INHOMOGENEOUS CONTACT-PROCESSES ON TREES

We consider an inhomogeneous contact process on a tree T-k of degree k , where the infection rate at any site is lambda, the death rate at an y site in S subset of T-k is delta (with 0 < delta less than or equal to 1) and that at any site in T-k - S is 1. Denote by lambda(c)(T-k) t he critical value for the homogeneous model (i.e., delta = 1) on T-k a nd by theta(delta, lambda) the survival probability of the inhomogeneo us model on T-k. We prove that when k > 4, if S = T-sigma, a subtree e mbedded in T-k, with 1 less than or equal to sigma less than or equal to root k, then there exists delta(c)(sigma) strictly between lambda(c )(T-k)/lambda(c)(T-sigma) and 1 such that theta(delta, lambda(c)(T-k)) = 0 when delta > delta(c)(sigma) and theta(delta, lambda(c)(T-k)) > 0 when delta < delta(c)(sigma); if S = {o}, the origin of T-k, then the ta(delta, lambda(c)(T-k)) = 0 for any delta is an element of (0, 1).