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. 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.