically stable. There exists an endemic equilibrium which is locally asymptotically stable if the basic reproduction number ℜ0γ>1. We also show the permanence of this SIR model. In addition, sufficient conditions are established for global asymptotic stability of the endemic equilibrium.


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## Body

## 1. Introduction

Epidemiology is the study of the spread of disease in time and space, aiming at tracing factors that give rise to their occurrence. Since Kermark and Mckendrick in [1] built up a system to study epidemiology in 1927, the concept of “compartment modeling” is widely used until now. From then on, many great epidemic models are proposed and researched [2–5], which assume that population lives in the fixed region, without travel. However, in fact, people usually travel among different regions; thus models involving dispersal are indispensable. To control the spread of an infectious disease, we have to know how the growth and spread of the disease affect its outbreak. And there are many factors that lead to the dynamics of an infectious disease of humans, such as human behavior as population dislocations, living styles, sexual practices, and rising international travel. On the other hand, climate change enables diseases and vectors to expand their range. Since the first AIDS case was reported in the United States in June 1981, the number of cases and deaths among persons with AIDS increased rapidly during the 1980s followed by substantial declines in new cases and deaths in the late 1990s. In 2003, SARS began in Guangdong province of China; however, it broke out at last in almost all parts of China and some other cities in the world due to dispersal [6]. Recently, some epidemic models have been proposed to understand the spread dynamics of infectious disease.Ahmed et al. in [7] introduced a model with travel between populations. In addition, Sattenspiel and Herring considered the same type of model but applied it to travel between populations in the Canadian subarctic, which can be thought of as a closed population where travel is easily quantified [8]. Ding et al. [9] and Sattenspiel et al. [10, 11] have also discussed other models for the spread of a disease among two patches and n patches. In [12], Wang and Mulone studied an SIS model with standard incidence rate on population dispersal among n patches. Wang and Zhao [13] proposed an SEIR epidemic model, assuming that the susceptible and exposed individuals have constant immigration rates. What is more, Wang and Zhao [14] formulated a general SEIRS for multispecies on multipatches, and the role of quarantine in the form of travel restriction was discussed.All these investigations ignore the possibility for the individuals to become infective during travel. In paper [15], Allen et al. have proposed the following SIS epidemic model to understand the effect of transport-related infection on disease spread for the first time:
(1)S˙1=a-βS1I1S1+I1-bS1+dI1-αS1+αS2-γαS2I2S2+I2,I˙1=βS1I1S1+I1-(c+d+α)I1+αI2+γαS2I2S2+I2,S˙2=a-βS2I2S2+I2-bS2+dI2-αS2+αS1-γαS1I1S1+I1,I˙2=βS2I2S2+I2-(c+d+α)I2+αI1+γαS1I1S1+I1.For many diseases (e.g., influenza, measles, chickenpox, etc.), after recovery, the individuals have immunity to the disease. Thus, an SIR or SIRS model is more suitable for this kind of disease. In this paper, we will study the effect of transport-related infection. Our results show that transport-related infection can make the disease endemic even if both the isolated regions are disease free.We consider a model with state variablesSi, Ii, Ri that represent the number of susceptible, infected, and removed individuals in city i(i=1,2), respectively. The basic assumptions underlying the dynamics of the system are as follows.(i)
We assume that both cities are identical.(ii)
All newborns, denoted bya, join into the susceptible class per unit time.(iii)
Natural death rate for susceptible, infected, and removed individuals is a constant per capita rateb.(iv)
Disease is transmitted with the standard form incidence rateβSiIi/Si+Ii+Ri,i=1,2 within city i. The transmission rate within a city is a constant β.(v)
We may assume that a susceptible individual goes into the infected part after infection.(vi)
Susceptible, infected, and removed individuals of every cityi leave for city j(i≠j,i,j=1,2) at a per capita rate α. We assume that two cities are connected by the direct transport such as airplanes or trains.(vii)
When the individuals in cityi travel to city j, disease is transmitted with the incidence rate γ(αSi)(αIi)/αSi+αIi=γαSiIi/Si+Ii+Ri,i=1,2 with a transmission rate γα.(viii)
The rate constant for recovery is denoted byd, and the per capita mortality rate for infected individual is c. Since this includes both natural and disease induced mortality, we have c>b.(ix)
We suppose that individuals who are traveling do not give birth and do not take death. Further we assume that removed individuals do not lose immunity during travel.These assumptions lead to a model of the following form:(2)S˙1=a-βS1I1S1+I1+R1-bS1-αS1+αS2-γαS2I2S2+I2+R2,I˙1=βS1I1S1+I1+R1-(c+α+b+d)I1+αI2+γαS2I2S2+I2+R2,R˙1=dI1-bR1-αR1+αR2,S˙2=a-βS2I2S2+I2-bS2-αS2+αS1-γαS1I1S1+I1+R1,I˙2=βS2I2S2+I2+R2-(c+α+b+d)I2+αI1+γαS1I1S1+I1+R1,R˙2=dI2-bR2-αR2+αR1.From the biological point of view, the termαSi represents the susceptible leaving city i and γαSiIi/(Si+Ii+Ri) denote individuals in αSi becoming infected during travel from city i to j. Hence, αSi-γαSiIi/(Si+Ii+Ri) should be nonnegative. Therefore, we always suppose 0≤γ≤1 in the following discussion.The paper is organized as follows. In next section, we will research the existence of equilibria and their local stability. In Section3, we will discuss permanence of the SIR model and some sufficient conditions for global stability of equilibrium in Section 4. In the final section, we will discuss our results and give some numerical simulations.


## 2. Local Stability

The assumption that both cities are identical, that is, demographic parameters are the same for each city, has enabled us to obtain an analytic expression for the equilibria. It is easy to check that system (2) has a disease-free equilibrium E0(S0,0,0,S0,0,0) for all parameter values, where S0=a/b. According to the concept of next generation matrix in Li et al. [16] and reproduction number presented in Liu et al. [17], we can define (3)F=(0000000β00γα00000000000000γα00β0000000),V=(b+αβ0-αγα00c+α+b+d00-α00-db+α00-α-αγα0b+αβ00-α00c+α+b+d000-α0-db+α).Hence the reproduction number for system (2) is
(4)ℜ0γ=ρ(FV-1)=β+γαb+d+c.
Here ρ(FV-1) represents the spectral radius of the matrix FV-1. When ℜ0γ>1, system (2) has unique positive equilibrium E+(S*,I*,R*,S*,I*,R*), where(5)I*=a(ℜ0γ-1)γα+β-c,S*=b+db(ℜ0γ-1)I*,R*=dbI*.The Jacobian matrix for the right hand of system (2) is given by
(6)J=(A1B1B2A2),
where (7)A1=(-βI1S1+I1+R1+βS1I1(S1+I1+R1)2-b-α-βS1S1+I1+R1+βS1I1(S1+I1+R1)2βS1I1(S1+I1+R1)2βI1S1+I1+R1-βS1I1(S1+I1+R1)2βS1S1+I1+R1-βS1I1(S1+I1+R1)2-c-α-b-d-βS1I1(S1+I1+R1)20d-b-α),(8)B1=(α-γαI2S2+I2+R2+γαS2I2(S2+I2+R2)2-γαS2S2+I2+R2+γαS2I2(S2+I2+R2)2γαS2I2(S2+I2+R2)2γαI2S2+I2+R2-γαS2I2(S2+I2+R2)2α+γαS2S2+I2+R2-γαS2I2(S2+I2+R2)2-γαS2I2(S2+I2+R2)200α),(9)B2=(α-γαI1S1+I1+R1+γαS1I1(S1+I1+R1)2-γαS1S1+I1+R1+γαS1I1(S1+I1+R1)2γαS1I1(S1+I1+R1)2γαI1S1+I1+R1-γαS1I1(S1+I1+R1)2α+γαS1S1+I1+R1-γαS1I1(S1+I1+R1)2-γαS1I1(S1+I1+R1)200α),(10)A2=(-βI2S2+I2+R2+βS2I2(S2+I2+R2)2-b-α-βS2S2+I2+R2+βS2I2(S2+I2+R2)2βS2I2(S2+I2+R2)2βI2S2+I2+R2-βS2I2(S2+I2+R2)2βS2S2+I2+R2-βS2I2(S2+I2+R2)2-c-α-b-d-βS2I2(S2+I2+R2)20d-b-α).Firstly, we study stability of the disease-free equilibriumE0.Theorem 1.
Ifℜ0γ<1, then E0 is locally asymptotically stable, and if ℜ0γ>1, then E0 is unstable.Proof.
Evaluating (6)–(10) at E0, we have the following Jacobian matrix:
(11)J(E0)=(ABBA),
where
(12)A=(-b-α-β00β-c-α-b-d00d-b-α),B=(α-γα00α+γα000α).
By Allen et al. [15], the eigenvalues of J(E0) are identical to those of A+B and A-B since(13)A+B=(-b-β-γα00β-c-b-d+γα00d-b),A-B=(-b-2α-β+γα00β-c-b-d-(2+γ)α00d-b-2α).
The eigenvalues of A+B are λ1=-b, λ2=-b, λ3=β-c-b-d+γα and the eigenvalues of A-B are λ1=-b-2α, λ2=-b-2α, λ3=β-c-b-d-2α-γα. By ℜ0γ<1, we have β+γα<b+d+c; we can conclude that all six eigenvalues of J(E0) are negative. When ℜ0γ>1, we have β+γα>b+d+c which implies λ3>0, so we obtain that at least one eigenvalue of J(E0) is positive. Hence E0 is locally asymptotically stable if ℜ0γ<1 and E0 is unstable if ℜ0γ>1. This completes the proof.Next, we research the stability ofE+. Evaluating (6)–(10) at E+ and using (5), we have the following Jacobian matrix for E+:
(14)J(E+)=(ABBA),
where
(15)A=(-βbb+d(1-1ℜ0γ)2-b-α-β(dℜ0γ+b)(b+d)ℜ0γ2bβ(ℜ0γ-1)(b+d)ℜ0γ2βbb+d(1-1ℜ0γ)2β(dℜ0γ+b)(b+d)ℜ0γ2-(c+α+b+d)-bβ(ℜ0γ-1)(b+d)ℜ0γ20d-b-α),B=(α-bγαb+d(1-1ℜ0γ)2-γα(dℜ0γ+b)(b+d)ℜ0γ2bγα(ℜ0γ-1)(b+d)ℜ0γ2bγαb+d(1-1ℜ0γ)2α+γα(dℜ0γ+b)(b+d)ℜ0γ2-bγα(ℜ0γ-1)(b+d)ℜ0γ200α).Similar to the proof of Theorem1, to calculate the eigenvalues of J(E+) is equivalent to calculating the eigenvalues of matrix A+B and A-B, where (16)A+B=(-b(β+γα)b+d(1-1ℜ0γ)2-b-(β+γα)(dℜ0γ+b)(b+d)ℜ0γ2b(β+γα)(ℜ0γ-1)(b+d)ℜ0γ2b(β+γα)b+d(1-1ℜ0γ)2(β+γα)(dℜ0γ+b)(b+d)ℜ0γ2-(c+b+d)-b(β+γα)(ℜ0γ-1)(b+d)ℜ0γ20d-b),A-B=(-b(β-γα)b+d(1-1ℜ0γ)2-b-2α-(β-γα)(dℜ0γ+b)(b+d)ℜ0γ2b(β-γα)(ℜ0γ-1)(b+d)ℜ0γ2b(β-γα)b+d(1-1ℜ0γ)2(β-γα)(dℜ0γ+b)(b+d)ℜ0γ2-(c+b+d+2α)-b(β-γα)(ℜ0γ-1)(b+d)ℜ0γ20d-b-2α).However, different from the case forE0, the eigenvalues of matrices A+B and A-B cannot be calculated explicitly. We will use the Routh-Hurwitz Theorem to study the stability of E+. Note that A+B and A-B have the same form as follows:
(17)J=(a11a12a13a21a22a230a32a33).
The characteristic polynomial of matrix J is λ3+A1λ2+A2λ+A3=0, where A1=-tr(J), A2=J1+J2+J3, and A3=-det(J) and J1=a11a22-a21a12, J2=a11a33, and J3=a22a33-a23a32. For convenience, we state the Routh-Hurwitz Theorem for above matrix.Lemma 2.
J is stable (i.e., each eigenvalue of J has negative real part) if and only if the following conditions hold:(i)
A1>0,(ii)
A3>0,(iii)
A1A2-A3>0.Using Lemma2, we have the following stability result for E+.Theorem 3.
Ifℜ0γ>1, then E+ is locally asymptotically stable.Proof.
Consider the matricesA and B in J(E+), the Jacobian matrix of system (2) at E+. It suffices to check that both A+B and A-B satisfy the conditions in Lemma 2. Firstly, we check them for A+B as the following three steps. For simplification, we will refer the entries of A+B as aij,i,j=1,2,3.
(i)A1>0. Obviously, a11<0 and a33<0. By (4),
(18)a22=(β+γα)(dℜ0γ+b)(b+d)ℜ0γ2-(c+b+d)=((β+γα)(dℜ0γ+bℜ0γ)+(β+γα)(b-bℜ0γ))×((b+d)ℜ0γ2)-1-(c+b+d)=b(β+γα)(1-ℜ0γ)(b+d)ℜ0γ2.
Since ℜ0γ>1, we have a22<0. Thus A1=-tr(A+B)=-(a11+a22+a33)>0.
(ii)Ji>0,i=1,2,3. Obviously, aii<0, i=1,2,3, a12<0, a21>0, a23<0, a32>0. So J1=a11a22-a21a12>0, J2=a11a33>0, J3=a22a33-a23a32>0.
(iii)A3>0,  A1A2-A3>0. Since a21>0, J3>0, we obtain
(19)A3=-det(A+B)=-(a11a22a33+a13a21a32-a12a21a33-a11a23a32)=bJ3+b(c+b+d)a21>0.
We also have
(20)A1A2-A3=-(a11+a22+a33)(J1+J2+J3)+det(A+B)=-a11(J1+J2)-a22(J1+J2+J3)-a33(J2+J3)+a13a21a32-a11a22a33.
Thus, A1A2-A3>0 by the fact that aii<0, Ji>0, i=1,2,3, and a13>0, a21>0, a32>0. Therefore, by Lemma 2, A+B is stable. Next we check A-B as follows. For convenience, we also refer the entries of A-B as aij,i,j=1,2,3.
(i)A1>0. Obviously, a33<0. Since 0<(1-1/ℜ0γ)2<1 when ℜ0γ>1 and also noting that 0≤γ≤1, we have
(21)a11=-b(β-γα)b+d(1-1ℜ0γ)2-b-2α=-bβb+d(1-1ℜ0γ)2-b-(2-bγb+d(1-1ℜ0γ)2)α<0;
the same to (18), one has
(22)a22=(β-γα)(dℜ0γ+b)(b+d)ℜ0γ2-(c+b+d+2α)≤(β+γα)(dℜ0γ+b)(b+d)ℜ0γ2-(c+b+d)<0.
Thus, A1=-tr(A-B)=-(a11+a22+a33)>0.
(ii)Ji>0, i=1,2,3. Since aii<0, i=1,2,3. Obviously J2=a11a33>0. For J1, we have
(23)J1=a11a22-a21a12=(a11+a21)a22-(a12+a22)a21=(b+2α)(c+b+d+2α)-(b+2α)(β-γα)(dℜ0γ+b)(b+d)ℜ0γ2+b(c+b+d+2α)(β-γα)b+d(1-1ℜ0γ)2.J1>0 can be shown as the following two cases.
Case 1 (β-γα>0). By (4), we obtain that
(24)b+d+c=(β+γα)1ℜ0γ=(β-γα+2γα)1ℜ0γ.
Thus,J1 can be rewritten as
(25)J1>(b+2α)[2α+(β-γα+2γα)1ℜ0γ]-(b+2α)(β-γα)1ℜ0γ+b(c+b+d+2α)(β-γα)b+d(1-1ℜ0γ)2=(b+2α)(2α+2γα1ℜ0γ)+b(c+b+d+2α)(β-γα)b+d(1-1ℜ0γ)2.
Since ℜ0γ>1, it is clear that J1>0.
Case 2  (β-γα≤0).  According to ℜ0γ>1, we have
(26)J1=(b+2α)(c+b+d+2α)-(b+2α)(β-γα)(dℜ0γ+b)b+d+b(c+b+d+2α)(β-γα)b+d-2b(c+b+d+2α)(β-γα)(b+d)ℜ0γ+b(c+b+d+2α)(β-γα)(b+d)ℜ0γ2>[bd+b2+2dα+bβ+bα(2-γ)](c+b+d+2α)b+d-(b+2α)(β-γα)(dℜ0γ+b)b+d-b(c+b+d+2α)(β-γα)(b+d)ℜ0γ2.
Since 0≤γ≤1, clearly we have J1>0. The same to above analysis, for J3, we also obtain that
(27)J3=a22a33-a23a32=-(b+2α)(β-γα)(dℜ0γ+b)(b+d)ℜ0γ2+(b+2α)(c+b+d+2α)+bd(β-γα)(ℜ0γ-1)(b+d)ℜ0γ2.J3>0 can be shown as the following two cases.
Case 1 (β-γα>0). Consider
(28)J3>(b+2α)[2α+(β-γα+2γα)1ℜ0γ]-(b+2α)(β-γα)1ℜ0γ+bd(β-γα)(ℜ0γ-1)(b+d)ℜ0γ2=(b+2α)(2α+2γα1ℜ0γ)+bd(β-γα)(ℜ0γ-1)(b+d)ℜ0γ2.
Since ℜ0γ>1, we have J3>0.
Case 2 (β-γα≤0). Consider
(29)J3=-(β-γα)bdℜ0γ+b2+2dαℜ0γ+2bα-bdℜ0γ+bd(b+d)ℜ0γ2+(b+2α)(c+b+d+2α)=-(β-γα)(b2+2dαℜ0γ+2bα+bd)(b+d)ℜ0γ2+(b+2α)(c+b+d+2α)>0.
(iii)A3>0, A1A2-A3>0(30)A3=-det(A-B)=-(a11a22a33+a13a21a32jyjg-a12a21a33-a11a23a32)=(b+2α)J3+b(β-γα)b+d(1-1ℜ0γ)2×(b+2α)(c+b+d+2α).
Case 1 (β-γα>0). Due to J3>0, so A3>0.
Case 2 (β-γα≤0).  According to ℜ0γ>1, we have
(31)A3=(b+2α)[-(β-γα)(b2+2dαℜ0γ+2bα+bd)(b+d)ℜ0γ2ffffffff+(b+2α)(c+b+d+2α)-(β-γα)(b2+2dαℜ0γ+2bα+bd)(b+d)ℜ0γ2]+b(b+2α)(c+b+d+2α)(β-γα)b+d-2b(b+2α)(c+b+d+2α)(β-γα)(b+d)ℜ0γ+b(b+2α)(c+b+d+2α)(β-γα)(b+d)ℜ0γ2>-(β-γα)(b+2α)(b2+2dαℜ0γ+2bα+bd)(b+d)ℜ0γ2+(b+2α)(c+b+d+2α)[b2+bd+2dα+bβkkkkkkkkkkkkkkkkkkkkkk+(2-γ)bα]×(b+d)-1-b(b+2α)(c+b+d+2α)(β-γα)(b+d)ℜ0γ2>0.
Since 0≤γ≤1, clearly one has A3>0. On the other hand,
(32)A1A2-A3=-(a11+a22+a33)(J1+J2+J3)+det(A-B)=-a11(J1+J2)-a22(J1+J2+J3)-a33(J2+J3)+a13a21a32-a11a22a33
because   a13a21a32=(b2d(β-γα)2(ℜ0γ-1)/(b+d)2ℜ0γ2)(1-(1/ℜ0γ))2>0. So A1A2-A3>0, since aii<0, Ji>0, i=1,2,3.
Therefore,A1A2-A3>0. By Lemma 2, A-B is also stable. Hence E+ is locally asymptotically stable. This completes the proof.


## 3. Permanence

Firstly, we consider permanence of the disease. Set initial conditions asSi(0)≥0,Ii(0)≥0,  and Ri(0)≥0 for i=1,2. It is easy to check that all solutions of system (2) are nonnegative (i.e., Si(t)≥0, Ii(t)≥0, and Ri(t)≥0 for t≥0 and i=1,2) under the assumption 0≤γ≤1. The following result shows that system (2) is ultimately bounded above.Theorem 4.
There exists anM>0 such that for any solution (S1(t),I1(t),R1(t),S2(t),I2(t),R2(t)) of system (2) with initial values Si(0)≥0, Ii(0)≥0, and Ri(0)≥0, i=1,2, there must be a t1>0 such that Si(t)≤M, Ii(t)≤M, and Ri(t)≤M for i=1,2 and t≥t1.Proof.
LetV(t)=S1(t)+I1(t)+R1(t)+S2(t)+I2(t)+R2(t). We have
(33)V˙(t)=2a-bS1-cI1-bI1-bR1-bS2-cI2-bI2-bR2=2a-bS1-(b+c)I1-bR1-bS2-(b+c)I2-bR2≤2a-bV.
Hence, by comparison theory of differential equations, it is easy to verify that there exists t1>0 such that V(t)≤M≜(2a/b)+ε, t≥t1 for ε>0. Then Si(t)≤V(t)≤M, Ii(t)≤V(t)≤M, and Ri(t)≤V(t)≤M for t≥t1. This completes the proof.Theorem 5.
Letℜ0γ>1. Then there exists an ε>0 such that every solution (S1(t),I1(t),R1(t),S2(t),I2(t),R2(t)) of system (2) with initial values Si(0)≥0, I1(0)+I2(0)>0 and Ri(0)≥0 for i=1,2 satisfies
(34)liminft→∞Si(t)≥ε,liminft→∞Ii(t)≥ε,liminft→∞Ri(t)≥ε,i=1,2.Proof.
By system (2) and the fact that 0≤γ≤1, we have
(35)S˙i≥a-(β+b+α)Si,i=1,2.
Hence, Si is always ultimately lower bounded by some positive constant; see, for example, ms=a/(2(β+b+α)), which is independent of initial values. And so is Ri if both I1 and I2 are ultimately lower bounded by some positive constant independent of initial values. Therefore, it suffices to prove that liminft→∞Ii(t)≥ε, i=1,2.
The result follows from an application of Theorem  4.6 in Arino et al. [18]. Define
(36)X={(S1,I1,R1,S2,I2,R2)∣Si≥0,Ii≥0,Ri≥0,i=1,2},X0={(S1,I1,R1,S2,I2,R2)∈X∣I1+I2>0,i=1,2},∂X0=X∖X0.
It then suffices to show that system (2) is uniformly persistent with respect to (X0,∂X0).
Next,X is positively invariant with respect to system (2). It is easy to verify that I1(t)+I2(t)>0, i=1,2 for t>0 if Si(0)≥0, Ri(0)≥0, and I1(0)+I2(0)>0 for i=1,2. This X0 is also positively invariant. Furthermore, by Theorem 4, there exists a compact set B in which all solutions of system (2) initiated in X will enter and remain forever after. The compactness condition (C4.2) in Arino et al. [18] is easily verified for this set B. Denote
(37)M∂={(S1(0),I1(0),R1(0),S2(0),I2(0),R2(0))∣dd(S1(t),I1(t),R1(t),S2(t),I2(t),R2(t))∈∂X0,dd∀t≥0}.
We now show that
(38)M∂={(S1,0,R1,S2,0,R2)∣Si≥0,Ri≥0,i=1,2}.
Suppose that (S1(0),I1(0),R1(0),S2(0),I2(0),R2(0))∈M∂. It suffices to show Ii(t)=0 for any t≥0 and i=1,2. If not, there exists t0≥0 such that I1(t0)>0 or I2(t0)>0. Hence, I1(t0)+I2(t0)>0 and (S1(t0),I1(t0),R1(t0),S2(t0),I2(t0),R2(t0))∈X0, which contradicts (S1(0),I1(0),R1(0),S2(0),I2(0),R2(0))∈M∂. This proves (38).
Denote the omega limit set of the solutions of system (2) starting in (S1(0),I1(0),R1(0),S2(0),I2(0),R2(0))∈X by ω(S1(0),I1(0),R1(0),S2(0),I2(0),R2(0)) (which exists by Theorem 4). Let
(39)Ω=∪{ω(S1(0),I1(0),R1(0),S2(0),I2(0),R2(0))∣aaaa(S1(0),I1(0),R1(0),S2(0),I2(0),R2(0))∈M∂}.
Restricting system (2) on M∂ gives
(40)S˙1=a-(b+α)S1+αS2,R˙1=-(b+α)R1+αR2,S˙2=a-(b+α)S2+αS1,R˙2=-(b+α)R2+αR1.
It is easy to verify that system (40) has a unique equilibrium E1(S0,0,S0,0), where S0=a/b. And thus E0(S0,0,0,S0,0,0) is the unique equilibrium of system (2) in M∂. It is easy to check that E1 is locally asymptotically stable. Hence, it is also globally asymptotically stable since system (40) is a linear system. Thus Ω={E0}. And E0 is a covering of Ω, which is isolated (since E0 is the unique equilibrium) and is acyclic (since there exists no solution in M∂ which links E0 to itself). Finally, the proof will be done if we show that E0 is a weak repeller for X0; that is,
(41)limsupt→∞dist((S1(t),I1(t),R1(t),S2(t),I2(t),R2(t)),E0)>0,
where  (S1(t),I1(t),R1(t),S2(t),I2(t),R2(t)) is the solution of system (2) with initial value (S1(0),I1(0),R1(0),S2(0),I2(0),R2(0))∈X0. By the proof of Lemma  3.5 in Cui et al. [19], (41) is valid if
(42)WS(E0)∩X0=ϕ,
where Ws(E0) denotes the stable manifold of E0. Suppose (42) is not valid; then there exists a solution (S1(t),I1(t),R1(t),S2(t),I2(t),R2(t))∈X0, t≥0 of system (2) with initial value (S1(0),I1(0),R1(0),S2(0),I2(0),R2(0))∈X0, such that
(43)Si(t)→S0,Ii(t)→0,Ri(t)→00000000000000000ast→∞,i=1,2.
Since ℜ0γ=(β+γα)/(b+d+c)>1, we can choose δ>0 which is small enough such that S0-δ>0 and
(44)mδ≜(β+γα)(S0-δ)S0-(b+d+c)>0.
Define V=I1+I2. For S0-δ>0, by (43) there exists t1>0 such that
(45)S0-δ<Si(t)<S0+δ,0<Ii(t)<δ,0<Ri(t)<δ,000000000000000000000000000000000t≥t1,i=1,2.
Hence by system (2),
(46)V˙=(β+γα)(S1I1S1+I1+R1+S2I2S2+I2+R2)-(c+b+d)(I1+I2)≥(β+γα)(S0-δS0-δ+δI1+S0-δS0-δ+δI2)-(c+d+b)(I1+I2)=mδV,
for t≥t1. Hence, V(t)≥V(t1)exp(mδ(t-t1)). By (44), we have V(t)→∞ as t→∞, which contradicts (43). Thus (42) holds, which completes the proof.


## 4. Global Stability

First of all, we will discuss the global stability ofE0 under the condition ℜ0γ≤1.Theorem 6.
Ifℜ0γ≤1, then E0 is globally asymptotically stable.Proof.
Consider the following function:(47)V(t)=I1+I2.
Its derivative along the solutions of system (2) is
(48)V˙(t)=(β+γα)(S1I1S1+I1+R1+S2I2S2+I2+R2)-(c+b+d)(I1+I2).
If ℜ0γ<1, then
(49)V˙(t)≤[(β+γα)-(c+b+d)](I1+I2)≤0.
If ℜ0γ=1, then
(50)V˙(t)=-(β+γα)[I1(I1+R1)S1+I1+R1+I2(I2+R2)S2+I2+R2]≤0.
When ℜ0γ≤1, we set
(51)L={(S1,I1,R1,S2,I2,R2)∣V˙(t)=0}={(S1,I1,R1,S2,I2,R2)∣I1=0,I2=0}.
Restricting system (2) on the set L, we have d(R1+R2)/dt=-b(R1+R2), then limt→∞Ri(t)=0 for i=1,2. d(S1+S2)/dt=2a-b(S1+S2), then limt→∞Si(t)=a/b for i=1,2. Therefore, M={E0} is the largest positively invariant subset of L. By Lyapunov-LaSalle theorem, E0 is global asymptotically stable provided ℜ0γ≤1. This completes the proof.Theorems4 and 5 imply that system (2) is permanent if ℜ0γ>1. Next, we consider the global asymptotic stability of E+. The set X0 defined in proof of Theorem 5 will be used.Theorem 7.
Suppose that(52)b+2α>max{92|β-γα|,d2+|β-γα|}.
Then the endemic equilibrium point E+ is globally asymptotically stable on X0 for ℜ0γ>1.Proof.
By Theorem let us consider the function:(53)V(t)={(S1(t)-S2(t))2+(I1(t)-I2(t))2vvvvgh+(R1(t)-R2(t))2}×(2)-1.
The time derivative of V(t) along the solution of system (2) becomes
(54)V˙=(S˙1-S˙2)(S1-S2)+(I˙1-I˙2)(I1-I2)+(R˙1-R˙2)(R1-R2)={(β-γα)(S2I2S2+R2+I2-S1I1S1+I1+R1)j-(b+2α)(S1-S2)(S2I2S2+R2+I2-S1I1S1+I1+R1)}(S1-S2)+{(β-γα)(S1I1S1+I1+R1-S2I2S2+R2+I2)jjk-(c+b+d+2α)(I1-I2)(S1I1S1+I1+R1-S2I2S2+R2+I2)}(I1-I2)+{d(I1-I2)-(b+2α)(R1-R2)}(R1-R2).
Note that
(55)S2I2S2+I2+R2-S1I1S1+I1+R1=S1S2I2+S2I1I2+S2I2R1-S1I1S2-S1I1S2-S1I1R2(S1+I1+R1)(S2+I2+R2)≤2|I2-I1|+2|S2-S1|+|R1-R2|,
which gives the following:
(56)V˙≤-(b+2α)(S1-S2)2-(c+2α+b+d)(I1-I2)2V˙≤-(b+2α)(R1-R2)2+d(I1-I2)(R1-R2)V˙≤+(β-γα)(S2I2S2+I2+R2-S1I1S1+I1+R1)V˙≤×(S1-S2-I1+I2)V˙≤-(b+2α)(S1-S2)2-(c+2α+b+d)(I1-I2)2V˙≤-(b+2α)(R1-R2)2+d|I1-I2||R1-R2|V˙≤+|β-γα|(2|I1-I2|+2|S1-S2|+|R1-R2|)V˙≤×(|S1-S2|+|I1-I2|)V˙≤-{(b+2α)-92|β-γα|}|S1-S2|2V˙≤-{(c+2α+b+12d)-92|β-γα|}|I1-I2|2V˙≤-{(b+2α)-|β-γα|-12d}|R1-R2|2.
The above quadratic form is negative definite if and only if
(57)b+2α>92|β-γα|,b+2α>d2+|β-γα|.
It is easy to check that the above conditions are satisfied if and only if (52) is satisfied. Hence we can find some positive constant λ=min{b+2α-(9/2)|β-γα|,b+2α-(d/2)-|β-γα|} satisfying
(58)V˙≤λ{(S1-S2)2+(I1-I2)2+(R1-R2)2}=-2λV(t),
which shows that for any solution  (S1(t),I1(t),R1(t),S2(t),I2(t),R2(t)) of system (2), we have
(59)limt→∞{S1(t)-S2(t)}=0,limt→∞{I1(t)-I2(t)}=0,limt→∞{R1(t)-R2(t)}=0.
By Lyapunov's theorem, we know that ω(x)∩ℝ+6 is contained in the set {x∈ℝ+6∣V˙=0}={(S1,I1,R1,S2,I2,R2)∈ℝ+6∣S1=S2,I1=I2,R1=R2}. Here ω(x) is the ω-limit set of the solution of system (2) with an initial value x and ℝ+6 is the state space {x∈ℝ6∣x≥0}. On the set Ω={(S1,I1,R1,S2,I2,R2)∈ℝ+6∣S1=S2,I1=I2,R1=R2}, we now consider the following system for S=Si, I=Ii, and R=Ri(i=1,2); that is,
(60)S˙=a-bS-(β+γα)SIS+I+R=f(S,I,R),I˙=(β+γα)SIS+I+R-(c+b+d)I=g(S,I,R),R˙=dI-bR=h(S,I,R).
It is trivial that the equilibrium (S*,I*,R*) of system (60) is globally asymptotically stable and the equilibrium (a/b,0,0) is unstable as ℜ0γ>1. This shows that the endemic equilibrium E+ of system (2) is globally asymptotically stable. This completes the proof.


## 5. Numerical Simulations and Discussion

In this paper, a two-city SIR epidemic model with transport-related infections is proposed. According to Theorems1 and  3, we obtain that there exist a disease-free equilibrium and an endemic equilibrium which are locally asymptotically stable if the basic reproduction number ℜ0γ≤1 and ℜ0γ>1, respectively. Theorems 4 and 5 provide the permanence of this SIR model. In addition, sufficient conditions are established in Theorems 6 and 7 for global asymptotic stability of the disease-free and the endemic equilibrium, severally. The following numerical simulations, we will present, are to explain the feasibility of our main results.If we seta=1, b=0.2, c=0.4, d=0.2, α=0.3, β=0.6, and γ=0.5, by a simple computation, we derive
(61)ℜ0γ=0.9375<1.
Obviously, the assumptions of Theorems 1 and 6 are satisfied, so disease-free equilibrium of system (2) is globally asymptotically stable. From Figure 1, it is easy to observe that if α is relatively small, then both isolated cities are disease free and the transport-related infection may not lead to the disease becoming endemic.(a) shows movement paths ofS1,I1, and R1 as functions of time t. (b) shows movement paths of S2,I2, and R2 as functions of time t. Here a=1, b=0.2, c=0.4, d=0.2, α=0.3, β=0.6, and γ=0.5. Initial data are 0.6,0.5,0.3,0.2,0.7,0.4.
(a)
(b)On the other hand, if we havea=1, b=0.2, c=0.4, d=0.2, α=0.3, β=0.6, and γ=0.5, we have
(62)ℜ0γ=1.2500>1.
The assumption of Theorem 3 is satisfied, and since
(63)b+2α-max{92|β-γα|,d2+|β-γα|}=0.9000>0,
it is not difficult to prove Theorem 7. Hence, the endemic equilibrium of system (2) is globally asymptotically stable. Figure 2 shows that if α is relatively large, then the disease is endemic in the two isolated cities and the transport-related infection will surely lead to the disease becoming endemic.(a) shows movement paths ofS1,I1, and R1 as functions of time t. (b) shows movement paths of S2,I2, and R2 as functions of time t. Here a=1, b=0.2, c=0.4, d=0.2, α=0.8,  β=0.6, and γ=0.5. Initial data are 0.6,0.5,0.3,0.2,0.7,0.4.
(a)
(b)



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*Source: 764278-2014-01-23.xml*