start method + finish RKI section

chapters/chap03/chap03.tex

@@ -5,26 +5,96 @@
 % Part:     Methods
 % Description:
 %         summary of the content in this chapter
-% Version:  01.01.2012
+% Version:  20.08.2024
 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \chapter{Methods   8}
 \label{chap:methods}
+This chapter provides the methods, that we employ to address the problem that we
+present in~\Cref{chap:introduction}.~\Cref{sec:preprocessing} outlines
+our approaches for preprocessing of the available data and has two
+sections. The first section describes the publicly available data provided by
+the \emph{Robert Koch Institute} (RKI)\footnote[1]{\url{https://www.rki.de/EN/Home/homepage_node.html}}.
+The second section outlines the techniques we use to process this data to fit
+our project's requirements. Subsequently, we give a theoretical overview of the
+PINN's that we employ. These latter sections, establish the foundation for the
+implementations described in~\Cref{sec:sir:setup} and~\Cref{sec:rsir:setup}.
 
 % -------------------------------------------------------------------
 
 \section{Data Preprocessing   3}
 \label{sec:preprocessing}
+For the PINN's to work with the data available to us, it must be in the form
+that is dictated by the epidemiological models, that are supposed to be solved.
+The PINN that solves the SIR model (see~\Cref{sec:pinn:dinn}), needs data points
+containing a time point $t^{(i)}$ and the corresponding true values of the
+compartments $\boldsymbol{S}^{(i)}, \boldsymbol{I}^{(i)}$ and
+$\boldsymbol{R}^{(i)}$ with $i\in\{1, ..., N_t\}$ and the number of training
+points $N_t$. For the reduced SIR model (see~\Cref{sec:pandemicModel:rsir}), we
+need pairs of $(t^{(i)}, \boldsymbol{I}^{(i)})$. This section, concentrates on
+the structure of the available data and the methods we employ to convert it to
+the correct structure.
 
 % -------------------------------------------------------------------
 
 \subsection{RKI Data   2}
 \label{sec:preprocessing:rki}
+The Robert Koch Institute works on monitoring and preventing diseases. As the
+central institution of the German government in the field of biomedicine, one of
+its tasks during the COVID-19 pandemic was it to track the number of infections
+and death cases in Germany. The data was collected by university hospitals,
+research facilities and laboratories, by conducting tests. Each new case must be
+reported after 24 hours at the latest to the respective state authority. Each
+state authority collects the cases for a day and must report them to the next
+working day to the RKI. The RKI refurbishes the data and releases statistics or
+update repositories holding the information for the public to access. For the
+work of this thesis we concentrate on two of these repositories.\\
+
+The first repository is called \emph{COVID-19-Todesfälle in Deutschland}\footnote{\url{https://github.com/robert-koch-institut/COVID-19-Todesfaelle_in_Deutschland.git}}. It
+holds data points with each having the date on which the respective data was
+collected. The dates reach from the ninth of march in 2020 to the present day.
+For every date, the total number of infection and death cases, the new death
+cases, and the case-fatality ratio. The total number of infection and death
+cases is the number of all cases that were reported until that day including the
+newly reported data. The dataset includes two more datasets, that hold the death
+case information for either the age groups or the individual states of Germany.\\
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=\textwidth]{dataset_visualization.pdf}
+    \caption{A visualization the total death case and infection case data for
+        each day from the data set \emph{COVID-19-Todesfälle in Deutschland}. Status
+        of the 20'th of August 2024.}
+    \label{fig:rki_data}
+\end{figure}
+
+The second repository is called \emph{SARS-CoV-2 Infektionen in Deutschland}
+This dataset holds detailed information about the infections of each county per
+day. The counties are encoded using the \emph{Community Identification Number}\footnote{\url{https://www.destatis.de/DE/Themen/Laender-Regionen/Regionales/Gemeindeverzeichnis/_inhalt.html}},
+wherein the first two digits denounce the state, the third number stand for the
+government district and the last two digits show the county. Each data point
+shows the gender, the age group, the death, infection and recovery cases and the
+reference and report date. The reference date marks the start of the individual
+feeling sick. If this is unknown, the reference date equals the report date.\\
+
+The RKI assumes the duration of the illness under normal conditions to 14 days,
+while the duration of severe cases is assumed to be 28 days. The recovery cases
+in the dataset are calculated using these assumptions, by adding the duration on
+the reference date if it is given. As written in the ReadMe, the recovery data
+should be used with caution. Since we need the recovery data for further
+calculations, the next section presents the solutions we employed to address
+this issue.
 
 % -------------------------------------------------------------------
 
-\subsection{Recovery Queue   1}
+\subsection{Recovery Queue and Recovery Rate   1}
 \label{sec:preprocessing:rq}
 
+In~\Cref{sec:preprocessing:rki} we present the data which the RKI provides.
+While containing the data of infections and death cases in form of accumulated
+or daily case. The data for the susceptible and removed compartments are absent.
+While this is not an issue for the reduced SIR model
+(see~\Cref{sec:pandemicModel:rsir}), we need the data. For com
+
 % -------------------------------------------------------------------
 
 \section{PINN for the SIR Model   3}