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

• Main Authors: Cordero, Fernando, Jorquera, Christian, Olivero, Héctor, Videla, Leonardo
• Format: Journal Article
• Language: English
• Published: 16-10-2024
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.2410.13107

We consider a population of hosts infected by a pathogen that exists in two
strains. We use a two-parameter family of Markov kernels on $[0,1]$ to describe
the discrete-time evolution of the pathogen type-composition within and across
individuals. First, we assume that there is no interaction between pathogen
populations across host individuals. For a particular class of parameters, we
establish moment duality between the type-frequency process and a process
reminiscent of the \emph{Ancestral Selection Graph}. We also show convergence,
under appropriate scaling of parameters and time, to a Wright-Fisher diffusion
with drift. Next, we assume that pathogen-type compositions are correlated
across hosts by their empirical measure. We show a propagation of chaos result
comparing the pathogen type-composition of a given host with the evolution of a
non-linear chain. Furthermore, we show that under appropriate scaling, the
non-linear chain converges to a McKean-Vlasov diffusion. To illustrate our
results, we consider a population affected by mutation rates that depend on the
instantaneous distribution across multiple hosts. For this example, we study
the uniform-in-time propagation of chaos and the ergodicity of the limiting
McKean-Vlasov SDE.