This tool allows an interactive visualization of data provided by the John Hopkins University from different countries (numbers of confirmed cases and numbers of deaths) as well as what the proposed epidemiological model predicts.

### Select a country

Select a country either :

- in the scroll-down menu,
- in the complete list of countries (select “Others” in the menu and type the name of the country in the text box).

### Plot the data

Plot the numbers of confirmed cases and the numbers of deaths. You can select between the cumulated counts and the daily counts.

The “confirmed” mortality rate is defined here as the number of deaths divided by the number of “confirmed cases”.

### SIR model fitting

A SIR-type model is used to jointly fit the series of confirmed cases and the series of deaths counts.

The model includes a weekly periodic component, as the counting procedures seem to vary according to the day of the week. Thus, some variations, announced as increases, are in fact explained by this periodic effect. It can be seen that the model fits the collected data extremely well.

**The tool can be used not only to describe the evolution of the pandemic in these different countries but also to predict - at least in the relatively short term - what is likely to happen in the coming days.**

The model and the method used for fitting the model to the data are described here: http://webpopix.org/covidix19.html

### (short time) Mortality forecasting

The purpose of this function is to find the best time-lag between the 2 series in order to superimpose them as well as possible.

This Shiny application has been developed by Marc Lavielle,

Inria Saclay & Ecole Polytechnique, Xpop team

April 9th, 2020

### (short time) Mortality forecasting

The purpose of this function is to find the best time-lag between the 2 series in order to superimpose them as well as possible.

Then you can select to plot:

The original series of death counts and the shifted and rescaled series of confirmed cases,

A linear prediction of the number of deaths based on the (smoothed) series of confirmed cases with a prediction interval.

This graph only allows you to see how the death toll series is likely to change over the next few days.

This short-term prediction is empirical. It is not based on any epidemiological model.

### Initial values

### rate constants

### Transmission rate parameters

### Periodic component

A SIR-type model is used to jointly fit the series of confirmed cases and the series of deaths counts.

\[\dot{I}_c(t) = \beta(t) \, I_c(t) - \gamma \, I_c(t) -\delta \, I_c(t) \] \[\dot{W_c}(t) = \beta(t) \, I_c(t) \] \[\dot{L}(t) = \delta \, I_c(t) - \lambda \, L(t) \] \[\dot{D}(t) = \lambda \, L(t) \] where \(W_c(t)\) and \(D(t)\) are, respectively, the cumulated numbers of confirmed cases and the total number of deaths at time \(t\).

The *effective reproduction number* is defined as

\[ R_{\rm eff}(t) = \frac{\beta(t)}{\gamma + \delta} \] Here, we use for \(\beta\) (and for \(R_{\rm eff}\)) a piecewise linear continuous function followed by an exponential decrease.

The observation model includes a weekly periodic component. Indeed, it appears that the reported numbers fluctuate depending on the day of the week.

Note that it is not the *true numbers* that fluctuate, but the *reported numbers*.

Then the predicted counts can selected as either

the predictions of the

*true numbers*, by removing the periodic component,the predictions of the

*reported numbers*, by including the periodic component.

Note that the prediction interval is a prediction interval for the *reported numbers*.

The model and the method used for fitting the model to the data are described here: http://webpopix.org/covidix19.html