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