Exact steady-state of the ε-SIS model on generic networks

Establish an exact, closed-form characterization of the non-equilibrium steady-state probability distribution for the susceptible–infected–susceptible (SIS) model with spontaneous infection (ε-SIS) on arbitrary finite networks. The ε-SIS process is a continuous-time Markov model with infection rate β along network edges, recovery rate γ (often normalized to 1), and spontaneous infection rate ε, which ensures a non-trivial steady state. Determine the exact steady-state distribution for generic network topologies rather than special highly symmetric graphs.

Background

The ε-SIS model introduces spontaneous infection into the standard SIS process, preventing relaxation to the absorbing all-healthy state and yielding a non-trivial steady-state distribution. While exact solutions are known for specific, highly symmetric graphs, the general case of arbitrary network topologies lacks an exact characterization.

The paper develops Matrix Product State (MPS) approximations for the ε-SIS steady-state on networks and demonstrates high accuracy and compressibility insights. An exact solution for generic networks would provide a benchmark for such approximations and clarify the fundamental structure of the steady-state distribution across diverse topologies.