A model for disease transmission in a patchy environment

For a spatially heterogeneous environment with patches in which travel rates between patches depend on disease status, a disease transmission model is formulated as a system of ordinary differential equations. An expression for the basic reproduction number $R_0$ is derived, and the disease free equilibrium is shown to be globally asymptotically stable for $R_0<1$. Easily computable bounds on $R_0$ are derived. For a disease with very short exposed and immune periods in an environment with two patches, the model is analyzed in more detail. In particular, it is proved that if susceptible and infectious individuals travel at the same rate, then $R_0$ gives a sharp threshold with the endemic equilibrium being globally asymptotically stable for $R_0>1$. If parameters are such that for isolated patches the disease is endemic in one patch but dies out in the other, then travel of infectious individuals from the patch with endemic disease may lead to the disease becoming endemic in both patches. However, if this rate of travel is increased, then the disease may die out in both patches. Thus travel of infectious individuals in a patchy environment can have an important influence on the spread of disease.