  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.
