  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.
