Using the same systems of transformed equations, we compute 140 years worth of time series for 500 realizations. Ignoring the first 40 years of transient solution, the data are used to create a histogram representing the probability density p SI of the S and I values. Figure 4͑a͒ shows the histogram associated with the complete stochastic system of transformed equations, while Fig. 4͑b͒ shows the histogram associated with the reduced system of equations with a replacement of the noise terms. The color-bar values in Figs. 4͑a͒ and 4͑b͒ have been normalized by 10 −3 .