**7. Discussion**

We proposed a delay model under the assumption that the infected individuals recover or die exactly after an average disease duration *τ*. Generally, in the case of COVID-19, we observed that the recovery and death distributions follow unimodal or bimodal gamma distributions [34,36]. Estimating these gamma distributions requires individuallevel data with the date of onset of disease and the date of recovery or death, which may be very difficult to gather for every country or province. Instead, the only information about the disease duration can help us to obtain sufficiently good results using the delay model. Furthermore, we developed a method to estimate the disease duration from the epidemiological data. It is important to mention here that one can use the Erlang distributions, instead of the gamma distributions, for a compartmental epidemic model with multi-phase disease transition [37].

Let us note that we consider only symptomatic individuals in the model. The influence of asymptomatic individuals is widely discussed in the COVID-19 literature. According to some estimates, they can constitute between 25% and 50% of the total number of cases [38,39]. On the other hand, the infectivity of asymptomatic individuals is much lower than the infectivity of symptomatic individuals because infectivity is proportional to the viral load in the upper respiratory tract [40] and symptoms correlate with viral load. Hence, in the first approximation, we can consider only symptomatic individuals. Further studies are needed to take into account asymptomatic individuals more precisely.

We noticed that during different peaks of COVID-19, the estimated value of *τ* was different. This difference might be due to different strains or the change of proportion of different infected compartments (such as asymptomatic, hospitalized) that we counted in the same compartment.

The presence of exposed individuals can be taken into account by means of timedependent infectivity rate *β*(*t* − *η*). For the individuals infected at time *η*, their infectivity at time *t* depends on the difference *t* − *η*. The function *β*(*t* − *η*) is small if the difference *t* − *η* is small, which is the case of exposed individuals. This case was studied in [32]. We did not consider the time-dependent infectivity rate in this work since its main objective is to compare the model with distributed recovery and death rates with the delay model.

We compared the final size of the epidemic and maximal number of infected obtained in the Formulas (15), (21), and (23), respectively. Note that these formulae depend on *τ* and *β*. Since the outbreak due to Omicron is the only one when the social distancing measures such as lockdowns or isolation were not strictly imposed, we can assume that the transmission rate *β* is approximately constant. We took the data of the Omicron outbreak in Italy from [35] from 10 November 2021 to 10 December 2021, and we fit the delay model to these 40 days of data and estimated the disease transmission rate *β* as 0.118. We took the value of disease duration *τ* = 11 days, which was obtained by using the method discussed in Section 5. Then, using the formula (15), we calculated the final size of the epidemic as 3.387 × 107. Using the formulae (21) and (23), we calculated the maximal number of infected as *Im* = 3.4139 × <sup>10</sup>6, whereas the maximal number of infected was *Im* = 2.7317 × <sup>10</sup><sup>6</sup> in the data. Thus, the formulae (21) and (23) give a reasonable estimate of the maximal number of infected. Similarly, we obtained an accurate estimate of the maximal number of infected for some other countries (not shown).

Let us finally note that the delay model presented in this work is simple and generic. It describes epidemic progression with two parameters *β* and *τ*, which can be easily estimated from the data. Our next goal will be to apply the proposed modeling approach to multicompartment models consisting of different groups of susceptible and/or infected and to immuno-epidemic models with time-varying recovery and death rates [32]. It is also interesting to check the applicability of the proposed model to other transmissible diseases.

**Author Contributions:** Model formulation, V.V.; Model analysis, S.G.; Model validation, M.B. All authors contributed equally to the interpretation and discussion of results. All authors have read and agreed to the published version of the manuscript.

**Funding:** Vitaly Volpert was supported by the Ministry of Science and Higher Education of the Russian Federation (Megagrant agreement no. 075-15-2022-1115).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All data sources are mentioned explicitly.

**Conflicts of Interest:** The authors declare no conflict of interest.
