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Stochastic Modeling of an Infectious Disease Part II: Simulation Experiments and Verification of the Analysis

In Part 1, we introduced a stochastic model of an infectious disease, based on the BDI (birth and death with immigration) process. We showed that random processes defined by this model can capture the essence of the stochastic, often erratic, behavior of the infection process. The most significant finding was that it is negative binomial distributed with small r, hence it is geometrically distributed with an exceedingly long tail. This leads to a much larger disparity in the epidemic patterns than has been known to the modeling community. In Part 2 we conduct simulation experiments by implementing an event-driven simulator. Several independent runs are presented to verify the findings reported in Part 1. The enormous variations among the sample paths obtained from several consecutive and independent runs with statistically identical conditions confirm our analysis. Note that the epidemic pattern that we observe is merely one sample path taken out of infinitely many paths. Thus, one such path cannot represent the ensemble of the underlying random process. By plotting the simulation runs in the semi-log scale, we can see that the probabilistic chances in the early phase of the infection process determine the behavior of the process. Once the process has reached a considerable number, the weak law of large numbers sets in, and the process behaves less erratically than in the early phase. One important implication of our findings is that it would be a futile effort to attempt to identify all plausible causes of the epidemic patterns. Mere luck may play a more significant role than most people believe. It would be worth remarking that the BDI process might be applicable to explain the disparity in wealth among individuals of similar earning power, expenditure, and investment portfolio.