The conceptual difference between equilibrium and non-equilibrium steady
state (NESS) is well established in physics and chemistry. This distinction,
however, is not widely appreciated in dynamical descriptions of biological
populations in terms of differential equations in which fixed point, steady
state, and equilibrium are all synonymous. We study NESS in a stochastic SIS
(susceptible-infectious-susceptible) system with heterogeneous individuals in
their contact behavior represented in terms of subgroups. In the infinite
population limit, the stochastic dynamics yields a system of deterministic
evolution equations for population densities; and for very large but finite
system a diffusion process is obtained. We report the emergence of a circular
dynamics in the diffusion process, with an intrinsic frequency, near the
endemic steady state. The endemic steady state is represented by a stable node
in the deterministic dynamics; As a NESS phenomenon, the circular motion is
caused by the intrinsic heterogeneity within the subgroups, leading to a broken
symmetry and time irreversibility.Comment: 29 pages, 5 figure
