Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

The contact process is a crude model for the spread of an epidemic in a graph. In this model, each "sick" vertex spreads the infection to a neighboring vertex at rate lambda>0 and becomes healthy at rate 1. This model undergoes a "metastability transition" over certain large finite graphs. For instance, in a large random d-regular graph with d>2 and n>>1 vertices, if all vertices are initially "sick," there exists a lambda_c such that the epidemic dies out in log n time when lambda<lambda_c, and survives for exp(Cn) time when lambda>lambda_c. This behavior is related to the phase transition of the model over the infinite d-regular tree, which provides a "local model" for the random d-regular graph.

This talk discusses what happens to the contact process over a time-evolving random graph. Specifically, we assume we have a dynamic graph evolving according to the "edge switching" Markov chain over d-regular graphs on n vertices. Assuming each edge changes at rate v>0, we show that exponential survival time still holds for lambda > lambda_c(v); however, the new critical parameter lambda_c(v) is strictly smaller than that of the static graph. At the heart of our proof is a "local model" for our process – that is, a contact process over a growing family of trees – that is of independent interest.

This is joint work with Gabriel Leite Baptista da Silva and Daniel Valesin (who are both at Groningen).