Fragility and robustness of self-sustained oscillations in an epidemiological model on small-world networks

We investigate the susceptible-infected-recovered-susceptible epidemic model, typical of mathematical epidemiology, with the diversity of the durations of infection and recovery of the individuals on small-world networks. Infection spreads from infected to healthy nodes, whose infection and recovery periods denoted by τ_I and τ_R, respectively, are either fixed or uniformly distributed around a specified mean. Whenever τ_I and τ_R are narrowly distributed around their mean values, the epidemic prevalence in the stationary state is found to reach its maximal level in the typical small-world region. This non-monotonic behavior of the final epidemic prevalence is thought to be similar to the efficient navigation in small worlds with cost minimization. Besides, pronounced oscillatory behavior of the fraction of infected nodes emerges when the number of shortcuts on the underlying network become sufficiently large. Remarkably, we find that the synchronized oscillation of infection incidences is quite fragile to the variability of the two characteristic time scales τ_I and τ_R. Specifically, even in the limit of a random network (where the amplest oscillations are expected to arise for fixed τ_I and τ_R), increasing the variability of the duration of the infectious period and/or that of the refractory period will push the system to change from a self-sustained oscillation to a fixed point with negligible fluctuations in the steady state. Interestingly, negative correlation between τ_I and τ_R can give rise to the robustness of the self-sustained oscillatory phenomenon. Our findings thus highlight the pivotal role of, apart from the external seasonal driving force and demographic stochasticity, the intrinsic characteristic of the system itself in understanding the cycle of outbreaks of recurrent epidemics.