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Pairwise approximation for SIR type network epidemics with non-Markovian recovery

We present the generalised mean-field and pairwise models for non-Markovian epidemics on networks with arbitrary recovery time distributions. First we consider a hyperbolic system, where the population of infective nodes and links are structured by age since infection. By solving the partial differential equations, the model is transformed into a system of integro-differential equations, which is analysed both from a mathematical and numerical point of view. We analytically study the asymptotic behaviour of the generalised model and provide a rigorous mathematical proof of the conjecture on the functional form of the final epidemic size and pairwise reproduction number. As an illustration of the applicability of the general model we recover known results for the exponentially distributed and fixed recovery time cases and obtain new pairwise models with gamma and uniformly distributed infectious period. The general framework that we proposed shows a more complete picture of the impact of non-Markovian recovery on network epidemics.