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% summary of the content in this chapter
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% Version: 01.01.2012
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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-\chapter{Experiments 10}
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+\chapter{Experiments}
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\label{chap:evaluation}
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-In the preceding chapters, we explained the methods (see~\Cref{chap:methods})
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-based the theoretical background, that we established in~\Cref{chap:background}.
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-In this chapter present the setups and results from the experiments and
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-simulations, we ran. First, we discuss the experiments dedicated to identify
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-the epidemiological parameters of $\beta$ and $\alpha$ in synthetic and
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-real-world data. Second, we examine the reproduction number in synthetic and
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-real-world data of Germany. Each section, is divided into a description of the
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-experimental setup and the results.
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+In ~\Cref{chap:methods}, we explain the methods based the theoretical
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+background, that we established in~\Cref{chap:background}. In this chapter, we
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+present the setups and results from the experiments and simulations. First, we
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+discuss the experiments dedicated to identify the epidemiological transition
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+rates of $\beta$ and $\alpha$ in synthetic and real-world data. Second, we
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+examine the reproduction number in synthetic and real-world data of Germany.
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% -------------------------------------------------------------------
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-\section{Identifying the Transition Rates on Real-World and Synthetic Data 5}
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+\section{Identifying the Transition Rates}
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\label{sec:sir}
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In this section, we aim to identify the transmission rate $\beta$ and the
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recovery rate $\alpha$ from either synthetic or preprocessed real-world data.
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The methodology that we employ to identify the transition rates is described
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in~\Cref{sec:pinn:sir}. Meanwhile, the methods we utilize to preprocess the
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-real-world data are detailed in~\Cref{sec:preprocessing:rq}.
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+real-world data are detailed in~\Cref{sec:preprocessing:rq}. In the first part
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+we present the setup of our experiments, then we provide the results including a
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+discussion.\\
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% -------------------------------------------------------------------
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-\subsection{Setup 1}
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+\subsection{Setup}
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\label{sec:sir:setup}
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-In this subsection, we present the configurations for the training of our
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-PINNs, which are designed to identify the transition parameters. This
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-encompasses the specific parameters for the preprocessing and the configuration
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-of the PINN themselves.\\
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-
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-In order to validate our method, we first generate a dataset of synthetic data.
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+\paragraph{Synthetic Data:}In order to validate our method, we first generate a dataset of synthetic data.
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We achieve this by solving~\Cref{eq:modSIR} for a given set of parameters.
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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 conduct the simulation over 150
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-days, resulting in a dataset of the form of~\Cref{fig:synthetic_SIR}.\\ In
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-order to process the real-world RKI data, it is necessary to preprocess the raw
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-data for each state and Germany separately. This is achieved by utilizing a
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-recovery queue with a recovery period of 14 days. With regard to population
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-size of each state, we set it to the respective value counted at the end of
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-2019\footnote{\url{https://de.statista.com/statistik/kategorien/kategorie/8/themen/63/branche/demographie/\#overview}}.
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+infectious individuals is $I_0 = 10$. We conduct the simulation over 150
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+days, resulting in a dataset of the form of~\Cref{fig:synthetic_SIR}.\\
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+
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+\paragraph{Real-World Data:}In order to process the real-world RKI data, it is
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+necessary to preprocess the raw data for each state and Germany separately.
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+This is achieved by utilizing a recovery queue with a recovery period of 14
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+days. With regard to population size of each state, we set it to the respective
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+value counted at the end of
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+2019\footnote{{\tiny \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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people on March 09. 2020 from the dataset. The data we extract spans from
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March 09. 2020 to June 22. 2023, encompassing a period of 1200 days and
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@@ -104,129 +101,128 @@ active and severe.
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\label{fig:datasets_sir}
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\end{figure}
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-The PINN that we utilize comprises of seven hidden layers with twenty neurons
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-each, and an activation function of ReLU. We employ the Adam optimizer and the
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-polynomial scheduler of the PyTorch library, for training, with a base learning rate
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-of $\expnumber{1}{-3}$. We train the model for 10000 epochs to extract the
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-parameters. For each set of parameters, we conduct five iterations to
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-demonstrate stability of the values. The configuration is similar to the
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-configuration, that Shaier \etal ~\cite{Shaier2021} use for their work aside
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-from the learning rate and the scheduler choice.\\
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-
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-The following section presents the results of the simulations conducted with the
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-setups that we describe in this section.
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+\paragraph{Training Parameters:}The PINN that we utilize comprises of seven
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+hidden layers with twenty neurons each, and an activation function of ReLU. We
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+follow the hyperparameter setting in~\cite{Shaier2021} but change the base
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+learning rate to $\expnumber{1}{-3}$. And employ a polynomial scheduler
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+implementation from the PyTorch library~\cite{Paszke2019} instead. We train the
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+model for 10000 epochs to extract the parameters. For each set of parameters, we
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+conduct five iterations to demonstrate stability of the values. For measuring the
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+accuracy, we calculate the error $e$, using the 2-Norm. Let $G$ be the set of
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+compartment training data the SIR model with $\boldsymbol{g}\in G$ and $\hat{\boldsymbol{g}}$ be the
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+corresponding model prediction, then,
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+\begin{equation}
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+ e_{G} = \frac{1}{|G|}\sum_{g\in G}^{}\frac{\Big\|\hat{\boldsymbol{g}} - \boldsymbol{g}\Big\|_2}{\Big\|\boldsymbol{g}\Big\|_2},
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+\end{equation}
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+is the average error across all three groups.
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% -------------------------------------------------------------------
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-\subsection{Results 4}
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+\subsection{Results}
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\label{sec:sir:results}
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-\begin{figure}[t]
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- \centering
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- \includegraphics[width=0.7\textwidth]{reproducability.pdf}
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- \caption{Visualization of all 5 predictions for the synthetic dataset,
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- compared to the true values of $\alpha = \nicefrac{1}{3}$ and $\beta = \nicefrac{1}{2}$}
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- \label{fig:reprod}
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-\end{figure}
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-
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-In this section, we present the results, that we obtain from the conducted
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-experiments, that we describe in the preceding section. We begin by examining
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-the results for the synthetic dataset, focusing the accuracy and
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-reproducibility. We then proceed to present and discuss the results for the
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-German states and Germany.\\
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+In this section, we start by examining the results for the synthetic dataset,
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+focusing the accuracy and reproducibility. We then proceed to present and
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+discuss the results for the German states and Germany.\\
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The results of the experiment regarding the synthetic data can be seen
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-in~\Cref{table:alpha_beta_synth} and in~\Cref{fig:reprod}.~\Cref{fig:reprod}
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-depicts the values of $\beta$ and $\alpha$ for each iteration in comparison to the true
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-values of $\beta=\nicefrac{1}{2}$ and $\alpha=\nicefrac{1}{3}$. In~\Cref{table:alpha_beta_synth}
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-we present the mean $\mu$ and standard deviation $\sigma$ of both values across
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-all five iterations.\\
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-
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+in~\Cref{table:alpha_beta_synth}. The error and the standard variation for both
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+parameters are negligible small. Taking the mean of the parameters across the
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+five iterations yields more accurate results.\\
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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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+ \caption{Simulation results for the synthetic data. The true values and
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+ the respective mean parameter is given.}
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+ \label{table:alpha_beta_synth}
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+ \begin{tabular}{ccccccccc}
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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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+ \multicolumn{2}{c}{$\alpha$} & \phantom{0} & \multicolumn{2}{c}{$\beta$} \\
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+ \cmidrule{1-2}\cmidrule{4-5}
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+ true & $\mu$ & \phantom{0} & true & $\mu$ & \phantom{0} & $e_{SIR}$ \\
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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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+ 0.333 & 0.333{\tiny$\pm 0.001$} & \phantom{0} & 0.500 & 0.500{\tiny$\pm 0.002$} & \phantom{0} & 0.004 \\
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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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- \label{table:alpha_beta_synth}
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\end{center}
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\end{table}
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-
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The results demonstrate that the model is capable of approximating the correct
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parameters for the small, synthetic dataset in each of the five iterations.
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-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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+The mean of the predicted values results in values with a sufficiently small
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+error. Thus, we argue that our selected method is well suited to analyze real
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+world pandemic data collected in Germany.\\
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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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-five iterations of our experiment. We can observe that the values of
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-\emph{Hamburg} have the highest standard deviation, while \emph{Mecklenburg Vorpommern}
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-has the lowest $\sigma$.\\
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+five iterations of our experiment. We can observe that the error $e_{SIR}$ is
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+the highest for \emph{Saxony} and the lowest for \emph{Lower Saxony}.
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+Furthermore, we include the distance $\Delta\beta_{\text{Germany}} = \beta_{\text{state}} - \beta_{\text{Germany}}$
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+and the percentage of people that have a basic immunity through vaccination
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+$\nu$ for each state provided by the Robert Koch Institute\footnote{{\tiny\url{https://impfdashboard.de/}}}.\\
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\begin{table}[h]
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\begin{center}
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+ \caption{Mean and standard deviation, error $e_{SIR}$ and the distance
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+ $\Delta\beta_{\text{Germany}} = \beta_{\text{state}} - \beta_{\text{Germany}}$
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+ across the 5 iterations, that we conducted for each German state and Germany
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+ as the whole country. Furthermore we include the vaccination percentage
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+ $\nu$ provided from the RKI.}
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+ \label{table:state_mean_std}
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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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+ state name & $\alpha$ & $\beta$ & $e_{SIR}$ & $\Delta\beta_{\text{Germany}}$ & $\nu$ [\%] \\
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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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+ Schleswig Holstein & 0.076{\tiny$\pm0.001$} & 0.095{\tiny$\pm 0.001$} & 0.085 & -0.013 & 79.5 \\
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+ Hamburg & 0.082{\tiny$\pm0.001$} & 0.104{\tiny$\pm 0.001$} & 0.095 & -0.004 & 84.5 \\
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+ Lower Saxony & 0.075{\tiny$\pm0.002$} & 0.097{\tiny$\pm 0.002$} & 0.077 & -0.011 & 77.6 \\
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+ Bremen & 0.058{\tiny$\pm0.002$} & 0.078{\tiny$\pm 0.002$} & 0.093 & -0.030 & 88.3 \\
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+ NRW & 0.079{\tiny$\pm0.001$} & 0.101{\tiny$\pm 0.001$} & 0.078 & -0.007 & 79.5 \\
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+ Hesse & 0.065{\tiny$\pm0.001$} & 0.085{\tiny$\pm 0.001$} & 0.102 & -0.023 & 75.8 \\
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+ Rhineland-Palatinate & 0.085{\tiny$\pm0.004$} & 0.108{\tiny$\pm 0.004$} & 0.090 & 0.001 & 75.6 \\
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+ Baden-Württemberg & 0.091{\tiny$\pm0.002$} & 0.118{\tiny$\pm 0.003$} & 0.080 & 0.010 & 74.5 \\
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+ Bavaria & 0.085{\tiny$\pm0.004$} & 0.116{\tiny$\pm 0.005$} & 0.095 & 0.008 & 75.1 \\
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+ Saarland & 0.075{\tiny$\pm0.002$} & 0.099{\tiny$\pm 0.003$} & 0.108 & -0.009 & 82.4 \\
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+ Berlin & 0.087{\tiny$\pm0.001$} & 0.109{\tiny$\pm 0.001$} & 0.067 & 0.001 & 78.1 \\
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+ Brandenburg & 0.087{\tiny$\pm0.003$} & 0.110{\tiny$\pm 0.003$} & 0.072 & 0.002 & 68.1 \\
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+ MV & 0.089{\tiny$\pm0.002$} & 0.114{\tiny$\pm 0.002$} & 0.054 & 0.006 & 74.7 \\
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+ Saxony & 0.075{\tiny$\pm0.002$} & 0.099{\tiny$\pm 0.002$} & 0.111 & -0.009 & 65.1 \\
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+ Saxony-Anhalt & 0.092{\tiny$\pm0.003$} & 0.119{\tiny$\pm 0.005$} & 0.079 & 0.011 & 74.1 \\
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+ Thuringia & 0.091{\tiny$\pm0.002$} & 0.119{\tiny$\pm 0.003$} & 0.084 & 0.011 & 70.3 \\
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+ \midrule
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+ Germany & 0.083{\tiny$\pm0.001$} & 0.108{\tiny$\pm 0.002$} & 0.080 & 0.000 & 76.4 \\
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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:state_mean_std}
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+
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\end{center}
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\end{table}
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\begin{figure}[t]
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\centering
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\includegraphics[width=\textwidth]{mean_std_alpha_beta_res.pdf}
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- \caption{Visualization of the mean $\mu$ and standard deviation $\sigma$ of
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- the transition rates $\alpha$ and $\beta$ for each state compared to the
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- mean values of $\alpha$ and $\beta$ for Germany.}
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+ \caption{Visualization of the mean and standard deviation of the transition
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+ rates $\alpha$ and $\beta$ for each state compared to the mean values of
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+ $\alpha$ and $\beta$ for Germany.}
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\label{fig:alpha_beta_mean_std}
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\end{figure}
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In~\Cref{fig:alpha_beta_mean_std}, we present a visual representation of the
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means and standard deviations in comparison to the national values. It is
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noteworthy that the states of Saxony-Anhalt and Thuringia have the highest
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-transmission rates of all states, while Bremen and Hessen have the lowest
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+transmission rates of all states, while Bremen and Hesse have the lowest
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values for $\beta$. The transmission rates of Hamburg, Baden Württemberg,
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Bavaria, and all eastern states lay above the national rate of transmission.
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Similarly, the recovery rate yields comparable outcomes. For the recovery rate,
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the same states that exhibit a transmission rate exceeding the national value,
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have a higher recovery rate than the national standard, with the exception of
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-Saxony.It is noteworthy that the recovery rates of all states exhibit a
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+Saxony. It is noteworthy that the recovery rates of all states exhibit a
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tendency to align with the recovery rate of $\alpha=\nicefrac{1}{14}$, which is
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-equivalent to a recovery period of 14 days.\\
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+equivalent to a recovery period of $D=\nicefrac{1}{\alpha}=14$ days. When
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+calculating the correlation coefficient between the predicted transmission rate
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+and the vaccination ratio, we get a value of $-0.5134$. The strong negative
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+correlation indicates that the transmission rate is high when the vaccination
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+ratio is low, and vice versa. This shows that the impact of the vaccines can be
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+witnessed in our results. \\
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It is evident that there is a correlation between the values of $\alpha$ and
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$\beta$ for each state. States with a high transmission rate tend to have a
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@@ -242,45 +238,41 @@ to the $R$ compartment 14 days after they were infected.\\
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This issue can be addressed by reducing the SIR model, thereby eliminating the
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significance of the $R$ compartment size. In the following section, we present
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-our experiments for the reduced SIR model with time-independent parameters.
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+our experiments for the reduced SIR model with time-dependent parameters.
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% -------------------------------------------------------------------
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-\section{Reduced SIR Model 5}
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+\section{Identifying the Reproduction Number}
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\label{sec:rsir}
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In this section we describe the experiments we conduct to identify the
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time-dependent reproduction number for both synthetic and real-world data.
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Similar to the previous section, we first describe the setup of our experiments
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-and afterwards present the results. The methods we employ for the preprocessing
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-are described in~\Cref{sec:preprocessing:rq} and for the PINN, that we use,
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-are described in~\Cref{sec:pinn:rsir}.
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+and afterwards present the results and a discussion. The methods we employ for
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+the preprocessing are described in~\Cref{sec:preprocessing:rq} and for the PINN,
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+that we use, are described in~\Cref{sec:pinn:rsir}.
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% -------------------------------------------------------------------
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-\subsection{Setup 1}
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+\subsection{Setup}
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\label{sec:rsir:setup}
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-This section outlines the selection of parameters and configuration for data
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-generation, preprocessing, and the neural networks. We employ these setups to
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-train the PINNs to identify the reproduction number on both synthetic and
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-real-world data.\\
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-
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-For the purposes of validation, we create a synthetic dataset, by setting the parameter
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+\paragraph{Synthetic Data:}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: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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+for a given time interval. As in the synthetic data for the aforementioned
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+experiments, we set $\alpha=\nicefrac{1}{3}$ and $\Rt$ to the values as can be
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+seen in~\Cref{fig:Rt_dataset} as well as the population size
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+$N=\expnumber{7.6}{6}$ and the initial amount of infected people to $I_0=10$.
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+Furthermore, we set our simulated time span to 150 days. We use this dataset to
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+demonstrate, that our method is working on a simple and minimal dataset.\\
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+\paragraph{Real-World Data:}To obtain a dataset of the infectious group, consisting of the
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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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+\emph{COVID-19-Todesfälle in Deutschland}~\cite{GHDead} 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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+ Infektionen in Deutschland}~\cite{GHInf}. In the preprocessing stage, we employ a constant
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rate for $\alpha$ to move individuals into the removed compartment. For each
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state we generate two datasets with a different recovery rate. First, we choose
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-$\alpha = \nicefrac{1}{14}$, which aligns with the time of recovery\footnote{\url{https://github.com/robert-koch-institut/SARS-CoV-2-Infektionen_in_Deutschland.git}}.
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+$\alpha = \nicefrac{1}{14}$, which aligns with the time of recovery~\cite{GHInf}.
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Second, we use $\alpha=\nicefrac{1}{5}$, as 5 days into the infection is the
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-point at which the infectiousness is at its peak\footnote{\url{https://www.infektionsschutz.de/coronavirus/fragen-und-antworten/ansteckung-uebertragung-und-krankheitsverlauf/}}.
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+point at which the infectiousness is at its peak~\cite{COVInfo}.
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As in~\Cref{sec:sir}, we set the population size $N$ of each state and Germany
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to the corresponding size at the end of 2019. Furthermore, for the same reason
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we restrict the data points to an interval of 1200 days, beginning on March 09.
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@@ -309,29 +301,29 @@ we restrict the data points to an interval of 1200 days, beginning on March 09.
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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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-architecture comprises of four hidden layers, each containing 100 neurons. The
|
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-activation function is the tangens hyperbolicus function. For the real-world
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-data, we weight the data loss by a factor of $\expnumber{1}{6}$, to the total
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-loss. The model is trained using a base learning rate of $\expnumber{1}{-3}$,
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-with the same scheduler and optimizer as we describe in~\Cref{sec:sir:setup}.
|
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-We train the model for 20000 epochs. To reduce the standard deviation, each
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-experiment is conducted 15 times.\\
|
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+\paragraph{Training Parameters:}In order to achieve the desired output, the
|
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+selected neural network architecture comprises of four hidden layers, each
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+containing 100 neurons. The activation function is the tangens hyperbolicus
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+function. For both the federal state and Germany, the physics loss is weighted
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+by a factor of $\expnumber{1}{-6}$, whereas the data loss belonging to Germany
|
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+is also weighted with a high factor of $\expnumber{1}{4}$, relative to the total
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+loss. We found this approach to yield the best results. The model is trained
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+using a base learning rate of $\expnumber{1}{-3}$, with the same scheduler and
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+optimizer as we describe in~\Cref{sec:sir:setup}. We train the model for the
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+states 20000 epochs and start the physics training after 10000 epochs, while we
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+train for Germany for 25000 and start the physics training after 15000 epochs.
|
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+To reduce the standard deviation, each experiment is conducted 15 times. For
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+evaluation, we use the error $e_G$ as we do in the subsequent section.\\
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|
% -------------------------------------------------------------------
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-\subsection{Results 4}
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+\subsection{Results}
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|
\label{sec:rsir:results}
|
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|
|
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|
-In this section we provide the results for our experiments considering the
|
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|
-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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-
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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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|
+The error for this is, $e_I = 0.0016$, which is of a negligible
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|
magnitude.\\
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|
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|
\begin{figure}[h]
|
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|
@@ -354,8 +346,7 @@ 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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+demonstrate remarkable precision.\\
|
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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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@@ -390,8 +381,9 @@ is shorter, but the peak value is higher.\\
|
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|
\label{fig:state_results}
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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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|
+ $\alpha = \nicefrac{1}{14}$ and $\alpha = \nicefrac{1}{5}$. Events~\cite{COVIDChronik} like
|
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|
+ the peak of an influential variant or the start of the vaccination of the public are marked horizontally. Further
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|
+ visualizations can be found in~\Cref{chap:appendix}.}
|
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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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@@ -406,6 +398,41 @@ 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.\\
|
|
|
|
|
|
+\begin{table}[t]
|
|
|
+ \begin{center}
|
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|
+ \caption{This table displays all average values of the error $e_{\text{I}}$
|
|
|
+ for all German states and Germany. The average is formed across all
|
|
|
+ 10 iteration.}
|
|
|
+ \label{table:state_error}
|
|
|
+ \begin{tabular}{lccccccc}
|
|
|
+ \toprule
|
|
|
+ & \multicolumn{2}{c}{$e_I$} & \phantom{0} & \multicolumn{2}{c}{days with $\Rt>1$} & \multicolumn{2}{c}{peak $\Rt$} \\
|
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|
+ \cmidrule{2-3}\cmidrule{5-6}\cmidrule{7-8}
|
|
|
+ state name & $\alpha=\frac{1}{14}$ & $\alpha=\frac{1}{5}$ & \phantom{0} & $\alpha=\frac{1}{14}$ & $\alpha=\frac{1}{5}$ & $\alpha=\frac{1}{14}$ & $\alpha=\frac{1}{5}$ \\
|
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|
+ \midrule
|
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|
+ Schleswig Holstein & 0.228 & 0.258 & \phantom{0} & 467.5 & 458.5 & 1.475 & 1.166 \\
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|
+ Hamburg & 0.265 & 0.330 & \phantom{0} & 424.3 & 409.8 & 1.500 & 1.297 \\
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|
+ Lower Saxony & 0.224 & 0.340 & \phantom{0} & 413.1 & 430.3 & 1.662 & 1.223 \\
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|
+ Bremen & 0.246 & 0.380 & \phantom{0} & 468.6 & 539.1 & 1.582 & 1.179 \\
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|
+ NRW & 0.185 & 0.252 & \phantom{0} & 486.3 & 602.0 & 1.573 & 1.205 \\
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|
+ Hesse & 0.302 & 0.346 & \phantom{0} & 553.0 & 511.2 & 1.409 & 1.157 \\
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|
+ Rhineland-Palatinate & 0.256 & 0.277 & \phantom{0} & 484.7 & 404.7 & 1.534 & 1.175 \\
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|
+ Baden-Württemberg & 0.198 & 0.284 & \phantom{0} & 469.2 & 590.0 & 1.457 & 1.180 \\
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|
+ Bavaria & 0.225 & 0.318 & \phantom{0} & 490.5 & 486.1 & 1.428 & 1.199 \\
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|
+ Saarland & 0.284 & 0.408 & \phantom{0} & 500.2 & 564.7 & 1.515 & 1.180 \\
|
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|
+ Berlin & 0.201 & 0.240 & \phantom{0} & 591.9 & 514.4 & 1.721 & 1.262 \\
|
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|
+ Brandenburg & 0.237 & 0.242 & \phantom{0} & 555.9 & 596.3 & 1.447 & 1.159 \\
|
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|
+ MV & 0.170 & 0.257 & \phantom{0} & 537.5 & 544.3 & 1.563 & 1.135 \\
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|
+ Saxony & 0.292 & 0.256 & \phantom{0} & 722.3 & 695.4 & 1.790 & 1.407 \\
|
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|
+ Saxony-Anhalt & 0.213 & 0.268 & \phantom{0} & 572.0 & 631.9 & 1.387 & 1.165 \\
|
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|
+ Thuringia & 0.180 & 0.222 & \phantom{0} & 732.1 & 730.6 & 1.586 & 1.249 \\
|
|
|
+ \midrule
|
|
|
+ Germany & 0.284 & 0.239 & \phantom{0} & 587.7 & 430.7 & 1.561 & 1.219 \\
|
|
|
+ \bottomrule
|
|
|
+ \end{tabular}
|
|
|
+ \end{center}
|
|
|
+\end{table}
|
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+
|
|
|
As illustrated in~\Cref{fig:state_results}, the training data is overlaid with the
|
|
|
corresponding prediction of the model. We can observe that the prediction, though
|
|
|
an exact reconstruction, accurately captures the general trajectory of the
|
|
|
@@ -414,8 +441,8 @@ peaks, exhibiting a tendency to ignore smaller changes. This suggests that the
|
|
|
prediction of the model is capable show the rough outline of the progression of COVID-19. In the
|
|
|
beginning, the majority of predictions below $\Rt=1$, indicating an outbreak.
|
|
|
As we observed in the synthetic data, the model exhibits a higher standard
|
|
|
-deviation at the boundaries. In the graphs, we mark the
|
|
|
-peaks of the most severe COVID-19 variants in Germany. While the peaks of the
|
|
|
+deviation at the boundaries. In the graphs, we mark the peaks of the most severe
|
|
|
+COVID-19 variants in Germany~\cite{COVIDChronik}. While the peaks of the
|
|
|
Alpha and Delta variants are clearly visible in the data, the model does not
|
|
|
learn these, and thus they are not reflected in the results. The peak of the
|
|
|
Omicron variant represents the culmination of the COVID-19 pandemic in Germany
|
|
|
@@ -424,46 +451,9 @@ value of the reproduction number across all states. This phenomenon can be expla
|
|
|
number of individuals infected by one infectious person reaching its peak. In
|
|
|
some states the peaks of other Omicron variants after the maximum peak are visible (see Thuringia).\\
|
|
|
|
|
|
-The experiments demonstrate, that our model encounteres difficulties in learning the data for the
|
|
|
+The experiments demonstrate, that our model encounters difficulties in learning the data for the
|
|
|
states and Germany and consequently in predicting the reproduction values for each dataset.
|
|
|
Nonetheless, the predictions illustrate the general trends of the most impactful
|
|
|
events of the COVID-19 pandemic.\\
|
|
|
|
|
|
-\begin{table}[t]
|
|
|
- \begin{center}
|
|
|
- \begin{tabular}{lcc}
|
|
|
- \toprule
|
|
|
- & \multicolumn{2}{c}{$e_I$} \\
|
|
|
- \cmidrule{2-3}
|
|
|
- state name & $\alpha=\nicefrac{1}{14}$ & $\alpha=\nicefrac{1}{5}$ \\
|
|
|
- \midrule
|
|
|
- Schleswig Holstein & 0.2005 & 0.2514 \\
|
|
|
- Hamburg & 0.3045 & 0.3357 \\
|
|
|
- Lower Saxony & 0.2140 & 0.3082 \\
|
|
|
- Bremen & 0.2370 & 0.3838 \\
|
|
|
- North Rhine-Westphalia & 0.1718 & 0.2460 \\
|
|
|
- Hesse & 0.2736 & 0.3172 \\
|
|
|
- Rhineland-Palatinate & 0.2442 & 0.2674 \\
|
|
|
- Baden-Württemberg & 0.1984 & 0.2958 \\\addlinespace
|
|
|
- Bavaria & 0.1928 & 0.2825 \\
|
|
|
- Saarland & 0.2554 & 0.4676 \\
|
|
|
- Berlin & 0.1885 & 0.2948 \\
|
|
|
- Brandenburg & 0.2023 & 0.2571 \\
|
|
|
- Mecklenburg-Vorpommern & 0.1518 & 0.3272 \\
|
|
|
- Saxony & 0.3382 & 0.2807 \\
|
|
|
- Saxony-Anhalt & 0.1959 & 0.2564 \\
|
|
|
- Thuringia & 0.1401 & 0.2221 \\\addlinespace
|
|
|
- Germany & 0.3371 & 0.2533 \\
|
|
|
- \bottomrule
|
|
|
- \end{tabular}
|
|
|
- \caption{This table displays all average values of the error $e_{\text{synth}}$
|
|
|
- for all German states and Germany. The average is formed across all
|
|
|
- 10 iteration.}
|
|
|
- \label{table:state_error}
|
|
|
- \end{center}
|
|
|
-\end{table}
|
|
|
-
|
|
|
-
|
|
|
-
|
|
|
-
|
|
|
% -------------------------------------------------------------------
|
Phillip Rothenbeck