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@@ -144,10 +144,14 @@ all five iterations.\\
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\begin{table}[h]
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\begin{center}
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- \begin{tabular}{ccc|ccc}
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- true $\alpha$ & $\mu(\alpha)$ & $\sigma(\alpha)$ & true $\beta$ & $\mu(\beta)$ & $\sigma(\beta)$ \\
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- \hline
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- 0.3333 & 0.3334 & 0.0011 & 0.5000 & 0.5000 & 0.0017 \\
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+ \begin{tabular}{ccc ccc}
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+ \toprule
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+ \multicolumn{3}{c}{$\alpha$} & \multicolumn{3}{c}{$\beta$} \\
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+ \cmidrule{1-3}\cmidrule{4-6}
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+ true & $\mu$ & $\sigma$ & true & $\mu$ & $\sigma$ \\
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+ \midrule
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+ 0.3333 & 0.3334 & 0.0011 & 0.5000 & 0.5000 & 0.0017 \\
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+ \bottomrule
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\end{tabular}
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\caption{The mean $\mu$ and standard deviation $\sigma$ across the 5
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independent iterations of training our PINNs with the synthetic dataset.}
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@@ -161,7 +165,7 @@ While the predicted value is not precisely accurate, the standard deviation is
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sufficiently small, and taking the mean of multiple iterations produces an
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almost perfect result.\\
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-In~\Cref{table:alpha_beta} we present the results of the training for the
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+In~\Cref{table:state_mean_std} we present the results of the training for the
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real-world data. The results are presented from top to bottom, in the order of
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the community identification number, with the last entry being Germany. Both
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the mean $\mu$ and the standard deviation $\sigma$ are calculated across all
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@@ -171,30 +175,34 @@ has the lowest $\sigma$.\\
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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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- Lower Saxony & 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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- North Rhine-Westphalia & 0.0780 & 0.0009 & 0.1001 & 0.0011 \\
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- Hesse & 0.0653 & 0.0016 & 0.0854 & 0.0020 \\
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- Rhineland-Palatinate & 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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- Bavaria & 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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- Saxony & 0.0797 & 0.0017 & 0.1073 & 0.0022 \\
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- Saxony-Anhalt & 0.0932 & 0.0019 & 0.1207 & 0.0027 \\
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- Thuringia & 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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+ \begin{tabular}{lccccc}
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+ \toprule
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+ & \multicolumn{2}{c}{$\alpha$} & \multicolumn{2}{c}{$\beta$} & \\
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+ \cmidrule{2-3}\cmidrule{4-5}
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+ state name & $\mu$ & $\sigma$ & $\mu$ & $\sigma$ & $e_{\text{synth}}$ \\
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+ \midrule
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+ Schleswig Holstein & 0.0771 & 0.0010 & 0.0966 & 0.0013 & 0.0849 \\
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+ Hamburg & 0.0847 & 0.0035 & 0.1077 & 0.0037 & 0.0948 \\
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+ Lower Saxony & 0.0735 & 0.0014 & 0.0962 & 0.0018 & 0.0774 \\
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+ Bremen & 0.0588 & 0.0018 & 0.0795 & 0.0025 & 0.0933 \\
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+ North Rhine-Westphalia & 0.0780 & 0.0009 & 0.1001 & 0.0011 & 0.0777 \\
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+ Hesse & 0.0653 & 0.0016 & 0.0854 & 0.0020 & 0.1017 \\
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+ Rhineland-Palatinate & 0.0808 & 0.0016 & 0.1036 & 0.0018 & 0.0895 \\
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+ Baden-Württemberg & 0.0862 & 0.0014 & 0.1132 & 0.0016 & 0.0796 \\\addlinespace
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+ Bavaria & 0.0809 & 0.0021 & 0.1106 & 0.0027 & 0.0952 \\
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+ Saarland & 0.0746 & 0.0021 & 0.0996 & 0.0024 & 0.1080 \\
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+ Berlin & 0.0901 & 0.0008 & 0.1125 & 0.0008 & 0.0667 \\
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+ Brandenburg & 0.0861 & 0.0008 & 0.1091 & 0.0010 & 0.0724 \\
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+ Mecklenburg-Vorpommern & 0.0910 & 0.0007 & 0.1167 & 0.0008 & 0.0540 \\
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+ Saxony & 0.0797 & 0.0017 & 0.1073 & 0.0022 & 0.1109 \\
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+ Saxony-Anhalt & 0.0932 & 0.0019 & 0.1207 & 0.0027 & 0.0785 \\
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+ Thuringia & 0.0952 & 0.0011 & 0.1248 & 0.0016 & 0.0837 \\\addlinespace
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+ Germany & 0.0803 & 0.0012 & 0.1044 & 0.0014 & 0.0804 \\
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+ \bottomrule
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\end{tabular}
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\caption{Mean and standard deviation 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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+ \label{table:state_mean_std}
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\end{center}
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\end{table}
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@@ -259,12 +267,12 @@ real-world data.\\
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For the purposes of validation, we create a synthetic dataset, by setting the parameter
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of $\alpha$ and the reproduction value each to a specific values, and solving~\Cref{eq:reduced_sir_ODE}
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for a given time interval. We set $\alpha=\nicefrac{1}{3}$ and $\Rt$ to the
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-values as can be seen in~\Cref{fig:synthetic_I_r_t} as well as the population
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+values as can be seen in~\Cref{fig:Rt_dataset} as well as the population
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size $N=\expnumber{7.6}{6}$ and the initial amount of infected people to
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$I_0=10$. Furthermore, we set our simulated time span to 150 days. We use this
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dataset to demonstrate, that our method is working on a simple and minimal
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dataset.\\ To obtain a dataset of the infectious group, consisting of the
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-real-world data, we we processed the data of the dataset
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+real-world data, we processed the data of the dataset
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\emph{COVID-19-Todesfälle in Deutschland} to extract the number of infections
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in Germany as a whole. For the German states, we use the data of \emph{SARS-CoV-2
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Infektionen in Deutschland}. In the preprocessing stage, we employ a constant
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@@ -298,6 +306,7 @@ we restrict the data points to an interval of 1200 days, beginning on March 09.
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infectious group (left) and the corresponding true reproduction value
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$\Rt$ (right) for the synthetic data. The lower graphic exemplary
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illustrates the different curves for Germany.}
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+ \label{fig:Rt_dataset}
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\end{figure}
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In order to achieve the desired output, the selected neural network
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@@ -319,13 +328,13 @@ reduced SIR model and the reproduction number $\Rt$. First, we present
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our findings for the synthetic dataset. Then, we provide and discuss the
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results for the real-world data.\\
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-\Cref{fig:synth_results} illustrates that the model successfully learns the
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-synthetic training data, with an error of $e_{\text{synth}} = 0.0016$. Meanwhile,
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-the prediction for the reproduction number $\Rt$ for the synthetic data, is accuracy,
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-while having by far the highest standard deviation in the first 30 days. The
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-error concerning the reproduction number is $e_{\Rt} = 0.0521$.
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+\Cref{fig:synth_results} illustrates the results of our experiments conducted on
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+the synthetic dataset, which can be seen in~\Cref{fig:Rt_dataset}. It is evident
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+that the model is capable of learning the infection data across all data points.
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+The error for this is, $e_{\text{synth}} = 0.0016$, which is of a negligible
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+magnitude.\\
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-\begin{figure}[t]
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+\begin{figure}[h]
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\centering
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{synthetic_I_prediction.pdf}
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@@ -341,34 +350,120 @@ error concerning the reproduction number is $e_{\Rt} = 0.0521$.
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standard deviation.}
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\end{figure}
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+An examination of the predictions for the representation value $\Rt$ reveals
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+that here as well, the model is capable of accurately delineating the value at
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+each time point. However, during the first 30 days, the standard deviation is
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+exhibits an upward trend, while during the final 120 days, the predictions
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+demonstrate remarkable precision. The overall prediction of $\Rt$ has an error
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+of $e_{\Rt} = 0.0521$.\\
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+
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+In~\Cref{fig:state_results}, we present the graphs of $\Rt$ for the state with
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+the highest value of $\beta$, namely Thuringia, and for the state with the lowest
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+transmission rate $\beta$, namely Bremen. Further visualizations of the results
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+can be found in~\Cref{chap:appendix}. In all datasets, the graphs with $\alpha =
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+ \nicefrac{1}{5}$ are of a smaller size than those with
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+$\alpha = \nicefrac{1}{14}$. This is due to the fact that the individuals are
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+being moved to the removed compartment at a faster rate. Resulting, it can be
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+observed that the value of $\Rt$ is constantly remaining closer to the threshold
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+of $\Rt=1$, while the reproduction number for datasets with $\alpha = \nicefrac{1}{14}$
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+reaches values of up to 1.6. In states with higher values of $\beta$, the period
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+during which the value of $\Rt$ is above the threshold of one 1 is longer, but
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+the peak is lower. In states with a lower transmission rate, the period above 1
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+is shorter, but the peak value is higher.\\
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+
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\begin{figure}[t]
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\centering
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\begin{subfigure}{0.45\textwidth}
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- \includegraphics[width=\textwidth]{Germany_R_t_statistics.pdf}
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+ \includegraphics[width=\textwidth]{I_prediction/Thueringen_I_prediction.pdf}
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\end{subfigure}
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\quad
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\begin{subfigure}{0.45\textwidth}
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- \includegraphics[width=\textwidth]{Germany_I_prediction.pdf}
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- \end{subfigure}
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- \vskip\baselineskip
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- \begin{subfigure}{0.45\textwidth}
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- \includegraphics[width=\textwidth]{R_t/Sachsen_Anhalt_R_t_statistics.pdf}
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+ \includegraphics[width=\textwidth]{I_prediction/Bremen_I_prediction.pdf}
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\end{subfigure}
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- \quad
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\begin{subfigure}{0.45\textwidth}
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\includegraphics[width=\textwidth]{R_t/Thueringen_R_t_statistics.pdf}
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\end{subfigure}
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- \vskip\baselineskip
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- \begin{subfigure}{0.45\textwidth}
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- \includegraphics[width=\textwidth]{R_t/Bremen_R_t_statistics.pdf}
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- \end{subfigure}
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\quad
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\begin{subfigure}{0.45\textwidth}
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- \includegraphics[width=\textwidth]{R_t/Hessen_R_t_statistics.pdf}
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+ \includegraphics[width=\textwidth]{R_t/Bremen_R_t_statistics.pdf}
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\end{subfigure}
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\label{fig:state_results}
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- \caption{text}
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+ \caption{Visualization of the prediction of the training and the graphs of
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+ $\Rt$ for Thuringia (left) and Bremen (right) with both
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+ $\alpha = \nicefrac{1}{14}$ and $\alpha = \nicefrac{1}{5}$. Events like
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+ the peak of an influential variant are marked horizontally.}
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\end{figure}
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+\Cref{table:state_error} presents data regarding the discrepancy between the
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+predicted and actual values from the dataset for compartment $I$. It is evident,
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+that the error for all experiments falls within a range of values that is not
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+negligible and will have an influence on the resulting reproduction values that
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+are learned while fitting the data. A comparison of the results for the various
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+values of $\alpha$ reveals that the errors associated with $\alpha = \nicefrac{1}{14}$
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+are consistently smaller, with the exception of Saxony and Germany. This can be
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+attributed to the differing sizes of infection counts, particularly in relation
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+to the normalization factor $C$. The model is unable to learn effectively if the
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+values of the data loss $\mathcal{L}_{\text{data}}$ are too large or too small
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+at the beginning.\\
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+
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+As illustrated in~\Cref{fig:state_results}, the training data is overlaid with the
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+corresponding prediction of the model. We can observe that the prediction, though
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+an exact reconstruction, accurately captures the general trajectory of the
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+pandemic. The model's prediction demonstrates an ability to capture larger
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+peaks, exhibiting a tendency to ignore smaller changes. This suggests that the
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+prediction of the model is capable show the rough outline of the progression of COVID-19. In the
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+beginning, the majority of predictions below $\Rt=1$, indicating an outbreak.
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+As we observed in the synthetic data, the model exhibits a higher standard
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+deviation at the boundaries. In the graphs, we mark the
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+peaks of the most severe COVID-19 variants in Germany. While the peaks of the
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+Alpha and Delta variants are clearly visible in the data, the model does not
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+learn these, and thus they are not reflected in the results. The peak of the
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+Omicron variant represents the culmination of the COVID-19 pandemic in Germany
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+and can be identified as the most prominent peak in the dataset. Immediately preceding this peak, we observe the highest
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+value of the reproduction number across all states. This phenomenon can be explained, by
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+number of individuals infected by one infectious person reaching its peak. In
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+some states the peaks of other Omicron variants after the maximum peak are visible (see Thuringia).\\
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+
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+The experiments demonstrate, that our model encounteres difficulties in learning the data for the
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+states and Germany and consequently in predicting the reproduction values for each dataset.
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+Nonetheless, the predictions illustrate the general trends of the most impactful
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+events of the COVID-19 pandemic.\\
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+
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+\begin{table}[t]
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+ \begin{center}
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+ \begin{tabular}{lcc}
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+ \toprule
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+ & \multicolumn{2}{c}{$e_I$} \\
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+ \cmidrule{2-3}
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+ state name & $\alpha=\nicefrac{1}{14}$ & $\alpha=\nicefrac{1}{5}$ \\
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+ \midrule
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+ Schleswig Holstein & 0.2005 & 0.2514 \\
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+ Hamburg & 0.3045 & 0.3357 \\
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+ Lower Saxony & 0.2140 & 0.3082 \\
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+ Bremen & 0.2370 & 0.3838 \\
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+ North Rhine-Westphalia & 0.1718 & 0.2460 \\
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+ Hesse & 0.2736 & 0.3172 \\
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+ Rhineland-Palatinate & 0.2442 & 0.2674 \\
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+ Baden-Württemberg & 0.1984 & 0.2958 \\\addlinespace
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+ Bavaria & 0.1928 & 0.2825 \\
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+ Saarland & 0.2554 & 0.4676 \\
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+ Berlin & 0.1885 & 0.2948 \\
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+ Brandenburg & 0.2023 & 0.2571 \\
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+ Mecklenburg-Vorpommern & 0.1518 & 0.3272 \\
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+ Saxony & 0.3382 & 0.2807 \\
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+ Saxony-Anhalt & 0.1959 & 0.2564 \\
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+ Thuringia & 0.1401 & 0.2221 \\\addlinespace
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+ Germany & 0.3371 & 0.2533 \\
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+ \bottomrule
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+ \end{tabular}
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+ \caption{This table displays all average values of the error $e_{\text{synth}}$
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+ for all German states and Germany. The average is formed across all
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+ 10 iteration.}
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+ \label{table:state_error}
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+ \end{center}
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+\end{table}
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+
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+
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+
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% -------------------------------------------------------------------
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Phillip Rothenbeck