Survival and extinction of epidemics on random graphs with general degree

The contact process with fitness on Galton-Watson trees

“…The second idea is to study the probability that the infection set of CP(T L ; 1 ρ ) goes deeper that a given height. More precisely, we prove that the probability that the infection travels deeper than a given heigh decays exponentially in the height (similarly as in [2]). The main strategy is to investigate the stationary distribution of another modification of the original contact process in finite Galton-Watson trees and relate it to the extinction time.…”

“…The proof of Theorem 2.2 is based on two main ideas which we adapt from [2], where it is used for the standard contact process. First, we use a recursive analysis on Galton-Watson trees that allows us to control the expected survival times.…”

“…To this end, we consider the contact process on the finite tree T L which corresponds to the restriction of T to the first L generations. The first goal is to show that, for small enough λ, the expected survival time of CP(T L ; 1 ρ ) is bounded from above uniformly in L. As in [2], we use a coupling, where we add an extra vertex only adjoined to the root that is always infected. In this way, the process on the subtrees rooted in the children of the root can by independence be compared to a the full process on a tree (with extra root) restricted to L − 1 vertices (see Lemma 4.1 below).…”

“…where in the last equality we use the definition of S ⊗ given in (2). This series can be computed explicitly.…”

“…Huang and Durret [9] showed that for the contact process on Galton-Watson trees with the root initially infected, the critical value for local survival is λ 2 = 0 if the offspring distribution L (ξ) is subexponential, i.e., if E[e cξ ] = ∞ for all c > 0. Shortly afterwards, Bhamidi et al [2] proved that on Galton-Watson trees, λ 1 > 0 if the offspring distribution L (ξ) has an exponential tail, i.e., if E[e cξ ] < ∞ for some c > 0. These two results give a complete characterisation on the existence of extinction phase on Galton-Watson trees.…”