Merge branch 'master' of https://git.inf-cv.uni-jena.de/rothenbeck/Thesis

chapters/chap03/chap03.tex

@@ -240,48 +240,49 @@ reproduction number $\Rt$ on the German data of the RKI.
 \section{Estimating the Reproduction Number using PINNs   2}
 \label{sec:pinn:rsir}
 
-The previous section, shows the methodology we utilize to ascertain the
-non-time-dependent transmission and recovery rates from a data set obtained from
-the COVID-19 pandemic in Germany. In this section we employ PINNs to identify
-the time-dependent reproduction number $\Rt$, while reducing the number of state
-variables and the reliance on assumptions, by reducing the system of ODEs
+The previous section illustrates the methodology we employ to detemine the
+constant transmission and recovery rates from a data set obtained from
+the COVID-19 pandemic in Germany. In this section, we utilize PINNs to identify
+the time-dependent reproduction number, $\Rt$, while reducing the number of
+state variables and the reliance on assumptions, by reducing the system of ODEs
 comprising the SIR model. The methodology presented in this section is based on
 the approach developed by Millevoi \etal~\cite{Millevoi2023}.\\
 
-In real-world pandemics the rate of infection is affected by a multitude of
-factors. Events like the rising awareness for the disease in the population, the
-implementation of non-pharmaceutical mitigations such as social distancing
-policies, and the emergence of a new variants have an impact on the transmission
-rate $\beta$. Accordingly, a transmission rate that is not time-dependent and
-constant across the whole duration of the pandemic may not accurately reflect
-the dynamics of the spread of a real-world disease correctly. Although we set
-the transmission rate to be time-dependent, the recovery time is assumed to be
-relatively constant in time. The Robert Koch
+In real-world pandemics, the rate of infection is influenced by a multitude of
+factors. Events such as the growing awareness for the disease among the general
+population, the introduction of non-pharmaceutical mitigations such as social
+distancing policies, and the emergence of a new variants have an impact on the
+transmission rate $\beta$. Accordingly, a transmission rate that is not
+time-dependent and constant across the entire duration of the pandemic may not
+accurately reflect the dynamics of the spread of a real-world disease correctly.
+Although we set the transmission rate to be time-dependent, the recovery time
+is assumed to be relatively constant over time. The Robert Koch
 Institute\footnote{\url{https://github.com/robert-koch-institut/SARS-CoV-2-Infektionen_in_Deutschland.git}}
 posits that the typical recovery period for the illness under normal conditions
-is 14 days, while those individuals with severe cases take about 28 days to
-recover. Given the negligible number of severe cases compared to the number of
-normal cases, we can set the recovery time to $D=14$ resulting in $\alpha = \nicefrac{1}{14}$.
-The reproduction number, $\Rt$ (see~\Cref{sec:pandemicModel:rsir}), represents
-the number of infections that occur as a result of one infectious individual. It
-indicates if a pandemic is emerging or if it is spreading rapidly through the
-susceptible population. By inserting the definition~\Cref{eq:repr_num}, into the
-system of ODEs of the SIR model, we can derive one~\Cref{eq:reduced_sir_ODE}. In
-order to solve this, we must identify a function that maps a time point to the
-size of the infectious compartment and the specific reproduction number.\\
+is 14 days, while those individuals with severe cases require approximately 28
+days to recover. In the light of the negligible number of severe cases in
+comparison to the number of normal cases, we can set the recovery time to
+$D=14$, which yields $\alpha = \nicefrac{1}{14}$. The reproduction number,
+$\Rt$ (see~\Cref{sec:pandemicModel:rsir}), represents the number of new
+infections that occur as a result of one infectious individual. It indicates
+whether a pandemic is emerging or if it is spreading rapidly through the susceptible
+population. By inserting the definition of~\Cref{eq:repr_num}, into the system
+of ODEs of the SIR model, we can derive one~\Cref{eq:reduced_sir_ODE}. In order
+to solve this, we must identify a function that maps a time point to the size
+of the infectious compartment and the specific reproduction number.\\
 
 As with the constant transition rates, we employ a data-driven approach for
 identifying the time-dependent reproduction number $\Rt$. The PINN approximates
-the size ,$\boldsymbol{I}$, with its model prediction $\hat{\boldsymbol{I}}$ by
+the size $\boldsymbol{I}$ with its model prediction $\hat{\boldsymbol{I}}$ by
 minimizing the term,
 \begin{equation}\label{eq:rSir_squared_err}
     \Big\|\hat{\boldsymbol{I}}^{(i)}-\boldsymbol{I}^{(i)}\Big\|^2,
 \end{equation}
 for each $i\in\{1,...,N_t\}$. In order to identify the reproduction number, the
 PINN minimizes the residuals of the ODE during the training process. The
-training process is analogous to the one of the PINN, which identifies $\beta$
-and $alpha$ (see~\Cref{sec:pinn:sir}). The distinction lies in the absence of
-trainable parameters and the inclusion of an additional state variable that
+training process is analogous to that of the PINN, which identifies $\beta$
+and $\alpha$ (see~\Cref{sec:pinn:sir}). However, there are two key differences. Firstly, the absence of
+trainable parameters. Secondly, the inclusion of an additional state variable that
 fluctuates in response to the input. While the state variable $\boldsymbol{I}$
 is approximated using the error between the training data and the predicted
 values, the state variable $\Rt$ is approximated exclusively based on the

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
             Thuringia              & 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 +248,121 @@ 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 considering the
+reduced SIR model and the reproduction number $\Rt$. First, we present
+our findings for the synthetic dataset. Then, we provide and discuss the
+results for the real-world data.\\
+
+In~\Cref{fig:synth_results}
+
+\begin{figure}[t]
+    \centering
+    \begin{subfigure}{0.45\textwidth}
+        \includegraphics[width=\textwidth]{synthetic_I_prediction.pdf}
+    \end{subfigure}
+    \quad
+    \begin{subfigure}{0.45\textwidth}
+        \includegraphics[width=\textwidth]{synthetic_R_t_statistics.pdf}
+    \end{subfigure}
+    \label{fig:synth_results}
+    \caption{text}
+\end{figure}
+
+\begin{figure}[t]
+    \centering
+    \begin{subfigure}{0.45\textwidth}
+        \includegraphics[width=\textwidth]{R_t/Sachsen_Anhalt_R_t_statistics.pdf}
+    \end{subfigure}
+    \quad
+    \begin{subfigure}{0.45\textwidth}
+        \includegraphics[width=\textwidth]{R_t/Thueringen_R_t_statistics.pdf}
+    \end{subfigure}
+
+    \vskip\baselineskip
+    \begin{subfigure}{0.45\textwidth}
+        \includegraphics[width=\textwidth]{R_t/Bremen_R_t_statistics.pdf}
+    \end{subfigure}
+    \quad
+    \begin{subfigure}{0.45\textwidth}
+        \includegraphics[width=\textwidth]{R_t/Hessen_R_t_statistics.pdf}
+    \end{subfigure}
+    \vskip\baselineskip
+    \begin{subfigure}{0.45\textwidth}
+        \includegraphics[width=\textwidth]{R_t/Mecklenburg_Vorpommern_R_t_statistics.pdf}
+    \end{subfigure}
+    \quad
+    \begin{subfigure}{0.45\textwidth}
+        \includegraphics[width=\textwidth]{R_t/Baden_Wuerttemberg_R_t_statistics.pdf}
+    \end{subfigure}
+    \label{fig:state_results}
+    \caption{text}
+\end{figure}
+
+
 % -------------------------------------------------------------------

chapters/conclusions/conclusions.tex

@@ -5,14 +5,88 @@
 % Part:     conclusions
 % Description:
 %         summary of the content in this chapter
-% Version:  01.01.2012
+% Version:  01.09.2024
 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \chapter{Conclusions  5}
 \label{chap:conclusions}
 
+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. Bremen has the highest ratio of vaccinated individuals,
+this might be a reason for the it having the lowest transmission of all states.\\
+
+% -------------------------------------------------------------------
+
 \section{Further Work}
 \label{sec:furtherWork}
+Our findings demonstrate that with our methods enable the quantification of the
+course of the COVID-19 pandemic in Germany using the data provided by the
+Robert Koch Institute. Additionally, we present the limitations of our work.
+The SIR model is subject to numerous limitations. For instance, it does not
+account for individuals, who may be immune due to the vaccination status or
+those who are not infectious due to quarantine. In this section, we explore
+epidemiological models that illustrate these dynamics observed in real-world
+pandemics and recommend further investigation for Germany. First, we examine
+extensions of the SIR models, then we focus on agent-based models (ABMs).
 
 % -------------------------------------------------------------------
 
-% insert further sections if necessary
+\subsection{Further Compartmental Models}
+As our results demonstrate, the SIR model is capable of approximating the
+dynamics of real-world pandemics. However, the model is not without
+limitations. As previously stated, the SIR model assumes that recovered
+individuals remain immune and does not account for the reduction of exposure of
+susceptible individuals through the introduction of non-pharmaceutical
+mitigation policies, such as social distancing policies. These shortcomings can
+be addressed by incorporating additional compartments and transmission rates
+into the model. For example, the SEIRD model incorporates an \emph{Exposed}
+group and subdivides the \emph{Removed} group into \emph{Dead} and
+\emph{Recovered} compartments. Furthermore, this adds four additional rates to
+the model: the contact rate, representing the average number of contacts
+between infectious and susceptible people with a high probability of infection;
+the manifestation index, indicating the proportion of individuals exposed to
+the disease who will become infectious; the incubation rate, measuring the time
+required for exposed individuals to become infectious; and the infection
+fatality rate, quantifying the fraction of individuals who succumb to the
+disease. As Doerre and Doblhammer~\cite{Doerre2022} show for Germany using a
+numerical approximation method, for an SIERD model that they specialize to be
+age- and gender-specific, that it shows the impact of non-pharmaceutical
+mitigation policies. In their work, Cooke and van den Driessche~\cite{Cooke1996}
+propose the SEIRS model with two delays. This is model is capable of
+approximating diseases, that have an immune period, after which the recovered
+individual becomes susceptible again. These are just a few examples of
+the numerous modifications of the basic SIR model that can be used to
+approximate and consequently quantify an pandemic.
+
+% -------------------------------------------------------------------
+
+\subsection{Agent based models}
+
+While compartmental models, such as the SIR model, look at the population as a
+divided group, with each group representing a specific characterization that
+all inhabitants of that group share, an \emph{Agent-Based Model} (ABM) sets its
+focus on the individual. Each individual, or agent, has specific attributes
+that determine its behavior and interactions with other agents during the
+simulation. As Gilbert~\cite{Gilbert2010} states, ABMs simulate the behavior of
+large groups, with each individual following simple rules. Kerr
+\etal~\cite{Kerr2021} put forth a simulation tool, \emph{Covasim}, which they
+base on an ABM. The ABM employs local data, including demographic data, disease
+incidence data from the region, and contact data for household, schools and
+workplaces, to define its simulation for a specific region. In their work,
+Maziarz and Zach~\cite{Maziarz2020} address the criticism levied against ABMs
+for simplifying the dynamics and lacking the empirical support for the
+assumptions it they make. The authors utilize an ABM and the data specific to
+Australia to demonstrate the efficacy of ABMs in portraying the dynamics of the
+COVID-19 pandemic. They further state that ABMs can serve as serve as a tool
+for assessing the impact of non-pharmaceutical mitigation policies. This
+illustrates that ABMs play a distinct role in analyzing the COVID-19 pandemic.
+As the data situation has evolved, it is imperative to investigate the
+potential of utilizing ABMs as a tool to assess the pandemic's course.
+
+% -------------------------------------------------------------------

header.tex

@@ -3,6 +3,7 @@
 % You may want to use these packages or edit some options
 % Feel free to insert more packages, if you need them
 
+\usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}
 
 % some usefull packages, if you do not need them, comment them out

thesis.bbl

@@ -1,4 +1,5 @@
-\begin{thebibliography}{RHW86}
+\newcommand{\etalchar}[1]{$^{#1}$}
+\begin{thebibliography}{KSM{\etalchar{+}}21}
 
 % this bibliography is generated by alphadin.bst [8.2] from 2005-12-21
 
@@ -22,6 +23,25 @@
 \newblock DOI 10.1186/s13662--022--03733--5. --
 \newblock ISSN 2731--4235
 
+\bibitem[CD96]{Cooke1996}
+\textsc{Cooke}, K.~L. ; \textsc{Driessche}, P. van~d.:
+\newblock Analysis of an SEIRS epidemic model with two delays.
+\newblock {In: }\emph{Journal of Mathematical Biology} 35 (1996), Dezember, Nr.
+  2, S. 240--260.
+\newblock \url{http://dx.doi.org/10.1007/s002850050051}. --
+\newblock DOI 10.1007/s002850050051. --
+\newblock ISSN 1432--1416
+
+\bibitem[DD22]{Doerre2022}
+\textsc{Doerre}, Achim ; \textsc{Doblhammer}, Gabriele:
+\newblock The influence of gender on COVID-19 infections and mortality in
+  Germany: Insights from age- and gender-specific modeling of contact rates,
+  infections, and deaths in the early phase of the pandemic.
+\newblock {In: }\emph{PLOS ONE} 17 (2022), Mai, Nr. 5, S. e0268119.
+\newblock \url{http://dx.doi.org/10.1371/journal.pone.0268119}. --
+\newblock DOI 10.1371/journal.pone.0268119. --
+\newblock ISSN 1932--6203
+
 \bibitem[Dem21]{Demtroeder2021}
 \textsc{Demtröder}, Wolfgang:
 \newblock \emph{Lehrbuch}. Bd.~1: {\emph{Experimentalphysik 1}}.
@@ -43,6 +63,14 @@
 \newblock MIT Press, 2016. --
 \newblock \url{http://www.deeplearningbook.org}
 
+\bibitem[Gil10]{Gilbert2010}
+\textsc{Gilbert}, G.~N.:
+\newblock \emph{Agent-based models}.
+\newblock 3. pr.
+\newblock Los Angeles [u.a.] : Sage Publ., 2010 (Quantitative applications in
+  the social sciences 153). --
+\newblock ISBN 978--1--4129--4964--4
+
 \bibitem[HSW89]{Hornik1989}
 \textsc{Hornik}, Kurt ; \textsc{Stinchcombe}, Maxwell  ; \textsc{White},
   Halbert:
@@ -62,6 +90,25 @@
 \newblock DOI 10.1098/rspa.1927.0118. --
 \newblock ISSN 2053--9150
 
+\bibitem[KSM{\etalchar{+}}21]{Kerr2021}
+\textsc{Kerr}, Cliff~C. ; \textsc{Stuart}, Robyn~M. ; \textsc{Mistry}, Dina ;
+  \textsc{Abeysuriya}, Romesh~G. ; \textsc{Rosenfeld}, Katherine ;
+  \textsc{Hart}, Gregory~R. ; \textsc{Núñez}, Rafael~C. ; \textsc{Cohen},
+  Jamie~A. ; \textsc{Selvaraj}, Prashanth ; \textsc{Hagedorn}, Brittany ;
+  \textsc{George}, Lauren ; \textsc{Jastrzębski}, Michał ; \textsc{Izzo},
+  Amanda~S. ; \textsc{Fowler}, Greer ; \textsc{Palmer}, Anna ;
+  \textsc{Delport}, Dominic ; \textsc{Scott}, Nick ; \textsc{Kelly}, Sherrie~L.
+  ; \textsc{Bennette}, Caroline~S. ; \textsc{Wagner}, Bradley~G. ;
+  \textsc{Chang}, Stewart~T. ; \textsc{Oron}, Assaf~P. ; \textsc{Wenger},
+  Edward~A. ; \textsc{Panovska-Griffiths}, Jasmina ; \textsc{Famulare}, Michael
+   ; \textsc{Klein}, Daniel~J.:
+\newblock Covasim: An agent-based model of COVID-19 dynamics and interventions.
+\newblock {In: }\emph{PLOS Computational Biology} 17 (2021), Juli, Nr. 7, S.
+  e1009149.
+\newblock \url{http://dx.doi.org/10.1371/journal.pcbi.1009149}. --
+\newblock DOI 10.1371/journal.pcbi.1009149. --
+\newblock ISSN 1553--7358
+
 \bibitem[LLF97]{Lagaris1997}
 \textsc{Lagaris}, I.~E. ; \textsc{Likas}, A.  ; \textsc{Fotiadis}, D.~I.:
 \newblock Artificial Neural Networks for Solving Ordinary and Partial
@@ -96,6 +143,16 @@
 \newblock \url{http://dx.doi.org/10.48550/ARXIV.2311.09944}. --
 \newblock DOI 10.48550/ARXIV.2311.09944
 
+\bibitem[MZ20]{Maziarz2020}
+\textsc{Maziarz}, Mariusz ; \textsc{Zach}, Martin:
+\newblock Agent‐based modelling for SARS‐CoV‐2 epidemic prediction and
+  intervention assessment: A methodological appraisal.
+\newblock {In: }\emph{Journal of Evaluation in Clinical Practice} 26 (2020),
+  August, Nr. 5, S. 1352--1360.
+\newblock \url{http://dx.doi.org/10.1111/jep.13459}. --
+\newblock DOI 10.1111/jep.13459. --
+\newblock ISSN 1365--2753
+
 \bibitem[OKF21]{Olumoyin2021}
 \textsc{Olumoyin}, K.~D. ; \textsc{Khaliq}, A. Q.~M.  ; \textsc{Furati}, K.~M.:
 \newblock Data-Driven Deep-Learning Algorithm for Asymptomatic COVID-19 Model

thesis.bib

@@ -294,4 +294,76 @@
   publisher = {MDPI AG},
 }
 
+@Article{Kerr2021,
+  author    = {Kerr, Cliff C. and Stuart, Robyn M. and Mistry, Dina and Abeysuriya, Romesh G. and Rosenfeld, Katherine and Hart, Gregory R. and Núñez, Rafael C. and Cohen, Jamie A. and Selvaraj, Prashanth and Hagedorn, Brittany and George, Lauren and Jastrzębski, Michał and Izzo, Amanda S. and Fowler, Greer and Palmer, Anna and Delport, Dominic and Scott, Nick and Kelly, Sherrie L. and Bennette, Caroline S. and Wagner, Bradley G. and Chang, Stewart T. and Oron, Assaf P. and Wenger, Edward A. and Panovska-Griffiths, Jasmina and Famulare, Michael and Klein, Daniel J.},
+  journal   = {PLOS Computational Biology},
+  title     = {Covasim: An agent-based model of COVID-19 dynamics and interventions},
+  year      = {2021},
+  issn      = {1553-7358},
+  month     = jul,
+  number    = {7},
+  pages     = {e1009149},
+  volume    = {17},
+  doi       = {10.1371/journal.pcbi.1009149},
+  editor    = {Marz, Manja},
+  publisher = {Public Library of Science (PLoS)},
+}
+
+@Article{Doerre2022,
+  author    = {Doerre, Achim and Doblhammer, Gabriele},
+  journal   = {PLOS ONE},
+  title     = {The influence of gender on COVID-19 infections and mortality in Germany: Insights from age- and gender-specific modeling of contact rates, infections, and deaths in the early phase of the pandemic},
+  year      = {2022},
+  issn      = {1932-6203},
+  month     = may,
+  number    = {5},
+  pages     = {e0268119},
+  volume    = {17},
+  doi       = {10.1371/journal.pone.0268119},
+  editor    = {Cheong, Siew Ann},
+  publisher = {Public Library of Science (PLoS)},
+}
+
+@Article{Cooke1996,
+  author    = {Cooke, K. L. and van den Driessche, P.},
+  journal   = {Journal of Mathematical Biology},
+  title     = {Analysis of an SEIRS epidemic model with two delays},
+  year      = {1996},
+  issn      = {1432-1416},
+  month     = dec,
+  number    = {2},
+  pages     = {240--260},
+  volume    = {35},
+  doi       = {10.1007/s002850050051},
+  publisher = {Springer Science and Business Media LLC},
+}
+
+@Book{Gilbert2010,
+  author    = {Gilbert, G. Nigel},
+  publisher = {Sage Publ.},
+  title     = {Agent-based models},
+  year      = {2010},
+  address   = {Los Angeles [u.a.]},
+  edition   = {3. pr.},
+  isbn      = {978-1-4129-4964-4},
+  number    = {153},
+  series    = {Quantitative applications in the social sciences},
+  pagetotal = {98},
+  ppn_gvk   = {1615580204},
+}
+
+@Article{Maziarz2020,
+  author    = {Maziarz, Mariusz and Zach, Martin},
+  journal   = {Journal of Evaluation in Clinical Practice},
+  title     = {Agent‐based modelling for SARS‐CoV‐2 epidemic prediction and intervention assessment: A methodological appraisal},
+  year      = {2020},
+  issn      = {1365-2753},
+  month     = aug,
+  number    = {5},
+  pages     = {1352--1360},
+  volume    = {26},
+  doi       = {10.1111/jep.13459},
+  publisher = {Wiley},
+}
+
 @Comment{jabref-meta: databaseType:bibtex;}