A phase transition for measure-valued SIR epidemic processes

We consider measure-valued processes X=(Xt) that solve the following martingale problem: for a given initial measure X0, and for all smooth, compactly supported test functions, Xt(.)=X0(.)+12.t0Xs(..)ds+..t0Xs(.)ds..t0Xs(Ls.)ds+Mt(.). Here Ls(x) is the local time density process associated with X, and Mt(.) is a martingale with quadratic variation [M(.)]t=.t0Xs(.2)ds. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values .c(d).(0,.) for dimensions d=2,3 such that if .>.c(d), then the solution survives forever with positive probability, but if .<.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 .. We also show that in dimensions d=2,3 the process dies out locally almost surely for any value of .; that is, for any compact set K, the process Xt(K)=0 eventually.