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 ${\delta }_0<{\delta }_1,$ 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.

Published in at http://dx.doi.org/10.1214/0091179606000001187 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Dynamical stability of percolation for some interacting particle systems and $\epsilon$-movability

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