# The French Covid-19 data

## The hospital data

This tool allows an interactive visualization of hospital data provided by Santé Publique France.

Data used are:

• the daily number of newly hospitalized patients,
• the daily number of patients newly admitted to intensive care units,
• the daily number of deceased patients,
• the daily number of discharded patients.

### The model

\begin{align} \ddot{I}_{\rm hosp}(t) & = k_{\rm hosp}(t) \, \dot{I}_{\rm hosp}(t) \\ \ddot{I}_{\rm icu}(t) & = k_{\rm icu}(t) \, \dot{I}_{\rm icu}(t) \\ \dot{H}(t) & = \dot{I}_{\rm hosp}(t) + \dot{I}_{\rm icu}(t) - k_{\rm deaths}(t)H(t) - k_{\rm out}H(t) \\ \dot{D}(t) & = k_{\rm deaths}(t)H(t) \\ \dot{O}(t) & = k_{\rm out}(t)H(t) \\ \end{align} where

• $$I_{\rm hosp}$$ is the total (i.e. cumulated) number of infected individuals who will be hospitalized
• $$I_{\rm icu}$$ is the total number of infected individuals who will be admitted to intensive care units
• $$H$$ is the number of individuals in hospital or in intensive care unit
• $$D$$ is the total number of deceased individuals.
• $$O$$ is the total number of individuals who have returned home (discharges).

Rate functions $$k_{\rm hosp}$$ and $$k_{\rm icu}$$ are continuous piecewise linear functions:

\begin{align} k_{\rm hosp}(t) &= k_{\rm hosp, 0} + a_{\rm hosp} t + \sum_{k=1}^{K-1} h_{{\rm hosp}, k} ( t - \tau_{{\rm hosp}, k}) \times {\Large 1} \{t\geq \tau_{{\rm hosp}, k} \} \\ k_{\rm icu}(t) &= k_{\rm icu, 0} + a_{\rm icu} t + \sum_{k=1}^{K-1} h_{{\rm icu}, k} ( t - \tau_{{\rm icu}, k}) \times {\Large 1} \{t\geq \tau_{{\rm icu}, k} \} \end{align}

The death rate $$k_{\rm deaths}$$ and the discharge rate $$k_{\rm out}$$ are (decreasing - increasing) functions of time:

$k_{\rm deaths}(t) = \left\{ \begin{array}{cc} \mu_1 e^{-\alpha_1 \, t } & {\rm if } \quad t \leq \gamma \\ \mu_1 e^{\alpha_2 t - (\alpha_1+\alpha_2) \gamma} & {\rm if } \quad t \geq \gamma \end{array} \right.$

$k_{\rm out}(t) = \left\{ \begin{array}{cc} \nu_1 e^{-\beta_1 \, t } & {\rm if } \quad t \leq \gamma \\ \nu_1 e^{\beta_2 t - (\beta_1+\beta_2) \gamma} & {\rm if } \quad t \geq \gamma \end{array} \right.$

According to the model, $$\dot{I}_{\rm hosp}(t_j)$$, $$\dot{I}_{\rm icu}(t_j)$$, $$\dot{D}(t_j)$$ and $$\dot{O}(t_j)$$ are the predicted numbers of, respectively, newly hospitalized, newly admitted to intensive care units, deceased patients and discharged patient, between time $$t_{j-1}$$ and time $$t_j$$, i.e. on day $$j$$.

The data are the daily numbers of deaths in the French EHPADs (residential care facilities for dependent elderly people). They are provided by the Ministère des Solidarités et de la Santé and can be downloaded here

# The global Covid-19 data

Coronavirus COVID-19 Global Cases by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University (JHU)

This Shiny application has been developed by Marc Lavielle,
Inria Saclay & Ecole Polytechnique, Xpop team
September 19th, 2020