Estimating finite-population reproductive numbers in heterogeneous populations

The basic reproductive number, [Formula: see text] , is one of the most important epidemiological quantities. [Formula: see text] provides a threshold for elimination and determines when a disease can spread or when a disease will die out. Classically, [Formula: see text] is calculated assuming an infinite population of identical hosts. Previous work has shown that heterogeneity in the host mixing rate increases [Formula: see text] in an infinite population. However, it has been suggested that in a finite population, heterogeneity in the mixing rate may actually decrease the finite-population reproductive numbers. Here, we outline a framework for discussing different types of heterogeneity in disease parameters, and how these affect disease spread and control. We calculate “finite-population reproductive numbers” with different types of heterogeneity, and show that in a finite population, heterogeneity has complicated effects on the reproductive number. We find that simple heterogeneity decreases the finite-population reproductive number, whereas heterogeneity in the intrinsic mixing rate (which affects both infectiousness and susceptibility) increases the finite-population reproductive number when [Formula: see text] is small relative to the size of the population and decreases the finite-population reproductive number when [Formula: see text] is large relative to the size of the population. Although heterogeneity has complicated effects on the finite-population reproductive numbers, its implications for control are straightforward: when [Formula: see text] is large relative to the size of the population, heterogeneity decreases the finite-population reproductive numbers, making disease control or elimination easier than predicted by [Formula: see text].