Mean-field interacting multi-type birth-death processes with a view to applications in phylodynamics

TL;DR

This paper introduces a mean-field interacting multi-type birth-death process to model complex evolutionary dynamics, allowing for interactions like carrying capacity and frequency-dependent selection, with implications for phylogenetic inference.

Contribution

It develops a novel mean-field interaction framework for multi-type birth-death processes, enabling modeling of interactions in evolutionary systems while maintaining tractable likelihoods.

Findings

Ensemble's interaction field converges to a deterministic trajectory as ensemble size grows.

The model captures both carrying capacity and frequency-dependent selection.

Provides a tractable likelihood framework for complex phylogenetic models.

Abstract

Multi-type birth-death processes underlie approaches for inferring evolutionary dynamics from phylogenetic trees across biological scales, ranging from deep-time species macroevolution to rapid viral evolution and somatic cellular proliferation. A limitation of current phylogenetic birth-death models is that they require restrictive linearity assumptions that yield tractable message-passing likelihoods, but that also preclude interactions between individuals. Many fundamental evolutionary processes -- such as environmental carrying capacity or frequency-dependent selection -- entail interactions, and may strongly influence the dynamics in some systems. Here, we introduce a multi-type birth-death process in mean-field interaction with an ensemble of replicas of the focal process. We prove that, under quite general conditions, the ensemble's stochastically evolving interaction field converges to a deterministic trajectory in the limit of an infinite ensemble. In this limit, the replicas effectively decouple, and self-consistent interactions appear as nonlinearities in the infinitesimal generator of the focal process. We investigate a special case that is rich enough to model both carrying capacity and frequency-dependent selection while yielding tractable message-passing likelihoods in the context of a phylogenetic birth-death model.