Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment

The ordinary contact process is used to model the spread of a disease in a population. In this model, each infected individual waits an exponentially distributed time with parameter 1 before becoming healthy. In this paper, we introduce and study the contact process in a randomly evolving environment. Here we associate to every individual an independent two-state, $\{0,1\},$ background process. Given $\delta_0<\delta_1,$ if the background process is in state $0,$ the individual (if infected) becomes healthy at rate $\delta_0,$ while if the background process is in state $1,$ it becomes healthy at rate $\delta_1.$ By stochastically comparing the contact process in a randomly evolving environment to the ordinary contact process, we will investigate matters of extinction and that of weak and strong survival. A key step in our analysis is to obtain stochastic domination results between certain point processes. We do this by starting out in a discrete setting and then taking continuous time limits.