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

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