## Annals of Probability

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

Erik I. Broman

#### 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.

#### Article information

Source
Ann. Probab., Volume 35, Number 6 (2007), 2263-2293.

Dates
First available in Project Euclid: 8 October 2007

https://projecteuclid.org/euclid.aop/1191860421

Digital Object Identifier
doi:10.1214/0091179606000001187

Mathematical Reviews number (MathSciNet)
MR2353388

Zentralblatt MATH identifier
1126.82024

#### Citation

Broman, Erik I. Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment. Ann. Probab. 35 (2007), no. 6, 2263--2293. doi:10.1214/0091179606000001187. https://projecteuclid.org/euclid.aop/1191860421

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