Analyzing stochastic processes over epidemic stages
An infection spreading through a population produces distinguishable patterns within the susceptible, infected, and recovered people. Early in the epidemic, when there are only a few infected individuals, the transmission is a stochastic process that can be best modeled by an individual-based stochastic model on a network. Once the infection has spread throughout the population, then a mean-field deterministic model is a more efficient approach to simulate the infection transmission accurately. In the final stages, when only a few people are infected or susceptible to infection, then a stochastic individual-based model is again needed to capture the random transmission dynamics. The standard errors for the model predictions of the number of people infected by a specific time are much smaller after the epidemic has transitioned to the deterministic stage. We analyze these three successive stages and identify distinguishing characteristics for the transition between the stages. We then characterize how the spread of the infection is affected by the structure of the population contact network for Erdos Renyi, Scale Free, and Small World networks. We focus specifically on the results from a SIR infection transmission model over a Scale-Free network structure. We find that the change in distributions of time it takes for the infection to infect the previous 1% of the population is a good indicator for when the epidemic reaches the deterministic stage and when it returns to a stochastic stage. Analyzing an emerging epidemic as three stages will provide a framework for more accurate predictions and will provide more useful predictions to help guide mitigation efforts.