The Basic Reproduction Number in a Multi-city Compartmental Epidemic Model

Abstract

A directed graph with cities as vertices and arcs determined by outgoing (or return) travel represents the mobility component in a population of individuals who travel between n cities. A model with 4 epidemiological compartments in each city that describes the propagation of a disease in this population is formulated as a system of 4n ² ordinary differential equations. Terms in the system account for disease transmission, latency, recovery, temporary immunity, birth, death, and travel between cities. The basic reproduction number $\mathcal{R}_0$ is determined as the spectral radius of a nonnegative matrix product, and easily computable bounds on $\mathcal{R}_0$ are obtained.