# Forecast of Omicron Wave Time Evolution

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## Abstract

**:**

## 1. Introduction

## 2. Results from the SIR-Model

#### 2.1. Exact Results

#### 2.2. Approximate Results

## 3. Consequences of Early 3-Day Doubling Time

## 4. Omicron Forecast in Individual Countries

- (1)
- the optimistic case with ${k}_{\mathrm{omicron}}={k}_{\beta}$ so that the increase in the ratio r is solely due to the increase in the stationary infection rate$${a}_{0}^{\mathrm{omicron}}=r{a}_{0}^{\beta}$$As noted earlier the larger the value of ${k}_{\mathrm{omicron}}$ the smaller the total cumulative number of infections ${J}_{\infty}$ and the maximum rate of new infections ${j}_{\mathrm{max}}$ will be. This justifies the classification of this case as optimistic.
- (2)
- the pessimistic scenario with ${a}_{0}^{\mathrm{omicron},\mathrm{pess}}={a}_{0}^{\beta}$ so that the increase in the ratio r is solely due to the decrease in the ratio k$${k}_{\mathrm{omicron},\mathrm{pess}}=1-r(1-{k}_{\beta})$$Clearly, with these small values of ${k}_{\mathrm{omicron}}$ the resulting total cumulative number of infections ${J}_{\infty}$ and the maximum rate of new infections ${j}_{\mathrm{max}}$ will be highest, justifying the classification of this case as pessimistic. In four countries (ITA, FRA, RUS, USA) the resulting ${k}_{\mathrm{omicron},\mathrm{pess}}$ is negative which cannot be. In these cases we use ${k}_{\mathrm{omicron},\mathrm{pess}}=0$ and ${a}_{0}^{\mathrm{omicron},\mathrm{pess}}=0.231$.
- (3)
- the intermediate case with$${a}_{0}^{\mathrm{omicron},\mathrm{inter}}=\frac{r}{2}{a}_{0}^{\beta},$$$${k}_{\mathrm{omicron},\mathrm{inter}}=2{k}_{\beta}-1$$

## 5. Medical Consequences for Germany

#### 5.1. Tolerable Maximum 7-Day Incidence Value

#### 5.2. Fatality Rates and Total Number of Fatalities

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) ${J}_{\infty}$, (

**b**) ${t}_{\mathrm{max}}-{t}_{0}$, and (

**c**) ${\dot{J}}_{\mathrm{max}}$ as a function of the only parameter ${k}_{\mathrm{omicron}}$ for different values of the initial fraction of infected persons $\eta $ in the case of an early 3-day doubling time.

**Figure 2.**Location of the considered values of ${a}_{0}$ and k for the omicron mutant in the investigated 12 countries with the adopted 3-day early doubling times. The symbols $\beta $ represent the values for the earlier beta mutant. The solid black line is Equation (23).

**Figure 3.**Time dependence of the daily rate of newly infected persons, $\dot{J}\left(t\right)$, as well as the cumulative fraction of infected persons, $J\left(t\right)$, for all 12 countries.

**Table 1.**Second wave parameters ${a}_{0}^{\beta}$ in days${}^{-1}$, ${k}_{\beta}$, initial fraction ${\eta}_{\beta}$, and the inferred second doubling time ${t}_{2}^{\beta}$ in days. For the omicron mutant in all countries we adopt ${t}_{2}^{\mathrm{omicron}}$ to calculate the ratio of the the two doubling times $r={t}_{2}^{\beta}/{t}_{2}^{\mathrm{omicron}}$.

Country | ${\mathit{a}}_{0}^{\mathit{\beta}}$ | ${\mathit{k}}_{\mathit{\beta}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{\eta}}_{\mathit{\beta}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{t}}_{2}^{\mathit{\beta}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{t}}_{2}^{\mathbf{omicron}}$ | $\mathit{r}={\mathit{t}}_{2}^{\mathit{\beta}}/{\mathit{t}}_{2}^{\mathbf{omicron}}$ |
---|---|---|---|---|---|---|

ITA | 0.13 | 0.823 | $1.1\times {10}^{-4}$ | 30.1 | 3.0 | 10.03 |

AUT | 0.43 | 0.898 | $1.8\times {10}^{-5}$ | 15.8 | 3.0 | 5.27 |

DNK | 2.48 | 0.972 | $3.6\times {10}^{-5}$ | 10.0 | 3.0 | 3.33 |

DEU | 0.45 | 0.907 | $1.1\times {10}^{-5}$ | 16.6 | 3.0 | 5.53 |

CHE | 0.44 | 0.892 | $2.2\times {10}^{-5}$ | 14.6 | 3.0 | 4.87 |

GBR | 0.44 | 0.874 | $4.6\times {10}^{-5}$ | 12.5 | 3.0 | 4.17 |

FRA | 0.17 | 0.868 | $1.0\times {10}^{-4}$ | 30.9 | 3.0 | 10.30 |

BEL | 0.53 | 0.893 | $1.8\times {10}^{-4}$ | 12.2 | 3.0 | 4.07 |

NLD | 0.37 | 0.926 | $3.4\times {10}^{-5}$ | 25.3 | 3.0 | 8.43 |

RUS | 0.03 | 0.801 | $6.9\times {10}^{-3}$ | 116.1 | 3.0 | 38.70 |

SWE | 0.58 | 0.919 | $8.0\times {10}^{-9}$ | 14.8 | 3.0 | 4.93 |

USA | 0.22 | 0.868 | $9.5\times {10}^{-4}$ | 23.9 | 3.0 | 7.97 |

**Table 2.**Forecast of the omicron mutant for the optimistic case, i.e., ${a}_{0}={a}_{0}^{\mathrm{omicron},\mathrm{optim}}$, $k={k}_{\mathrm{omicron},\mathrm{optim}}$, and initial fraction $\eta ={\eta}_{\beta}$ from Table 1 for this table. Columns list the final cumulative fraction ${J}_{\infty}$ of infected persons, the maximum (dimensionless) rate ${j}_{\mathrm{max}}$ of new infections, the cumulative fraction ${J}_{0}$ of infected persons at peak time, the reduced peak time ${\tau}_{\mathrm{max}}$, the peak time ${t}_{\mathrm{max}}-{t}_{0}$ in days, and the SDI, the maximum 7-day incidence per ${10}^{5}$ persons. Country names are abbreviated by their ${\alpha}_{3}$ codes.

Optimistic Scenario | |||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{0}}$ | $\mathit{k}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{J}}_{\mathbf{\infty}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{j}}_{\mathbf{max}}\phantom{\rule{0.277778em}{0ex}}$ | ${\dot{\mathit{J}}}_{\mathbf{max}}$ | ${\mathit{J}}_{\mathbf{0}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{\tau}}_{\mathbf{max}}$ | $\phantom{\rule{-0.166667em}{0ex}}{\mathit{t}}_{\mathbf{max}}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}{\mathit{t}}_{\mathbf{0}}$ | SDI |

ITA | 1.304 | 0.823 | 0.33 | 0.0139 | 0.018 | 0.160 | 35.6 | 27days | 12,725 |

AUT | 2.265 | 0.898 | 0.20 | 0.0049 | 0.011 | 0.097 | 68.9 | 30days | 7716 |

DNK | 8.267 | 0.972 | 0.06 | 0.0004 | 0.004 | 0.028 | 130.1 | 16days | 2424 |

DEU | 2.490 | 0.907 | 0.18 | 0.0041 | 0.010 | 0.089 | 79.0 | 32days | 7090 |

CHE | 2.141 | 0.892 | 0.21 | 0.0054 | 0.012 | 0.102 | 64.0 | 30days | 8148 |

GBR | 1.833 | 0.874 | 0.24 | 0.0073 | 0.013 | 0.118 | 51.3 | 28days | 9392 |

FRA | 1.751 | 0.868 | 0.25 | 0.0080 | 0.014 | 0.123 | 43.5 | 25days | 9853 |

BEL | 2.155 | 0.893 | 0.21 | 0.0055 | 0.012 | 0.101 | 44.3 | 21days | 8250 |

NLD | 3.120 | 0.926 | 0.14 | 0.0026 | 0.008 | 0.071 | 77.4 | 25days | 5751 |

RUS | 1.161 | 0.801 | 0.39 | 0.0219 | 0.025 | 0.173 | 10.7 | 9days | 17,773 |

SWE | 2.861 | 0.919 | 0.16 | 0.0031 | 0.009 | 0.078 | 175.6 | 61days | 6216 |

USA | 1.753 | 0.868 | 0.26 | 0.0087 | 0.015 | 0.122 | 26.0 | 15days | 10,650 |

**Table 3.**Forecast of the omicron mutant for the pessimistic case, i.e., ${a}_{0}={a}_{0}^{\mathrm{omicron},\mathrm{pess}}$ and $k={k}_{\mathrm{omicron},\mathrm{pess}}$.

Pessimistic Scenario | |||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{0}}$ | $\mathit{k}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{J}}_{\mathsf{\infty}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{j}}_{\mathbf{max}}\phantom{\rule{0.277778em}{0ex}}$ | ${\dot{\mathit{J}}}_{\mathbf{max}}$ | ${\mathit{J}}_{\mathbf{0}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{\tau}}_{\mathbf{max}}$ | $\phantom{\rule{-0.166667em}{0ex}}{\mathit{t}}_{\mathbf{max}}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}{\mathit{t}}_{\mathbf{0}}$ | SDI |

ITA | 0.231 | 0.000 | 1.000 | 0.2500 | 0.058 | 0.500 | 9.2 | 40days | 40,414 |

AUT | 0.430 | 0.463 | 0.836 | 0.0984 | 0.042 | 0.379 | 18.8 | 44days | 29,621 |

DNK | 2.480 | 0.907 | 0.181 | 0.0041 | 0.010 | 0.089 | 65.6 | 27days | 7148 |

DEU | 0.450 | 0.485 | 0.812 | 0.0918 | 0.041 | 0.369 | 20.4 | 45days | 28,911 |

CHE | 0.440 | 0.474 | 0.824 | 0.0950 | 0.042 | 0.374 | 18.7 | 42days | 29,259 |

GBR | 0.440 | 0.475 | 0.823 | 0.0949 | 0.042 | 0.374 | 17.3 | 39days | 29,208 |

FRA | 0.231 | 0.000 | 1.000 | 0.2500 | 0.058 | 0.500 | 9.2 | 40days | 40,414 |

BEL | 0.530 | 0.565 | 0.721 | 0.0696 | 0.037 | 0.331 | 17.0 | 32days | 25,819 |

NLD | 0.370 | 0.376 | 0.911 | 0.1249 | 0.046 | 0.412 | 15.5 | 42days | 32,336 |

RUS | 0.231 | 0.000 | 1.000 | 0.2500 | 0.058 | 0.500 | 5.0 | 22days | 40,422 |

SWE | 0.580 | 0.600 | 0.675 | 0.0602 | 0.035 | 0.312 | 43.1 | 74days | 24,408 |

USA | 0.231 | 0.000 | 1.000 | 0.2500 | 0.058 | 0.500 | 7.0 | 30days | 40,419 |

**Table 4.**Forecast of the omicron mutant for the intermediate case, i.e., ${a}_{0}={a}_{0}^{\mathrm{omicron},\mathrm{inter}}$ and $k={k}_{\mathrm{omicron},\mathrm{inter}}$.

Intermediate Scenario | |||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{0}}$ | $\mathit{k}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{J}}_{\mathsf{\infty}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{j}}_{\mathbf{max}}\phantom{\rule{0.277778em}{0ex}}$ | ${\dot{\mathit{J}}}_{\mathbf{max}}$ | ${\mathit{J}}_{\mathbf{0}}\phantom{\rule{0.277778em}{0ex}}$ | ${\mathit{\tau}}_{\mathbf{max}}$ | $\phantom{\rule{-0.166667em}{0ex}}{\mathit{t}}_{\mathbf{max}}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}{\mathit{t}}_{\mathbf{0}}$ | SDI |

ITA | 0.652 | 0.646 | 0.613 | 0.0489 | 0.032 | 0.286 | 21.4 | 33days | 22,307 |

AUT | 1.132 | 0.796 | 0.378 | 0.0181 | 0.021 | 0.182 | 41.0 | 36days | 14,328 |

DNK | 4.133 | 0.944 | 0.110 | 0.0015 | 0.006 | 0.054 | 90.9 | 22days | 4462 |

DEU | 1.245 | 0.814 | 0.347 | 0.0152 | 0.019 | 0.168 | 46.7 | 38days | 13,263 |

CHE | 1.071 | 0.784 | 0.398 | 0.0201 | 0.022 | 0.191 | 38.2 | 36days | 15,060 |

GBR | 0.917 | 0.748 | 0.458 | 0.0267 | 0.024 | 0.218 | 31.0 | 34days | 17,109 |

FRA | 0.876 | 0.736 | 0.477 | 0.0290 | 0.025 | 0.226 | 26.8 | 31days | 17,797 |

BEL | 1.078 | 0.786 | 0.396 | 0.0199 | 0.021 | 0.189 | 28.6 | 27days | 14,979 |

NLD | 1.560 | 0.852 | 0.281 | 0.0099 | 0.016 | 0.137 | 48.0 | 31days | 10,832 |

RUS | 0.580 | 0.602 | 0.678 | 0.0627 | 0.036 | 0.307 | 8.8 | 15days | 25,466 |

SWE | 1.431 | 0.838 | 0.305 | 0.0117 | 0.017 | 0.148 | 96.2 | 67days | 11,748 |

USA | 0.876 | 0.736 | 0.479 | 0.0295 | 0.026 | 0.226 | 18.3 | 21days | 18,110 |

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**MDPI and ACS Style**

Schlickeiser, R.; Kröger, M.
Forecast of Omicron Wave Time Evolution. *COVID* **2022**, *2*, 216-229.
https://doi.org/10.3390/covid2030017

**AMA Style**

Schlickeiser R, Kröger M.
Forecast of Omicron Wave Time Evolution. *COVID*. 2022; 2(3):216-229.
https://doi.org/10.3390/covid2030017

**Chicago/Turabian Style**

Schlickeiser, Reinhard, and Martin Kröger.
2022. "Forecast of Omicron Wave Time Evolution" *COVID* 2, no. 3: 216-229.
https://doi.org/10.3390/covid2030017