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The recent COVID-19 pandemic has shown that when the reproduction number ishigh and there are no proper measurements in place, the number of infectedpeople can increase dramatically in a short time, producing a phenomenon thatmany stochastic SIR-like models cannot describe: overdispersion of the numberof infected people (i.e., the variance of the number of infected people duringany interval is very high compared to the average). To address this issue, inthis paper we explore the possibility of modeling the total number ofinfections as a state dependent self-exciting point process. In this way,infections are not independent among themselves, but any infection willincrease the likelihood of a new infection while also the number of currentlyinfected and recovered individuals are included into determining the likelihoodof new infections, Since long term simulation is extremely computationallyintensive, exact expressions for the moments of the processes determining thenumber of infected and recovered individuals are computed, while alsosimulation algorithms for these state-dependent processes are provided.

A Stochastic Model for the Early Stages of Highly Contagious Epidemics by using a State-Dependent Point Process