Approximating quasi-stationary behaviour in network-based SIS dynamics

TL;DR

This paper introduces a new system of differential equations that accurately approximate the quasi-stationary distribution of network-based SIS epidemic models, especially near the epidemic threshold, improving the connection between stochastic and deterministic models.

Contribution

The authors develop a novel ODE system that approximates the QSD in SIS models on arbitrary networks, enhancing understanding of stable epidemic behaviour near thresholds.

Findings

QSD-based models match stochastic dynamics at high epidemic levels

Models deviate near the epidemic threshold, aligning with stochastic behaviour

Existing methods approach the all susceptible state near the threshold

Abstract

Deterministic approximations to stochastic Susceptible-Infectious-Susceptible models typically predict a stable endemic steady-state when above threshold. This can be hard to relate to the underlying stochastic dynamics, which has no endemic steady-state but can exhibit approximately stable behaviour. Here we relate the approximate models to the stochastic dynamics via the definition of the quasi-stationary distribution (QSD), which captures this approximately stable behaviour. We develop a system of ordinary differential equations that approximate the number of infected individuals in the QSD for arbitrary contact networks and parameter values. When the epidemic level is high, these QSD approximations coincide with the existing approximation methods. However, as we approach the epidemic threshold, the models deviate, with these models following the QSD and the existing methods approaching the all susceptible state. Through consistently approximating the QSD, the proposed methods provide a more robust link to the stochastic models.