  Stochastic epidemic models on networks are inherently high-dimensional and
the resulting exact models are intractable numerically even for modest network
sizes. Mean-field models provide an alternative but can only capture average
quantities, thus offering little or no information about variability in the
outcome of the exact process. In this paper we conjecture and numerically prove
that it is possible to construct PDE-limits of the exact stochastic SIS
epidemics on regular and Erd\H{o}s-R\'enyi networks. To do this we first
approximate the exact stochastic process at population level by a
Birth-and-Death process (BD) (with a state space of $O(N)$ rather than
$O(2^N)$) whose coefficients are determined numerically from Gillespie
simulations of the exact epidemic on explicit networks. We numerically
demonstrate that the coefficients of the resulting BD process are
density-dependent, a crucial condition for the existence of a PDE limit.
Extensive numerical tests for Regular and Erd\H{o}s-R\'enyi networks show
excellent agreement between the outcome of simulations and the numerical
solution of the Fokker-Planck equations. Apart from a significant reduction in
dimensionality, the PDE also provides the means to derive the epidemic outbreak
threshold linking network and disease dynamics parameters, albeit in an
implicit way. Perhaps more importantly, it enables the formulation and
numerical evaluation of likelihoods for epidemic and network inference as
illustrated in a worked out example.
