# Determination of a Key Pandemic Parameter of the SIR-Epidemic Model from Past COVID-19 Mutant Waves and Its Variation for the Validity of the Gaussian Evolution

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## Abstract

**:**

## 1. Introduction

## 2. SIR Model

#### 2.1. Basic Equations

#### 2.2. Key Parameter

#### 2.3. Limiting Case $J\left(t\right)\ll 1$

## 3. Condition for the Validity of the Gaussian Evolution

- (i)
- at early times $t\ll {t}_{\mathrm{max}}$, the Gaussian ratio increases linearly starting from ratio values less than unity;
- (ii)
- at times near maximum, i.e., close to ${t}_{\mathrm{max}}$ near the maximum of $\dot{J}\left(t\right)$, the Gaussian ratio exhibits a dip, which is more pronounced for smaller values of ${a}_{0}$ and which is also indicated by Equation (25) as the third linear term is inversely proportional to ${a}_{0}$;
- (iii)
- at late times beyond ${t}_{\mathrm{max}}$, the Gaussian ratio resumes its linear increase with time.

## 4. Determination of Ratio (16) from Monitored Infection Rates of COVID-19 Waves

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The Gaussian ratio (24) (full curves) and its approximation (25) (dashed curves) plotted for the three cases (A) ${a}_{0}=1$ days${}^{-1}$, ${J}_{\infty}=0.3$, $\Delta =10$ days, (B) ${a}_{0}=0.5$ days${}^{-1}$, ${J}_{\infty}=0.3$, $\Delta =30$ days and (C) ${a}_{0}=0.5$ days${}^{-1}$, ${J}_{\infty}=0.1$, $\Delta =20$ days, respectively. See text for details.

**Figure 2.**

**Left:**differential infection rate, $\dot{J}\left(t\right)$. The raw data (in grey) are smoothed (black curve) in order to infer the second derivative, $\ddot{J}\left(t\right)$.

**Right:**the derived ratio, $k\left(t\right)$ (see Equation (20)) for different values of the stationary infection rate, ${a}_{0}$. The data are inferred from the reported death rates [43] adopting a fatality rate of 0.005 for (

**a**) Germany, from the first corona wave during days 70–200 (March 10–July 18) in the year 2020, (

**b**) Switzerland, during days 60–150, (

**c**) the Netherlands, during days 80–180, (

**d**) the United States, during days 80–180, and (

**e**) Sweden, during days 80–220, where day 1 is 1 January 2020 in (

**b**–

**e**).

**Figure 3.**(

**a**) Differential infection rate, $\dot{J}\left(t\right)$, in the United States, from the first three corona waves during days 80–450 (20 March 2020–24 March 2021) inferred from the reported death rates [43] adopting a fatality rate of 0.005. The raw data (in grey) are smoothed (black curve) in order to infer the second derivative, $\ddot{J}\left(t\right)$. (

**b**) The derived ratio, $k\left(t\right)$ (see Equation (20)), for different values of the stationary infection rate, ${a}_{0}$.

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**MDPI and ACS Style**

Schlickeiser, R.; Kröger, M.
Determination of a Key Pandemic Parameter of the SIR-Epidemic Model from Past COVID-19 Mutant Waves and Its Variation for the Validity of the Gaussian Evolution. *Physics* **2023**, *5*, 205-214.
https://doi.org/10.3390/physics5010016

**AMA Style**

Schlickeiser R, Kröger M.
Determination of a Key Pandemic Parameter of the SIR-Epidemic Model from Past COVID-19 Mutant Waves and Its Variation for the Validity of the Gaussian Evolution. *Physics*. 2023; 5(1):205-214.
https://doi.org/10.3390/physics5010016

**Chicago/Turabian Style**

Schlickeiser, Reinhard, and Martin Kröger.
2023. "Determination of a Key Pandemic Parameter of the SIR-Epidemic Model from Past COVID-19 Mutant Waves and Its Variation for the Validity of the Gaussian Evolution" *Physics* 5, no. 1: 205-214.
https://doi.org/10.3390/physics5010016