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

The SIR model towards he data. One year of Covid-19 pandemic in Italy case study and plausible "real" numbers