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

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 numberR 0γ ≤ 1, there exists a disease-free equilibriumwhich is globally asymptotically stable.There exists an endemic equilibrium which is locally asymptotically stable if the basic reproduction numberR 0γ > 1. We also show the permanence of this SIRmodel. In addition, sufficient conditions are established for global asymptotic stability of the endemic equilibrium.


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][3][4][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  patches.In [12], Wang and Mulone studied an SIS model with standard incidence rate on population dispersal among  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 2 Journal of Applied Mathematics understand the effect of transport-related infection on disease spread for the first time: 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 transportrelated infection can make the disease endemic even if both the isolated regions are disease free.
We consider a model with state variables   ,   ,   that represent the number of susceptible, infected, and removed individuals in city  ( = 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 by , join into the susceptible class per unit time.
(iii) Natural death rate for susceptible, infected, and removed individuals is a constant per capita rate .
The transmission rate within a city is a constant .
(v) We may assume that a susceptible individual goes into the infected part after infection.
(vi) Susceptible, infected, and removed individuals of every city  leave for city  ( ̸ = , ,  = 1, 2) at a per capita rate .We assume that two cities are connected by the direct transport such as airplanes or trains.
(vii) When the individuals in city  travel to city , disease is transmitted with the incidence rate (  )(  )/  +   =     /  +   +   ,  = 1, 2 with a transmission rate .
(viii) The rate constant for recovery is denoted by , and the per capita mortality rate for infected individual is .Since this includes both natural and disease induced mortality, we have  > .
(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: ( From the biological point of view, the term   represents the susceptible leaving city  and     /(  +   +   ) denote individuals in   becoming infected during travel from city  to .Hence,   −     /(  +   +   ) should be nonnegative.Therefore, we always suppose 0 ≤  ≤ 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 Section 3, 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.
Proof.Evaluating ( 6)-( 10) at  0 , we have the following Jacobian matrix: where By Allen et al. [15], the eigenvalues of ( 0 ) are identical to those of  +  and  −  since ) . ( The eigenvalues of  +  are we have  +  <  +  + ; we can conclude that all six eigenvalues of ( 0 ) are negative.When R 0 > 1, we have  +  >  +  +  which implies  3 > 0, so we obtain that at least one eigenvalue of ( 0 ) is positive.Hence  0 is locally asymptotically stable if R 0 < 1 and  0 is unstable if R 0 > 1.This completes the proof.
Next, we research the stability of  + .Evaluating ( 6)-( 10) at  + and using (5), we have the following Jacobian matrix for  + : where Similar to the proof of Theorem 1, to calculate the eigenvalues of ( + ) is equivalent to calculating the eigenvalues of matrix  +  and  − , where However, different from the case for  0 , the eigenvalues of matrices  +  and  −  cannot be calculated explicitly.We will use the Routh-Hurwitz Theorem to study the stability of  + .Note that  +  and  −  have the same form as follows: ) .
The characteristic polynomial of matrix  is Using Lemma 2, we have the following stability result for  + .Theorem 3. If R 0 > 1, then  + is locally asymptotically stable.
Proof.Consider the matrices  and  in ( + ), the Jacobian matrix of system (2) at  + .It suffices to check that both  +  and − satisfy the conditions in Lemma 2. Firstly, we check them for + as the following three steps.For simplification, we will refer the entries of  +  as   , ,  = 1, 2, 3.
Case 1 ( −  > 0).By (4), we obtain that Thus,  1 can be rewritten as Since 0 ≤  ≤ 1, clearly we have  1 > 0. The same to above analysis, for  3 , we also obtain that 3 > 0 can be shown as the following two cases.
Then there exists an  > 0 such that every solution Proof.By system (2) and the fact that 0 ≤  ≤ 1, we have Hence,   is always ultimately lower bounded by some positive constant; see, for example,   = /(2( +  + )), which is independent of initial values.And so is   if both  1 and  2 are ultimately lower bounded by some positive constant independent of initial values.Therefore, it suffices to prove that lim inf  → ∞   () ≥ ,  = 1, 2.

Numerical Simulations and Discussion
In this paper, a two-city SIR epidemic model with transportrelated infections is proposed.According to Theorems 1 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 R 0 ≤ 1 and R 0 > 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.