Wright--Fisher kernels: from linear to non-linear dynamics, ergodicity and McKean--Vlasov scaling limits
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| Format: | Preprint |
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2024
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| _version_ | 1866915624418017280 |
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| author | Cordero, Fernando Jorquera, Christian Olivero, Héctor Videla, Leonardo |
| author_facet | Cordero, Fernando Jorquera, Christian Olivero, Héctor Videla, Leonardo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_13107 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wright--Fisher kernels: from linear to non-linear dynamics, ergodicity and McKean--Vlasov scaling limits Cordero, Fernando Jorquera, Christian Olivero, Héctor Videla, Leonardo Probability 60K35, 92D10, 92D15 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. |
| title | Wright--Fisher kernels: from linear to non-linear dynamics, ergodicity and McKean--Vlasov scaling limits |
| topic | Probability 60K35, 92D10, 92D15 |
| url | https://arxiv.org/abs/2410.13107 |