SIR epidemics on graphs with given degrees
Seminar/Forum
We study the susceptibleinfectiverecovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model, infective vertices infect each of their susceptible neighbours, and recover, at a constant rate. Suppose that initially there are only a few infective vertices. We prove that there is a threshold for a parameter involving the rates and vertex degrees below which only a small number of infections occur. Above the threshold a large outbreak may occur. We prove that, conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time. In contrast to earlier results for this model, our results only require basic regularity conditions and a uniformly bounded second moment of the degree of a random vertex. (Joint work with Svante Janson and Peter Windridge, RSA 2014.)
We also study the regime just above the threshold: we determine the probability that a large epidemic occurs and the size of a large epidemic. We also mention the consequences for the size of the giant component in the configuration model just above criticality. (This is joint work with Thomas House, Svante Janson, Peter Windridge, JMB 2018; see also van der Hofstad, Janson and Luczak, RSA 2019, to appear.)
Presenter

Professor Malwina Luczak, The University of Melbourne