Wright--Fisher kernels: from linear to non-linear dynamics, ergodicity and McKean--Vlasov scaling limits

Abstract

We study the evolution of a pathogen with two allelic types infecting a population of hosts, where within-host type frequencies evolve in discrete time. Our framework is built on a two-parameter family of transition kernels on [0,1], which describe one-step updates of type frequencies. In the absence of host interaction, the single-host type-frequency process admits, for a broad class of parameters, a moment dual with a branching-coalescing structure reminiscent of the Ancestral Selection Graph. Under suitable parameter and time scalings, it converges to a Wright--Fisher diffusion with drift. To incorporate interactions among hosts, we introduce a mean-field mechanism whereby within-host dynamics depend on the empirical type distribution across the population. We prove uniform-in-time propagation of chaos, comparing the dynamics in a typical host with a corresponding non-linear Markov chain. Under appropriate scaling, this non-linear chain converges to a McKean--Vlasov Wright--Fisher diffusion. As an illustration, we analyse a model where mutation rates depend on the current type distribution across hosts and establish uniform-in-time propagation of chaos together with ergodicity of the limiting McKean--Vlasov equation.