Predict the growth rate at horizon \(k\) using the past during the last \(l\) days of growth rate

The motivation for using a functional regression model to predict CoViD19 cases arises from the classic SIR epidemiological model, which in the part specifically dealing with infections proposes the equation: \[\frac{dI}{dt}=\beta S I-\gamma I\] where \(\beta\) is the infection rate, \(\gamma\) is the recovery rate, \(S\) is the population susceptible to infection and \(I\)is the number of infections. Rewriting the above equation we obtain \[GR=\frac{dI/dt}{I}=\beta S - \gamma\] which signals that the growth rate of infections (\(GR\)) is a function of \(S\). If we extend this idea and discretize the above equation, we can pose the following functional regression model: \[ GR_{t+h}=f(S_{t-l}^t,GR_{t-l}^t,I_{t-l}^t,\ldots)+\epsilon_{t+h} \] where the notation \(X_{t-l}^t\) refers to process \(X\) in the interval \([t-l,t]\) and \[GR_{t+h}=\frac{I_{t+h}-I_{t}}{I_{t}+c}\] (divided by \(I_{t}+c\) to not divide by zero) with \(h\) as the prediction horizon.

The function \(f\) is the way to link the scalar response with the functional covariates for which the literature on functional data provides several possibilities.

- World country dataset from: John Hopkins University Center for System Science and Engineering John Hopkins University dataset, which is updated daily in DATA1. The name of the latest time series (since 22/3):

- time_series_covid19_confirmed_global.csv for cumulative confirmed cases.
- time_series_covid19_deaths_global.csv for cumulative deaths.
- time_series_covid19_recovered_global.csv for cumulative recovered cases.

Spanish region dataset. Confirmed, hospitalised, Intensive care units (ICU), deaths and recovered cases by Autonomous Community of Spain available at Situation of COVID-19 in Spain from Instituto de Salud Carlos III. Data updated daily in DATA2. The structure of this file is not stable over time. The current variables are: CCAA, FECHA, CASOS, PCR+, TestAc+, Hospitalizados, UCI, Fallecidos, Recuperados. Please read the notes at the end of the CSV.

Italian region dataset. Confirmed, hospitalised, Intensive care units (ICU), deaths and recovered cases by regions of Italy available at COVID-19 Italia - Monitoraggio situazioneDipartimento della Protezione Civile from Presidenza del Consiglio dei Ministri - Dipartimento della Protezione Civile. Data updated daily in DATA3.

Catalonia region dataset. These data come from the RSAcovid19 record from the Health Department and show data from the accumulated positive cases, which are those that tested positive on some diagnostic test (PCR or fast test). It also includes data from the accumulated suspicious cases corresponding to people who presented symptoms at some point and a sanitary professional has classified them as a possible case, but they do not have a diagnostic test (PCR or fast test) with a positive result. The surveillance service activated all the cases and they identified the person’s residence zone indicated on each sanitary card. Information is updated in open data daily at Dades obertes de Catalunya.

Madrid region dataset Portal de Datos Abiertos de la Comunidad de Madrid and new app

The availability of quality updated data conditions the selection of the training sample as well as the resolution at which the prediction/estimation/forecast may be made.

- Other datasets

- Cumulative cases at day \(t\): \(x_t^{(j)}\) with \(j\in \{1,...,5\}\) being, respectively for, confirmed, deaths, hospitalized, ICU and recovered cases.
- New cases at day \(t\): \(x_t^{(j)} - x_{t-1}^{(j)}\)
- Growth Rate of cases - H\(_k\): \(r_{k}^{(j)}(t)=\frac{x_{t+k}^{(j)} - x_{t}^{(j)}}{x_{t}^{(j)} + 1}\) for \(t=...,t_0-1\) and \(k=1,\ldots,5\)
- Active cases at day \(t\): \(a_t = x_{t}^{(1)} - x_{t}^{(2)} - x_{t}^{(5)}\).

- Hospitalised and ICU cases are only available for regions of Spain,

Note: new active cases can be negative for some days, if on this day there were more new recoveries \(+\) deaths cases than there were new confirmed cases.

Related with the idea of “flattening the curve”, we consider the curve (\(r_{1}^{(j)}(t)\)) that captures how growth rate changes over time. Besides, we smooth this signal to avoid the effect of sudden changes in notification (such as the weekend effect).

Objective: Predict the growth rate at horizon \(k\) using the past during the last 15 days of growth rate H\(_1\):

\[R_{1}(0)=\{r_1^{(j)}(-14),\ldots,r_1^{(j)}(0)\}\]

- Some data from certain regions are banned by certain inconsistency on the records: “Diamond Princess”,“Iran”,“Japan”,“Bahrain” and “Qatar”
- For \(r_{t+k}^{(1)}\) response (confirmed cases), we uses the countries or regions with more than 200 confirmed cases at time \(t\).
- For \(r_{t+k}^{(2)}\) response (deaths cases), we uses the countries or regions with more than 30 deaths at time \(t\).

**Data Incidences only from Instituto de Salud Carlos III (ISCIII)**

The file obtained from Instituto de Salud Carlos III (ISCIII) has suffer changes along time in the units of the variables. Typically, the historical data is not reconstructed.

- Apr, 4th, 2020. Hospitalized - Extremadura. Adjustment (-36)
- Apr, 8th, 2020. ICU - C. Valenciana. Cumulative instead prevalence.
- Apr, 11th, 2020. Hospitalized - Castilla La Mancha. Cumulative instead prevalence.
- Apr, 12th, 2020. ICU - Castilla La Mancha. Cumulative instead prevalence.
- Apr, 16th, 2020. ICU - Castilla y León. Cumulative instead prevalence.
- Apr, 16th, 2020. ICU - Aragón. Adjustment (-51)
- Apr, 17-18th, 2020. Recovered - Galicia. Missing data.
- Apr, 23rd, 2020. ICU - Extremadura. Adjustment.
- Apr, 26th, 2020. ICU and Hospitalized - Madrid. Cumulative instead prevalence.
- Apr, 28th, 2020. ICU - Galicia - Cumulative instead prevalence. (+235)
- Apr, 28th, 2020. Recovered - Galicia. Increased the number of recovered people at home (+3552)
- Apr, 28th, 2020. Hospitalized - Galicia. Adjustment (-22)
- Apr, 29th, 2020. Confirmed - Galicia. Adjustment (-769)
- May, 21th, 2020. Due to the new surveillance and control strategy there is a change in the notification of the Spanish regions (CCAA). The predictions for Spain and its regions will be updated when this information becomes again available, source: ISCIII.

All these models are implemented in the **fda.usc** packages (Febrero-Bande and Oviedo de la Fuente 2012)

**fregre.lm**, Lineal Model (FLM)(Cardot, Ferraty, and Sarda 1999): The linear operator is used into functional space \(f(X_{a}^b,Y_{c}^d,\ldots)=\alpha+\int_{a}^b\beta_X(t)X(t)dt+\int_c^d\beta_Y(t)Y(t)dt+\ldots\)**fregre.gsam**, Spectral Additive Model (FSAM)(Műller and Yao 2008): Given one finite representation of the curves on a basis of Hilbert space \(X_{t-l+1}^t\approx\sum_{k=1}^{K_X} c_k^X\phi_k^X\) then \(f(X_{a}^b,Y_{c}^d,\ldots)=\alpha+\sum_{k=1}^{K_X}f_k^X(c_k^X)+\sum_{k=1}^{K_Y}f_k^Y(c_k^Y)+\ldots\) where the functions \(f_k^X(c_k^X)\) e \(f_k^Y(c_k^Y)\) of the scalar coefficients of the basis representation are smooth.**fregre.gkam**, Additive Kernel Model (FKAM)(Febrero-Bande and González-Manteiga 2013): \(f(X_{a}^b,Y_{c}^d,\ldots)=\alpha+f_X(X_{a}^b)+f_Y(Y_{c}^d)+\ldots\) where the function \(f_X\)(resp. \(f_Y\)) is estimated using a functional Kernel.Other models

**fregre.gls**, Functional GLS model (Oviedo de la Fuente et al. 2018)**fregre.basis.fr**, Functional Response Model (Chiou et al. 2004)

- Re-estimate Functional Models (Step 2) when new data is available (all countries and regions of Data1, Data2 and Data2).
- Reconstruct the expected number of accumulated cases and deduce the new cases to each horizon (confirmed , deaths and actives).
- Calculate the incidence rate by contry or region.

More information available in Informest

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This work has been supported by Project MTM2016-76969-P from Ministerio de Economía y Competitividad - Agencia Estatal de Investigación and European Regional Development Fund (ERDF) and IAP network StUDyS from Belgian Science Policy.

Cardot, Hervé, Frédéric Ferraty, and Pascal Sarda. 1999. “Functional Linear Model.” *Statistics & Probability Letters* 45 (1): 11–22.

Chiou, Jeng-Min, Hans-Georg Muller, Jane-Ling Wang, and others. 2004. “Functional Response Models.” *Statistica Sinica* 14 (3): 675–94.

Febrero-Bande, Manuel, and Wenceslao González-Manteiga. 2013. “Generalized Additive Models for Functional Data.” *Test* 22 (2): 278–92. http://dx.doi.org/10.1007/s11749-012-0308-0.

Febrero-Bande, Manuel, and M Oviedo de la Fuente. 2012. “Statistical Computing in Functional Data Analysis: The R Package fda.usc.” *J. Statist. Software* 51 (4): 1–28.

Műller, HG, and F Yao. 2008. “Functional Additive Model.” *J Am Stat Assoc* 103: 1534–44.

Oviedo de la Fuente, Manuel, Manuel Febrero-Bande, Marı́a Pilar Muñoz, and Àngela Domı́nguez. 2018. “Predicting Seasonal Influenza Transmission Using Functional Regression Models with Temporal Dependence.” *PloS One* 13 (4): e0194250.