  We numerically study the dynamics of the SIR disease model on small-world
networks by using a large-deviation approach. This allows us to obtain the
probability density function of the total fraction of infected nodes and of the
maximum fraction of simultaneously infected nodes down to very small
probability densities like $10^{-2500}$. We analyze the structure of the
disease dynamics and observed three regimes in all probability density
functions, which correspond to quick mild, quick extremely severe and sustained
severe dynamical evolutions, respectively. Furthermore, the mathematical rate
functions of the densities are investigated. The results indicate that the so
called large-deviation property hold for the SIR model. Finally, we measured
correlations with other quantities like the duration of an outbreak or the peak
position of the fraction of infections, also in the rare regions which are not
accessible by standard simulation techniques.
