Synchronization and extinction in a high-infectivity spatial SIRS with long-range links

A numerical study of synchronization and extinction is done for a SIRS model with fixed infective and refractory periods, in the regime of high infectivity, on one- and two-dimensional networks for which the connectivity probability decays as $r^{-\alpha}$ with distance. In both one and two dimensions, a long-lasting synchronized state is reached when $\alpha < d$ but not when $\alpha > d$. Three dynamical stages are identified for small $\alpha$, respectively: a short period of initial synchronization, followed by a long oscillatory stage of random duration, and finally a third phase of rapid increase in synchronization that invariably leads to dynamical extinction. For large $\alpha$, the second stage is not synchronized, but is instead a long-lasting endemic state of incoherent activity. Dynamical extinction is in this case still preceded by a short third stage of rapidly intensifying synchronized oscillations. A simple model of noise-induced escape from a potential barrier is introduced, that explains the main characteristics of the observed three-stage dynamical structure before extinction. This model additionally provides specific predictions regarding the size-scaling of the different timescales for the observed dynamical stages, which are found to be consistent with our numerical results.