Complete convergence theorem for a two-level contact process

We study a two-level contact process. We think of fleas living on a species of animals. The animals are a supercritical contact process in $\mathbb{Z}^d$. The contact process acts as the random environment for the fleas. The fleas do not affect the animals, give birth at rate $\mu$ when they are living on a host animal, and die at rate $\delta$ when they do not have a host animal. The main result is that if the contact process is supercritical and the fleas survive then the complete convergence theorem holds. This is done using a block construction so as a corollary we conclude that the fleas die out at their critical value.