A phase transition for a spatial host-parasite model with extreme host immunities on $\mathbb{Z}^d$ and $\mathbb{T}_d$
Sascha Franck
TL;DR
The paper introduces SIMI, a spatial host-parasite system with immobile hosts and immune responses, and proves a phase transition in parasite survival as the host-immunity parameter $p$ varies. It develops two robust constructions (parasite-wise and vertex-wise) to study infection dynamics and couples the process to supercritical site percolation on lattices and trees, yielding rigorous survival results on $ obreak \\mathbb{Z}^d$ ($d\ge2$) and $ obreak \\mathbb{T}_d$ ($d\ge3$). A key finding is that the critical threshold $p_c(G,A)$ is strictly between 0 and 1 under mild assumptions, with explicit bounds and asymptotics, and that the origin is almost surely not recurrent for $p<1$ on vertex-transitive graphs with finite mean offspring. The methods blend random-walk infection dynamics, coupling with percolation, and a frog-model shape-theorem-inspired framework to derive phase-transition and recurrence results. These results deepen understanding of spatial spreading with immune obstacles and provide tools for analyzing similar interacting particle systems on lattices and trees.
Abstract
We investigate a model of a parasite population invading spatially distributed immobile hosts. Each host has an unbreakable immunity against infection with a certain probability $p$. We show that, on $\mathbb{Z}^d$ with $d\ge 2$ and the $d$-regular tree $\mathbb{T}_d$ with $d\ge 3$, the survival probability of parasites undergoes a phase transition in the probability $p$ of a host to be immune. Also we show that on vertex-transitive graphs a fixed vertex is only visited finitely often by a parasite almost surely under mild assumptions on the parasites offspring distribution.