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

Abstract

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 δ₀<δ₁, if the background process is in state 0, the individual (if infected) becomes healthy at rate δ₀, while if the background process is in state 1, it becomes healthy at rate δ₁. 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.