PDE-limits of stochastic SIS epidemics on networks

TL;DR

This paper develops PDE-based limits for stochastic SIS epidemic models on networks, enabling efficient analysis of epidemic thresholds and inference, surpassing mean-field models in capturing variability.

Contribution

It introduces a PDE-limit framework for stochastic SIS epidemics on networks by approximating the process with a density-dependent Birth-and-Death process and deriving Fokker-Planck equations.

Findings

Excellent agreement between PDE solutions and simulations

PDE framework captures epidemic thresholds

Enables likelihood-based epidemic inference

Abstract

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.