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 2n2 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