In the interactive graph shown in Fig. 1, we compare the national cumulative curves of confirmed cases and deaths. The available commands allow the user to act two operations on the confirmed case curve: a forward translation (Shift) forward in time and multiplication by a factor (Factor). These operations allow the user to check whether, to what extent, and with what parameters the combination of the two operations causes the two curves to overlap. The Shift “S” applied to bring the curves to the overlap corresponds to the average delay between diagnosis the eventual death. The factor “F” corresponds to the fatality rate found within the sample of confirmed cases. If the average delay between diagnosis and death and fatality rate were constant during the outbreak and if the data contain no flaws, the two curves could be brought, beyond statistical fluctuations, to almost perfect overlap. During the first wave, a decent overall agreement between the curves, as shown here, could be found with S = 4 or 5 days and with F about 15%. This extremely value of the case fatality ratio CFR remained approximately constant until summer, when the CFR greatly reduced as a consequence of increased testing. At the end of year 2020 the cumulative CFR has refuced, and in the month of December F = 4%. The flat shape of the curve makes the shift S not measurable. If should be noticed that, due to the extended time period of this epidemics, this analyis of the cumulative curve becomes less significant.


Further details on the estimation of the fatality rate

Fatality rate estimates are affected by both a major error and considerable variability. “True” fatality rate, i.e. the one hypothetically calculated on the entire sample of the infected, is much lower than the official statistics, since the sample of confirmed cases has a much higher percentage of severely symptomatic patients than the overall one. By increasing the intensity of sampling (increase of tests) the sample of confirmed cases is enriched with asymptomatic or pauci-symptomatic patients, resulting in lower measured mortality. In addition, assuming the average time between infection and eventual death is constant, earlier diagnoses increase the value of S. Varying over time the policy of sampling, F and S will not remain constant during the outbreak. In particular, a tests number increase carried out over time, such as the one in Italy, will reduce the apparent F fatality rate. In fact, optimizing the agreement only for the initial part of the curve, when the intensity of the samples was very low, the fatality rate obtained is above 20%.
The so-called “case fatality ratio” CFR, or the estimate of fatality rate made by dividing directly, day by day, the number of deaths by the number of confirmed cases, is a misleading indicator misleading. It is very low at the beginning of the epidemic and grows over time. We discuss the reason in the modelling section and in more detail in this work.

In the interactive graph shown in Fig. 2, we compare the curves of daily increases in confirmed cases and deaths. As in the previous case, the available controls allow the user to perform two operations on the confirmed case curve: a forward translation (Shift) forward in time and multiplication by a factor (Factor). The optimal values to overlap the two curves, as shown here, are S = 4 days and F =15%, in good agreement with the data from the interactive graph of Fig. 1. The value of S is extremely low compared to both the average delay between infection and death and the delay between symptoms and death, confirming the lateness of many diagnoses. It can be usefully compared with the data reported here and taken from the ISS statistics that presumably only take into account hospitalized patients. In late December this numer are largely modified. The best parameter to overlap the curves are today S = 15 days and F = 2.5%, conferming that testing is today much more extended and much more timely.