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

Fernando Cordero, Christian Jorquera, Héctor Olivero, Leonardo Videla

TL;DR

A mean-field mechanism whereby within-host dynamics depend on the empirical type distribution across the population, and uniform-in-time propagation of chaos together with ergodicity of the limiting McKean--Vlasov equation is introduced.

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.

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

TL;DR

A mean-field mechanism whereby within-host dynamics depend on the empirical type distribution across the population, and uniform-in-time propagation of chaos together with ergodicity of the limiting McKean--Vlasov equation is introduced.

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.

Paper Structure

This paper contains 42 sections, 28 theorems, 190 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Notations and preliminaries
  4. Main results
  5. Ergodic properties of the Wright--Fisher kernels
  6. The single host level
  7. Non-linear chains and McKean--Vlasov limits
  8. The deme level
  9. A case study: self-stabilizing MVWF diffusion
  10. Ergodic properties of the Wright--Fisher kernels
  11. Updating functions
  12. Proof of Theorem \ref{['cor:unique']} and properties of the invariant measures
  13. The single host level: duality and scaling limits
  14. Some aspects of the dynamics
  15. Moment duality in fittest-type wins kernels with mutation
  16. ...and 27 more sections

Key Result

Theorem 2.1

For any $\delta \in (0,1]$ and ${p} \in [0,1]$, the kernel $P_{\delta, {p}}$ admits a unique invariant distribution, denoted by $\beta_{\delta, {p}}$. Moreover, for any $\mu\in\mathcal{P}$, $\mu (P_{\delta,p})^n$ converges in distribution as $n \to \infty$ to $\beta_{\delta, {p}}$, and the rate of c

Figures (3)

  • Figure 1: The action of the Wright--Fisher kernel $P_{\delta, p}$.
  • Figure 2: Invariant densities for the self-stabilizing MVWF model ($\theta_0=0.8$, $\theta_1=0.6$) for several values of $\gamma$: $\gamma=0$ (light blue), $\gamma=3$ (pink), and $\gamma=30$ (green). Darker curves show the theoretical invariant densities. Histograms are based on $10^5$ simulations of the rescaled Markovian particle system with scale parameter $N=600$, started from i.i.d. uniform initial conditions and sampled at time $T=10$.
  • Figure 3: Building blocks of the particle representation of the dual process $\widehat{M}$. Each panel depicts a one-step transition of the dual, with an intermediate configuration included to illustrate that every transition can be interpreted as the outcome of two successive events. In the dual picture, coalescence events may be followed by selective branchings or by killings of ancestral lines (or of the entire process). Forward in time, these correspond respectively to mutation or selective reproduction events followed by neutral reproduction events.

Theorems & Definitions (71)

  • Theorem 2.1: Ergodicity of Wright--Fisher kernels
  • Theorem 2.2: Moment duality
  • Theorem 2.4: Diffusion limit
  • Remark 2.5
  • Definition 2.6: $(L_1,L_2)$-Lipschitz
  • Theorem 2.7: Ergodicity of the non-linear dynamics
  • Theorem 2.9: McKean--Vlasov limit
  • Theorem 2.10: Propagation of chaos I
  • Theorem 2.11: Propagation of chaos II
  • Theorem 2.12: Ergodicity
  • ...and 61 more