The contact process on a graph adapting to the infection

Abstract

We find a non-trivial phase transition for the contact process, a simple model for infection without immunity, on a network which reacts dynamically to prevent an epidemic. This network is initially blue distributed as an Erdős–Rényi graph, but is made adaptive via updating in only the infected neighbourhoods, at constant rate. Adaptive dynamics are new to the mathematical contact process literature—in adaptive dynamics the presence of infection can help to prevent the spread and thus monotonicity-based techniques fail. We show, further, that the phase transition in the fast adaptive model occurs at larger infection rate than in the non-adaptive model.