  The dramatic outbreak of the coronavirus disease 2019 (COVID-19) pandemics
and its ongoing progression boosted the scientific community's interest in
epidemic modeling and forecasting. The SIR (Susceptible-Infected-Removed) model
is a simple mathematical model of epidemic outbreaks, yet for decades it evaded
the efforts of the community to derive an explicit solution. The present work
demonstrates that this is a non-trivial task. Notably, it is proven that the
explicit solution of the model requires the introduction of a new
transcendental special function, related to the Wright's Omega function. The
present manuscript reports new analytical results and numerical routines
suitable for parametric estimation of the SIR model. The manuscript introduces
iterative algorithms approximating the incidence variable, which allows for
estimation of the model parameters from the numbers of observed cases. The
numerical approach is exemplified with data from the European Centre for
Disease Prevention and Control (ECDC) for several European countries in the
period Jan 2020 -- Jun 2020.
