Coalescent models for populations with time-varying population sizes and arbitrary offspring distributions

Abstract

The coalescent has been used to infer from gene genealogies the population dynamics of biological systems, such as the prevalence of an infectious disease. The offspring distribution affects the relationship between population dynamics and the genealogy, and for infectious diseases, the offspring distribution is often highly overdispersed. Here, we provide a general formula for the coalescent rate for populations with time-varying sizes and any offspring distribution. The formula is valid in the same large population limit as Kingmans original derivation. By relating our derivation to existing formulations of the coalescent, we show that differences in the coalescent rate derived for many population models may be explained by differences in the offspring distribution. The coalescent derivations presented here could be used to quantify the overdispersion in the offspring distribution of infectious diseases, which is useful for accurate modelling disease outbreaks.