The contact process with dynamic edges on $\mathbb {Z}$

Abstract

We study the contact process running in the one-dimensional lattice undergoing dynamical percolation, where edges open at rate $vp$ and close at rate $v(1-p)$. Our goal is to explore how the speed of the environment, $v$, affects the behavior of the process. Among our main results we find that: 1. For small enough $v$ the process dies out, while for large $v$ the process behaves like a contact process on $\mathbb {Z}$ with rate $\lambda p$, where $\lambda $ is the birth rate of each particle, so in particular it survives if $\lambda $ is large. 2. For fixed $v$ and small enough $p$ the network becomes immune, in the sense that the process dies out for any infection rate $\lambda $, while if $p$ is sufficiently close to $1$ then for all $v>0$ survival is possible for large enough $\lambda $. 3. Even though the first two points suggest that larger values of $v$ favor survival, this is not necessarily the case for small $v$: when the number of initially infected sites is large enough, the infection survives for a larger expected time in a static environment than in the case of $v$ positive but small. Some of these results hold also in the setting of general (infinite) vertex-transitive regular graphs.