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 number , 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 . We also show the permanence of this SIR model. In addition, sufficient conditions are established for global asymptotic stability of the endemic equilibrium. 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 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