  In stochastic modeling of infectious diseases, it has been established that
variations in infectivity affect the probability of a major outbreak, but not
the shape of the curves during a major outbreak, which is predicted by
deterministic models [Diekmann et al., 2012]. However, such conclusions are
derived under idealized assumptions such as the population size tending to
infinity, and the individual degree of infectivity only depending on variations
in the infectiousness period. In this paper we show that the same conclusions
hold true in a finite population representing a medium size city, where the
degree of infectivity is determined by the offspring distribution, which we try
to make as realistic as possible for SARS-CoV-2. In particular, we consider
distributions with fat tails, to incorporate the existence of super-spreaders.
We also provide new theoretical results on convergence of stochastic models
which allows to incorporate any offspring distribution with a finite variance.
