**5. Determination of Disease Duration from Data**

In this section, we determine the disease duration *τ* from the epidemiological data for the daily number of infected *J*(*t*) and the total number of infected individuals *I*(*t*), using the equation

$$\frac{dI(t)}{dt} = J(t) - J(t - \tau).$$

Let *I*(*t*) have the maximum at *t* = *tm*. Set *I*(*tm*) = *Im*. Then, *J*(*tm*) = *J*(*tm* − *τ*), i.e., the daily number of infected is the same at two different time points *t* = *tm* and *t* = *tm* − *τ*. From the real data of the infected individuals *I*(*t*), we can find the day on which the daily number of active cases is maximal, and it determines *tm*. From the data of daily reported cases *J*(*t*), we can observe that *J*(*t*) crosses its maximum at some time before *tm*. Now, we have to find the value of *J*(*t*) such that *J*(*tm*) will be equal to *J*(*tm* − *τ*), which in turn determines the disease duration *τ*. Hence, considering the delay model, using the real data of daily new cases *J*(*t*) and active cases *I*(*t*) around a peak, we can find the disease duration *τ*.

We illustrate this method using the data of *J*(*t*) and *I*(*t*) taken from [35] for COVID-19 in Italy. We collected the daily new reported data *J*(*t*) and active case data *I*(*t*) for Italy from 21 February 2020 to 31 May 2021 (which capture the first three peaks in Italy) and from 10 November 2021 to 28 February 2022 (which capture the peak due to Omicron in Italy). To have smoother data, we used the 7-day moving average, the data on the

*j*-th day replaced by the average data from the (*j* − 3)th day to the (*j* + 3)th day. As the concerned method is focused on the peaks, the error at the beginning and end of the time interval is not essential. In Italy, during the first peak (in April 2020), the peak of *I*(*t*) is attained at *tm* = 51 (Figure 6a) and the peak of *J*(*t*) is attained before on *t* = 51, which is less than *tm* (Figure 6b). First, we find that *<sup>J</sup>*(*tm* = <sup>51</sup>) = 4.17 × <sup>10</sup><sup>3</sup> and then find *<sup>J</sup>*(32) = 4.15 × <sup>10</sup><sup>3</sup> ≈ *<sup>J</sup>*(*tm* = <sup>51</sup>). This implies *<sup>J</sup>*(*tm* − *<sup>τ</sup>*) = *<sup>J</sup>*(32), and consequently, we can calculate *τ* = 19 as the disease duration during the first peak. Similarly, during the second peak (in November 2020) and third peak (in March 2021), we estimated the disease duration as *τ* = 20 days and *τ* = 14 days, respectively. The peak of epidemic progression due to Omicron in January 2022 is shown in Figure 6c,d, and we estimated the value of *τ* as 11 days. Similarly, the value of *τ* is estimated for some other countries (Table 2).

**Figure 6.** Estimation of the disease duration *τ* using the data around different peaks of COVID-19 in Italy. (**a**,**b**) Time *t* = 0 corresponds to 21 February 2020. The obtained value of *τ* is 19 days for the first peak, 20 days for the second peak, and 14 days for the last peak. (**c**,**d**) Time *t* = 0 corresponds to 10 November 2021 (which corresponds to the Omicron outbreak), and the obtained value of *τ* corresponding to the Omicron outbreak is 11 days.


**Table 2.** Estimated value of *τ* for different countries during different outbreaks of the COVID-19 epidemic. The months indicated in the table correspond to the time when the corresponding peak appeared.

**Remark 1.** *It is important to note that in the case of the combination of strains, the estimated value of τ corresponds to the weighted average of the strain-specific disease duration. Furthermore, note that this method depends on how the daily cases of infection were reported. However, in many cases, there is a dominant variant during epidemic outbreaks. Thus, the estimated value of τ can be considered as the disease duration corresponding to the dominant variant during a specific epidemic wave.*

**Remark 2.** *Note that the delay model (9) does not take into account the difference in duration to recovery and the duration to death. If we assume that the death cases are relatively rare (e.g., approximately* ≤ 2% *in the case of COVID-19), then this difference may not be very essential.*

**Remark 3.** *Note that this method of the estimation of τ remains the same for time-varying β* = *β*(*t*)*. Thus, the method discussed above does not depend on whether β is a constant or β varies with respect to time. Since the time-varying β*(*t*) *is capable of incorporating the effect of all possible infected compartments (i.e., exposed E*(*t*)*, asymptomatic IA*(*t*)*, symptomatic IS*(*t*)*, hospitalized H*(*t*)*, etc.) on disease transmission, we can use this method to obtain an estimation of τ for any infectious disease with available data.*
