Exact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold

TL;DR

This paper derives exact stochastic equations for SIR and SIS epidemic models on networks without approximations, providing a new Laplacian-based expression for the maximum infected fraction and an upper bound for the epidemic threshold.

Contribution

It presents the first exact stochastic equations for SIR and SIS models on networks and introduces a Laplacian-based formula for the maximum infected fraction and epidemic threshold bounds.

Findings

Exact stochastic equations for SIR and SIS models derived

Maximum infected fraction expressed as a quadratic form of the Laplacian

Upper bound for the epidemic SIS threshold established

Abstract

Exploiting the power of the expectation operator and indicator (or Bernoulli) random variables, we present the exact governing equations for both the SIR and SIS epidemic models on \emph{networks}. Although SIR and SIS are basic epidemic models, deductions from their exact stochastic equations \textbf{without} making approximations (such as the common mean-field approximation) are scarce. An exact analytic solution of the governing equations is highly unlikely to be found (for any network) due to the appearing pair (and higher order) correlations. Nevertheless, the maximum average fraction $y_{I}$ of infected nodes in both SIS and SIR can be written as a quadratic form of the graph's Laplacian. Only for regular graphs, the expression for the maximum of $y_{I}$ can be simplied to exhibit the explicit dependence on the spectral radius. From our new Laplacian expression, we deduce a general \textbf{upper} bound for the epidemic SIS threshold in any graph.