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The ordinary contact process is used to model the spread of a disease in apopulation. In this model, each infected individual waits an exponentiallydistributed time with parameter 1 before becoming healthy. In this paper, weintroduce 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 instate $0,$ 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 randomlyevolving environment to the ordinary contact process, we will investigatematters of extinction and that of weak and strong survival. A key step in ouranalysis is to obtain stochastic domination results between certain pointprocesses. We do this by starting out in a discrete setting and then takingcontinuous time limits.

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