**6. Model Validation with Epidemiological Data**

In order to validate the delay model (9) and the method of estimation of *τ*, we compared the results with the epidemiological data. We used the estimated value of *τ* obtained by the method described above during different peaks of COVID-19.

In Italy, the estimated value of *τ* is 19 days, 20 days, 14 days, and 11 days during the first peak (April 2020), the second peak (November 2020), the third peak (March 2021), and the fourth peak (January 2022), respectively. We assumed that the disease duration *τ* remains *τ* = 19 days from 21 February 2020 to 15 August 2020; *τ* = 20 days from 16 August 2020 to 19 February 2021; *τ* = 14 days from 20 February 2021 to 21 May 2021; and 11 days from 10 November 2021 to 28 February 2022 (during Omicron in Italy).

Once these parameter values were determined, we took the number *J*(*t*) of daily infected individuals from the epidemiological data [35] and found the sum of daily recoveries and deaths from the expression

$$
\Sigma(t) = J(t - \pi). \tag{24}
$$

These results were compared with the sum of recoveries and deaths in the data. Figure 7 shows the result of such a comparison for Italy from 21 February 2020 to 21 May 2021 and from 10 November 2021 to 28 February 2022, with the data from [35] (7-day moving average). Recoveries and deaths can also be determined as a proportion of active cases *σ*(*t*)=(*r*<sup>0</sup> + *d*0)*I*(*t*), as is done in the SIR model. Here, *I*(*t*) is taken from the data and *r*<sup>0</sup> + *d*<sup>0</sup> = 1/*τ*, and we observed that the SIR model overestimates the sum of recovered and dead. Thus, the delay model (9) gives a good description of the recovery and death compared with the epidemiological data, while the SIR model overestimates them.

**Figure 7.** The blue curves show the number Σ(*t*) of recovered and dead in the delay model; the magenta curves correspond to *σ*(*t*) in the SIR model; the black dots correspond to the 7-day moving average of daily recoveries and death in Italy.
