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We study the susceptible-infective-recovered (SIR) epidemic on a random graphchosen uniformly subject to having given vertex degrees. In this modelinfective vertices infect each of their susceptible neighbours, and recover, ata constant rate.  Suppose that initially there are only a few infective vertices. We provethere is a threshold for a parameter involving the rates and vertex degreesbelow which only a small number of infections occur. Above the threshold alarge outbreak occurs with probability bounded away from zero. Our main resultis that, conditional on a large outbreak, the evolutions of certain quantitiesof interest, such as the fraction of infective vertices, converge todeterministic functions of time.  We also consider more general initial conditions for the epidemic, and derivecriteria for a simple vaccination strategy to be successful.  In contrast to earlier results for this model, our approach only requiresbasic regularity conditions and a uniformly bounded second moment of the degreeof a random vertex.  En route, we prove analogous results for the epidemic on the configurationmodel multigraph under much weaker conditions. Essentially, our main resultrequires only that the initial values for our processes converge, i.e. it isthe best possible.

Law of large numbers for the SIR epidemic on a random graph with given degrees