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@@ -44,7 +44,7 @@ The parameters are set to $\alpha = \nicefrac{1}{3}$ and $\beta = \nicefrac{1}{2
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The size of the population is $N = \expnumber{7.6}{6}$ and the initial amount of
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infectious individuals of is $I_0 = 10$. We simulate over 150 days and get a
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dataset of the form of~\Cref{fig:synthetic_SIR}.\\For the real-world RKI data we
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-preprocess the row data data of each state and Germany separately using a
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+preprocess the raw data of each state and Germany separately using a
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recovery queue with a recovery period of 14 days. As for the population size of
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each state we set it to the respective value counted at the end of 2019\footnote{\url{https://de.statista.com/statistik/kategorien/kategorie/8/themen/63/branche/demographie/\#overview}}.
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The initial number of infectious individuals is set to the number of infected
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@@ -117,30 +117,34 @@ setups that we describe in this section.
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\subsection{Results 4}
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\label{sec:sir:results}
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-\begin{center}
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- \begin{tabular}{c|cc|cc}
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- & $\alpha$ & $\sigma(\alpha)$ & $\beta$ & $\sigma(\beta)$ \\
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- \hline
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- Schleswig Holstein & 0.0771 & 0.0010 & 0.0966 & 0.0013 \\
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- Hamburg & 0.0847 & 0.0035 & 0.1077 & 0.0037 \\
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- Niedersachsen & 0.0735 & 0.0014 & 0.0962 & 0.0018 \\
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- Bremen & 0.0588 & 0.0018 & 0.0795 & 0.0025 \\
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- Nordrhein-Westfalen & 0.0780 & 0.0009 & 0.1001 & 0.0011 \\
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- Hessen & 0.0653 & 0.0016 & 0.0854 & 0.0020 \\
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- Rheinland-Pfalz & 0.0808 & 0.0016 & 0.1036 & 0.0018 \\
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- Baden-Württemberg & 0.0862 & 0.0014 & 0.1132 & 0.0016 \\
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- Bayern & 0.0809 & 0.0021 & 0.1106 & 0.0027 \\
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- Saarland & 0.0746 & 0.0021 & 0.0996 & 0.0024 \\
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- Berlin & 0.0901 & 0.0008 & 0.1125 & 0.0008 \\
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- Brandenburg & 0.0861 & 0.0008 & 0.1091 & 0.0010 \\
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- Mecklenburg Vorpommern & 0.0910 & 0.0007 & 0.1167 & 0.0008 \\
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- Sachsen & 0.0797 & 0.0017 & 0.1073 & 0.0022 \\
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- Sachsen-Anhalt & 0.0932 & 0.0019 & 0.1207 & 0.0027 \\
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- Thüringen & 0.0952 & 0.0011 & 0.1248 & 0.0016 \\
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- Germany & 0.0803 & 0.0012 & 0.1044 & 0.0014 \\
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- \end{tabular}
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-\end{center}
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-
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+\begin{table}[h]
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+ \begin{center}
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+ \begin{tabular}{c|cc|cc}
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+ & $\mu(\alpha)$ & $\sigma(\alpha)$ & $\mu(\beta)$ & $\sigma(\beta)$ \\
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+ \hline
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+ Schleswig Holstein & 0.0771 & 0.0010 & 0.0966 & 0.0013 \\
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+ Hamburg & 0.0847 & 0.0035 & 0.1077 & 0.0037 \\
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+ Niedersachsen & 0.0735 & 0.0014 & 0.0962 & 0.0018 \\
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+ Bremen & 0.0588 & 0.0018 & 0.0795 & 0.0025 \\
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+ Nordrhein-Westfalen & 0.0780 & 0.0009 & 0.1001 & 0.0011 \\
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+ Hessen & 0.0653 & 0.0016 & 0.0854 & 0.0020 \\
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+ Rheinland-Pfalz & 0.0808 & 0.0016 & 0.1036 & 0.0018 \\
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+ Baden-Württemberg & 0.0862 & 0.0014 & 0.1132 & 0.0016 \\
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+ Bayern & 0.0809 & 0.0021 & 0.1106 & 0.0027 \\
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+ Saarland & 0.0746 & 0.0021 & 0.0996 & 0.0024 \\
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+ Berlin & 0.0901 & 0.0008 & 0.1125 & 0.0008 \\
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+ Brandenburg & 0.0861 & 0.0008 & 0.1091 & 0.0010 \\
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+ Mecklenburg Vorpommern & 0.0910 & 0.0007 & 0.1167 & 0.0008 \\
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+ Sachsen & 0.0797 & 0.0017 & 0.1073 & 0.0022 \\
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+ Sachsen-Anhalt & 0.0932 & 0.0019 & 0.1207 & 0.0027 \\
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+ Thüringen & 0.0952 & 0.0011 & 0.1248 & 0.0016 \\
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+ Germany & 0.0803 & 0.0012 & 0.1044 & 0.0014 \\
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+ \end{tabular}
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+ \caption{Mean and standard variation across the 5 iterations, that we
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+ conducted for each German state and Germany as the whole country.}
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+ \label{table:alpha_beta}
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+ \end{center}
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+\end{table}
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\begin{figure}[h]
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\centering
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\includegraphics[width=\textwidth]{mean_std_alpha_beta_res.pdf}
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@@ -164,7 +168,84 @@ are described in~\Cref{sec:pinn:rsir}.
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\label{sec:rsir:setup}
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In this section we describe the choice of parameters and configuration for data
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generation, preprocessing and the neural networks. We use these setups to train
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-the PINNs to find the reproduction number on both synthetic and real-world data.
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+the PINNs to find the reproduction number on both synthetic and real-world data.\\
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+
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+For validation reasons we create a synthetic dataset, by setting the parameters
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+of $\alpha$ and $\beta$ each to a specific value, and solving~\Cref{eq:modSIR}
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+for a given time interval. We set $\alpha=\nicefrac{1}{3}$ and
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+$\beta=\nicefrac{1}{2}$ as well as the population size $N=\expnumber{7.6}{6}$
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+and the initial amount of infected people to $I_0=10$. Furthermore, we set our
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+simulated time span to 150 days.We will use this dataset to show, that our
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+method is working on a simple and minimal dataset.\\ For the real-world data we
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+we processed the data of the dataset \emph{COVID-19-Todesfälle in Deutschland}
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+to extract the number of infections in the whole of Germany, while we used the
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+data of \emph{SARS-CoV-2 Infektionen in Deutschland} for the German states. For
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+the preprocessing we use a constant rate for $\alpha$ to move individual into
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+the removed compartment. First we choose $\alpha = \nicefrac{1}{14}$ as this is
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+covers the time of recovery\footnote{\url{https://github.com/robert-koch-institut/SARS-CoV-2-Infektionen_in_Deutschland.git}}.
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+Second we use $\alpha=\nicefrac{1}{5}$ since the peak of infectiousness is
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+reached right in front or at 5 days into the infection\footnote{\url{https://www.infektionsschutz.de/coronavirus/fragen-und-antworten/ansteckung-uebertragung-und-krankheitsverlauf/}}.
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+Just as in~\Cref{sec:sir} we set the population size $N$ of each state and
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+Germany to the corresponding size at the end of 2019. Also, for the same reason
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+we restrict the data points to an interval of 1200 days starting from March 09.
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+2020.
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+\begin{figure}[h]
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+ %\centering
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+ \setlength{\unitlength}{1cm} % Set the unit length for coordinates
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+ \begin{picture}(12, 9.5) % Specify the size of the picture environment (width, height)
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+ \put(1.5, 4.5){
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+ \begin{subfigure}{0.3\textwidth}
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+ \centering
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+ \includegraphics[width=\textwidth]{SIR_synth.pdf}
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+ \label{fig:synthetic_I}
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+ \end{subfigure}
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+ }
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+ \put(8, 4.5){
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+ \begin{subfigure}{0.3\textwidth}
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+ \centering
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+ \includegraphics[width=\textwidth]{datasets_states/Germany_SIR_14.pdf}
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+ \label{fig:germany_I}
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+ \end{subfigure}
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+ }
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+ \put(0, 0){
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+ \begin{subfigure}{0.3\textwidth}
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+ \centering
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+ \includegraphics[width=\textwidth]{datasets_states/Schleswig_Holstein_SIR_14.pdf}
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+ \label{fig:schleswig_holstein_I}
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+ \end{subfigure}
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+ }
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+ \put(4.75, 0){
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+ \begin{subfigure}{0.3\textwidth}
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+ \centering
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+ \includegraphics[width=\textwidth]{datasets_states/Berlin_SIR_14.pdf}
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+ \label{fig:berlin_I}
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+ \end{subfigure}
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+ }
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+ \put(9.5, 0){
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+ \begin{subfigure}{0.3\textwidth}
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+ \centering
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+ \includegraphics[width=\textwidth]{datasets_states/Thueringen_SIR_14.pdf}
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+ \label{fig:thüringen_I}
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+ \end{subfigure}
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+ }
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+
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+ \end{picture}
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+ \caption{Synthetic and real-world training data. The synthetic data is
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+ generated with $\alpha=\nicefrac{1}{3}$ and $\beta=\nicefrac{1}{2}$
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+ and~\Cref{eq:modSIR}. The Germany data is taken from the death case
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+ data set. Exemplatory we show illustrations of the datasets of Schleswig
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+ Holstein, Berlin, and Thuringia. For the other states see~\Cref{chap:appendix} }
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+ \label{fig:datasets}
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+\end{figure}
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+
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+For this task the chosen architecture of the neural network consists of 4 hidden
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+layers with each 100 neurons. The activation function is the tangens
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+hyperbolicus function tanh. We weight the data loss with a weight of
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+$\expnumber{1}{6}$ into the total loss. The model is trained using a base
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+learning rate of $\expnumber{1}{-3}$ with the same scheduler and optimizer as
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+we use in~\Cref{sec:sir:setup}. We train the model for 20000 epochs. Also, we
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+conduct each experiment 15 times to reduce the standard deviation.
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+
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% -------------------------------------------------------------------
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