A Phase Transition for Measure-valued SIR Epidemic Processes

Details A Phase Transition for Measure-valued SIR Epidemic Processes A Phase Transition for Measure-valued SIR Epidemic Processes

2011-11-28

arxiv.org/abs/1111.6451v1

Mathematics

Probability

Scientific paper

We consider measure-valued processes X_t that solve the following martingale problem: For a given initial measure X_0, and for all smooth, compactly supported test functions phi, X_t(phi)= X_0 (phi)+ (1/2)\int_0^t X_s(Delta phi)ds + theta\int_0^t X_s(phi) ds - \int_0^t X_s(L_s phi) ds + M_t(phi). Here L_s(x) is the local time density process associated with X_t, and M_t(phi) is a martingale with quadratic variation [M(phi)]_t=\int_0^t X_s(phi^2) ds. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values theta_c(d) \in (0,\infty) for dimensions d=2,3 such that if theta> theta_c(d), then the solution survives forever with positive probability, but if theta< theta_c(d), then the solution dies out in finite time with probability 1. For d=1 we prove that the solution dies out almost surely for all values of theta. We also show that in dimensions d=2,3 the process dies out locally almost surely for any value of theta, that is, for any compact set K, the process X_t (K)=0 eventually.

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