% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Author:   Phillip Rothenbeck
% Title:    Investigating the Evolution of the COVID-19 Pandemic in Germany Using Physics-Informed Neural Networks
% File:     chap04/chap04.tex
% Part:     Experiments
% Description:
%         summary of the content in this chapter
% Version:  01.01.2012
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Experiments   10}
\label{chap:evaluation}
In the previous chapters we explained the methods (see~\Cref{chap:methods})
based the theoretical background, that we established in~\Cref{chap:background}.
In this chapter, we present the setups and results from the experiments and
simulations, we ran. First, we tackle the experiments dedicated to find the
epidemiological parameters of $\beta$ and $\alpha$ in synthetic and real-world
data. Second, we identify the reproduction number in synthetic and real-world
data of Germany. Each section, is divided in the setup and the results of the
experiments.

% -------------------------------------------------------------------

\section{Identifying the Transition Rates on Real-World and Synthetic Data  5}
\label{sec:sir}
In this section we seek to find the transmission rate $\beta$ and the recovery
rate $\alpha$ from either synthetic or preprocessed real-world data. The
methodology that we employ to identify the transition rates is described
in~\Cref{sec:pinn:sir}. Meanwhile, the methods we use to preprocess the
real-world data is to be found in~\Cref{sec:preprocessing:rq}.

% -------------------------------------------------------------------

\subsection{Setup   1}
\label{sec:sir:setup}

In this section we show the setups for the training of our PINNs, that are
supposed to find the transition parameters. This includes the specific
parameters for the preprocessing and the configuration of the PINN their
selves.\\

In order to validate our method we first generate a dataset of synthetic data.
We conduct this by solving~\Cref{eq:modSIR} for a given set of parameters.
The parameters are set to $\alpha = \nicefrac{1}{3}$ and $\beta = \nicefrac{1}{2}$.
The size of the population is $N = \expnumber{7.6}{6}$ and the initial amount of
infectious individuals of is $I_0 = 10$. We simulate over 150 days and get a
dataset of the form of~\Cref{fig:synthetic_SIR}.\\For the real-world RKI data we
preprocess the row data data of each state and Germany separately using a
recovery queue with a recovery period of 14 days. As for the population size of
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}}.
The initial number of infectious individuals is set to the number of infected
people on March 09. 2020 from the dataset. The data we extract spans from
March 09. 2020 to June 22. 2023, which is a span of 1200 days and covers the time
in which the COVID-19 disease was the most active and severe.

\begin{figure}[h]
    %\centering
    \setlength{\unitlength}{1cm} % Set the unit length for coordinates
    \begin{picture}(12, 9.5) % Specify the size of the picture environment (width, height)
        \put(1.5, 4.5){
            \begin{subfigure}{0.3\textwidth}
                \centering
                \includegraphics[width=\textwidth]{SIR_synth.pdf}
                \label{fig:synthetic_SIR}
            \end{subfigure}
        }
        \put(8, 4.5){
            \begin{subfigure}{0.3\textwidth}
                \centering
                \includegraphics[width=\textwidth]{datasets_states/Germany_SIR_14.pdf}
                \label{fig:germany_sir}
            \end{subfigure}
        }
        \put(0, 0){
            \begin{subfigure}{0.3\textwidth}
                \centering
                \includegraphics[width=\textwidth]{datasets_states/Schleswig_Holstein_SIR_14.pdf}
                \label{fig:schleswig_holstein_sir}
            \end{subfigure}
        }
        \put(4.75, 0){
            \begin{subfigure}{0.3\textwidth}
                \centering
                \includegraphics[width=\textwidth]{datasets_states/Berlin_SIR_14.pdf}
                \label{fig:berlin_sir}
            \end{subfigure}
        }
        \put(9.5, 0){
            \begin{subfigure}{0.3\textwidth}
                \centering
                \includegraphics[width=\textwidth]{datasets_states/Thueringen_SIR_14.pdf}
                \label{fig:thüringen_sir}
            \end{subfigure}
        }

    \end{picture}
    \caption{Synthetic and real-world training data. The synthetic data is
        generated with $\alpha=\nicefrac{1}{3}$ and $\beta=\nicefrac{1}{2}$
        and~\Cref{eq:modSIR}. The Germany data is taken from the death case
        data set. Exemplatory we show illustrations of the datasets of Schleswig
        Holstein, Berlin, and Thuringia. For the other states see~\Cref{chap:appendix} }
    \label{fig:datasets}
\end{figure}

The PINN that we employ consists of seven hidden layers with twenty neurons
each and an activation function of ReLU. For training, we use the Adam optimizer
and the polynomial scheduler of the pytorch library with a base learning rate
of $\expnumber{1}{-3}$. We train the model for 10000 epochs to extract the
parameters. For each set of parameters we do 5 iterations to show stability of
the values. Our configuration is similar to the configuration, that Shaier
\etal.~\cite{Shaier2021} use for their work aside from the learning rate and the
scheduler choice.\\

In the next section we present the results of the simulations conducted with the
setups that we describe in this section.

% -------------------------------------------------------------------

\subsection{Results    4}
\label{sec:sir:results}
\begin{center}
    \begin{tabular}{c|cc|cc}
                               & $\alpha$ & $\sigma(\alpha)$ & $\beta$ & $\sigma(\beta)$ \\
        \hline
        Schleswig Holstein     & 0.0771   & 0.0010           & 0.0966  & 0.0013          \\
        Hamburg                & 0.0847   & 0.0035           & 0.1077  & 0.0037          \\
        Niedersachsen          & 0.0735   & 0.0014           & 0.0962  & 0.0018          \\
        Bremen                 & 0.0588   & 0.0018           & 0.0795  & 0.0025          \\
        Nordrhein-Westfalen    & 0.0780   & 0.0009           & 0.1001  & 0.0011          \\
        Hessen                 & 0.0653   & 0.0016           & 0.0854  & 0.0020          \\
        Rheinland-Pfalz        & 0.0808   & 0.0016           & 0.1036  & 0.0018          \\
        Baden-Württemberg      & 0.0862   & 0.0014           & 0.1132  & 0.0016          \\
        Bayern                 & 0.0809   & 0.0021           & 0.1106  & 0.0027          \\
        Saarland               & 0.0746   & 0.0021           & 0.0996  & 0.0024          \\
        Berlin                 & 0.0901   & 0.0008           & 0.1125  & 0.0008          \\
        Brandenburg            & 0.0861   & 0.0008           & 0.1091  & 0.0010          \\
        Mecklenburg Vorpommern & 0.0910   & 0.0007           & 0.1167  & 0.0008          \\
        Sachsen                & 0.0797   & 0.0017           & 0.1073  & 0.0022          \\
        Sachsen-Anhalt         & 0.0932   & 0.0019           & 0.1207  & 0.0027          \\
        Thüringen              & 0.0952   & 0.0011           & 0.1248  & 0.0016          \\
        Germany                & 0.0803   & 0.0012           & 0.1044  & 0.0014          \\
    \end{tabular}
\end{center}

\begin{figure}[h]
    \centering
    \includegraphics[width=\textwidth]{mean_std_alpha_beta_res.pdf}
    \label{fig:alpha_beta_mean_std}
\end{figure}

% -------------------------------------------------------------------

\section{Reduced SIR Model   5}
\label{sec:rsir}
In this section we describe the experiments we conduct to identify the
time-dependent reproduction number for both synthetic and real-world data.
Similar to the previous section, we first describe the setup of our experiments
and afterwards present the results. The methods we employ for the preprocessing
are described in~\Cref{sec:preprocessing:rq} and for the PINN, that we use,
are described in~\Cref{sec:pinn:rsir}.

% -------------------------------------------------------------------

\subsection{Setup    1}
\label{sec:rsir:setup}
In this section we describe the choice of parameters and configuration for data
generation, preprocessing and the neural networks. We use these setups to train
the PINNs to find the reproduction number on both synthetic and real-world data.


% -------------------------------------------------------------------

\subsection{Results   4}
\label{sec:rsir:results}

% -------------------------------------------------------------------
