Epidemics on networks with preventive rewiring

A stochastic SIR (susceptible -> infective -> recovered) model is considered for the spread of an epidemic on a network, described initially by an Erdos-Renyi random graph, in which susceptible individuals connected to infectious neighbors may drop or rewire such connections. A novel construction of the model is used to derive a deterministic model for epidemics started with a positive fraction initially infected and prove convergence of the scaled stochastic model to that deterministic model as the population size n ->infinity. For epidemics initiated by a single infective that take off, we prove that for part of the parameter space, in the limit as n ->infinity, the final fraction infected tau(lambda) is discontinuous in the infection rate lambda at its threshold lambda c, thus not converging to 0 as lambda down arrow lambda c. The discontinuity is particularly striking when rewiring is necessarily to susceptible individuals in that tau(lambda) jumps from 0 to 1 as lambda passes through lambda c.