# A simple function to fit the time evolution of daily reported COVID-19 deaths in EU countries

Tristan Beau, Julien Browaeys, Olivier Dadoun

<figure><img src="image_1.png" /></figure>

## To cite this version:

Tristan Beau, Julien Browaeys, Olivier Dadoun. A simple function to fit the time evolution of daily reported COVID-19 deaths in EU countries. 2020. hal-02567266

HAL Id: hal-02567266

https://hal.science/hal-02567266

Preprint submitted on 7 May 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
A SIMPLE FUNCTION TO FIT THE TIME EVOLUTION OF DAILY
REPORTED COVID-19 DEATHS IN EU COUNTRIES

Tristan Beau¹, Julien Browaeys², Olivier Dadoun¹

¹ : Laboratoire de Physique Nuclaire et des Hautes Énergies, CNRS UMR 7585,
Sorbonne Université, Université de Paris, CNRS/IN2P3, 4 place Jussieu, 75005 Paris, France.

² : Laboratoire Matière et Systèmes Complexes, CNRS UMR 7057, Université de Paris,
10 rue Alice Domon et Léonie Duquet, 75013 Paris, France

We present a simple five-parameters function that is able to reasonably describe the evolution
of the number of daily death due to COVID-19 in every country of the European Union.
By extrapolation, assuming current lockdown conditions were to remain in place, this *ad hoc*
fitting model is forecasting a total of about 200 000 deaths in the EU. Following this estimate,
the current (May 3rd) number of death is expected to amount to two third of the total future
casualty count.

# 1 Introduction

The current Covid-19 pandemia is a major worldwide concern since its outbreak¹ in December 2019. Although all countries are publishing their statistics in term of deaths, confirmed or recovered cases, the epidemiological modeling remains complex.

The realization of a major epidemic crisis happening in the European Union (EU) came in March 2020, and most EU countries consequently instituted some form of lockdown in a matter of days. Since then, at least in Europe, the epidemics has receded enough so that these countries have planned to lift the lockdown, at least partially, in the second week of May.

We wish to contribute to the evaluation of the incidence of such a decision based on current data. We fit an effective physical model to public Covid-19 deaths data in all 28 EU countries (including UK) in order to forecast the evolution of the epidemic if lockdown were to be maintained. Assuming that hypothesis, we estimate the total number of death due to COVID-19 at the end of the epidemic.

# 2 Data and method

The Johns Hopkins University database⁵ provides a daily report of deaths, confirmed and recovered Covid-19 patients all over the world, since January 22nd, 2020; this date is considered to be our time reference *t* = 0. Because of large variations in testing policies in different countries, in this analysis we limit ourselves to the study of daily deaths as a function of time. We are aware that this remains an imperfect proxy of the epidemic as counting policies can also vary from one country to the other.

In order to be able to compare countries, we normalize data by country population. This information is taken from the latest *United Nations Population Division* estimates⁴. Databases extraction of data and their initial analysis is ensured by the open source software CoCoA²,³.
## 3 Analysis and extrapolation

The modeling of data is done through a standard least-square fit of a Crystal Ball function⁶.

$$f(t; \alpha, n, t_0, \sigma) \propto \begin{cases} \exp\left(-\frac{(t-t_0)^2}{2\sigma^2}\right), & \text{for } \frac{t-t_0}{\sigma} > -\alpha \\ A \cdot \left(B - \frac{t-t_0}{\sigma}\right)^{-n}, & \text{for } \frac{t-t_0}{\sigma} \le -\alpha \end{cases}$$

The Crystal Ball function consists of a smoothly joined Gaussian part and a power-law tail at a certain cut-off. It has 5 parameters: N is the full integral of the function over the time, n the power law parameter, α a normalized cutoff where the power-law replaces the Gaussian part, t₀ the position of the peak, σ the time scale. We are only reporting the useful parameters N and t₀, which respectively describe the total number of death (assuming the daily death rate to follow this function) and the time at which the peak of the epidemic occurred.

Commonly used in high-energy physics, this function is named after a neutral particle detector initially used at the Stanford Linear Accelerator Center. The reason for choosing this function is two-fold. First we needed to describe properly the asymmetrical bell shape curve of the data (see figures below). Second we wanted a minimum number of adjustable parameters to avoid over-fitting.

For every EU countries (and UK), we represent daily deaths as a function of time since t=0, together with the fitted Crystal Ball function. It is represented over 200 days to better visualize our forecast. The best fit parameters N and t₀ are given on each figure.

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>
<figure><img src="image_9.png" /><figcaption>Estonia, N=82, t0=74</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Finland, N=344, t0=90</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>France, N=30266, t0=78</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Germany, N=8158, t0=84</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Greece, N=216, t0=71</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Hungary, N=414, t0=89</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Ireland, N=1537, t0=91</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Italy, N=43326, t0=67</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Latvia, N=19, t0=92</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Lithuania, N=55, t0=81</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Luxembourg, N=96, t0=80</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Malta, N=6, t0=78</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Netherlands, N=7608, t0=75</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Poland, N=878, t0=90</figcaption></figure>

<figure><img src="image_23.png" /></figure>
<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

The same procedure has also been applied to European Union (including UK) as a whole.

<figure><img src="image_31.png" /></figure>
# 4 Discussion

Our modeling yields an estimation of the expected overall number of deaths for each country. We provide these numbers in the table below, and compare them with the total number of deaths reported until May 3rd, 2020, to infer the progress of the epidemic at this date. We normalize the figures by population to facilitate comparison between countries.

<table><thead><tr><th>Country</th><th>Population</th><th>Deaths (May 3rd)</th><th>Total forecast N</th><th>Peak date t<sub>0</sub></th><th>Deaths (May 3rd) per 100k pop</th><th>Total forecast per 100k pop</th><th>Extent (May 3rd) in %</th></tr></thead><tbody><tr><td>UE + UK</td><td>513 136 525</td><td>137 796</td><td>204 586</td><td>78</td><td>26.9</td><td>39.9</td><td>67.4</td></tr><tr><td>Germany</td><td>83 783 942</td><td>6 993</td><td>8 158</td><td>84</td><td>8.3</td><td>9.7</td><td>85.7</td></tr><tr><td>United Kingdom</td><td>67 886 011</td><td>28 809</td><td>34 630</td><td>85</td><td>42.4</td><td>51.0</td><td>83.2</td></tr><tr><td>France</td><td>65 273 511</td><td>25 204</td><td>30 266</td><td>78</td><td>38.6</td><td>46.4</td><td>83.3</td></tr><tr><td>Italy</td><td>60 461 826</td><td>29 079</td><td>43 326</td><td>67</td><td>48.1</td><td>71.7</td><td>67.1</td></tr><tr><td>Spain</td><td>46 754 778</td><td>25 428</td><td>36 168</td><td>69</td><td>54.4</td><td>77.4</td><td>70.3</td></tr><tr><td>Poland</td><td>37 846 611</td><td>698</td><td>878</td><td>90</td><td>1.8</td><td>2.3</td><td>79.5</td></tr><tr><td>Romania</td><td>19 237 691</td><td>818</td><td>1 013</td><td>89</td><td>4.3</td><td>5.3</td><td>80.8</td></tr><tr><td>Netherlands</td><td>17 134 872</td><td>5 098</td><td>7 608</td><td>75</td><td>29.8</td><td>44.4</td><td>67.0</td></tr><tr><td>Belgium</td><td>11 589 623</td><td>7 924</td><td>10 007</td><td>82</td><td>68.4</td><td>86.3</td><td>79.2</td></tr><tr><td>Czech Rep.</td><td>10 708 981</td><td>252</td><td>372</td><td>77</td><td>2.4</td><td>3.5</td><td>67.7</td></tr><tr><td>Greece</td><td>10 423 054</td><td>146</td><td>216</td><td>71</td><td>1.4</td><td>2.1</td><td>67.6</td></tr><tr><td>Portugal</td><td>10 196 709</td><td>1 063</td><td>1 564</td><td>82</td><td>10.4</td><td>15.3</td><td>68.0</td></tr><tr><td>Sweden</td><td>10 099 265</td><td>2 769</td><td>3 267</td><td>88</td><td>27.4</td><td>32.3</td><td>84.8</td></tr><tr><td>Hungary</td><td>9 660 351</td><td>351</td><td>414</td><td>89</td><td>3.6</td><td>4.3</td><td>84.8</td></tr><tr><td>Austria</td><td>9 006 398</td><td>600</td><td>863</td><td>76</td><td>6.7</td><td>9.6</td><td>69.5</td></tr><tr><td>Bulgaria</td><td>6 948 445</td><td>78</td><td>92</td><td>89</td><td>1.1</td><td>1.3</td><td>84.8</td></tr><tr><td>Denmark</td><td>5 792 202</td><td>493</td><td>726</td><td>75</td><td>8.5</td><td>12.5</td><td>67.9</td></tr><tr><td>Finland</td><td>5 540 720</td><td>240</td><td>344</td><td>90</td><td>4.3</td><td>6.2</td><td>69.8</td></tr><tr><td>Slovakia</td><td>5 459 642</td><td>25</td><td>27</td><td>90</td><td>0.5</td><td>0.5</td><td>92.6</td></tr><tr><td>Ireland</td><td>4 937 786</td><td>1 319</td><td>1 537</td><td>91</td><td>26.7</td><td>31.1</td><td>85.8</td></tr><tr><td>Croatia</td><td>4 105 267</td><td>80</td><td>109</td><td>93</td><td>1.9</td><td>2.7</td><td>73.4</td></tr><tr><td>Lithuania</td><td>2 722 289</td><td>46</td><td>55</td><td>81</td><td>1.7</td><td>2.0</td><td>83.6</td></tr><tr><td>Slovenia</td><td>2 078 938</td><td>97</td><td>144</td><td>77</td><td>4.7</td><td>6.9</td><td>67.4</td></tr><tr><td>Latvia</td><td>1 886 198</td><td>16</td><td>19</td><td>92</td><td>0.8</td><td>1.0</td><td>84.2</td></tr><tr><td>Estonia</td><td>1 326 535</td><td>55</td><td>82</td><td>74</td><td>4.1</td><td>6.2</td><td>67.1</td></tr><tr><td>Cyprus</td><td>1 207 359</td><td>15</td><td>22</td><td>67</td><td>1.2</td><td>1.8</td><td>68.2</td></tr><tr><td>Luxembourg</td><td>625 978</td><td>96</td><td>96</td><td>80</td><td>15.3</td><td>15.3</td><td>100.0</td></tr><tr><td>Malta</td><td>441 543</td><td>4</td><td>6</td><td>78</td><td>0.9</td><td>1.4</td><td>66.7</td></tr></tbody></table>

With merely five free parameters, our model fits reasonably well all data for the 28 EU countries (and UK). Fitting seems all the more accurate that the peak can be easily identified, and that the total number of daily death is high.

Some countries plan lifting the lockdown in early May 2020, such as Germany (May 4th), or France (May 11th). One can assume that without lifting the lockdown, the evolution of the daily rate of death could have followed the trend we have exhibited.

On May 3rd, EU countries (and UK) have recorded a total of 136 391 deaths due to Covid-19. Our effective model is projecting 204 586 deaths if all conditions remain as current. In this hypothesis, 2/3 of the total number of Covid-19 victims have already perished. It will be interesting to compare these figures with the total number of death at the end of the epidemic.

We should stress that our proposal is prospective and not predictive, in the sense that it
does *not* rely on epidemiological modeling. We merely describe data through *ad hoc* fitting of
mathematical functions, a methodology similar to what is used in economic forecasts.

# 5 Conclusion

We provide an minimal model, within the CoCoA framework, that reasonably fits the daily
rate of death as a function of time with only five free parameters. Assuming lockdown is not
lifted, we estimate that at the beginning of May 2020 about two third of deaths have already
occurred in Europe before the end of the epidemic. This implies a final total death rate of 1
over 2 500 people. In the future, lockdown lifting policies could be evaluated with regard to our
projection. In any case, our model provides a simple tool to reasonably describe the evolution
from inception to the current stage of the epidemic.

1. “About the Virus.” World Health Organization, May 4, 2020. http://www.euro.who.int/en/health-topics/health-emergencies/coronavirus-covid-19/novel-coronavirus-2019-ncov.

2. Tristan Beau, Julien Browaeys, Olivier Dadoun. *CoCoA, a free Collaborative Covid Analysis tool*. Preprint under preparation.

3. Tristan Beau, Olivier Dadoun, Julien Browaeys. GitHub CoCoA Official Repository (version 0.1). Python, 2020. https://github.com/tjbtjbtjb/CoCoA.

4. “Population by Country (2020) - Worldometer.” Accessed May 4, 2020. https://www.worldometers.info/world-population/population-by-country/.

5. CSSEGISandData/COVID-19 by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University, 2020. https://github.com/CSSEGISandData/COVID-19.

6. Skwarnicki, T. A study of the radiative CASCADE transitions between the Upsilon-Prime and Upsilon resonances. (PhD Thesis, Cracow, INP, 1986). https://inspirehep.net/literature/230779