# Key Epidemic Parameters of the SIRV Model Determined from Past COVID-19 Mutant Waves

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## Abstract

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## 1. Introduction

## 2. SIRV Model

#### 2.1. Starting Equations

#### 2.2. Key Parameter

#### 2.3. Comparison with the SIR Model Limit

#### 2.4. Real Time Dependence

## 3. Cumulative Vaccination Fraction

## 4. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Solution to the (

**a**) SIR and (

**b**) SIRV models versus reduced time $\tau $ for $k\left(\tau \right)=0.5+0.2tanh\left[0.05\right(\tau -30\left)\right]$, $\eta ={10}^{-8}$ and $b\left(\tau \right)=0$ and $b\left(\tau \right)=0.01$ in (

**a**) and (

**b**), respectively. Shown are $S\left(\tau \right)$, $I\left(\tau \right)$, $R\left(\tau \right)$, $V\left(\tau \right)$ as well as $j\left(\tau \right)=S\left(\tau \right)I\left(\tau \right)$. (

**c**) Assuming that the data in (

**b**) had been measured, $k\left(\tau \right)$ and $b\left(\tau \right)$ are correctly reconstructed from Equations (18) and (16). For comparison, we show ${k}_{\mathrm{SIR}}\left(\tau \right)$ obtained from Equation (19). The displayed values of I, j and b were multiplied by factors of 4, 4 and 10, respectively, in order to avoid overlarge figures.

**Figure 2.**(

**a**) Reported data [38] for Germany for the estimated cumulative fraction $I\left(t\right)=200\phantom{\rule{0.166667em}{0ex}}D\left(t\right)$ of infected people, where $D\left(t\right)$ is the reported cumulative fraction of fatalities and $V\left(t\right)$ is the cumulative fraction of vaccinated persons, as well as their sum. We assume here that vaccinated and infected fractions belong to disjunct compartments, the SIRV model therefore breaks down as soon as the reported $I\left(t\right)+V\left(t\right)$ exceeds unity, manifested by a sign change of $k\left(t\right)$. (

**b**) Ratios $k\left(t\right)$ (solid) and ${k}_{\mathrm{SIR}}\left(t\right)$ (dashed) according to Equation (24) with and without $V\left(t\right)$, respectively.

**Figure 3.**(

**a**) Daily fraction $\dot{J}\left(t\right)$ and (

**b**) cumulative fraction $J\left(t\right)$ of infected persons in Australia (AUS), Switzerland (CHE), Germany (DEU), France (FRA), Italy (ITA), Sweden (SWE) and the United States (USA). Up to the end of 2021, the number of truly infected persons is estimated from the number of fatalities, using a fatality rate of $0.005$. Afterwards, due to a not precisely known change in the fatality rate, the estimated $\dot{J}\left(t\right)$ is not considered in this work. The raw data were retrieved from [38].

**Figure 4.**(

**a**) Daily fraction and (

**b**) cumulative fraction of fully vaccinated persons in Australia (AUS), Switzerland (CHE), Germany (DEU), France (FRA), Italy (ITA), Sweden (SWE) and the United States (USA). The raw data were retrieved from [38].

**Figure 5.**Dimensionless ratios (a) $b\left(t\right)$ and (b) $k\left(t\right)$ evaluated using Equation (24) and the data shown in Figure 3 and Figure 4. As in Figure 2, regimes with $k\left(t\right)<0$, here also $b\left(t\right)$, are indicated by light color. The raw data were retrieved from [38].

Country | ${\mathit{\alpha}}_{3}$ Code | ${\mathit{t}}_{\mathit{A}}$ | $\mathit{\tau}$ | ${\mathit{V}}_{\mathit{\infty}}$ |
---|---|---|---|---|

Australia | AUS | 570 | 78 days | 0.90 |

Switzerland | CHE | 479 | 86 days | 0.72 |

Germany | DEU | 491 | 83 days | 0.76 |

France | FRA | 494 | 92 days | 0.80 |

Italy | ITA | 492 | 88 days | 0.79 |

Sweden | SWE | 503 | 85 days | 0.77 |

United States | USA | 407 | 122 days | 0.70 |

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**MDPI and ACS Style**

Schlickeiser, R.; Kröger, M.
Key Epidemic Parameters of the SIRV Model Determined from Past COVID-19 Mutant Waves. *COVID* **2023**, *3*, 592-600.
https://doi.org/10.3390/covid3040042

**AMA Style**

Schlickeiser R, Kröger M.
Key Epidemic Parameters of the SIRV Model Determined from Past COVID-19 Mutant Waves. *COVID*. 2023; 3(4):592-600.
https://doi.org/10.3390/covid3040042

**Chicago/Turabian Style**

Schlickeiser, Reinhard, and Martin Kröger.
2023. "Key Epidemic Parameters of the SIRV Model Determined from Past COVID-19 Mutant Waves" *COVID* 3, no. 4: 592-600.
https://doi.org/10.3390/covid3040042