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

## TL;DR

This paper investigates how the speed of a dynamic environment influences the survival of the contact process on a one-dimensional lattice, revealing thresholds for extinction and survival based on environmental parameters.

## Contribution

It introduces a model of the contact process on a dynamically changing network and characterizes its phase transition behavior depending on the environment's speed and openness.

## Key findings

- Process dies out for small environment speed v
- Process survives for large v with high infection rate λ
- Network immunity occurs when v and p are small

## 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. We show in particular that for small enough $v$ the process dies out, while for large $v$ the process behaves like a contact process on $\mathbb{Z}$ with rate $\lambda p$, so it survives if $\lambda$ is large. We also show that if $v$ and $p$ are small then 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 $\lambda$.