Using a total population size of N =10ϫ 10 6 , we have performed a Markov simulation of the system. After completing the Markov simulation, we divided s, e, and i by N to find S, E, and I. Figure 9͑a͒ shows a time series, after the transients have decayed, of the fraction of the population that is infected with a disease I. The results reflect both the mean and the frequency of the deterministic system. Performing the simulation for 500 realizations allows us to create a histogram representing the probability density p SI of the S and I values. This histogram is shown in Fig. 9͑b͒ , and one can see that the probability density reflects the amplitude, which var- ies with the population size of S and I. The color-bar values in Fig. 9͑b͒ have been normalized by 10 −4