Intrinsic circular motions in stochastic pairwise epidemic models

The main purpose of the paper is to show that there exists intrinsically stochastic circular motion in a pairwise epidemic model, which does not exist in the classical mean-field susceptible–infective–susceptible (SIS) models. Here a basic pairwise SIS epidemic model is adopted. By the method of scale-separation in the case of a large population, we can get a deterministic dynamical system—which represents the temporal evolution of averaged densities of the system; plus a diffusion process, which is centered at the orbit of the aforementioned deterministic system. It is discovered that there is a mode of circular motion in the diffusion process. We consider that this intrinsic circular motion must originate from the fact that the original stochastic pairwise process is time irreversible, since the intrinsic periods, which are calculated from the two systems respectively, have similar forms.