Abstract
In mathematical epidemiology, a well-known formula describes the impact of heterogeneity on the basic reproductive number for situations in which transmission is separable and for which there is one source of variation in susceptibility and one source of variation in infectiousness. This formula is written in terms of the magnitudes of the heterogeneities, as quantified by their coefficients of variation, and the correlation between them. A natural question to ask is whether analogous results apply when there are multiple sources of variation in susceptibility and/or infectiousness. In this paper we demonstrate that under three or more coupled heterogeneities, the basic reproductive number depends on details of the distribution of the heterogeneities in a way that is not seen in the well-known simpler situation. We provide explicit results for the cases of multivariate normal and multivariate log-normal distributions, showing that the basic reproductive number can again be expressed in terms of the magnitudes of the heterogeneities and the pairwise correlations between them. The results, however, differ between the two multivariate distributions, demonstrating that no formula of this type applies generally when there are three or more coupled heterogeneities. We see that the results are approximately equal when heterogeneities are relatively small and show that an earlier result in the literature (Koella, 1991) should be viewed in this light. We provide numerical illustrations of our results.
