title: Generalization of Pairwise Models to non-Markovian Epidemics on Networks.
abstract: 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, which shows excellent agreement with results based on stochastic simulations. Furthermore, we analytically compute a new R_{0}-like threshold quantity and an analytical relation between this and the final epidemic size. Additionally, we show that the pairwise model and the analytic results can be generalized to an arbitrary distribution of the infectious times, using integro-differential equations, and this leads to a general expression for the final epidemic size. By showing the rigorous link between non-Markovian dynamics and pairwise delay differential equations, we provide the framework for a more systematic understanding of non-Markovian dynamics.
authors: Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom.
authors: Istvan Z
authors: IZ
authors: Kiss
authors: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary.
authors: Gergely
authors: G
authors: Röst
authors: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary.
authors: Zsolt
authors: Z
authors: Vizi