  We investigate the problem of estimating the distribution of the individual
reproduction number governing the COVID-19 pandemic. Under the assumption that
this random variable follows a Negative Binomial distribution, we focus on
constructing estimators of the parameters of this distribution using reported
infection data and taking into account issues like under-reporting or the time
behavior of the infection and of the reporting processes. To this end, we
extract information from regionally dissaggregated data reported by German
health authorities, in order to estimate not only the mean but also the
variance of the distribution of the individual reproduction number. In contrast
to the mean, the latter parameter also depends on the unknown under-reporting
rate of the pandemic. The estimates obtained allow not only for a better
understanding of the time-varying behavior of the expected value of the
individual reproduction number but also of its dispersion, for the construction
of bootstrap confidence intervals and for a discussion of the implications of
different policy interventions. Our methodological investigations are
accompanied by an empirical study of the development of the COVID-19 pandemic
in Germany, which shows a strong overdispersion of the individual reproduction
number.
