Temporal duration and event size distribution at the epidemic threshold

The epidemic event, seen as a nonequilibrium dynamic process, is studied through a simple stochastic system reminiscent of the classical SIR model. The system is described in terms of global and local variables and was mainly treated by means of Monte Carlo simulation; square lattices N×N, with N=23, 51, 100, 151, and 211 were used. Distinct extensive runs were performed and then classified as corresponding to epidemic or non-epidemic phase. They were examined with detail through the analysis of the event duration and event size; illustrations, such as density-like plots in the space of the model's parameters, are provided. The epidemic/non-epidemic phase presents smaller/larger relative fluctuations, whereas closer to the threshold the uncertainty reaches its highest values. Far enough from the threshold, the distribution φ(t) of the events time duration t shows a step-like appearance. However at the threshold line it shows an exponential behavior of the form φ (t) ∼ exp (-ωt); the same behavior is observed for the event size distribution. These results help to explain why the approach to epidemic threshold would be hard to anticipate with standard census data.