Exact results for a simple epidemic model on a directed network: Explorations of a system in a non-equilibrium steady state

TL;DR

This paper derives exact results for a susceptible-infected epidemic model on a directed network, revealing insights into non-equilibrium steady states and persistent probability currents using analytical and Monte Carlo methods.

Contribution

It provides the first exact probability distribution for the SIS model on a directed network in a non-equilibrium steady state.

Findings

Exact probability distribution derived for the model

Identification of persistent probability currents

Insights into non-equilibrium statistical mechanics phenomena

Abstract

Motivated by fundamental issues in non-equilibrium statistical mechanics (NESM), we study the venerable susceptible-infected (SIS) model of disease spreading in an idealized, simple setting. Using Monte Carlo and analytic techniques, we consider a fully connected, uni-directional network of odd number of nodes, each having an equal number of in- and out-degrees. With the standard SIS dynamics at high infection rates, this system settles into an active non-equilibrium steady state. We find the exact probability distribution and explore its implications for NESM, such as the presence of persistent probability currents.