corr no. 1

abstract.tex

@@ -28,7 +28,7 @@ $\beta$, 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
+approach to solve the differential equations utilizing 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,

chapters/chap01-introduction/chap01-introduction.tex

@@ -19,21 +19,21 @@ 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
+Germany had recorded over 38,400,000 infection cases and 174,000 deaths, by
 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
+The dynamics 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
+tools that allow 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
+in 1927. The SIR model is a compartmental 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, $R$.
 In the context of the SIR model, the constant parameters of the transmission
@@ -41,7 +41,7 @@ rate $\beta$ and the recovery rate $\alpha$ serve to quantify and determine the
 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
+reproduction numbers $\Rt$. 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
@@ -56,7 +56,7 @@ 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
+$\alpha$ and $\beta$, 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)\footnote{\url{https://www.rki.de/EN/Home/homepage_node.html}} has compiled data on both reported cases and
 associated moralities from the beginning of the outbreak in Germany to the
@@ -64,21 +64,20 @@ 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 PINN learning the data, which consists
+\etal~\cite{Shaier2021}, which utilizes a PINN learning the data, that 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 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
-the peaks of impactful variants to demonstrate the relevance of these numbers.
-This analysis demonstrates the relevance of our findings and reveals a
-correlation between our results and real-world developments, thus supporting the
-effectiveness of our approach.\\
+system of differential equations. Additionally, we apply the methodology by
+Millevoi \etal~\cite{Millevoi2023} to estimate the time-dependent reproduction
+number, $\Rt$, over a 1200-day period for each German federal state and Germany
+as a whole 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 the peaks of impactful variants. This analysis demonstrates the
+relevance of our findings and reveals a correlation between our results and
+real-world developments, thus supporting the effectiveness of our approach.\\
 
 
 % -------------------------------------------------------------------
@@ -89,36 +88,36 @@ In this section, we categorize our work into the context of existing literature
 on the topic of solving the epidemiological models for real-world data. The
 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)
-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 PINNs to find the constant epidemiological parameters
-and the reproduction number for Germany and its states.\\
+$\beta(t)$. 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) 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, as demonstrated on both simulated data and real-world data
+from the 1918 influenza pandemic and the Ebola epidemic. In contrast, we
+utilize PINNs to find the constant epidemiological parameters and the
+reproduction number for Germany and its states.\\
 
-Some related works similar to our approach apply PINN approaches to COVID-19 and
+Some related works similar to our method 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
+Specifically Shaier \etal~\cite{Shaier2021} put forth a data-driven method
 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
+work, they demonstrate the capacity of PINNs 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.\\
+model. Finally, they present that this method is a robust and effective means
+of identifying the parameters of a SIR model.\\
 
 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
-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
+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 utilize
+both forecasting methods to project the trajectory of COVID-19 from mid-April
+2023 onwards. Berkhahn and Ehrhard find that PINNs are able to adapt to varying
 vaccination rates and emerging variants.\\
 
 Furthermore, Olumoyin \etal~\cite{Olumoyin2021} employ an alternative
@@ -128,39 +127,40 @@ approach they introduce, utilizes the cumulative and daily reported infection
 cases and symptomatic recovered cases, to demonstrate the effect of different
 mitigation measures and to ascertain the proportion of non-symptomatic
 individuals and asymptomatic recovered individuals. With this they can
-illustrate the influence of vaccination and a set non-pharmaceutical mitigation
-methods on the transmission of COVID-19 on data from Italy, South Korea, the
-United Kingdom, and the United States.\\
+illustrate the influence of vaccinations and a set non-pharmaceutical
+mitigation methods on the transmission of COVID-19 on data from Italy, South
+Korea, the United Kingdom, and the United States.\\
 
 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 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.
+reproduction number $\Rt$ 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
+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 epidemiological of $\alpha$ and
+$\beta$ and the reproduction number $\Rt$ for Germany for the entirety of the
+span from early March in 2020 to late June in 2023.
 
 % -------------------------------------------------------------------
 
 \section{Overview}
 
 This thesis is comprised of four chapters. \Cref{chap:background}
-presents with the theoretical overview of mathematical modeling in epidemiology,
+starts 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 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
+methodology employed in this thesis. First, we present the data, that was
+collected by the RKI and our preprocessing. 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} provides the
+setups and results of the experiments that we conduct. This chapter is divided
+into two sections. The first section shows and discusses the results concerning
+the epidemiological parameters of $\alpha$ and $\beta$. 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.
+\Cref{chap:conclusions}, give a conclusion of our work and provide an overview
+of potential further work.
 
 % -------------------------------------------------------------------

chapters/chap02/chap02.tex

@@ -325,7 +325,7 @@ and the recovery rate $\alpha$ influence the height and time of the peak of $I$.
 When the number of infections exceeds the number of recoveries, the peak of $I$
 will occur early and will be high. On the other hand, if recoveries occur at a
 faster rate than new infections the peak will occur later and will be low. Thus,
-it is crucial to know both $\beta$ and $\alpha$, as these parameters
+it is crucial to know both $\alpha$ and $\beta$, as these parameters
 characterize how the pandemic evolves.\\
 
 The SIR model is based on a number of assumptions that are intended to reduce
@@ -346,14 +346,14 @@ next~\Cref{sec:pandemicModel:rsir}.
 \subsection{Reduced SIR Model and the Reproduction Number}
 \label{sec:pandemicModel:rsir}
 The~\Cref{sec:pandemicModel:sir} presents the classical SIR model. This model
-contains two scalar parameters $\beta$ and $\alpha$, which describe the course
+contains two scalar parameters $\alpha$ and $\beta$, which describe the course
 of a pandemic over its duration. This is beneficial when examining the overall
 pandemic; however, in the real world, disease behavior is dynamic, and the
-values of the parameters $\beta$ and $\alpha$ change throughout the course of
+values of the parameters $\alpha$ and $\beta$ change throughout the course of
 the disease. The reason for this is due to events such as the implementation of
 countermeasures that reduce the contact between the infectious and susceptible
 individuals, the emergence of a new variant of the disease that increases its
-infectivity or deadliness, or the administration of a vaccination that provides
+infectiousness or deadliness, or the administration of a vaccination that provides
 previously susceptible individuals with immunity without ever being infected.
 As these fine details of disease progression are missed in the constant rates,
 Liu and Stechlinski~\cite{Liu2012}, and Setianto and Hidayat~\cite{Setianto2023},
@@ -362,7 +362,7 @@ number to address this issue. Millevoi \etal~\cite{Millevoi2023} present a
 reduced version of the SIR model.\\
 
 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,
+they alter the definition of $\alpha$ and $\beta$ to be time-dependent,
 \begin{equation}
   \beta: \mathcal{T}\rightarrow\mathbb{R}_{\geq0}, \quad\alpha: \mathcal{T}\rightarrow\mathbb{R}_{\geq0}.
 \end{equation}
@@ -389,7 +389,7 @@ defined as,
 on the time interval $\mathcal{T}$ and the population site $N$. This definition
 includes the epidemiological parameters for information about the spread of the disease
 and information of the decrease of the ratio of susceptible individuals in the
-population. In contrast to $\beta$ and $\alpha$, $\Rt$ is not a parameter but
+population. In contrast to $\alpha$ and $\beta$, $\Rt$ is not a parameter but
 a state variable in the model, which gives information about the reproduction of the disease
 for each day. As Millevoi \etal~\cite{Millevoi2023} show, $\Rt$ enables the
 following reduction of the SIR model.\\
@@ -683,8 +683,8 @@ differential equations models these relations. Shaier \etal~\cite{Shaier2021}
 propose a method to solve the equations of the SIR model using a PINN, which
 they call a \emph{Disease-Informed Neural Network} (DINN).\\
 
-To solve~\Cref{eq:sir} we need to find the transmission rate $\beta$ and the
-recovery rate $\alpha$. As Shaier \etal~\cite{Shaier2021} point out, there are
+To solve~\Cref{eq:sir} we need to find the recovery rate $\alpha$ and the
+transmission rate $\beta$. As Shaier \etal~\cite{Shaier2021} point out, there are
 different approaches to solve this set of equations. For instance, building on
 the assumption, that at the beginning one infected individual infects $-n$ other
 people, concluding in $\frac{dS(0)}{dt} = -n$. Then,
@@ -698,7 +698,7 @@ infection and the start of isolation $d$, $\alpha = \frac{1}{d}$. The analytical
 solutions to the SIR models often use heuristic methods and require knowledge
 like the sizes $S_0$ and $I_0$. A data-driven approach such as the one that
 Shaier \etal~\cite{Shaier2021} propose does not suffer from these problems. Since the
-model learns the parameters $\beta$ and $\alpha$ while learning the training
+model learns the parameters $\alpha$ and $\beta$ while learning the training
 data consisting of the time points $\boldsymbol{t}$,  and the corresponding
 measured sizes of the groups $\boldsymbol{S}, \boldsymbol{I}, \boldsymbol{R}$.
 Let $\hat{\boldsymbol{S}}, \hat{\boldsymbol{I}}, \hat{\boldsymbol{R}}$ be the

chapters/chap03/chap03.tex

@@ -147,7 +147,7 @@ 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 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
+to identify the SIR parameters $\alpha$ and $\beta$ 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.\\
 
@@ -187,7 +187,7 @@ 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, $\hat{\beta}$ and
+$\alpha$ and $\beta$ 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. \\
@@ -230,11 +230,11 @@ Thus,
 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 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 $\alpha$
+to simultaneously learn the values of the parameters $\alpha$ and $\beta$
 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
+$\alpha$ and $\beta$, the next section will show our approach of finding the
 reproduction number $\Rt$ on the German data of the RKI.
 
 % -------------------------------------------------------------------

chapters/chap04/chap04.tex

@@ -13,7 +13,7 @@ In ~\Cref{chap:methods}, we explain the methods based the theoretical
 background, that we established in~\Cref{chap:background}. In this chapter, we
 present the setups and results from the experiments and simulations. First, we
 discuss the experiments dedicated to identify the epidemiological transition
-rates of $\beta$ and $\alpha$ in synthetic and real-world data. Second, we
+rates of $\alpha$ and $\beta$ in synthetic and real-world data. Second, we
 examine the reproduction number in synthetic and real-world data of Germany.
 
 % -------------------------------------------------------------------
@@ -82,7 +82,7 @@ is the average error across all three compartments.
 
 % -------------------------------------------------------------------
 
-\subsection{Results}
+\subsection{Results and Discussion}
 \label{sec:sir:results}
 
 In this section, we start by examining the results for the synthetic dataset,
@@ -290,7 +290,7 @@ evaluation, we use the error $e_G$ as we do in the subsequent section.\\
 
 % -------------------------------------------------------------------
 
-\subsection{Results}
+\subsection{Results and Discussion}
 \label{sec:rsir:results}
 
 \Cref{fig:synth_results} illustrates the results of our experiments conducted on