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

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.

20 pages, 2 figures