Link between the Birth-Death Process and the Kingman Coalescent-Applications to Phylogenetic Epidemiology

The two most popular tree models used in phylogenetics are the birth-death process (BD) and the Kingman coalescent (KC). These two models differ in several respects, notably: (i) the curve of the population size through time is a stochastic process in the BD, versus a parametrized curve in the KC, (ii) the BD makes assumptions about the way samples are collected, while the KC conditions on the number of samples and the collection times, thus bypassing the need to describe the sampling procedure. These two models have been applied to different contexts: the BD in macroevolutionary studies of clades of species, and the KC for populations. The exception is the field of phylogenetic epidemiology which uses both models. This then asks the question of how such different models can be used in the same context. In this paper, we study large-population limits of the BD, in a search for a mathematical link between the BD and the KC. We show that the KC is the large-population limit of a BD conditioned on a given population trajectory, and we provide the formula for the parameter θ of the limiting KC. This formula appears in earlier studies, but the present article is the first to show formally how the correspondence arises as a large-population limit, and that the BD needs to be conditioned for the KC to arise. Besides these fundamentally mathematical results, we demonstrate how our findings can be used practically in phylogenetic inference. In particular, we propose a new method for phylogenetic epidemiology, called CalicoBird, ensuing from our results. We conjecture that this new method, used in conjunction with auxiliary data (e.g. prevalence or incidence data), should allow estimating important epidemiological parameters (e.g. the prevalence and the effective reproduction number), in a way that is robust to the data-generating model and the sampling procedure. Future studies will be needed to put our claims to the test.