Exact results for a simple epidemic model on a directed network:   Explorations of a system in a non-equilibrium steady state
  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. 
Maxim S. Shkarayev and R. K. P. Zia
cond-mat.stat-mech