# Using Early Data to Estimate the Actual Infection Fatality Ratio from COVID-19 in France

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. Mechanistic-Statistical Model

^{®}ode45 solver.

#### 2.3. Statistical Inference

^{®}function fmincon. In order to find a global maximum of $\mathcal{L}$, we apply this method starting from random initial values for $\alpha ,{t}_{0},\kappa $ drawn uniformly in the following intervals: $\alpha \in (0,1)$, ${t}_{0}\in (1,31)$, (1–31 January) and $\kappa \in (0,1)$. The minimisation algorithm is applied to 10,000 random initial values of the parameters.

^{®}codes are available as Supplementary Materials.

## 3. Results

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IFR | Infection fatality ratio |

CFR | Case fatality rate |

ODE | Ordinary differential equation |

ARS | Agence Régionale de Santé |

WHO | World Health Organization |

MLE | Maximum likelihood estimator |

DREES | Direction de la recherche, des études, de l’évaluation et des statistiques |

## Appendix A

- The joint posterior distributions of the three pairs of parameters $(\alpha ,\kappa )$, $({t}_{0},\alpha )$ and $({t}_{0},\kappa )$ are depicted in Figure A1.
- To check the robustness of our results with respect to the choice of the prior distribution, we also considered the case of a more informative prior. Namely, we assumed the following uniform prior distributions:
- ∘
- $\alpha \in (0.14,0.65)$, corresponding to $\beta \times {R}_{0}$ with $\beta =1/10$ and ${R}_{0}$ values ranging between 1.4 and 6.49 (the range described in [18]);
- ∘
- ${t}_{0}\in (20,31)$ corresponding to an introduction during late January;
- ∘
- $\kappa \in (0,{10}^{-2})$, corresponding to a small probability of being tested for the susceptible cases, compared to the infected cases.

We obtained the posterior distributions shown in Figure A2, based on two independent chains with ${10}^{6}$ iterations (only the second half of the iterations are used to generate the posterior). Overall, these distributions are relatively similar to those displayed on Figure A1 and obtained with the prior distributions defined in the main text. - The dynamics of the estimated distribution of the IFR are depicted in Figure A3.
- The marginal posterior distribution of ${R}_{0}$ is depicted in Figure A4.

**Figure A1.**Joint posterior distributions of $(\alpha ,\kappa )$, $({t}_{0},\alpha )$ and $({t}_{0},\kappa )$.

**Figure A2.**Joint posterior distributions of $(\alpha ,\kappa )$, $({t}_{0},\alpha )$ and $({t}_{0},\kappa )$ obtained with a more informative prior.

**Figure A3.**Dynamics of the IFR in France. Solid line: average value obtained from the posterior distribution of the parameters. Dotted curves: 0.025 and 0.975 pointwise quantiles.

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**Figure 1.**Expected number of observed cases associated with the MLE vs. number of cases actually detected (total cases). The curve corresponds to cumulated values of the expected observation ${n}_{t}\phantom{\rule{0.166667em}{0ex}}{p}_{t}^{\ast}$ given by the model, and the crosses correspond to the data (cumulated values of ${\widehat{\delta}}_{t}$).

**Figure 2.**Distribution of the cumulated number of infected cases ($I\left(t\right)+R\left(t\right)$) across time. Solid line: average value obtained from the posterior distribution of the parameters. Dotted curves: 0.025 and 0.975 pointwise posterior quantiles. Blue crosses: data (cumulated values of ${\widehat{\delta}}_{t}$).

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Roques, L.; Klein, E.K.; Papaïx, J.; Sar, A.; Soubeyrand, S. Using Early Data to Estimate the Actual Infection Fatality Ratio from COVID-19 in France. *Biology* **2020**, *9*, 97.
https://doi.org/10.3390/biology9050097

**AMA Style**

Roques L, Klein EK, Papaïx J, Sar A, Soubeyrand S. Using Early Data to Estimate the Actual Infection Fatality Ratio from COVID-19 in France. *Biology*. 2020; 9(5):97.
https://doi.org/10.3390/biology9050097

**Chicago/Turabian Style**

Roques, Lionel, Etienne K Klein, Julien Papaïx, Antoine Sar, and Samuel Soubeyrand. 2020. "Using Early Data to Estimate the Actual Infection Fatality Ratio from COVID-19 in France" *Biology* 9, no. 5: 97.
https://doi.org/10.3390/biology9050097