John Fernley (RényiJohn Fernley (Rényi Institute): The contact process on a graph adapting to infection density

We prove a phase transition for the contact process, a simple model for infection without immunity, on a homogeneous random graph which reacts dynamically to the infection to try to prevent an epidemic. This graph initially has the distribution of an Erdős-Rényi network, but is made adaptive via updating in only the infected neighbourhoods, at constant rate. Under these graph dynamics, the presence of infection can help to prevent the spread and so many monotonicity-based techniques fail; instead we show an upper bound by coupling to the local limit of the graph around an outbreak which is a dynamic forest, and putting a particular reversible stochastic upper bound for the infection on each local tree.

Joint with Peter Mörters (Cologne) and Marcel Ortgiese (Bath).