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 Erdős–Rényi 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 → ∞ . For epidemics initiated by a single infective that take off, we prove that for part of the parameter space, in the limit as n → ∞ , the final fraction infected τ ( λ ) is discontinuous in the infection rate λ at its thresholdλ c, thus not converging to 0 as λ ↓ λ c. The discontinuity is particularly striking when rewiring is necessarily to susceptible individuals in that τ ( λ ) jumps from 0 to 1 as λ passes throughλ c.