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We numerically study the dynamics of the SIR disease model on small-worldnetworks by using a large-deviation approach. This allows us to obtain theprobability density function of the total fraction of infected nodes and of themaximum fraction of simultaneously infected nodes down to very smallprobability densities like $10^{-2500}$. We analyze the structure of thedisease dynamics and observed three regimes in all probability densityfunctions, which correspond to quick mild, quick extremely severe and sustainedsevere dynamical evolutions, respectively. Furthermore, the mathematical ratefunctions of the densities are investigated. The results indicate that the socalled large-deviation property hold for the SIR model. Finally, we measuredcorrelations with other quantities like the duration of an outbreak or the peakposition of the fraction of infections, also in the rare regions which are notaccessible by standard simulation techniques.

Large-deviations of the SIR model around the epidemic threshold