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 $λp$, so it survives if $λ$ 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 $λ$, while if $p$ is sufficiently close to $1$ then for all $v>0$ survival is possible for large enough $λ$.

14 pages, 4 figures. New results were added about the extinction time of the process and about extensions to general vertex-transitive graphs. To appear in the Electronic Journal of Probability