correct section 4.1

chapters/chap04/chap04.tex

@@ -9,48 +9,51 @@
 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \chapter{Experiments   10}
 \label{chap:evaluation}
-In the previous chapters we explained the methods (see~\Cref{chap:methods})
+In the preceding 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.
+In this chapter present the setups and results from the experiments and
+simulations, we ran. First, we discuss the experiments dedicated to identify
+the epidemiological parameters of $\beta$ and $\alpha$ in synthetic and
+real-world data. Second, we examine the reproduction number in synthetic and
+real-world data of Germany. Each section, is divided into a description of the
+experimental setup and the results.
 
 % -------------------------------------------------------------------
 
 \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}.
+In this section, we aim to identify 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 utilize to preprocess the
+real-world data are detailed 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 this subsection, we present the configurations for the training of our
+PINNs, which are designed to identify the transition parameters. This
+encompasses the specific parameters for the preprocessing and the configuration
+of the PINN themselves.\\
 
-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.
+In order to validate our method, we first generate a dataset of synthetic data.
+We achieve 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 raw 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}}.
+infectious individuals of is $I_0 = 10$. We conduct the simulation over 150
+days, resulting in a dataset of the form of~\Cref{fig:synthetic_SIR}.\\ In
+order to process the real-world RKI data, it is necessary to preprocess the raw
+data for each state and Germany separately. This is achieved by utilizing a
+recovery queue with a recovery period of 14 days. With regard to 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.
+March 09. 2020 to June 22. 2023, encompassing a period of 1200 days and
+representing the time span during which the COVID-19 disease was the most
+active and severe.
 
 \begin{figure}[h]
     %\centering
@@ -101,16 +104,16 @@ in which the COVID-19 disease was the most active and severe.
     \label{fig:datasets_sir}
 \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
+The PINN that we utilize comprises of seven hidden layers with twenty neurons
+each, and an activation function of ReLU. We employ the Adam optimizer and the
+polynomial scheduler of the PyTorch library, for training, 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.\\
+parameters. For each set of parameters, we conduct five iterations to
+demonstrate stability of the values. The 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
+The following section presents the results of the simulations conducted with the
 setups that we describe in this section.
 
 % -------------------------------------------------------------------
@@ -126,17 +129,18 @@ setups that we describe in this section.
     \label{fig:reprod}
 \end{figure}
 
-In this section we describe the results, that we obtain from the conducted
-experiments, that we describe in the preceding section. First we show the
-results for the synthetic dataset and look at the accuracy and reproducibility.
-Then we present and discuss the results for the German states and Germany.\\
+In this section, we present the results, that we obtain from the conducted
+experiments, that we describe in the preceding section. We begin by examining
+the results for the synthetic dataset, focusing the accuracy and
+reproducibility. We then proceed to present and discuss the results for the
+German states and Germany.\\
 
 The results of the experiment regarding the synthetic data can be seen
 in~\Cref{table:alpha_beta_synth} and in~\Cref{fig:reprod}.~\Cref{fig:reprod}
-shows the values of $\beta$ and $\alpha$ of each iteration compared to the true
+depicts the values of $\beta$ and $\alpha$ for each iteration in comparison to the true
 values of $\beta=\nicefrac{1}{2}$ and $\alpha=\nicefrac{1}{3}$. In~\Cref{table:alpha_beta_synth}
-we present the mean $\mu$ and standard variation $\sigma$ of both values across
-all 5 iterations.\\
+we present the mean $\mu$ and standard deviation $\sigma$ of both values across
+all five iterations.\\
 
 \begin{table}[h]
     \begin{center}
@@ -145,23 +149,25 @@ all 5 iterations.\\
             \hline
             0.3333        & 0.3334        & 0.0011           & 0.5000       & 0.5000       & 0.0017          \\
         \end{tabular}
-        \caption{The mean $\mu$ and standard variation $\sigma$ across the 5
+        \caption{The mean $\mu$ and standard deviation $\sigma$ across the 5
             independent iterations of training our PINNs with the synthetic dataset.}
         \label{table:alpha_beta_synth}
     \end{center}
 \end{table}
-From the results we can see that the model is able to approximate the correct
-parameters for the small, synthetic dataset in each of the 5 iterations. Even
-though the predicted value is never exactly correct, the standard deviation is
-negligible small and taking the mean of multiple iterations yields an almost
-perfect result.\\
+
+The results demonstrate that the model is capable of approximating the correct
+parameters for the small, synthetic dataset in each of the five iterations.
+While the predicted value is not precisely accurate, the standard deviation is
+sufficiently small, and taking the mean of multiple iterations produces an
+almost perfect result.\\
 
 In~\Cref{table:alpha_beta} we present the results of the training for the
-real-world data. These are presented from top to bottom, in the order of the
-community identification number, with the last entry being Germany. $\mu$ and
-$\sigma$ are both calculated across all 5 iterations of our experiment. We can
-see that the values of \emph{Hamburg} have the highest standard deviation, while
-\emph{Mecklenburg Vorpommern} has the smallest $\sigma$.\\
+real-world data. The results are presented from top to bottom, in the order of
+the community identification number, with the last entry being Germany. Both
+the mean $\mu$ and the standard deviation $\sigma$ are calculated across all
+five iterations of our experiment. We can observe that the values of
+\emph{Hamburg} have the highest standard deviation, while \emph{Mecklenburg Vorpommern}
+has the lowest $\sigma$.\\
 
 \begin{table}[h]
     \begin{center}
@@ -186,39 +192,47 @@ see that the values of \emph{Hamburg} have the highest standard deviation, while
             Thüringen              & 0.0952        & 0.0011           & 0.1248       & 0.0016          \\
             Germany                & 0.0803        & 0.0012           & 0.1044       & 0.0014          \\
         \end{tabular}
-        \caption{Mean and standard variation across the 5 iterations, that we
+        \caption{Mean and standard deviation across the 5 iterations, that we
             conducted for each German state and Germany as the whole country.}
         \label{table:alpha_beta}
     \end{center}
 \end{table}
 
-In~\Cref{fig:alpha_beta_mean_std} we visualize the means and standard variations
-in contrast to the national values. The states with the highest transmission rate
-values are Thuringia, Saxony Anhalt and Mecklenburg West-Pomerania. It is also,
-visible that all six of the eastern states have a higher transmission rate than
-Germany. These results may be explainable with the ratio of vaccinated individuals\footnote{\url{https://impfdashboard.de/}}.
-The eastern state have a comparably low complete vaccination ratio, accept for
-Berlin. While Berlin has a moderate vaccination ratio, it is also a hub of
-mobility, which means that contact between individuals happens much more often. This is also a reason for Hamburg being a state with an above national standard rate of transmission.
-\\
-
-
-
-We visualize these numbers in~\Cref{fig:alpha_beta_mean_std},
-where all means and standard variations are plotted as points, while the values
-for Germany are also plotted as lines to make a classification easier. It is
-visible that Hamburg, Baden-Württemberg, Bayern and all six of the states that
-lie in the eastern part of Germany have a higher transmission rate $\beta$ than
-overall Germany. Furthermore, it can be observed, that all values for the
-recovery $\alpha$ seem to be correlating to the value of $\beta$, which can be
-explained with the assumption that we make when we preprocess the data using the
-recovery queue by setting the recovery time to 14 days.
-\begin{figure}[h]
+\begin{figure}[t]
     \centering
     \includegraphics[width=\textwidth]{mean_std_alpha_beta_res.pdf}
     \label{fig:alpha_beta_mean_std}
 \end{figure}
 
+In~\Cref{fig:alpha_beta_mean_std}, we present a visual representation of the
+means and standard deviations in comparison to the national values. It is
+noteworthy that the states of Saxony-Anhalt and Thuringia have the highest
+transmission rates of all states, while Bremen and Hessen have the lowest
+values for $\beta$. The transmission rates of Hamburg, Baden Württemberg,
+Bavaria, and all eastern states lay above the national rate of transmission.
+Similarly, the recovery rate yields comparable outcomes. For the recovery rate,
+the same states that exhibit a transmission rate exceeding the national value,
+have a higher recovery rate than the national standard, with the exception of
+Saxony.It is noteworthy that the recovery rates of all states exhibit a
+tendency to align with the recovery rate of $\alpha=\nicefrac{1}{14}$, which is
+equivalent to a recovery period of 14 days.\\
+
+It is evident that there is a correlation between the values of $\alpha$ and
+$\beta$ for each state. States with a high transmission rate tend to have a
+high recovery rate, and vice versa. The correlation between $\alpha$ and
+$\beta$ can be explained by the implicate definition of $\alpha$ using a
+recovery queue with a constant recovery period of 14 days. This might result to
+the PINN not learning $\alpha$ as a standalone parameter but rather as a
+function of the transmission rate $\beta$. This phenomenon occurs because the
+transmission rate determines the number of individuals that get infected per
+day, and the recovery queue moves a proportional number of people to the
+removed compartment. Consequently, a number of people defined by $\beta$ move
+to the $R$ compartment 14 days after they were infected.\\
+
+This issue can be addressed by reducing the SIR model, thereby eliminating the
+significance of the $R$ compartment size. In the following section, we present
+our experiments for the reduced SIR model with time-independent parameters.
+
 % -------------------------------------------------------------------
 
 \section{Reduced SIR Model   5}
@@ -234,128 +248,82 @@ 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.\\
-
-For validation reasons we create a synthetic dataset, by setting the parameters
-of $\alpha$ and $\beta$ each to a specific value, and solving~\Cref{eq:modSIR}
-for a given time interval. We set $\alpha=\nicefrac{1}{3}$ and
-$\beta=\nicefrac{1}{2}$ as well as the population size $N=\expnumber{7.6}{6}$
-and the initial amount of infected people to $I_0=10$. Furthermore, we set our
-simulated time span to 150 days.We will use this dataset to show, that our
-method is working on a simple and minimal dataset.\\ For the real-world data we
-we processed the data of the dataset \emph{COVID-19-Todesfälle in Deutschland}
-to extract the number of infections in the whole of Germany, while we used the
-data of \emph{SARS-CoV-2 Infektionen in Deutschland} for the German states. For
-the preprocessing we use a constant rate for $\alpha$ to move individual into
-the removed compartment. First we choose $\alpha = \nicefrac{1}{14}$ as this is
-covers the time of recovery\footnote{\url{https://github.com/robert-koch-institut/SARS-CoV-2-Infektionen_in_Deutschland.git}}.
-Second we use $\alpha=\nicefrac{1}{5}$ since the peak of infectiousness is
-reached right in front or at 5 days into the infection\footnote{\url{https://www.infektionsschutz.de/coronavirus/fragen-und-antworten/ansteckung-uebertragung-und-krankheitsverlauf/}}.
-Just as in~\Cref{sec:sir} we set the population size $N$ of each state and
-Germany to the corresponding size at the end of 2019. Also, for the same reason
-we restrict the data points to an interval of 1200 days starting from March 09.
-2020.
-\begin{figure}[h]
-    %\centering
-    \setlength{\unitlength}{1cm} % Set the unit length for coordinates
-    \begin{picture}(12, 14.5) % Specify the size of the picture environment (width, height)
-        \put(0, 10){
-            \begin{subfigure}{0.3\textwidth}
-                \centering
-                \includegraphics[width=\textwidth]{I_synth.pdf}
-                \caption{Synthetic data}
-                \label{fig:synthetic_I}
-            \end{subfigure}
-        }
-        \put(4.75, 10){
-            \begin{subfigure}{0.3\textwidth}
-                \centering
-                \includegraphics[width=\textwidth]{datasets_states/Germany_I_14.pdf}
-                \caption{Germany with $\alpha=\nicefrac{1}{14}$}
-                \label{fig:germany_I_14}
-            \end{subfigure}
-        }
-        \put(9.5, 10){
-            \begin{subfigure}{0.3\textwidth}
-                \centering
-                \includegraphics[width=\textwidth]{datasets_states/Germany_I_5.pdf}
-                \caption{Germany with $\alpha=\nicefrac{1}{5}$}
-                \label{fig:germany_I_5}
-            \end{subfigure}
-        }
-        \put(0, 5){
-            \begin{subfigure}{0.3\textwidth}
-                \centering
-                \includegraphics[width=\textwidth]{datasets_states/Nordrhein_Westfalen_I_14.pdf}
-                \caption{NRW with $\alpha=\nicefrac{1}{14}$}
-                \label{fig:schleswig_holstein_I_14}
-            \end{subfigure}
-        }
-        \put(4.75, 5){
-            \begin{subfigure}{0.3\textwidth}
-                \centering
-                \includegraphics[width=\textwidth]{datasets_states/Hessen_I_14.pdf}
-                \caption{Hessen with $\alpha=\nicefrac{1}{14}$}
-                \label{fig:berlin_I_14}
-            \end{subfigure}
-        }
-        \put(9.5, 5){
-            \begin{subfigure}{0.3\textwidth}
-                \centering
-                \includegraphics[width=\textwidth]{datasets_states/Thueringen_I_14.pdf}
-                \caption{Thüringen with $\alpha=\nicefrac{1}{14}$}
-                \label{fig:thüringen_I_14}
-            \end{subfigure}
-        }
-        \put(0, 0){
-            \begin{subfigure}{0.3\textwidth}
-                \centering
-                \includegraphics[width=\textwidth]{datasets_states/Nordrhein_Westfalen_I_5.pdf}
-                \caption{NRW with $\alpha=\nicefrac{1}{5}$}
-                \label{fig:schleswig_holstein_I_5}
-            \end{subfigure}
-        }
-        \put(4.75, 0){
-            \begin{subfigure}{0.3\textwidth}
-                \centering
-                \includegraphics[width=\textwidth]{datasets_states/Hessen_I_5.pdf}
-                \caption{Hessen with $\alpha=\nicefrac{1}{5}$}
-                \label{fig:berlin_I_5}
-            \end{subfigure}
-        }
-        \put(9.5, 0){
-            \begin{subfigure}{0.3\textwidth}
-                \centering
-                \includegraphics[width=\textwidth]{datasets_states/Thueringen_I_5.pdf}
-                \caption{Thüringen with $\alpha=\nicefrac{1}{5}$}
-                \label{fig:thüringen_I_5}
-            \end{subfigure}
-        }
+This section outlines the selection of parameters and configuration for data
+generation, preprocessing, and the neural networks. We employ these setups to
+train the PINNs to identify the reproduction number on both synthetic and
+real-world data.\\
+
+For the purposes of validation, we create a synthetic dataset, by setting the parameter
+of $\alpha$ and the reproduction value each to a specific values, and solving~\Cref{eq:reduced_sir_ODE}
+for a given time interval. We set $\alpha=\nicefrac{1}{3}$ and $\Rt$ to the
+values as can be seen in~\Cref{fig:synthetic_I_r_t} as well as the population
+size $N=\expnumber{7.6}{6}$ and the initial amount of infected people to
+$I_0=10$. Furthermore, we set our simulated time span to 150 days. We use this
+dataset to demonstrate, that our method is working on a simple and minimal
+dataset.\\ To obtain a dataset of the infectious group, consisting of the
+real-world data, we we processed the data of the dataset
+\emph{COVID-19-Todesfälle in Deutschland} to extract the number of infections
+in Germany as a whole. For the German states, we use the data of \emph{SARS-CoV-2
+    Infektionen in Deutschland}. In the preprocessing stage, we employ a constant
+rate for $\alpha$ to move individuals into the removed compartment. For each
+state we generate two datasets with a different recovery rate. First, we choose
+$\alpha = \nicefrac{1}{14}$, which aligns with the time of recovery\footnote{\url{https://github.com/robert-koch-institut/SARS-CoV-2-Infektionen_in_Deutschland.git}}.
+Second, we use $\alpha=\nicefrac{1}{5}$, as 5 days into the infection is the
+point at which the infectiousness is at its peak\footnote{\url{https://www.infektionsschutz.de/coronavirus/fragen-und-antworten/ansteckung-uebertragung-und-krankheitsverlauf/}}.
+As in~\Cref{sec:sir}, we set the population size $N$ of each state and Germany
+to the corresponding size at the end of 2019. Furthermore, for the same reason
+we restrict the data points to an interval of 1200 days, beginning on March 09.
+2020.\\
 
-    \end{picture}
-    \caption{Visualization of the datasets for the training process.
-        Illustration (a) is the synthetic data. For the real-world data we use a
-        dataset with $\alpha=\nicefrac{1}{14}$ and $\alpha=\nicefrac{1}{5}$ each.
-        (b) and (c) for Germany, (d) and (g) for Nordrhein-Westfalen (NRW), (e) and (h)
-        for Hessen, and (f) and (i) for Thüringen.}
-    \label{fig:i_datasets}
-\end{figure}
-
-For this task the chosen architecture of the neural network consists of 4 hidden
-layers with each 100 neurons. The activation function is the tangens
-hyperbolicus function tanh. We weight the data loss with a weight of
-$\expnumber{1}{6}$ into the total loss. The model is trained using a base
-learning rate of $\expnumber{1}{-3}$ with the same scheduler and optimizer as
-we use in~\Cref{sec:sir:setup}. We train the model for 20000 epochs. Also, we
-conduct each experiment 15 times to reduce the standard deviation.
+\begin{figure}[t]
+    \centering
+    \begin{subfigure}{0.3\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{I_synth.pdf}
+        \caption{Synthetic data}
+        \label{fig:synthetic_I}
+    \end{subfigure}
+    \quad
+    \begin{subfigure}{0.3\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{I_synth_r_t.pdf}
+        \caption{Synthetic data}
+        \label{fig:synthetic_I_r_t}
+    \end{subfigure}
+    \vskip\baselineskip
+    \begin{subfigure}{0.67\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{datasets_states/Germany_datasets.pdf}
+        \caption{}
+        \label{fig:germany_I_14}
+    \end{subfigure}
 
+\end{figure}
 
+In order to achieve the desired output, the selected neural network
+architecture comprises of four hidden layers, each containing 100 neurons. The
+activation function is the tangens hyperbolicus function. For the real-world
+data, we weight the data loss by a factor of $\expnumber{1}{6}$, to the total
+loss. The model is trained using a base learning rate of $\expnumber{1}{-3}$,
+with the same scheduler and optimizer as we describe in~\Cref{sec:sir:setup}.
+We train the model for 20000 epochs. To reduce the standard deviation, each
+experiment is conducted 15 times.\\
 
 % -------------------------------------------------------------------
 
 \subsection{Results   4}
 \label{sec:rsir:results}
 
+In this section we provide the results for our experiments. First, we present
+our findings for the synthetic dataset. Then, we provide and discuss the
+results for the real-world data.\\
+
+\begin{figure}
+    \centering
+    \includegraphics[width=\textwidth]{synthetic_R_t_statistics.pdf}
+    \caption{text}
+    \label{fig:synth_r_t_results}
+\end{figure}
+
+
 % -------------------------------------------------------------------