# Epidemic Dynamics via Wavelet Theory and Machine Learning with Applications to Covid-19

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Epidemic Modelling via Wavelet Theory and Machine Learning

#### 2.1. Wavelets

**Definition**

**1**

**Lemma**

**1**

**Theorem**

**1**

#### 2.2. Epidemic-Fitted Wavelets and Modelling

**Definition**

**2.**

**Proposition**

**1**

## 3. Epidemic-Fitted (EF) Wavelets

#### 3.1. Gaussian EF Wavelets

#### 3.2. Log-Normal EF Wavelets

#### 3.3. Further Examples of EF Wavelets

#### 3.4. Choosing Suitable EF Wavelets

## 4. Data-Driven Numerical Forecasts

#### 4.1. The Log-Normal Wavelet Model

#### 4.2. Data and Smoothing

#### 4.3. Projections and Validations for the Czech Republic, France, Germany and Italy

#### 4.3.1. Projections from 25 October 2020

#### 4.3.2. Updated Projections from 9 November 2020

#### 4.4. Projections for Federal States in the United States

#### Updated Projections for Florida and New York from 10 November 2020

## 5. Comparing with Other Methods

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Czechia: fitting and forecasting (green curve) from 25 October with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 4.**Czechia: fitting and forecasting from 19 October with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 5.**France: fitting and forecasting from 25 October with 5 wavelets. Our model predicts a new wave starting from October 2020.

**Figure 6.**France: fitting and forecasting from 19 October with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 7.**France: fitting and forecasting from 26/09 with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 8.**Germany: fitting and forecasting from 25 October with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 9.**Germany: fitting and forecasting from 19 October with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 10.**Italy: fitting and forecasting from 25 October with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 11.**Italy: fitting and forecasting from 19 October with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 12.**Italy: fitting and forecasting from 9 November with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 13.**France: fitting and forecasting from 9 November with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 14.**Germany: fitting and forecasting from 9 November with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 15.**Czechia: fitting and forecasting from 10 November with 5 wavelets. The green curve is the combination of other curves which are EF wavelets.

**Figure 16.**Florida: fitting and forecasting from 25 October. The green curve is the combination of other curves which are EF wavelets.

**Figure 17.**New York: fitting and forecasting from 25 October. The green curve is the combination of other curves which are EF wavelets.

**Figure 18.**Florida: fitting and forecasting from 10 November 2020. The green curve is the combination of other curves which are EF wavelets.

**Figure 19.**New York: fitting and forecasting from 10 November 2020. The green curve is the combination of other curves which are EF wavelets.

**Figure 20.**Forecasting 20 days from 30 March, using a wavelet model (green curve) which is combined from EF wavelets, SMA (blue curve), ARMA model (cyan curve) and ARIMA model (yellow curve).

**Figure 21.**Forecasting 20 days from 06 April, using a wavelet model (green curve) which is combined from EF wavelets, SMA (blue curve), ARMA model (cyan curve) and ARIMA model (yellow curve).

**Table 1.**Prediction by log-normal wavelet model for Czechia, Germany, Italy from 20 October to 25 October.

Czechia | ||||
---|---|---|---|---|

Day | Real Data | Smoothing | Prediction | Error |

20 October | 11,984 | 11,173 | 10,730 | 3.96% |

21 October | 14,969 | 11,710 | 11,161 | 4.68% |

22 October | 14,150 | 12,030 | 11,564 | 3.87% |

23 October | 15,258 | 12,689 | 11,934 | 5.95% |

24 October | 12,474 | 12,830 | 12,269 | 4.37% |

25 October | 7300 | 12,295 | 12,564 | 2.18% |

Germany | ||||

Day | Real Data | Smoothing | Prediction | Error |

20 October | 8523 | 9472 | 8346 | 11.88% |

21 October | 12,331 | 10,019 | 8763 | 12.53% |

22 October | 5952 | 9861 | 9164 | 7.06% |

23 October | 22,236 | 10,105 | 9545 | 5.54% |

24 October | 8688 | 10,421 | 9902 | 4.98% |

25 October | 2900 | 9944 | 10,231 | 2.88% |

Italy | ||||

Day | Real Data | Smoothing | Prediction | Error |

20 October | 10,871 | 13,322 | 13,000 | 2.41% |

21 October | 15,199 | 14,567 | 14,080 | 3.34% |

22 October | 16,078 | 15,934 | 15,203 | 4.58% |

23 October | 19,143 | 17,034 | 16,364 | 3.93% |

24 October | 19,640 | 18,266 | 17,557 | 3.88% |

25 October | 21,273 | 19,033 | 18,777 | 1.34% |

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**MDPI and ACS Style**

Tat Dat, T.; Frédéric, P.; Hang, N.T.T.; Jules, M.; Duc Thang, N.; Piffault, C.; Willy, R.; Susely, F.; Lê, H.V.; Tuschmann, W.;
et al. Epidemic Dynamics via Wavelet Theory and Machine Learning with Applications to Covid-19. *Biology* **2020**, *9*, 477.
https://doi.org/10.3390/biology9120477

**AMA Style**

Tat Dat T, Frédéric P, Hang NTT, Jules M, Duc Thang N, Piffault C, Willy R, Susely F, Lê HV, Tuschmann W,
et al. Epidemic Dynamics via Wavelet Theory and Machine Learning with Applications to Covid-19. *Biology*. 2020; 9(12):477.
https://doi.org/10.3390/biology9120477

**Chicago/Turabian Style**

Tat Dat, Tô, Protin Frédéric, Nguyen T. T. Hang, Martel Jules, Nguyen Duc Thang, Charles Piffault, Rodríguez Willy, Figueroa Susely, Hông Vân Lê, Wilderich Tuschmann,
and et al. 2020. "Epidemic Dynamics via Wavelet Theory and Machine Learning with Applications to Covid-19" *Biology* 9, no. 12: 477.
https://doi.org/10.3390/biology9120477