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Deterministic approximations to stochastic Susceptible-Infectious-Susceptiblemodels typically predict a stable endemic steady-state when above threshold.This can be hard to relate to the underlying stochastic dynamics, which has noendemic steady-state but can exhibit approximately stable behaviour. Here werelate the approximate models to the stochastic dynamics via the definition ofthe quasi-stationary distribution (QSD), which captures this approximatelystable behaviour. We develop a system of ordinary differential equations thatapproximate the number of infected individuals in the QSD for arbitrary contactnetworks and parameter values. When the epidemic level is high, these QSDapproximations coincide with the existing approximation methods. However, as weapproach the epidemic threshold, the models deviate, with these modelsfollowing the QSD and the existing methods approaching the all susceptiblestate. Through consistently approximating the QSD, the proposed methods providea more robust link to the stochastic models.

Approximating quasi-stationary behaviour in network-based SIS dynamics