Mathematics – Probability
Scientific paper
2007-11-22
Annals of Probability 2007, Vol. 35, No. 6, 2263-2293
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/0091179606000001187 the Annals of Probability (http://www.imstat.org/aop/) by the In
Scientific paper
10.1214/0091179606000001187
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.
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