A Multi-city Epidemic Model

Some analytical results are given for a model that describes the propagation of a disease in a population of individuals who travel between n cities. The model is formulated as a system of 2n ordinary differential equations with terms accounting for disease transmission, recovery, birth, death, and travel between cities. The mobility component is represented as a directed graph with cities as vertices and arcs determined by outgoing (or return) travel. An explicit formula that can be used to compute the basic reproduction number R0 is obtained, and explicit bounds on R0 are determined in the case of homogeneous contacts between individuals within each city. Numerical simulations indicate that R0 is a sharp threshold, with the disease dying out if R0 < 1 and reaching an endemic level in all connected cities if R0 > 1.