A multi-city epidemic model

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Abstract
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 2 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, {\cal R}_0 , is obtained, and explicit bounds on {\cal R}_0 are determined in the case of homogeneous contacts between individuals within each city. Numerical simulations indicate that {\cal R}_0 is a sharp threshold, with the disease dying out if {\cal R}_0 1 .