The SIR epidemic on a dynamic Erd\H{o}s-R\'enyi random graph

We investigate the SIR epidemic on a dynamic inhomogeneous Erd\H{o}s-R\'enyi random graph, in which vertices are of one of $k$ types and in which edges appear and disappear independently of each other. We establish a functional law of large numbers for the susceptible, infected, and recovered ratio curves after a random time shift, and demonstrate that, under a variety of possible scaling limits of the model parameters, the epidemic curves are solutions to a system of ordinary differential equations. In most scaling regimes, these equations coincide with the classical SIR epidemic equations. In the regime where the average degree of the network remains constant and the edge-flipping dynamics remain on the same time scale as the infectious contact process, however, a novel set of differential equations emerges. This system contains additional quantities related to the infectious edges, but somewhat surprisingly, contains no quantities related to higher-order local network configurations. To the best of our knowledge, this study represents the first thorough and rigorous analysis of large population epidemic processes on dynamic random graphs, although our findings are contingent upon conditioning on a (possibly strict) subset of the event of an epidemic outbreak.