A General Birth-Death-Sampling Model for Epidemiology and Macroevolution

Birth-death stochastic processes are the foundation of many phylogenetic models and are widely used to make inferences about epidemiological and macroevolutionary dynamics. There are a large number of birth-death model variants that have been developed; these impose different assumptions about the temporal dynamics of the parameters and about the sampling process. As each of these variants was individually derived, it has been difficult to understand the relationships between them as well as their precise biological and mathematical assumptions. Without a common mathematical foundation, deriving new models is non-trivial. Here we unify these models into a single framework, prove that many previously developed epidemiological and macroevolutionary models are all special cases of a more general model, and illustrate the connections between these variants. This framework centers around a technique for deriving likelihood functions for arbitrarily complex birth-death(-sampling) models that will allow researchers to explore a wider array of scenarios than was previously possible. We then use this frame-work to derive general model likelihoods for both the "single-type" case in which all lineages diversify according to the same process and the "multi-type" case, where there is variation in the process among lineages. By re-deriving existing single-type birth-death sampling models we clarify and synthesize the range of explicit and implicit assumptions made by these models.