A phase transition for a spatial host-parasite model with extreme host immunities on $\mathbb{Z}^d$ and $\mathbb{T}_d$

TL;DR

This paper studies a spatial host-parasite model with extreme host immunities, revealing a phase transition in parasite survival probability on certain graphs and analyzing how graph modifications affect this transition.

Contribution

It introduces a modified frog model with host immunities, establishing phase transition results on $ abla^d$ and $ abla_d$ graphs and examining the impact of graph alterations.

Findings

Parasite survival exhibits a phase transition at a critical immunity probability $p_c$.

Adding vertices or edges can raise or lower $p_c$, affecting parasite persistence.

On quasi-vertex-transitive graphs, a fixed vertex is visited finitely often by parasites with probability 1.

Abstract

We investigate a model of a parasite population invading spatially distributed immobile hosts on a graph, which is a modification of the frog model. Each host has an unbreakable immunity against infection with a certain probability $1-p$ and parasites move as simple symmetric random walks attempting to infect any host they encounter and subsequently reproduce themselves. 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 exhibits a phase transition at a critical value of $p_c\in(0,1)$. Also, we show that adding vertices and edges to the underlying graph can, in general, both increase or decrease the value of $p_c$. Finally, we show that on quasi-vertex-transitive graphs, with probability $1$, a fixed vertex is only visited finitely often by a parasite under mild assumptions on the offspring distribution of parasites.