  In this letter, a generalization of pairwise models to non-Markovian
epidemics on networks is presented. For the case of infectious periods of fixed
length, the resulting pairwise model is a system of delay differential
equations (DDEs), which shows excellent agreement with results based on
explicit stochastic simulations of non-Markovian epidemics on networks.
Furthermore, we analytically compute a new R0-like threshold quantity and an
implicit analytical relation between this and the final epidemic size. In
addition we show that the pairwise model and the analytic calculations can be
generalized in terms of integro-differential equations to any distribution of
the infectious period, and we illustrate this by presenting a closed form
expression for the final epidemic size. By showing the rigorous mathematical
link between non-Markovian network epidemics and pairwise DDEs, we provide the
framework for a deeper and more rigorous understanding of the impact of
non-Markovian dynamics with explicit results for final epidemic size and
threshold quantities.
