Introduction

This repositiory provides code and some explanation for the ILM-prop model that participates in the German COVID-19 NowcastHub.

Method

To predict the number of hospitalisations we consider the reporting process of both reported COVID-19 cases and reported hospitalisations. Recall that the reporting date of a COVID-19 case is shared for both the case and its hospitalisation, i.e. the case and hospitalisation are linked through this date.

As hospitalisations are only available as $7$-day rolling sums we use $7$-day rolling sums for daily reported incidences as well. To avoid dealing with the double weekday effect of both reporting date of the case and reporting date of the hospitalisation we divide the future hospitalisations we wish to predict into chunks of one week, which gets rid of the weekday effect for the hospitalisations. This is depicted in the figure below. Our prediction of each of these weekly chunks then consists of the fraction of hospitalisations of reported cases in the past.

Decomposition of the daily reported hospitalisation incidences into the known incidences $\color{#FF7900}H_{t,d}$, i.e. the reporting triangle, and the future weekly increments $\color{#00747A} H_{t, d + 7 (k + 1)} - H_{t, d + 7 k}$. The last increment might not be a weekly one, but we expect few cases to occur for such long delays.

More formally, denote by $h_{t,d}$ the number of hospitalisations with reporting date $t$ that are known $d$ days later. Unfortunately we only observe $$H_{t,d} = \sum_{s = t - 6}^{t} h_{s,d + (t - s)},$$ i.e. a weekly sum of reported hospitalisations. On day $T$ our goal is to predict $H_{t,D}$ for large delays $D$ and days $t \leq T$, for which it clearly suffices to predict $H_{t, D} - H_{t, T - t}$ and add the known $H_{t, T - t}$ to this prediction. We rewrite this into a weekly telescoping sum

$$ H_{t,D} - H_{t,d} = \left(H_{t, d + 7} - H_{t,d}\right) + \left(H_{t, d + 14} - H_{t, d + 7}\right) + \dots + \left(H_{t,D} - H_{t, d + 7 K}\right), $$

where $K = \lfloor (D -d) / 7 \rfloor$, reducing the task at hand to predict hospitalisations in the $k$-th week ahead, $H_{t, d + 7k} - H_{t, d + 7\cdot(k - 1)}$ for $k = 1, \dots, K$.

To leverage known reported incidences, rewrite this as

$$ \frac{H_{t, d + 7k} - H_{t, d + 7\cdot(k - 1)}}{I_{t,d}} I_{t, d} = p_{t,d,k} I_{t,d} $$

where $I_{t,d}$ is the $7$-day case incidence with reporting date $t$ known at time $t + d$, i.e. the incidenct case analouge of $H_{t,d}$.

Assuming that the proportions $p_{t,d,k}$ change slowly over time $t$ we estimate them by

$$ \widehat {p_{t,d,k}} = \frac{H_{t - 7k, d + 7k} - H_{t - 7k, d + 7\cdot(k - 1)}}{I_{t - 7k,d}} = p_{t - 7k,d,k} $$

and finally predict

$$ \widehat{H_{t,D}} = H_{t,d} + \left(\widehat{p_{t,d,1}} + \dots + \widehat{p_{t,d,K}}\right) I_{t,d}. $$

In essence, this model is a regression of reported hospitalisations on reported cases.

As hospitalisation is affected by age, we perform this procedure for all available age groups separately and finally aggregate over all age groups to obtain a nowcast for all age groups combined.

This describes our point nowcast for $7$-day hospitalisations. To obtain uncertainty intervals we fit a normal (age groups 00-04 and 05-14) or lognormal (all other age groups) distribution to the past performance of our model. We chose these distributions based on explorative analysis and believe that these should be seen as heuristics rather than as a matter of fact, which is in line with the philosophy of our model to be as simple as possible.

Denote by $\hat H_{t,D,s}$ the nowcast made for date $t$ on date $s \geq t$. Starting with date $t + D$ the definite $H_{t,D}$ is known and we can estimate the absolute prediction error $\varepsilon_{t,s} = H_{t,D} - \hat H_{t,D,s}$ and the relative prediction error $\eta_{t,s} = \log \left( H_{t,D} - H_{t, s - t}\right) - \log \left( \hat H_{t,D,s} - H_{t, s- t} \right)$. For the nowcast for date $t$ made on date $s$ we estimate the standard deviation $\hat\sigma$ of $\varepsilon_{t - D - i, s - D - i}$ or $\eta_{t - D - i, s - D - i}$ (age groups 00-04, 05-14 and others respectively), $i = 0, \dots, 27$ by its empirical counterpart. The estimated predictive distribution which informs our prediction intervals is then $\mathcal N (\hat H_{t,D,s}, \hat\sigma^2)$ (age groups 00-04 and 05-14) or $\mathcal{LN} \left( \log \left(\hat H_{t,D,s} - H_{t, s - t}\right), \hat\sigma^2 \right) + H_{t, s - t}$ (all other age groups).

Reproducing Results

Clone this repository and run

make dependencies # install necessary R packages
make data # prepare data from RKI case and hospitalisations
make submissions # re-create submissions

in a terminal. You'll need to have R installed to run this model.

Building all submissions may take a while, even on a decently powered machine.

The result will be stored in the data/processed directory in the file submissions-ILM-prop.csv which consists of all submissions stacked on top of one another. See the NowcastHub wiki for more information about the data format.

Note that there might be small deviations from the submitted nowcasts because every retrospective change of the data affects (due to the long horizon of 12 weeks) potentially a lot of submissions and we only provide the data at release of this repository rather than a full history of the data.