add comments

abstract.tex

@@ -1,37 +1,38 @@
 \begin{center}{\sectfont\LARGE \"{U}berblick}\end{center}
 
-Deutschland wurde, wie viele andere Länder, von der schweren COVID-19-Pandemie
-betroffen, die im Jahr 2020 begann und bis 2023 andauerte. Die hohe Zahl der
-Infektionen und Todesfälle erfordert eine Analyse. Das kompartimentäre
-SIR-Modell bietet eine Reihe von Metriken für eine solche Analyseaufgabe,
-darunter die Übertragungsrate $\beta$, die Erholungsrate $\alpha$ und die
-Reproduktionszahl $\Rt$. Diese Werte zeigen die Ausbreitung einer Krankheit an
-und können durch die Lösung des maßgeblichen Systems von
-Differentialgleichungen des SIR-Modells ermittelt werden. Ziel dieser Arbeit
-ist es also, diese Parameter und Werte für Deutschland zu finden, indem ein
-datengesteuerter Ansatz zur Lösung der Differentialgleichungen unter Verwendung
-eines physikalisch informierten neuronalen Netzes verwendet wird. Dazu
-nutzen wir die vom Robert-Koch-Institut erhobenen Daten und bereiten sie für
-unsere Zwecke auf. Schließlich zeigen wir, dass unsere Herangehensweise an
-diese Aufgabe zu einem erfolgreichen Ergebnis führt. Trotz der unvollkommenen
-Genauigkeit unserer Methode finden wir eine Korrelation zwischen unseren
-Ergebnissen und den realen Ereignissen während der COVID-19 Pandemie in
-Deutschland, was die Effektivität unserer Methode unterstreicht.
+Deutschland war, wie zahlreiche andere Länder, von der im Jahr 2019
+ausgebrochenen und bis 2023 andauernden COVID-19-Pandemie betroffen. Aufgrund der hohen Zahl der
+Infektionen und Todesfälle ist eine Analyse erfordert. Das kompartimentelle SIR-Modell
+bietet eine Reihe von Metriken für eine solche Analyseaufgabe, darunter die
+Übertragungsrate $\beta$, die Erholungsrate $\alpha$ und die Reproduktionszahl
+$\Rt$. Diese Werte zeigen die Ausbreitung einer Krankheit an und lassen sich durch
+die Lösung des dem SIR-Modell grundlegenden Systems von Differentialgleichungen
+ermitteln. Ziel dieser Arbeit ist es also, diese Parameter und Werte für
+Deutschland zu finden. Dazu wird ein datengesteuerter Ansatz zur Lösung der
+Differentialgleichungen unter Verwendung eines physikalisch informierten
+neuronalen Netzes genutzt. Zu diesem Zweck verwenden wir die vom
+Robert-Koch-Institut gesammelten Daten und bereiten sie für unsere Ziele auf.
+Mit unserem Modell sind wir in der Lage, sowohl die Pandemiedaten als auch das
+Gleichungssystem des SIR-Modells so zu rekonstruieren, dass wir entsprechende
+epidemiologische Parameter und Reproduktionszahlen finden. Diese korrelieren mit den realen
+Ereignissen während der COVID-19 Pandemie in Deutschland, was die
+Wirksamkeit unserer Methode unterstreicht.
 
 \begin{center}{\sectfont\LARGE Abstract}\end{center}
 
 Germany, like many other countries, was hit by the severe COVID-19 pandemic
 that began in 2020 and continued through 2023. The amounted infection and death
-counts call for an analysis. The compartmental SIR model provides a number of
-metrics for such an analysis task, including the transmission rate $\beta$,
-the recovery rate $\alpha$, and the reproduction number $\Rt$. These values
-demonstrate the propagation of a disease and can be identified by solving the governing system
-of differential equations of the SIR model. Thus the objective of this thesis is
-find these parameters and values for Germany, by employing a data-driven
-approach to solve differential equations utilizing physics-informed neural
-network. Therefor, we use the data which was collected by the Robert Koch
-Institute and preprocess it for our means. Finally, we show, that our approach
-to this task yields successful result. Despite the imperfect accuracy of our
-method, we find a correlation between our results and the real-world events
-during COVID-19 in Germany, which highlights the efficacy of our method.
+counts call for an in-depth analysis. The compartmental SIR model provides a
+number of metrics for such an analysis task, including the transmission rate
+$\beta$, the recovery rate $\alpha$, and the reproduction number $\Rt$. These
+values demonstrate the propagation of a disease and can be identified by solving
+the governing system of differential equations of the SIR model. In this thesis,
+we find these parameters and values for Germany, by employing a data-driven
+approach to solve the differential equations employing physics-informed neural
+network. Towards this objective, we use the data collected by the Robert Koch
+Institute and preprocess it for our means. Utilizing our model, we are able to
+fit both the pandemic data as well as the governing system of equations. Hence,
+we are able to find corresponding epidemiological parameters and reproduction
+numbers, which correlate with the real-world events during COVID-19 in Germany,
+which highlights the efficacy of out method.
 \cleardoublepage

chapters/appendix/appendix.tex

@@ -8,30 +8,46 @@
 % Version:  19.07.2024
 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\chapter{Appendix}
+\chapter{Additional Results}
 \label{chap:appendix}
 
-\begin{figure}[t]
+Here, we show the results of our experiments which we describe in~\Cref{chap:evaluation}.
+Additionally, we show visualizations of the underlying dataset.
+
+\subsection{SIR Datasets}
+\label{sec:sir_datasets}
+In this section, we present the datasets utiized for~\Cref{sec:sir}.~\Cref{fig:SIR_state_results_1}
+and ~\Cref{fig:SIR_state_results_2} show the datasets, which we use for finding
+the epidemiological parameters of $\alpha$ and $\beta$.
+
+\begin{figure}[h!]
     \centering
     \includegraphics[width=\textwidth]{state_sir_cluster_1.pdf}
-    \caption{Part 1 of the results}
+    \caption{Part 1 of the datasets}
     \label{fig:SIR_state_results_1}
 \end{figure}
 
-\begin{figure}[t]
+\begin{figure}[h!]
     \centering
     \begin{subfigure}{\textwidth}
         \includegraphics[width=\textwidth]{state_sir_cluster_2.pdf}
     \end{subfigure}
     \quad
-    \begin{subfigure}{0.4\textwidth}
+    \begin{subfigure}{0.35\textwidth}
         \includegraphics[width=\textwidth]{germany_single_sir.pdf}
     \end{subfigure}
-    \caption{Part 1 of the results}
+    \caption{Part 1 of the datasets}
     \label{fig:SIR_state_results_2}
 \end{figure}
 
-\begin{figure}[t]
+\subsection{$\Rt$ Results}
+\label{sec:r_t_results}
+Here, we present the results from our experiments in ~\Cref{sec:rsir}.
+For each federal state and Germany we provide the results of the reproduction
+number in the left and the corresponding dataset and the model prediction on the
+right.
+
+\begin{figure}[h!]
     \centering
     \begin{subfigure}{0.45\textwidth}
         \includegraphics[width=\textwidth]{r_t_cluster_0.pdf}
@@ -43,8 +59,7 @@
     \label{fig:I_state_results_1}
 \end{figure}
 
-
-\begin{figure}[t]
+\begin{figure}[h!]
     \centering
     \begin{subfigure}{0.45\textwidth}
         \includegraphics[width=\textwidth]{r_t_cluster_1.pdf}
@@ -56,7 +71,7 @@
     \label{fig:I_state_results_2}
 \end{figure}
 
-\begin{figure}[t]
+\begin{figure}[h!]
     \centering
     \begin{subfigure}{0.45\textwidth}
         \includegraphics[width=\textwidth]{r_t_cluster_2.pdf}
@@ -68,7 +83,7 @@
     \label{fig:I_state_results_3}
 \end{figure}
 
-\begin{figure}[t]
+\begin{figure}[h!]
     \centering
     \begin{subfigure}{0.45\textwidth}
         \includegraphics[width=\textwidth]{r_t_cluster_3.pdf}

chapters/chap01-introduction/chap01-introduction.tex

@@ -11,65 +11,67 @@
 \chapter{Introduction}
 \label{chap:introduction}
 
-In the early months of 2020, Germany, like many other countries, was struck by the novel
-\emph{Coronavirus Disease} (COVID-19)~\cite{WHO}. The pandemic, which originates in
-Wuhan, China, had a profound impact on the global community, paralyzing it for
-over two years. In response to the pandemic, the German government employed a
-multifaceted approach~\cite{RKI}, encompassing the introduction of vaccines and
-non-pharmaceutical mitigation policies such as lockdowns. Between mitigation
-policies and varying strains of COVID-19, which have exhibited varying degrees
-of infectiousness and lethality~\cite{RKIa}, Germany had recorded over 38,400,000 infection
-cases and 174,000 deaths, as of the end of June in 2023~\cite{SRD}. In light of these
-figures the need for an analysis arises.\\
+In the early months of 2020, Germany, like many other countries, was struck by
+the novel \emph{Coronavirus Disease} (COVID-19)~\cite{WHO}. The pandemic, which
+originates in Wuhan, China, had a profound impact on the global community,
+paralyzing it for over two years. In response to the pandemic, the German
+government employed a multifaceted approach~\cite{RKI}, encompassing the
+introduction of vaccines and non-pharmaceutical mitigation policies such as
+lockdowns. Between mitigation policies and varying strains of COVID-19, which
+have exhibited varying degrees of infectiousness and lethality~\cite{RKIa},
+Germany had recorded over 38,400,000 infection cases and 174,000 deaths, as of
+the end of June in 2023~\cite{SRD}. In light of these figures the need for an
+analysis arises.\\
 
 The dynamics of the spread of disease transmission in the real-world are
 complex. A multitude of factors influence the course of a disease, and it is
 challenging to gain a comprehensive understanding of these factors and develop
-tools that allows for the comparison of disease courses across different diseases
-and time points. The common approach in epidemiology to address this is the
-utilization of epidemiological models that approximate the dynamics by focusing
-on specific factors and modeling these using mathematical tools. These models
-provide epidemiological parameters that determine the behavior of a disease
-within the boundaries of the model. A seminal epidemiological model, is the
-\emph{SIR model}, which was first proposed by Kermack and McKendrick~\cite{1927}
+tools that allows for the comparison of disease courses across different
+diseases and time points. The common approach in epidemiology to address this is
+the utilization of epidemiological models that approximate the dynamics by
+focusing on specific factors and modeling these using mathematical tools. These
+models provide epidemiological parameters that determine the behavior of a
+disease within the boundaries of the model. A seminal epidemiological model is
+the \emph{SIR model}, which was first proposed by Kermack and McKendrick~\cite{1927}
 in 1927. The SIR model is a compartmentalized model that divides the entire
-population into three distinct groups: the \emph{susceptible} compartment, $S$; the
-\emph{infectious} compartment, $I$; and the \emph{removed} compartment, $R$.
+population into three distinct groups: the \emph{susceptible} compartment, $S$;
+the \emph{infectious} compartment, $I$; and the \emph{removed} compartment, $R$.
 In the context of the SIR model, the constant parameters of the transmission
 rate $\beta$ and the recovery rate $\alpha$ serve to quantify and determine the
-course of a pandemic. However, pandemic is not a static entity, therefor, Liu
-and Stechlinski~\cite{Liu2012}, and Setianto and Hidayat~\cite{Setianto2023},
-propose an SIR model with time-dependent epidemiological parameters and reproduction number $\Rt$. The SIR model
-is defined by a system of differential equations, that incorporate
-the parameters $\alpha$ and $\beta$, thereby depicting the fluctuation between the three
-compartments. For a given set of data, the epidemiological parameters can be identified by
-solving the set of differential systems. Recently, the data-driven approach of
-\emph{physics-informed neural networks} (PINN) has gained attention due to its
-capability of finding solutions to differential equations by fitting its
-predictions to both given data and the governing system of differential
-equations. By employing this methodology, Shaier \etal~\cite{Shaier2021} were
-able to find the epidemiological parameters on data for different diseases. Additionally,
-Millevoi \etal~\cite{Millevoi2023} were able to identify the reproduction number
-$\Rt$ for both synthetic and Italian COVID-19 data using an approach based on a
-reduced version of the SIR model.\\
+course of a pandemic. However, a pandemic is not a static entity, therefore Liu
+and Stechlinski~\cite{Liu2012}, and Setianto and Hidayat~\cite{Setianto2023}
+propose an SIR model with time-dependent epidemiological parameters and
+reproduction number $\Rt$. The SIR model is defined by a system of differential
+equations, that incorporate the parameters $\alpha$ and $\beta$, thereby
+depicting the fluctuation between the three compartments. For a given set of
+data, the epidemiological parameters can be identified by solving the set of
+differential systems. Recently, the data-driven approach of \emph{Physics-Informed Neural Networks}
+(PINN) has gained attention due to its capability of finding solutions to
+differential equations by fitting its predictions to both given data and the
+governing system of differential equations. By employing this methodology,
+Shaier \etal~\cite{Shaier2021} were able to find the epidemiological parameters
+on data for different diseases. Additionally, Millevoi \etal~\cite{Millevoi2023}
+were able to identify the reproduction number $\Rt$ for both synthetic and
+Italian COVID-19 data using an approach based on a reduced version of the SIR
+model.\\
 
-The objective of this thesis is to identify the epidemiological parameters $\beta$ and
-$\alpha$, as well as the reproduction number $\Rt$ of COVID-19 over the first
-1200 days of recorded data in Germany and its federal states. The Robert Koch
-Institute (RKI) has compiled data on both reported cases and associated
-moralities from the beginning of the outbreak in Germany to the present. We
-utilize and preprocess this data according to the required format of our
-approaches. As the raw data lacks information on recovery incidence, we
+The objective of this thesis is to identify the epidemiological parameters
+$\beta$ and $\alpha$, as well as the reproduction number $\Rt$ of COVID-19 over
+the first 1200 days of recorded data in Germany and its federal states. The
+Robert Koch Institute (RKI) has compiled data on both reported cases and
+associated moralities from the beginning of the outbreak in Germany to the
+present. We utilize and preprocess this data according to the required format of
+our approaches. As the raw data lacks information on recovery incidence, we
 introduce the recovery queue that simulates a recovery period. To estimate the
-epidemiological parameters we adopt the approach of Shaier \etal~\cite{Shaier2021}, which
-utilizes a physics-informed neural network learning the data, which consists of
-time point with their respective sizes of  the $S, I$ and $R$ compartments, to
-predict the epidemiological parameters based on the data and the governing system of
-differential equations. Moreover, we utilize the methodology proposed by
-Millevoi \etal~\cite{Millevoi2023} that estimates the reproduction number for
+epidemiological parameters we adopt the approach of Shaier
+\etal~\cite{Shaier2021}, which utilizes a PINN learning the data, which consists
+of time points with their respective sizes of  the $S, I$ and $R$ compartments,
+to predict the epidemiological parameters based on the data and the governing
+system of differential equations. Moreover, we utilize the methodology proposed
+by Millevoi \etal~\cite{Millevoi2023} that estimates the reproduction number for
 each day across the 1200-day span for each German state and Germany as a whole,
-in reduced SIR model. Thus needing only the size of the $I$ group for each time
-step. To validate the effectiveness of these methods, we first conduct
+in the reduced SIR model. Thus needing only the size of the $I$ group for each
+time step. To validate the effectiveness of these methods, we first conduct
 experiments on a small synthetic dataset before applying the techniques to
 real-world data. We then analyze the plausibility of our results by comparing
 them to real-world events and data such as vaccination ratios of each region or
@@ -89,35 +91,35 @@ first work, by Smirnova \etal~\cite{Smirnova2017}, endeavors to identify a
 stochastic methodology for estimating the time-dependent transmission rate
 $\beta(t)$. They achieve this by projecting the time-dependent transmission rate
 onto a finite subspace, that is defined by Legendre polynomials. Subsequently,
-they compare the three regularization techniques of variational (Tikhonov’s)
+they compare the three regularization techniques of variational (Tikhonov's)
 regularization, truncated singular value decomposition (TSVD), and modified TSVD
 to ascertain the most reliable method for forecasting with limited data. Their
 findings indicate that modified TSVD provides the most stable forecasts on
 limited data, as demonstrated on both simulated data and real-world data from
 the 1918 influenza pandemic and the Ebola epidemic. In contrast, we
-utilize physics-informed neural networks (PINN) to find the constant epidemiological parameters
-and the reproduction number for Germany and its states\\
+utilize PINNs to find the constant epidemiological parameters
+and the reproduction number for Germany and its states.\\
 
-Some related works similarly to us apply PINN approaches to COVID-19 and other
-real-world disease data such as~\cite{Shaier2021,Berkhahn2022,Olumoyin2021,Millevoi2023}.
-Specifically in~\cite{Shaier2021}, Shaier \etal put forth a data-driven
-approach which they refer to as disease informed neural networks (DINN). In their work,
-they demonstrate the capacity of DINNs to forecast the trajectory of epidemics
-and pandemics. They underpin the efficacy of their approach by applying it to 11
-diseases, that have previously been modeled. In their experiments they employ
-the SIDR (susceptible, infectious, dead, recovered) model. Finally, they present
-that this method is a robust and effective means of identifying the parameters
-of a SIR model.\\
+Some related works similar to our approach apply PINN approaches to COVID-19 and
+other real-world disease examples~\cite{Shaier2021,Millevoi2023,Berkhahn2022,Olumoyin2021}.
+Specifically Shaier \etal~\cite{Shaier2021} put forth a data-driven approach
+which they refer to as \emph{Disease-Informed Neural Networks} (DINN). In their
+work, they demonstrate the capacity of DINNs to forecast the trajectory of
+epidemics and pandemics. They underpin the efficacy of their approach by
+applying it to 11 diseases, that have previously been modeled. In their
+experiments they employ the SIDR (susceptible, infectious, dead, recovered)
+model. Finally, they present that this method is a robust and effective means of
+identifying the parameters of a SIR model.\\
 
-Similarly in~\cite{Berkhahn2022}, Berkhahn and Ehrhard employ the susceptible,
+Similarly  Berkhahn and Ehrhard~\cite{Berkhahn2022}, employ the susceptible,
 vaccinated, infectious, hospitalized and removed (SVIHR) model. The proposed
 PINN methodology initially estimates the SVIHR model parameters for German
 COVID-19 data, covering the time span from the inceptions of the outbreak to the
 end of 2021. For comparative purposes, Berkhahn and Ehrhard employ the method of
-non-standard finite differences (NSFD) as well.  The authors employ both methods
-the two forecasting methods project the trajectory of COVID-19 from mid-April
-2023 onwards. Berkhahn and Ehrhard find that the PINN is able to adapt to
-varying vaccination rates and emerging variants.\\
+non-standard finite differences (NSFD) as well.  The authors employ both
+forecasting methods project the trajectory of COVID-19 from mid-April 2023
+onwards. Berkhahn and Ehrhard find that the PINN is able to adapt to varying
+vaccination rates and emerging variants.\\
 
 Furthermore, Olumoyin \etal~\cite{Olumoyin2021} employ an alternative
 methodology for identifying the time-dependent transmission rate of an
@@ -134,12 +136,12 @@ Finally, Millevoi \etal~\cite{Millevoi2023} address the issue of the changes in
 the transmission rate due to the dynamics of a pandemic.  The authors employ the
 reproduction number to reduce the system of differential equations to a single
 equation and introduce a reduced-split version of the PINN, which initially
-trains on the data and then trains to minimize the residual of the ODE. They
-test their approach on five synthetic and two real-world scenarios from the
-early stages of the COVID-19 pandemic in Italy. This method yields an increase
-in both accuracy and training speed. In contrast, to these works, we estimate
-the rates and the reproduction number for Germany for the entirety of the span
-from early March in 2020 to late June in 2023.
+trains on the data and then trains to minimize the residual of the ordinary
+differential equation. They test their approach on five synthetic and two
+real-world scenarios from the early stages of the COVID-19 pandemic in Italy.
+This method yields an increase in both accuracy and training speed. In contrast,
+to these works, we estimate the rates and the reproduction number for Germany
+for the entirety of the span from early March in 2020 to late June in 2023.
 
 % -------------------------------------------------------------------
 
@@ -148,17 +150,17 @@ from early March in 2020 to late June in 2023.
 This thesis is comprised of four chapters. \Cref{chap:background}
 presents with the theoretical overview of mathematical modeling in epidemiology,
 with a particular focus on the SIR model. Subsequently, it shifts its focus to
-neural networks, specifically on the background of physics-informed neural
-networks (PINN) and their use in solving ordinary differential equations.~\Cref{chap:methods}
-outlines the methodology employed in this thesis. First
-we present the data, that was collected by the Robert Koch Institute (RKI). Then
-we present the PINN approaches, which are inspired by the work of Shaier \etal~\cite{Shaier2021}
-and Millevoi \etal~\cite{Millevoi2023}.~\Cref{chap:evaluation}
-presents the setups and results of the experiments that we conduct. This chapter
-is divided into two sections. The first section presents and discusses the
-results concerning the epidemiological parameters of $\beta$ and $\alpha$. The subsequent
-section presents the results concerning the reproduction value $\Rt$. Finally,
-in \Cref{chap:conclusions}, we connect our results with the events of the
+neural networks, specifically on the background of PINNs and their use in
+solving ordinary differential equations.~\Cref{chap:methods} outlines the
+methodology employed in this thesis. First we present the data, that was
+collected by the RKI. Then we present the PINN approaches, which are inspired by
+the work of Shaier \etal~\cite{Shaier2021} and Millevoi
+\etal~\cite{Millevoi2023}.~\Cref{chap:evaluation} presents the setups and
+results of the experiments that we conduct. This chapter is divided into two
+sections. The first section presents and discusses the results concerning the
+epidemiological parameters of $\beta$ and $\alpha$. The subsequent section
+presents the results concerning the reproduction value $\Rt$. Finally, in
+\Cref{chap:conclusions}, we connect our results with the events of the
 real-world and give an overview of potential further work.
 
 % -------------------------------------------------------------------

chapters/chap02/chap02.tex

@@ -18,7 +18,7 @@ Rudin~\cite{Rudin2007} and a book about ordinary differential equations by
 Tenenbaum and Pollard~\cite{Tenenbaum1985}. Subsequently, we employ this
 knowledge to examine various pandemic models in~\Cref{sec:epidemModel}. Finally,
 we address the topic of neural networks with a focus on the multilayer
-perceptron in~\Cref{sec:mlp} and physics informed neural networks
+perceptron in~\Cref{sec:mlp} and physics-informed neural networks
 in~\Cref{sec:pinn}.
 
 % -------------------------------------------------------------------
@@ -63,9 +63,10 @@ interval of real numbers. The expression
   m = \frac{f(b) - f(a)}{a-b}
 \end{equation}
 gives the average rate of change. While the average rate of change is useful in
-many cases, the momentary rate of change is more accurate. To calculate this, \todo{look up in Rudin - cite (wordly)}
-we need to narrow down, the interval to an infinitesimal. For each $x\in[a, b]$
-we calculate
+many cases, the momentary rate of change is more accurate. To calculate the
+momentary rate of change at $x$, we let the value $t$ approach $x$ thereby
+narrowing down the interval to an infinitesimal. For each $x\in[a, b]$ we
+calculate
 \begin{equation} \label{eqn:differential}
   \frac{df}{dx} = \lim_{t\to x} \frac{f(t) - f(x)}{t-x},
 \end{equation}
@@ -90,16 +91,16 @@ of financial derivatives, such as options, over time; epidemiology with the SIR
 Model~\cite{1927}; and beyond.\\
 
 In the context of functions, it is possible to have multiple domains, meaning
-that function has more than one parameter. To illustrate, consider a function
+that a function has more than one parameter. To illustrate, consider a function
 operating in two-dimensional space, wherein each parameter represents one axis.
 Another example would be a function, that maps its inputs of a location variable
 and a time variable on a height. The term \emph{partial differential equations}
-(\emph{PDE}'s) describes differential equations of such functions, which contain
-partial derivatives with respect to each individual domain. In contrast, \emph{ordinary differential
-  equations} (\emph{ODE}'s) are the single derivatives for a function having only
-one domain~\cite{Tenenbaum1985}. In this thesis, we restrict ourselves to ODE's.\\
-
-A \emph{system of differential equations} is the name for a set of differential
+(PDE) describes differential equations of such functions, which contain
+partial derivatives with respect to each individual domain. In contrast,
+\emph{ordinary differential equations} (ODE) are the single derivatives for a
+function having only one domain~\cite{Tenenbaum1985}. In this thesis, we
+restrict ourselves to ODE's. Furthermore, a
+\emph{system of differential equations} is the name for a set of differential
 equations. The derivatives in a system of differential equations each have their
 own codomain, which is part of the problem, while they all share the same
 domain.\\
@@ -136,15 +137,15 @@ models.
 \section{Epidemiological Models}
 \label{sec:epidemModel}
 
-Pandemics, like \emph{COVID-19}, which have resulted in a significant
+Pandemics, like \emph{COVID-19}, have resulted in a significant
 number of fatalities. Hence, the question arises: How should we analyze a
 pandemic effectively? It is essential to study whether the employed
 countermeasures are efficacious in combating the pandemic. Given the unfavorable
-public response to measures such as lockdowns, it is imperative to investigate
-that their efficacy remains commensurate with the costs incurred to those
-affected. In the event that alternative and novel technologies were in use, such
-as the mRNA vaccines in the context of COVID-19, it is needful to test the
-effect and find the optimal variant. In order to shed light on the
+public response~\cite{Jaschke2023} to measures such as lockdowns, it is
+imperative to investigate if their efficacy remains commensurate with the costs
+incurred to those affected. In the event that alternative and novel technologies
+were in use, such as the mRNA vaccines in the context of COVID-19, it is needful
+to test the effect and find the optimal variant. In order to shed light on the
 aforementioned events, we need a method to quantify the pandemic along with its
 course of progression.\\
 
@@ -193,10 +194,10 @@ The model makes another assumption by stating that recovered people are immune
 to the illness and infectious individuals can not infect them. The individuals
 in the $R$ group are either recovered or deceased, and thus unable to transmit
 or carry the disease.
-\begin{figure}[h]
+\begin{figure}[t]
   \centering
   \includegraphics[width=\textwidth]{sir_graph.pdf}
-  \caption{A visualization of the SIR model, illustrating $N$ being split in the
+  \caption{A visualization of the SIR model~\cite{1927}, illustrating $N$ being split in the
     three groups $S$, $I$ and $R$.}
   \label{fig:sir_model}
 \end{figure}
@@ -234,8 +235,8 @@ and May~\cite{Anderson1991} propose a modified model:
     \frac{dR}{dt} &= \alpha I.
   \end{split}
 \end{equation}
-In~\Cref{eq:modSIR} $\beta SI$ gets normalized by $N$, which is more correct in
-a real world aspect~\cite{Anderson1991}.\\
+By normylizing $\beta SI$ by $N$  the~\Cref{eq:modSIR} is more correct in a
+real world~aspect~\cite{Anderson1991}.\\
 
 The initial phase of a pandemic is characterized by the infection of a small
 number of individuals, while the majority of the population remains susceptible.
@@ -252,7 +253,7 @@ the number of infected individuals at the beginning of the disease. Then,
 describes the initial configuration of a system in which a disease has just
 emerged.\\
 
-\begin{figure}[h]
+\begin{figure}[t]
   %\centering
   \setlength{\unitlength}{1cm} % Set the unit length for coordinates
   \begin{picture}(12, 9.5) % Specify the size of the picture environment (width, height)
@@ -302,14 +303,16 @@ emerged.\\
       \end{subfigure}
     }
   \end{picture}
-  \caption{Synthetic data, using~\Cref{eq:modSIR} and $N=7.9\cdot 10^6$, $I_0=10$ with different sets of parameters.
-    We visualize the case with the reference parameters in (a). In (b) and (c) we keep $\alpha$ constant, while varying
-    the value of $\beta$. In contrast, (d) and (e) have varying values of $\alpha$.}
+  \caption{Synthetic data, using~\Cref{eq:modSIR} and $N=7.9\cdot 10^6$,
+    $I_0=10$ with different sets of parameters. We visualize the case with the
+    reference parameters in (a). In (b) and (c) we keep $\alpha$ constant, while
+    varying the value of $\beta$. In contrast, (d) and (e) have varying values
+    of $\alpha$.}
   \label{fig:synth_sir}
 \end{figure}
 
 In the SIR model the temporal occurrence and the height of the peak (or peaks)
-of  the infectious group are of paramount importance for understanding the
+of  the infectious group are of great importance for understanding the
 dynamics of a pandemic. A low peak occurring at a late point in time indicates
 that the disease is unable to keep pace with the rate of recovery, resulting
 in its demise before it can exert a significant influence on the population. In
@@ -358,8 +361,8 @@ introduce time-dependent epidemiological parameters and the time-dependent repro
 number to address this issue. Millevoi \etal~\cite{Millevoi2023} present a
 reduced version of the SIR model.\\
 
-First, they alter the definition of $\beta$ and $\alpha$ to be dependent on the time interval
-$\mathcal{T} = [t_0, t_f]\subseteq \mathbb{R}_{\geq0}$,
+For the time interval, $\mathcal{T} = [t_0, t_f]\subseteq \mathbb{R}_{\geq0}$,
+they alter the definition of $\beta$ and $\alpha$ to be time-dependent,
 \begin{equation}
   \beta: \mathcal{T}\rightarrow\mathbb{R}_{\geq0}, \quad\alpha: \mathcal{T}\rightarrow\mathbb{R}_{\geq0}.
 \end{equation}
@@ -476,12 +479,12 @@ output layer, resulting in a scalar loss. The alternating structure of linear
 and nonlinear calculation enables MLP's to approximate any function. As Hornik
 \etal~\cite{Hornik1989} proves, MLP's are universal approximators.\\
 
-\begin{figure}[h]
+\begin{figure}[t]
   \centering
   \includegraphics[width=\textwidth]{MLP.pdf}
-  \caption{A illustration of an MLP with two hidden layers. Each neuron of a layer
-    is connected to every neuron of the neighboring layers. The arrow indicates
-    the direction of the forward propagation.}
+  \caption{A illustration of an MLP~\cite{Rumelhart1986} with two hidden layers.
+    Each neuron of a layer is connected to every neuron of the neighboring
+    layers. The arrow indicates the direction of the forward propagation.}
   \label{fig:mlp_example}
 \end{figure}
 
@@ -542,12 +545,12 @@ solutions to differential systems.
 
 % -------------------------------------------------------------------
 
-\section{Physics Informed Neural Networks}
+\section{Physics-Informed Neural Networks}
 \label{sec:pinn}
 
 In~\Cref{sec:mlp}, we describe the structure and training of MLP's, which are
 wildely recognized tools for approximating any kind of function. In 1997
-Lagaris \etal~\cite{Lagaris1997} provided a method, that utilizes gradient
+Lagaris \etal~\cite{Lagaris1998} provided a method, that utilizes gradient
 descent to solve ODEs and PDEs. Building on this approach, Raissi
 \etal~\cite{Raissi2019} introduced the methodology with the name
 \emph{Physics-Informed Neural Network} (PINN) in 2017. In this approach, the
@@ -556,7 +559,7 @@ leveraging the available knowledge about the problem in the form of a system of
 differential equations.\\
 
 In contrast to standard MLP models, PINNs are not solely data-driven. The differential
-equation,\todo{Sai Paper cite? https://www.mdpi.com/1424-8220/23/21/8665}
+equation,
 \begin{equation}
   \boldsymbol{y}=\mathcal{D}(\boldsymbol{x}),
 \end{equation}
@@ -608,8 +611,8 @@ this procedure must be repeated.\\
 Above we present a method by Raissi et al.~\cite{Raissi2019} for approximating
 functions through the use of systems of differential equations. As previously
 stated, we want to find a
-solution for systems of differential equations. In problems, where we must solve
-an ODE or PDE, we have to find a set of parameters, that satisfies the system
+solution for systems of differential equations. In problems where we must solve
+an ODE or PDE, we have to find a set of parameters that satisfies the system
 for any input $\boldsymbol{x}$. In the context of PINNs, this is an inverse
 problem. We have training data from measurements and the corresponding
 differential equations, but the parameters of these equations are unknown. To
@@ -621,7 +624,7 @@ training phase the optimizer aims to minimize the physics loss, which should
 ultimately yield an approximation of the true parameter value fitting the
 observations.\\
 
-\begin{figure}[h]
+\begin{figure}[t]
   \centering
   \includegraphics[width=\textwidth]{oscilator.pdf}
   \caption{Illustration of of the movement of an oscillating body in the
@@ -629,7 +632,7 @@ observations.\\
   \label{fig:spring}
 \end{figure}
 In order to illustrate the working of a PINN, we use the example of a
-\emph{damped harmonic oscillator} taken from~\cite{Moseley}. In this problem, we\todo{cite Raissi?}
+\emph{damped harmonic oscillator} taken from~\cite{Moseley}. In this problem, we
 displace a body, which is attached to a spring, from its resting position. The
 body is subject to three forces: firstly, the inertia exerted by the
 displacement $u$, which points in the direction of the displacement; secondly,
@@ -656,21 +659,23 @@ not know the value of the friction $\mu$. In this case the loss function,
 \begin{equation}
   \begin{split}
     \mathcal{L}_{\text{osc}}(\boldsymbol{x}, \boldsymbol{u}, \hat{\boldsymbol{u}}) = & (u^{(1)}-1)+\frac{du(0)}{dt}+ \frac{1}{N_t}\sum_{i=1}^{N_t} ||\hat{\boldsymbol{u}}^{(i)}-\boldsymbol{u}^{(i)}||^2 \\
-    +                                                                                & ||m\frac{d^2u}{dx^2}+\mu\frac{du}{dx}+ku||^2,
+    +                                                                                & ||m\frac{d^2u}{dx^2}+\hat{\mu}\frac{du}{dx}+ku||^2,
   \end{split}
 \end{equation}
-includes the border conditions, the residual, in which $\mu$ is a learnable
-parameter and the data loss. This shows the methodology by which PINNs are capable
-of learning the parameters of physical systems, such as the damped harmonic oscillator.
-In the following section, we present the approach of Shaier \etal~\cite{Shaier2021}
-to find the transmission rate and recovery rate of the SIR model using PINNs.
+includes the border conditions, the residual, in which $\hat{\mu}$ is a learnable
+parameter and the data loss. By minimizing $\mathcal{L}_{\text{osc}}$ and
+solving the inverse problem the PINN is able to find the missing parameter
+$\mu$. This shows the methodology by which PINNs are capable of learning the
+parameters of physical systems, such as the damped harmonic oscillator. In the
+following section, we present the approach of Shaier \etal~\cite{Shaier2021} to
+find the transmission rate and recovery rate of the SIR model using PINNs.
 
 % -------------------------------------------------------------------
 
-\subsection{Disease Informed Neural Networks}
+\subsection{Disease-Informed Neural Networks}
 \label{sec:pinn:dinn}
 In the preceding section, we present a data-driven methodology, as described by Lagaris
-\etal~\cite{Lagaris1997}, for solving systems of differential equations by employing
+\etal~\cite{Lagaris1998}, for solving systems of differential equations by employing
 PINNs. In~\Cref{sec:pandemicModel:sir}, we describe the SIR model, which models
 the relations of susceptible, infectious and removed individuals and simulates
 the progress of a disease in a population with a constant size. A system of

chapters/chap03/chap03.tex

@@ -23,54 +23,56 @@ implementations described in~\Cref{sec:sir:setup} and~\Cref{sec:rsir:setup}.
 
 \section{Epidemiological Data}
 \label{sec:preprocessing}
-In this thesis we want to analyze the COVID-19 pandemic In Germany utilizing
+In this thesis we want to analyze the COVID-19 pandemic in Germany utilizing
 the SIR model and PINNs. For a PINN to learn the parameters of the SIR model,
 we need pandemic data in the correct format for the approach. Let $N_t$ be the
 number of training points, then let $i\in\{1, ..., N_t\}$
 be the index of the training points. The data required by the PINN for solving
 the SIR model (see~\Cref{sec:pinn:dinn}), consists of pairs
-$(\boldsymbol{t}^{(i)}, (\boldsymbol{S}^{(i)}, \boldsymbol{I}^{(i)}, \boldsymbol{R}^{(i)}))$.
-Given that the system of differential equations representing the reduced SIR
-model (see~\Cref{sec:pandemicModel:rsir}) consists of a single differential
-equation for $I$, it is necessary to obtain pairs of the form
-$(\boldsymbol{t}^{(i)}, \boldsymbol{I}^{(i)})$. This section, focuses on the
-structure of the available data and the methods we employ to transform it into
-the correct structure.
+$(\boldsymbol{t}^{(i)}, (\boldsymbol{S}^{(i)}, \boldsymbol{I}^{(i)}, \boldsymbol{R}^{(i)}))$,
+with $\boldsymbol{t}^{(i)}$ representing the time in days since the first
+measurement and $\boldsymbol{S}^{(i)}, \boldsymbol{I}^{(i)}$, and $\boldsymbol{R}^{(i)}$
+the corresponding size of the compartments. Given that the system of
+differential equations representing the reduced SIR model
+(see~\Cref{sec:pandemicModel:rsir}) consists of a single differential equation
+for $I$, it is necessary to obtain pairs of the form $(\boldsymbol{t}^{(i)}, \boldsymbol{I}^{(i)})$.
+This section, focuses on the structure of the available data and the methods we
+employ to transform it into the correct structure.
 
 % -------------------------------------------------------------------
 
 \subsection{RKI Data}
 \label{sec:preprocessing:rki}
-The Robert Koch Institute is a biomedical institute in Germany responsible for
+The RKI is a biomedical institute in Germany responsible for
 the on monitoring and prevention of 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
+COVID-19 pandemic was to track the number of infections and death cases in
 Germany. The data was collected by university hospitals, research facilities
-and laboratories through the conduction of tests. Each new case must be
+and laboratories through the conduction of tests. Each new case had to be
 reported within a period of 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 RKI by the following working day. The RKI then refines the data and
-releases statistics and updates its repositories holding the information for
-the public to access. For the purposes of this thesis we concentrate on two of
-these repositories.\\
+authority. Each state authority collects the cases for a day and had to report
+them to the RKI by the following working day~\cite{GHDead}. The RKI then refines
+the data and releases statistics and updates its repositories holding the
+information for the public to access. For the purposes of this thesis we
+concentrate on two of these repositories.\\
 
 The first repository is called \emph{COVID-19-Todesfälle in Deutschland}~\cite{GHDead}.
 The dataset comprises discrete data points, each with a date indicating the
 point in time at which the respective data was collected. The dates span from
-March 9, 2020, to the present day. For each date, the dataset provides the total
+2020-03-09, to the present day. For each date, the dataset provides the total
 number of infection and death cases, the number  of new deaths, and the
 case-fatality ratio. The total number of infection and death cases represents
 the sum of all cases reported up to that date, including the newly reported
-data. The dataset includes two additional datasets, that contain the death case
+data. The dataset includes two additional subsets, that contain the death case
 information organized by age group or by the individual states within Germany on
 a weekly basis.\\
 
-\begin{figure}[h]
+\begin{figure}[t]
     \centering
     \includegraphics[width=\textwidth]{dataset_visualization.pdf}
     \caption{A visualization of 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.}
+        of 2024-08-20.}
     \label{fig:rki_data}
 \end{figure}
 
@@ -120,7 +122,7 @@ that recovery takes $D$ days, we present the recovery queue, a data structure
 that holds the number of infections for a given day, retains them for $D$ days,
 and releases them into the removed group $D$ days later.\\
 
-\begin{figure}[h]
+\begin{figure}[t]
     \centering
     \includegraphics[width=\textwidth]{recovery_queue.pdf}
     \caption{The recovery queue takes in the infected individuals for the $k$'th
@@ -130,12 +132,12 @@ and releases them into the removed group $D$ days later.\\
 
 In order to solve the reduced SIR model, we employ a similar algorithm to that
 used for the SIR model. However, in contrast to the recovery queue, we utilize
-the set recovery rate $\alpha$ to transfer a portion $\alpha\boldsymbol{I}^{(i)}$
-of infections, which have recovered on the $i$ and put them into the
-$\boldsymbol{R}^{(i)}$ compartment, which is irrelevant to our purposes. \\
-
-The transformed data for both the SIR model and the reduced SIR model are then
-employed by the PINN models, which we describe in the subsequent section.
+a set recovery rate $\alpha$ to transfer a portion $\alpha\boldsymbol{I}^{(i)}$
+of infections, which have recovered or died on the $i$'th day and put them into
+the $\boldsymbol{R}^{(i+1)}$ compartment of the next day, which is irrelevant to
+our purposes. The transformed data for both the SIR model and the reduced SIR
+model are then employed by the PINN models, which we describe in the subsequent
+section.
 
 % -------------------------------------------------------------------
 
@@ -144,92 +146,92 @@ employed by the PINN models, which we describe in the subsequent section.
 
 In the preceding section, we present the methods we employ to preprocess and
 format the data from the RKI in accordance with the specifications required for
-the work of this thesis. In this section, we will present the method we employ
-to identify the SIR parameters $\beta$ and $\alpha$ for the
-data. As a foundation for our work, we draw upon the work of Shaier et
-al.~\cite{Shaier2021}, to solve the SIR system of ODEs using PINNs.\\
-
-In order to conduct an analysis of a pandemic, it is necessary to have a quantifiable measure
-that indicates whether the disease in question has the capacity to spread rapidly through a
-population or is it not successful in infecting a significant number of
-individuals. We employ the SIR model to construct an abstraction of the complex
-relations inherent to real-world pandemics. The SIR model divides the population into three
-compartments. It is accompanied by a with system of ODEs that encapsulates the
-fluctuations and relationships between these compartments (see~\Cref{eq:sir}).
-The transmission rate $\beta$ and the recovery rate $\alpha$ work as the
-aforementioned quantifiers. We obtain data from the preprocessing stage. It
-provides insight into the progression of the COVID-19 pandemic in Germany.
-The objective is to identify a function that solves the system of differential
-equations of the SIR model, by returning the size of each compartment at a
-specific point in time. This function is supposed to be able to reconstruct the
-training data and is defined by the values of the epidemiological parameters $\beta$ and
-$\alpha$. From a mathematical and semantic perspective, it is essential to
-determine these values of the parameter.\\
-
-In order to ascertain the transmission rate $\beta$ and the recovery rate $\alpha$
-from the preprocessed RKI data of $(\boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R})$
-for a given set of time points, it is necessary to employ a data-driven approach that outputs
-a model prediction of $(\hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}})$
+the application in this thesis. Here, we will present the method we employ
+to identify the SIR parameters $\beta$ and $\alpha$ for our data. As a
+foundation for our work, we draw upon the work of Shaier \etal~\cite{Shaier2021},
+to solve the SIR system of ODEs using PINNs.\\
+
+In order to conduct an analysis of a pandemic, it is necessary to have a
+quantifiable measure that indicates whether the disease in question has the
+capacity to spread rapidly through a population or is it not successful in
+infecting a significant number of individuals. In~\Cref{sec:pandemicModel:sir},
+we provide an in-depth discussion of the SIR model, and show, that the
+transmission rate $\beta$ and the recovery rate $\alpha$ work as the
+aforementioned quantifiers in this model. The specific values of these
+epidemiological parameters belonging to the training data define a function that
+solves the system of differential equations of the SIR model. This function is
+able to return the size of each compartment at a specific point in time. Thus,
+from a mathematical and semantic perspective, it is essential to determine the
+corresponding values governing the development of the pandemic.\\
+
+In order to ascertain the transmission rate $\beta$ and the recovery rate
+$\alpha$ from the preprocessed RKI data of $\Psi=(\boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R})$
+for a given set of time points, it is necessary to employ a data-driven approach
+that outputs a model prediction of $\hat{\Psi}=(\hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}})$
 for a set of time points, with the aim of minimizing the term,
 \begin{equation}\label{eq:SIR_obs_term}
     \Big\|\hat{\boldsymbol{S}}^{(i)}-\boldsymbol{S}^{(i)}\Big\|^2  + \Big\|\hat{\boldsymbol{I}}^{(i)}-\boldsymbol{I}^{(i)}\Big\|^2 + \Big\|\hat{\boldsymbol{R}}^{(i)}-\boldsymbol{R}^{(i)}\Big\|^2,
 \end{equation}
-for each data point in the set of training dataset of a cardinality $N_tt$ and with
+for each data point in the set of training dataset of a cardinality $N_t$ and with
 $i\in\{1, ..., N_t\}$. Moreover, the aforementioned parameters must satisfy the system
 of differential equations that govern the SIR model. For this reason, Shaier
 \etal~\cite{Shaier2021} utilize a PINN framework to satisfy both requirements.
-Their approach, which they refer to as the \emph{disease-informed neural network}
+Their approach, which they refer to as the \emph{Disease-Informed Neural Network}
 (see~\Cref{sec:pinn:dinn}), takes epidemiological data as the input and returns
-the two epidemiological parameters $\alpha$ and $\beta$. This method
+the two epidemiological parameters $\alpha$ and $\beta$. Their method
 achieves this by finding an approximate solution of to the inverse problem of
-physics-informed neural networks (see~\Cref{sec:pinn}). In terms of the terms of
-the SIR model, a PINN addresses the inverse problem in two ways. First, it minimizes the mean of~\Cref{eq:SIR_obs_term}
-by bringing the model predictions $(\boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R})$
-closer to the actual values $(\hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}})$
+physics-informed neural networks (see~\Cref{sec:pinn}). In terms of the SIR
+model, a PINN addresses the inverse problem in two ways. First, it minimizes the mean of~\Cref{eq:SIR_obs_term}
+by bringing the model predictions $\hat{\Psi}$
+closer to the actual values $\Psi$
 for each time point. Second, it reduces the residuals of the ODEs that
 constitute the SIR model. While the forward problem concludes at this point, the
 inverse problem presets that a parameter is unknown. Thus, we designate the parameters
-$\beta$ and $\alpha$ as free, learnable parameters, $\widehat{\beta}$ and
-$\widehat{\alpha}$. These separate trainable parameters are values that are
+$\beta$ and $\alpha$ as free, learnable parameters, $\hat{\beta}$ and
+$\hat{\alpha}$. These separate trainable parameters are values that are
 optimized during the training process and must fit the equations of the set of
-ODEs. Assuming that the values of the epidemiological parameters stay below
+ODEs. \\
+
+Assuming that the values of the epidemiological parameters stay below
 1~\cite{Shaier2021}, we force the value of both rates to be in a
-range of $[-1, 1]$. Therefor, we regularize the parameters using the
+range of $[-1, 1]$. Therefore, we regularize the parameters using the
 \emph{tangens hyperbolicus}. This results in the terms,
 \begin{equation}
-    \widehat{\beta} = \tanh(\tilde{\beta}),\quad \widehat{\alpha} = \tanh(\tilde{\alpha}),
+    \tilde{\beta} = \tanh(\hat{\beta}),\quad \tilde{\alpha} = \tanh(\hat{\alpha}),
 \end{equation}
-where $\tilde{\beta}$ and $\tilde{\alpha}$ are the predicted values of the model
-and $\widehat{\beta}$ and $\widehat{\alpha}$ are regularized model predictions.\\
+where $\tilde{\alpha}$ are regularized model predictions.\\
 
 The input data must include the time point $\boldsymbol{t}^{(i)}$ and its
-corresponding measured true values of $(\boldsymbol{S}^{(i)}, \boldsymbol{I}^{(i)}, \boldsymbol{R}^{(i)})$.
+corresponding measured true values of $\Psi^{(i)}$.
 In its forward path, the PINN receives the time point $\boldsymbol{t}^{(i)}$ as its input, from which it
-calculates its model prediction $(\hat{\boldsymbol{S}}^{(i)}, \hat{\boldsymbol{I}}^{(i)}, \hat{\boldsymbol{R}}^{(i)})$
+calculates its model prediction $\hat{\Psi}^{(i)}$
 based on its model parameters $\theta$. Subsequently, the model computes the loss function. It calculates the data loss by taking the
 mean squared error of~\Cref{eq:SIR_obs_term} over all $N_t$ training samples.
-Therefore, the term for the data loss is,
+Therefore,
 \begin{equation}
-    \mathcal{L}_{\text{data}}(\boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R}, \hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}}) = \frac{1}{N_t}\sum_{i=1}^{N_t} \Big\|\hat{\boldsymbol{S}}^{(i)}-\boldsymbol{S}^{(i)}\Big\|^2  + \Big\|\hat{\boldsymbol{I}}^{(i)}-\boldsymbol{I}^{(i)}\Big\|^2 + \Big\|\hat{\boldsymbol{R}}^{(i)}-\boldsymbol{R}^{(i)}\Big\|^2,
+    \mathcal{L}_{\text{data}}(\Psi, \hat{\Psi}) = \frac{1}{N_t}\sum_{i=1}^{N_t} \Big\|\hat{\boldsymbol{S}}^{(i)}-\boldsymbol{S}^{(i)}\Big\|^2  + \Big\|\hat{\boldsymbol{I}}^{(i)}-\boldsymbol{I}^{(i)}\Big\|^2 + \Big\|\hat{\boldsymbol{R}}^{(i)}-\boldsymbol{R}^{(i)}\Big\|^2,
 \end{equation}
-is the term for the data loss. Given superior performance in practical applications
-relative to the ODEs of~\Cref{eq:sir}, we utilize the ODEs of~\Cref{eq:modSIR}
-in our physics loss. In order for the model to learn the system of differential,
-it is necessary to obtain the residual of each ODE. The mean square error of the residuals constitutes
-the physics loss $\mathcal{L}_{\text{physics}}(\boldsymbol{t}, \boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R}, \hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}})$.
-The residuals are calculated using the model predictions $(\hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}})$ and the regularized model predictions of the parameters $\widehat{\beta}$ and $\widehat{\alpha}$. The residuals are given by,
+is the term for the data loss. We find the ODEs of~\Cref{eq:modSIR} perform best
+in our setup. Hence, we utilize them in our physics loss. In order for the model
+to learn the system of differential equations, it is necessary to obtain the
+residual of each ODE. The mean square error of the residuals constitutes the
+physics loss
+$\mathcal{L}_{\text{physics}}(\boldsymbol{t}, \Psi, \hat{\Psi})$.
+The residuals are calculated using the model predictions $\hat{\Psi}$
+and the regularized model predictions of the parameters, $\tilde{\beta}$ and $\tilde{\alpha}$.
+The residuals are given by,
 \begin{equation}
-    0=\frac{d\hat{\boldsymbol{S}}}{d\boldsymbol{t}}+ \widehat{\beta}\frac{\hat{\boldsymbol{S}}\hat{\boldsymbol{I}}}{N}, \quad 0=\frac{d\hat{\boldsymbol{I}}}{d\boldsymbol{t}} - \widehat{\beta}\frac{\hat{\boldsymbol{S}}\hat{\boldsymbol{I}}}{N} + \widehat{\alpha}\hat{\boldsymbol{I}}, \quad 0=\frac{d\hat{\boldsymbol{R}}}{d\boldsymbol{t}} + \widehat{\alpha}\hat{\boldsymbol{I}}.
+    0=\frac{d\hat{\boldsymbol{S}}}{d\boldsymbol{t}}+ \tilde{\beta}\frac{\hat{\boldsymbol{S}}\hat{\boldsymbol{I}}}{N}, \quad 0=\frac{d\hat{\boldsymbol{I}}}{d\boldsymbol{t}} - \tilde{\beta}\frac{\hat{\boldsymbol{S}}\hat{\boldsymbol{I}}}{N} + \tilde{\alpha}\hat{\boldsymbol{I}}, \quad 0=\frac{d\hat{\boldsymbol{R}}}{d\boldsymbol{t}} + \hat{\alpha}\hat{\boldsymbol{I}}.
 \end{equation}
 Thus,
 \begin{equation}
-    \mathcal{L}_{\text{SIR}}(\boldsymbol{t}, \boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R}, \hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}}) = \mathcal{L}_{\text{physics}}(\boldsymbol{t}, \boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R}, \hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}}) + \mathcal{L}_{\text{data}}(\boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R}, \hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}})
+    \mathcal{L}_{\text{SIR}}(\boldsymbol{t}, \Psi, \hat{\Psi}) = \mathcal{L}_{\text{physics}}(\boldsymbol{t}, \Psi, \hat{\Psi}) + \mathcal{L}_{\text{data}}(\Psi, \hat{\Psi})
 \end{equation}
 is the multi-objective loss equation encapsuling both the physics loss and the
-data loss for our approach. By minimizing these loss terms our model learn the
+data loss for our approach. By minimizing these loss terms our model learns the
 given training data but also the physics of the system. This enables our model
 to simultaneously learn the values of the parameters $\beta$ and $\alpha$
-during training. \\
+during training.\\
 
 As this section concentrates on the finding of the time constant parameters
 $\beta$ and $\alpha$, the next section will show our approach of finding the
@@ -240,36 +242,31 @@ reproduction number $\Rt$ on the German data of the RKI.
 \section{Estimating the Reproduction Number using PINNs}
 \label{sec:pinn:rsir}
 
-The previous section illustrates the methodology we employ to detemine the
+The previous section illustrates the methodology we employ to determine 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}.\\
+state variables and the reliance on assumptions, by decreasing the number of ODEs
+in the system of differential equations of 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 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
+distancing policies, and the emergence of a new variant 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~\cite{GHInf}
-posits that the typical recovery period for the illness under normal conditions
-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.\\
+accurately reflect the dynamics of the spread of a real-world disease. In~\Cref{sec:pandemicModel:rsir},
+we provide, following Millevoi \etal~\cite{Millevoi2023}, the definition of the
+time-dependent $\beta(t)$ and subsequently that of the reproduction number,
+$\Rt$ which 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 epidemiological parameters, we employ a data-driven approach for
 identifying the time-dependent reproduction number $\Rt$. The PINN approximates
@@ -280,9 +277,9 @@ minimizing the term,
 \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 that of the PINN, which identifies $\beta$
+training process is analogous to 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
+free, 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
@@ -309,7 +306,15 @@ Then we train on composite loss function given by,
 \end{equation}
 to achieve a better solution.\\
 
-The process of determining the reproduction number, along with the other
-techniques, that this chapter presents find application in the following chapter.
+Although we set the transmission rate to be time-dependent, we define the
+recovery time constant over time to reduce the complexity of the problem. The
+RKI~\cite{GHInf} posits that the typical recovery period for the illness under
+normal conditions is 14 days, while those individuals with severe cases require
+approximately 28 days to recover. As we assume the case with normal condition,
+we can set the recovery time to $D=14$, which yields $\alpha = \nicefrac{1}{14}$.\\
+
+We perform extensive empirical evaluations of the methodology employed to
+determine the reproduction number, along with the other techniques, that this
+chapter presents in the next chapter.
 
 % -------------------------------------------------------------------

chapters/chap04/chap04.tex

@@ -38,7 +38,7 @@ 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 is $I_0 = 10$. We conduct the simulation over 150
-days, resulting in a dataset of the form of~\Cref{fig:synthetic_SIR}.\\
+days, resulting in a dataset of the form of~\Cref{fig:datasets_sir}.\\
 
 \paragraph{Real-World Data:}In order to process the real-world RKI data, it is
 necessary to preprocess the raw data for each state and Germany separately.
@@ -47,25 +47,27 @@ days. With regard to population size of each state, we set it to the respective
 value counted at the end of
 2019\footnote{{\tiny \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, encompassing a period of 1200 days and
+people on 2020-03-09 from the dataset. The data we extract spans from
+2020-03-09 to 2023-06-22, 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]
+\begin{figure}[t]
     \centering
     \includegraphics[width=\textwidth]{in_text_SIR.pdf}
     \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} }
+        data set. Exemplatory we show illustrations of the datasets of
+        Schleswig-Holstein, Berlin, and Thuringia. Mind that this visualization
+        does not have standardized y-axes. For all other states
+        with standardized y-axes see~\Cref{sec:sir_datasets}.}
     \label{fig:datasets_sir}
 \end{figure}
 
 \paragraph{Training Parameters:}The PINN that we utilize comprises of seven
-hidden layers with twenty neurons each, and an activation function of ReLU. We
-follow the hyperparameter setting in~\cite{Shaier2021} but change the base
+hidden layers with twenty neurons each, and an activation function of ReLU~\cite{Fukushima1969}.
+We follow the hyperparameter setting in~\cite{Shaier2021} but change the base
 learning rate to $\expnumber{1}{-3}$. And employ a polynomial scheduler
 implementation from the PyTorch library~\cite{Paszke2019} instead. We train the
 model for 10000 iterations to extract the parameters. For each set of parameters, we
@@ -73,7 +75,7 @@ conduct five runs to demonstrate stability of the values. For measuring the
 accuracy, we calculate the \emph{Relative L2 Error} $e$. Let $G$ be the set of
 compartment training data the SIR model with $\boldsymbol{g}\in G$ and $\hat{\boldsymbol{g}}$ be the
 corresponding model prediction, then,
-\begin{equation}
+\begin{equation}\label{eq:error}
     e_{G} = \frac{1}{|G|}\sum_{g\in G}^{}\frac{\Big\|\hat{\boldsymbol{g}} - \boldsymbol{g}\Big\|_2}{\Big\|\boldsymbol{g}\Big\|_2},
 \end{equation}
 is the average error across all three compartments.
@@ -91,22 +93,25 @@ The results of the experiment regarding the synthetic data can be seen
 in~\Cref{table:alpha_beta_synth}. The error and the standard variation for both
 parameters are negligible small. Taking the mean of the parameters across the
 five iterations yields more accurate results.\\
-\begin{table}[h]
+
+\begin{table}[t]
     \begin{center}
         \caption{Simulation results for the synthetic data. The true values and
-            the respective mean parameter is given.}
+            the respective mean parameter and standard deviation is given. We
+            calculate the error $e_{\text{SIR}}$ with~\Cref{eq:error}.}.
         \label{table:alpha_beta_synth}
         \begin{tabular}{ccccccccc}
             \toprule
-            \multicolumn{2}{c}{$\alpha$} & \phantom{0}             & \multicolumn{2}{c}{$\beta$}                                                             \\
+            \multicolumn{2}{c}{$\alpha$} & \phantom{0}             & \multicolumn{2}{c}{$\beta$}                                                                    \\
             \cmidrule{1-2}\cmidrule{4-5}
-            true                         & $\mu$                   & \phantom{0}                 & true  & $\mu$                   & \phantom{0} & $e_{SIR}$ \\
+            true                         & $\mu$                   & \phantom{0}                 & true  & $\mu$                   & \phantom{0} & $e_{\text{SIR}}$ \\
             \midrule
-            0.333                        & 0.333{\tiny$\pm 0.001$} & \phantom{0}                 & 0.500 & 0.500{\tiny$\pm 0.002$} & \phantom{0} & 0.004     \\
+            0.333                        & 0.333{\tiny$\pm 0.001$} & \phantom{0}                 & 0.500 & 0.500{\tiny$\pm 0.002$} & \phantom{0} & 0.004            \\
             \bottomrule
         \end{tabular}
     \end{center}
 \end{table}
+
 The results demonstrate that the model is capable of approximating the correct
 parameters for the small, synthetic dataset in each of the five iterations.
 The mean of the predicted values results in values with a sufficiently small
@@ -117,42 +122,43 @@ In~\Cref{table:state_mean_std} we present the results of the training for the
 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 error $e_{SIR}$ is
+five iterations of our experiment. We can observe that the error $e_{\text{SIR}}$ is
 the highest for \emph{Saxony} and the lowest for \emph{Lower Saxony}.
 Furthermore, we include the distance $\Delta\beta_{\text{Germany}} = \beta_{\text{state}} - \beta_{\text{Germany}}$
 and the percentage of people that have a basic immunity through vaccination
 $\nu$ for each state provided by the Robert Koch Institute~\cite{FMH}.\\
 
-\begin{table}[h]
+\begin{table}[t]
     \begin{center}
-        \caption{Mean and standard deviation, error $e_{SIR}$ and the distance
+        \caption{Mean and standard deviation, the error $e_{\text{SIR}}$ which we
+            calculate with~\Cref{eq:error} and the distance
             $\Delta\beta_{\text{Germany}} = \beta_{\text{state}} - \beta_{\text{Germany}}$
-            across the 5 iterations, that we conducted for each German state and Germany
-            as the whole country. Furthermore we include the vaccination percentage
-            $\nu$ provided from the RKIe~\cite{FMH}.}
+            across the 5 iterations, that we conducted for each German state (Mecklenburg-Western Pomerania=MWP, North Rhine-Westphalia=NRW) and Germany
+            as the whole country. Furthermore, we include the vaccination percentage
+            $\nu$ provided from the RKI~\cite{FMH}.}
         \label{table:state_mean_std}
         \begin{tabular}{lccccc}
             \toprule
-            state name           & $\alpha$               & $\beta$                 & $e_{SIR}$ & $\Delta\beta_{\text{Germany}}$ & $\nu$ [\%] \\
+            state name           & $\alpha$               & $\beta$                 & $e_{\text{SIR}}$ & $\Delta\beta_{\text{Germany}}$ & $\nu$ [\%] \\
             \midrule
-            Schleswig Holstein   & 0.076{\tiny$\pm0.001$} & 0.095{\tiny$\pm 0.001$} & 0.085     & -0.013                         & 79.5       \\
-            Hamburg              & 0.082{\tiny$\pm0.001$} & 0.104{\tiny$\pm 0.001$} & 0.095     & -0.004                         & 84.5       \\
-            Lower Saxony         & 0.075{\tiny$\pm0.002$} & 0.097{\tiny$\pm 0.002$} & 0.077     & -0.011                         & 77.6       \\
-            Bremen               & 0.058{\tiny$\pm0.002$} & 0.078{\tiny$\pm 0.002$} & 0.093     & -0.030                         & 88.3       \\
-            NRW                  & 0.079{\tiny$\pm0.001$} & 0.101{\tiny$\pm 0.001$} & 0.078     & -0.007                         & 79.5       \\
-            Hesse                & 0.065{\tiny$\pm0.001$} & 0.085{\tiny$\pm 0.001$} & 0.102     & -0.023                         & 75.8       \\
-            Rhineland-Palatinate & 0.085{\tiny$\pm0.004$} & 0.108{\tiny$\pm 0.004$} & 0.090     & 0.001                          & 75.6       \\
-            Baden-Württemberg    & 0.091{\tiny$\pm0.002$} & 0.118{\tiny$\pm 0.003$} & 0.080     & 0.010                          & 74.5       \\
-            Bavaria              & 0.085{\tiny$\pm0.004$} & 0.116{\tiny$\pm 0.005$} & 0.095     & 0.008                          & 75.1       \\
-            Saarland             & 0.075{\tiny$\pm0.002$} & 0.099{\tiny$\pm 0.003$} & 0.108     & -0.009                         & 82.4       \\
-            Berlin               & 0.087{\tiny$\pm0.001$} & 0.109{\tiny$\pm 0.001$} & 0.067     & 0.001                          & 78.1       \\
-            Brandenburg          & 0.087{\tiny$\pm0.003$} & 0.110{\tiny$\pm 0.003$} & 0.072     & 0.002                          & 68.1       \\
-            MV                   & 0.089{\tiny$\pm0.002$} & 0.114{\tiny$\pm 0.002$} & 0.054     & 0.006                          & 74.7       \\
-            Saxony               & 0.075{\tiny$\pm0.002$} & 0.099{\tiny$\pm 0.002$} & 0.111     & -0.009                         & 65.1       \\
-            Saxony-Anhalt        & 0.092{\tiny$\pm0.003$} & 0.119{\tiny$\pm 0.005$} & 0.079     & 0.011                          & 74.1       \\
-            Thuringia            & 0.091{\tiny$\pm0.002$} & 0.119{\tiny$\pm 0.003$} & 0.084     & 0.011                          & 70.3       \\
+            Schleswig-Holstein   & 0.076{\tiny$\pm0.001$} & 0.095{\tiny$\pm 0.001$} & 0.085            & -0.013                         & 79.5       \\
+            Hamburg              & 0.082{\tiny$\pm0.001$} & 0.104{\tiny$\pm 0.001$} & 0.095            & -0.004                         & 84.5       \\
+            Lower Saxony         & 0.075{\tiny$\pm0.002$} & 0.097{\tiny$\pm 0.002$} & 0.077            & -0.011                         & 77.6       \\
+            Bremen               & 0.058{\tiny$\pm0.002$} & 0.078{\tiny$\pm 0.002$} & 0.093            & -0.030                         & 88.3       \\
+            NRW                  & 0.079{\tiny$\pm0.001$} & 0.101{\tiny$\pm 0.001$} & 0.078            & -0.007                         & 79.5       \\
+            Hesse                & 0.065{\tiny$\pm0.001$} & 0.085{\tiny$\pm 0.001$} & 0.102            & -0.023                         & 75.8       \\
+            Rhineland-Palatinate & 0.085{\tiny$\pm0.004$} & 0.108{\tiny$\pm 0.004$} & 0.090            & 0.001                          & 75.6       \\
+            Baden-Württemberg    & 0.091{\tiny$\pm0.002$} & 0.118{\tiny$\pm 0.003$} & 0.080            & 0.010                          & 74.5       \\
+            Bavaria              & 0.085{\tiny$\pm0.004$} & 0.116{\tiny$\pm 0.005$} & 0.095            & 0.008                          & 75.1       \\
+            Saarland             & 0.075{\tiny$\pm0.002$} & 0.099{\tiny$\pm 0.003$} & 0.108            & -0.009                         & 82.4       \\
+            Berlin               & 0.087{\tiny$\pm0.001$} & 0.109{\tiny$\pm 0.001$} & 0.067            & 0.001                          & 78.1       \\
+            Brandenburg          & 0.087{\tiny$\pm0.003$} & 0.110{\tiny$\pm 0.003$} & 0.072            & 0.002                          & 68.1       \\
+            MWP                  & 0.089{\tiny$\pm0.002$} & 0.114{\tiny$\pm 0.002$} & 0.054            & 0.006                          & 74.7       \\
+            Saxony               & 0.075{\tiny$\pm0.002$} & 0.099{\tiny$\pm 0.002$} & 0.111            & -0.009                         & 65.1       \\
+            Saxony-Anhalt        & 0.092{\tiny$\pm0.003$} & 0.119{\tiny$\pm 0.005$} & 0.079            & 0.011                          & 74.1       \\
+            Thuringia            & 0.091{\tiny$\pm0.002$} & 0.119{\tiny$\pm 0.003$} & 0.084            & 0.011                          & 70.3       \\
             \midrule
-            Germany              & 0.083{\tiny$\pm0.001$} & 0.108{\tiny$\pm 0.002$} & 0.080     & 0.000                          & 76.4       \\
+            Germany              & 0.083{\tiny$\pm0.001$} & 0.108{\tiny$\pm 0.002$} & 0.080            & 0.000                          & 76.4       \\
             \bottomrule
         \end{tabular}
 
@@ -172,7 +178,7 @@ 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 Hesse have the lowest
-values for $\beta$. The transmission rates of Hamburg, Baden Württemberg,
+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,
@@ -242,7 +248,7 @@ Second, we use $\alpha=\nicefrac{1}{5}$, as 5 days into the infection is the
 point at which the infectiousness is at its peak~\cite{COVInfo}.
 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.
+we restrict the data points to an interval of 1200 days, beginning on 2020-03-09.
 2020.\\
 
 \begin{figure}[t]
@@ -293,7 +299,7 @@ that the model is capable of learning the infection data across all data points.
 The error for this is, $e_I = 0.0016$, which is of a negligible
 magnitude.\\
 
-\begin{figure}[h]
+\begin{figure}[t]
     \centering
     \begin{subfigure}{0.45\textwidth}
         \includegraphics[width=\textwidth]{synthetic_I_prediction.pdf}
@@ -311,7 +317,7 @@ magnitude.\\
 
 An examination of the predictions for the representation value $\Rt$ reveals
 that here as well, the model is capable of accurately delineating the value at
-each time point. However, during the first 30 days, the standard deviation is
+each time point. However, during the first 30 days, the standard deviation
 exhibits an upward trend, while during the final 120 days, the predictions
 demonstrate remarkable precision.\\
 
@@ -342,7 +348,7 @@ is shorter, but the peak value is higher.\\
         $\Rt$ for Thuringia (upper) and Bremen (lower) with both
         $\alpha = \nicefrac{1}{14}$ and $\alpha = \nicefrac{1}{5}$. Events~\cite{COVIDChronik} like
         the peak of an influential variant or the start of the vaccination of the public are marked horizontally. Further
-        visualizations can be found in~\Cref{chap:appendix}.}
+        visualizations can be found in~\Cref{sec:r_t_results}.}
 \end{figure}
 
 \Cref{table:state_error} presents data regarding the discrepancy between the
@@ -359,8 +365,12 @@ at the beginning.\\
 
 \begin{table}[t]
     \begin{center}
-        \caption{This table displays all average values of the error $e_{\text{I}}$
-            for all German states and Germany. The average is formed across all
+        \caption{For both $\alpha=\nicefrac{1}{14}$ and $\alpha=\nicefrac{1}{5}$
+            this table presents the error $e_{\text{I}}$, calculated with~\Cref{eq:error},
+            the average number of days with $\Rt > 1$, and
+            the average peak values of $\Rt$ for all German states
+            (Mecklenburg-Western Pomerania=MWP, North Rhine-Westphalia=NRW) and
+            Germany. The average is formed across all
             10 iteration.}
         \label{table:state_error}
         \begin{tabular}{lccccccc}
@@ -369,7 +379,7 @@ at the beginning.\\
             \cmidrule{2-3}\cmidrule{5-6}\cmidrule{7-8}
             state name           & $\alpha=\frac{1}{14}$     & $\alpha=\frac{1}{5}$ & \phantom{0}                           & $\alpha=\frac{1}{14}$          & $\alpha=\frac{1}{5}$ & $\alpha=\frac{1}{14}$ & $\alpha=\frac{1}{5}$ \\
             \midrule
-            Schleswig Holstein   & 0.228                     & 0.258                & \phantom{0}                           & 467.5                          & 458.5                & 1.475                 & 1.166                \\
+            Schleswig-Holstein   & 0.228                     & 0.258                & \phantom{0}                           & 467.5                          & 458.5                & 1.475                 & 1.166                \\
             Hamburg              & 0.265                     & 0.330                & \phantom{0}                           & 424.3                          & 409.8                & 1.500                 & 1.297                \\
             Lower Saxony         & 0.224                     & 0.340                & \phantom{0}                           & 413.1                          & 430.3                & 1.662                 & 1.223                \\
             Bremen               & 0.246                     & 0.380                & \phantom{0}                           & 468.6                          & 539.1                & 1.582                 & 1.179                \\
@@ -381,7 +391,7 @@ at the beginning.\\
             Saarland             & 0.284                     & 0.408                & \phantom{0}                           & 500.2                          & 564.7                & 1.515                 & 1.180                \\
             Berlin               & 0.201                     & 0.240                & \phantom{0}                           & 591.9                          & 514.4                & 1.721                 & 1.262                \\
             Brandenburg          & 0.237                     & 0.242                & \phantom{0}                           & 555.9                          & 596.3                & 1.447                 & 1.159                \\
-            MV                   & 0.170                     & 0.257                & \phantom{0}                           & 537.5                          & 544.3                & 1.563                 & 1.135                \\
+            MWP                  & 0.170                     & 0.257                & \phantom{0}                           & 537.5                          & 544.3                & 1.563                 & 1.135                \\
             Saxony               & 0.292                     & 0.256                & \phantom{0}                           & 722.3                          & 695.4                & 1.790                 & 1.407                \\
             Saxony-Anhalt        & 0.213                     & 0.268                & \phantom{0}                           & 572.0                          & 631.9                & 1.387                 & 1.165                \\
             Thuringia            & 0.180                     & 0.222                & \phantom{0}                           & 732.1                          & 730.6                & 1.586                 & 1.249                \\

chapters/conclusions/conclusions.tex

@@ -10,47 +10,50 @@
 \chapter{Conclusions}
 \label{chap:conclusions}
 The severe COVID-19 pandemic~\cite{WHO} infected millions of people, while hundreds of thousands
-succumbed to it in Germany alone~\cite{SRD}. Across three years the pandemic
+succumbed to it in Germany alone~\cite{SRD}. Over three years the pandemic
 changed through the influence of various mitigation policies and numerous
-emerging variant. In order to get a hold of the complex situation the necessity
+emerging variants. In order to get a hold of the complex situation the necessity
 for analysis arises. Therefore, the objective of this thesis is to measure the
 COVID-19 pandemic in Germany and its 16 federal states by identifying several
 epidemiological parameters that describe the spread of the disease. \\
 
-We use the SIR model~\cite{1927}
-to describe the dynamics of the disease over time, offering an approximation of
-reality. In this model, the transmission rate $\beta$ and recovery rate $\alpha$
+We use the SIR model~\cite{1927} to describe the dynamics of the COVID-19
+infection over time, offering an approximation of reality. In this model, the
+transmission rate $\beta$ and recovery rate $\alpha$
 describe the infectiousness and development of the disease that the respective
 population experience. These rates serve as global evaluation measures
 throughout the entire duration of the pandemic. Meanwhile, the time-dependent
 reproduction number indicates the number of individuals infected by a single infectious
-individual. The relation between parameters and values is defined in the system
-of differential equations which governs the the SIR model. In order to obtain these values
+individual. The relations between parameters are defined in the system
+of differential equations which governs the SIR model.\\
+
+In order to obtain these epidemiological parameters and the reproduction number
 for Germany, it is necessary to solve the system of ordinary differential equations (ODEs)
-for real-world pandemic data which was recorded in each state and in Germany as a whole.
-The data-driven approach of \emph{Physics-Informed Neural Networks} (PINN)~\cite{Raissi2019}
-to solve systems of differential equations has gained  attention in the last
-years. These integrate the knowledge in form of physical
-models, while they learn the solution by fitting data points. We adapt previous
-epidemiological PINN approaches~\cite{Shaier2021,Millevoi2023} to solve the
-ODE's. The data on which we train is collected by the Robert Koch Institute and
-made publicly available on GitHub~\cite{GHInf,GHDead}. After a
-preprocessing, we solve the inverse problem posed by the SIR model utilizing
-PINNs in order to find the epidemiological parameters and the reproduction number
-for the given data. Using this we conduct experiments on synthetic data and on
-the data for the federal states and Germany itself. The results for the
-synthetic data demonstrate the efficacy of our approach on small datasets.\\
+for real-world pandemic data recorded in each state and in Germany as a whole.
+One method that has gained significant attention in recent years for solving
+systems of differential equations is the data-driven approach known as
+\emph{Physics-Informed Neural Networks} (PINN)~\cite{Raissi2019}. PINNs
+integrate knowledge in form of physical models, while learning an approximation
+the solution by fitting data points. We adapt previous epidemiological PINN
+approaches~\cite{Shaier2021,Millevoi2023} to solve the set of ODEs of the SIR
+model. The data for training is collected by the Robert Koch Institute and made
+publicly available on GitHub~\cite{GHDead,GHInf}. After preprocessing, we
+solve the inverse problem posed by the SIR model utilizing PINNs in order to find the
+epidemiological parameters and the reproduction number for the given data. Using
+this we conduct experiments on synthetic data and on the data for the federal
+states and Germany itself. The results for the synthetic data yield a small
+error, which demonstrates the efficacy of our approach on small datasets.\\
 
-We divide our analysis of the real-world data into two groups. First we have
+We divide our analysis of the real-world data into two groups. First, we have
 the time-constant epidemiological parameters $\alpha$ and $\beta$, which
 provide insight into the overall trajectory of the pandemic in a given region.
 Given the assumed constant recovery period (see~\Cref{sec:preprocessing:rq}),
 there is a dependency between the two parameters. Therefore, we focus our analysis on the
 transmission rate $\beta$. The states with the highest estimated transmission rate values
-are Thuringia, Saxony-Anhalt and Mecklenburg-Vorpommern which means that on
-average these states had a high number of infections during the pandemic.
-Furthermore, it is evident the six eastern states exhibit a higher transmission
-rate than the overall German rate(see~\Cref{fig:alpha_beta_mean_std}).
+are Thuringia, Saxony-Anhalt, and Mecklenburg-Western Pomerania, which means that
+these states had a higher average number of infections during the pandemic.
+Furthermore, it is evident that the six eastern states exhibit a higher transmission
+rate than the overall German rate (see~\Cref{fig:alpha_beta_mean_std}).
 Our results align with similarly observed differences in vaccination rates~\cite{FMH}
 and highlight perceived discrepancies between the eastern and western federal
 states~\cite{FMH,Desson2022}. We further substantiate this observation by
@@ -64,32 +67,31 @@ Our results indicate a tendency for states with a high $\beta$ to experience
 longer periods with $\Rt>1$. Furthermore, we can identify the time point on
 which the most impactful events happened during the pandemic in Germany such as
 the peak of the omicron variant~\cite{COVIDChronik} at around 700 days after
-the start of data collection on March 9. 2020.\\
+the start of data collection on 2020-03-09.\\
 
 In conclusion, our approach has proven effective in yielding meaningful results
 for the epidemiological parameters of $\alpha$ and $\beta$, as well as the
-reproduction number $\Rt$ for Germany and its federal states. Despite some
-limitations during training, there is a clear connection between the results
-and real-world data and events are evident. We hope that this work will prove
-useful in the analysis of the events of the COVID-19 pandemic in Germany.
+reproduction number $\Rt$ for Germany and its federal states. Despite the SIR
+model being an approximation of reality, there is a clear connection between the
+results and real-world data and events. We hope that this work will prove useful
+in the analysis of the events of the COVID-19 pandemic in Germany.
 
 % -------------------------------------------------------------------
 
 \section{Further Work}
 \label{sec:furtherWork}
 
-Our findings demonstrate that with our methods enable the quantification of the
+Our findings demonstrate that our methods enable the quantification of the
 course of the COVID-19 pandemic in Germany using the data provided by the
 Robert Koch Institute~\cite{GHDead,GHInf}. Here we present some limitations of
-our work and propose future directions to remedy these point. First we find
-that our model does not reconstruct the input data as precisely as possible.
-To address this, we propose a comprehensive hyperparameter search to find the
-best configuration. Furthermore, 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.
+our work and propose future directions to address these points. First, we find
+that our model does not accurately reconstruct the input data to the desired
+level of precision. To address this, we propose a comprehensive hyperparameter
+search to find the best configuration. Moreover, the SIR model 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 incorporate such dynamics observed in real-world
+pandemics and recommend further investigation for Germany.
 
 % -------------------------------------------------------------------
 
@@ -97,21 +99,21 @@ investigation for Germany.
 As our results demonstrate, the SIR model is capable of approximating the
 dynamics of real-world pandemics. However, the model is not without
 limitations. The SIR model assumes that recovered
-individuals remain immune and does not account for the reduction of exposure of
+individuals remain immune and does not account for the reduction of exposure to
 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~\cite{Korolev2021} 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, the manifestation index, the incubation rate, and
-the infection fatality rate. As Doerre and Doblhammer~\cite{Doerre2022} show
-for Germany using a numerical approximation method, for a SIERD model that they
-specialize to be age- and gender-specific, that it shows the impact of
-non-pharmaceutical mitigation policies.\\
+into the model. For example, the SEIRD model~\cite{Korolev2021} incorporates an
+\emph{Exposed} group and subdivides the \emph{Removed} group into \emph{Dead}
+and \emph{Recovered} compartments. Furthermore, the model is extended with four
+additional parameters: the contact rate, the manifestation index, the incubation
+rate, and the infection fatality rate. Doerre and Doblhammer~\cite{Doerre2022}
+introduce an approach utilizing a SIERD model that they specialize to be age-
+and gender-specific. For Germany, they show the impact of non-pharmaceutical
+mitigation policies, solving the model using a numerical approximation method.\\
 
-In their work, Cooke and van den Driessche~\cite{Cooke1996}
-propose the SEIRS model with two delays. This is model is capable of
+Additionally, Cooke and van den Driessche~\cite{Cooke1996}
+propose the SEIRS model with two delays. This 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 display the dynamics
@@ -122,26 +124,27 @@ and consequently quantify a 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
+Compartmental models, such as the SIR model, look at the population as a
+divided group. Each group represents 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
+large groups. Each individual follows simple rules which leads to the emergence
+of complex and stochastic behaviour on the mascroscopic level of the
+system~\cite{Bodine2020}. With regard to COVID-19, 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. Maziarz and Zach~\cite{Maziarz2020} address
+the criticism levied against ABMs for simplifying the dynamics and lacking the
+empirical support for the assumptions 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 state that ABMs can 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
+As the quantity of data has evolved, it is imperative to investigate the
 potential of utilizing ABMs as a tool to assess the pandemic's course for
-Germany.
+Germany in greater detail.
 
 % -------------------------------------------------------------------

header.tex

@@ -13,7 +13,7 @@
 \usepackage{amssymb}
 \usepackage{amsthm}
 \usepackage{graphicx}
-\usepackage[hang]{caption}
+\usepackage[hang,font=footnotesize]{caption}
 \usepackage{subcaption}
 \usepackage{booktabs}
 \usepackage{makeidx}

thesis.bbl

@@ -1,20 +1,20 @@
 \begin{thebibliography}{10}
 
 \bibitem{WHO}
-WHO.
+{World Health Organization}.
 \newblock Coronavirus disease (covid-19).
 \newblock \url{https://www.who.int/health-topics/coronavirus#tab=tab_1}.
 \newblock {Accessed: 2024-09-06}.
 
 \bibitem{RKI}
-RKI.
+{Robert Koch Institute}.
 \newblock Covid-19-strategiepapiere und nationaler pandemieplan.
 \newblock
   \url{https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/ZS/Pandemieplan_Strategien.html}.
 \newblock {Accessed: 2024-09-06}.
 
 \bibitem{RKIa}
-RKI.
+{Robert Koch Institute}.
 \newblock Sars-cov-2: Virologische basisdaten sowie virusvarianten im zeitraum
   von 2020 - 2022.
 \newblock
@@ -51,13 +51,14 @@ Setianto Setianto and Darmawan Hidayat.
 \bibitem{Shaier2021}
 Sagi Shaier, Maziar Raissi, and Padmanabhan Seshaiyer.
 \newblock Data-driven approaches for predicting spread of infectious diseases
-  through dinns: Disease informed neural networks, 2021.
+  through dinns: Disease informed neural networks.
+\newblock {\em Letters in Biomathematics}, 9(1):71--105, 2023.
 
 \bibitem{Millevoi2023}
 Caterina Millevoi, Damiano Pasetto, and Massimiliano Ferronato.
 \newblock A physics-informed neural network approach for compartmental
   epidemiological models.
-\newblock 2023.
+\newblock 20(9):e1012387, September 2023.
 
 \bibitem{Smirnova2017}
 Alexandra Smirnova, Linda deCamp, and Gerardo Chowell.
@@ -108,6 +109,13 @@ Bernt Oksendal.
   5th ed. edition, 2000.
 \newblock Description based on publisher supplied metadata and other sources.
 
+\bibitem{Jaschke2023}
+Philipp Jaschke, Sekou Keita, Ehsan Vallizadeh, and Simon Kühne.
+\newblock Satisfaction with pandemic management and compliance with public
+  health measures: Evidence from a german household survey on the covid-19
+  crisis.
+\newblock {\em PLOS ONE}, 18(2):e0281893, February 2023.
+
 \bibitem{EdelsteinKeshet2005}
 Leah Edelstein-Keshet.
 \newblock {\em Mathematical Models in Biology}.
@@ -147,11 +155,11 @@ Kurt Hornik, Maxwell Stinchcombe, and Halbert White.
 \newblock Multilayer feedforward networks are universal approximators.
 \newblock {\em Neural Networks}, 2(5):359--366, January 1989.
 
-\bibitem{Lagaris1997}
-I.~E. Lagaris, A.~Likas, and D.~I. Fotiadis.
+\bibitem{Lagaris1998}
+Isaac Lagaris, Aristidis Likas, and Dimitrios Fotiadis.
 \newblock Artificial neural networks for solving ordinary and partial
   differential equations.
-\newblock 1997.
+\newblock {\em IEEE Transactions on Neural Networks}, 9:987--1000, 09 1998.
 
 \bibitem{Raissi2019}
 M.~Raissi, P.~Perdikaris, and G.E. Karniadakis.
@@ -168,27 +176,35 @@ Ben Moseley.
 \newblock {Accessed: 2024-09-08}.
 
 \bibitem{GHDead}
-RKI.
+{Robert Koch Institute}.
 \newblock Github covid-19-todesfälle in deutschland.
 \newblock
   \url{https://github.com/robert-koch-institut/COVID-19-Todesfaelle_in_Deutschland}.
 \newblock {Accessed: 2024-09-05}.
 
 \bibitem{GHInf}
-RKI.
+{Robert Koch Institute}.
 \newblock Github sars-cov-2 infektionen in deutschland.
 \newblock
   \url{https://github.com/robert-koch-institut/SARS-CoV-2-Infektionen_in_Deutschland}.
 \newblock {Accessed: 2024-09-05}.
 
+\bibitem{Fukushima1969}
+Kunihiko Fukushima.
+\newblock Visual feature extraction by a multilayered network of analog
+  threshold elements.
+\newblock {\em IEEE Transactions on Systems Science and Cybernetics},
+  5(4):322--333, 1969.
+
 \bibitem{Paszke2019}
 Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory
   Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban
-  Desmaison, Andreas Köpf, Edward Yang, Zach DeVito, Martin Raison, Alykhan
+  Desmaison, Andreas K\"{o}pf, Edward Yang, Zach DeVito, Martin Raison, Alykhan
   Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu~Fang, Junjie Bai, and Soumith
   Chintala.
-\newblock Pytorch: An imperative style, high-performance deep learning library,
-  2019.
+\newblock {\em PyTorch: an imperative style, high-performance deep learning
+  library}.
+\newblock Curran Associates Inc., Red Hook, NY, USA, 2019.
 
 \bibitem{FMH}
 {Federal Ministry of Health}.

thesis.tex

@@ -72,8 +72,7 @@
 %--------------------------------------------------
 %--------------------------------------------------
 
-%\appendix
-
+\appendix
 % if you do not have appendix sections, comment this include command out
 \include{./chapters/appendix/appendix}