  In this work, the SIR epidemiological model is reformulated so to highlight
the important {\em effective reproduction number}, as well as to account for
the {\em generation time}, inverse of the {\em incidence rate}, and the {\em
infectious period} (or {\em removal period}), inverse of the {\em removal
rate}. The aim is to check whether the relationships the model poses among the
various observables are actually found in the data. The study case of the
second through the third wave of the Covid-19 pandemic in Italy is taken. Given
its scale invariance, initially the model is tested with reference to the curve
of swab-confirmed infectious individuals only. It is found to match the data if
the given curve of the {\em removed} (that is healed or deceased) individuals
is assumed underestimated by a factor of about 3 together with other related
curves. Contextually, the {\em generation time} and the {\em removal period},
as well as the {\em effective reproduction number}, are obtained fitting the
SIR equations to the data; the outcomes prove to be in good agreement with
those of other works. Then, using knowledge of the proportion of Covid-19
transmissions likely occurring from individuals who didn't develop symptoms,
thus mainly undetected, an estimate of the {\em "true numbers"} of the epidemic
is obtained, looking also in good agreement with results from other, completely
different works. The line of this work is new and the procedures are
computationally really inexpensive.
