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@@ -25,64 +25,147 @@ experiments.
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In this section we seek to find the transmission rate $\beta$ and the recovery
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rate $\alpha$ from either synthetic or preprocessed real-world data. The
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methodology that we employ to identify the transition rates is described
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-in~\Cref{sec:pinn:sir}.
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+in~\Cref{sec:pinn:sir}. Meanwhile, the methods we use to preprocess the
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+real-world data is to be found in~\Cref{sec:preprocessing:rq}.
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% -------------------------------------------------------------------
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\subsection{Setup 1}
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\label{sec:sir:setup}
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+In this section we show the setups for the training of our PINNs, that are
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+supposed to find the transition parameters. This includes the specific
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+parameters for the preprocessing and the configuration of the PINN their
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+selves.\\
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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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+We conduct 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 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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+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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+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, which is a span of 1200 days and covers the time
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+in which the COVID-19 disease was the most active and severe.
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+
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\begin{figure}[h]
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- \centering
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- \includegraphics[width=0.3\textwidth]{SIR_synth.pdf}
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- \caption{Visualization of the synthetic data for the SIR model, parameters for creation:
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- $\alpha=\nicefrac{1}{3}$, $\beta=\nicefrac{1}{2}$}
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- \label{fig:synthetic_SIR}
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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_SIR}
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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_sir}
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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_sir}
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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_sir}
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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_sir}
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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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+The PINN that we employ consists of seven hidden layers with twenty neurons
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+each and an activation function of ReLU. For training, we use the Adam optimizer
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+and the polynomial scheduler of the pytorch library 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 do 5 iterations to show stability of
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+the values. Our configuration is similar to the configuration, that Shaier
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+\etal.~\cite{Shaier2021} use for their work aside from the learning rate and the
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+scheduler choice.\\
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+
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+In the next section we present the results of the simulations conducted with the
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+setups that we describe in this section.
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+
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% -------------------------------------------------------------------
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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|c c c c c}
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- \hline
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- & Schleswig Holstein & Hamburg & Niedersachsen & Bremen & Nordrhein-Westfalen \\
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- \hline
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- $\alpha$ & 0.0739 & 0.0774 & 0.0681 & 0.0548 & 0.0725 \\
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- $\beta$ & 0.0931 & 0.0995 & 0.0894 & 0.0744 & 0.0939 \\
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- \hline
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- \end{tabular}
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-
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- \begin{tabular}{c|c c c c c c}
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- \hline
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- & Hessen & Rheinland-Pfalz & Baden Württemberg & Bayern & Saarland & Berlin \\
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- \hline
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- $\alpha$ & 0.0598 & 0.0754 & 0.0803 & 0.0767 & 0.0655 & 0.0845 \\
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- $\beta$ & 0.0787 & 0.0971 & 0.1061 & 0.1045 & 0.0888 & 0.1050 \\
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- \hline
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- \end{tabular}
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-
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- \begin{tabular}{c|c c c c c c}
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- \hline
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- & Brandenburg & Mecklenburg-Vorpommern & Sachsen & Sachsen-Anhalt & Thüringen & Germany \\
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- \hline
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- $\alpha$ & 0.0796 & 0.0864 & 0.0705 & 0.0843 & 0.0852 & 0.0821 \\
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- $\beta$ & 0.1010 & 0.1111 & 0.0951 & 0.1095 & 0.1120 & 0.1066 \\
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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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+\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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+ \label{fig:alpha_beta_mean_std}
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+\end{figure}
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% -------------------------------------------------------------------
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\section{Reduced SIR Model 5}
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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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% -------------------------------------------------------------------
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\subsection{Setup 1}
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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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+
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% -------------------------------------------------------------------
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Phillip Rothenbeck