The Final Size of an Epidemic and Its Relation to the Basic Reproduction Number

Abstract

We study the final size equation for an epidemic in a subdivided population with general mixing patterns among subgroups. The equation is determined by a matrix with the same spectrum as the next generation matrix and it exhibits a threshold controlled by the common dominant eigenvalue, the basic reproduction number R 0: There is a unique positive solution giving the size of the epidemic if and only if R 0 exceeds unity. When mixing heterogeneities arise only from variation in contact rates and proportionate mixing, the final size of the epidemic in a heterogeneously mixing population is always smaller than that in a homogeneously mixing population with the same basic reproduction number R 0. For other mixing patterns, the relation may be reversed. © 2011 Society for Mathematical Biology.