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\chapter{Introduction 5}
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\label{chap:introduction}
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+In the early months of 2020, Germany, like many other countries, was struck by the novel
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+\emph{Coronavirus Disease} (COVID-19). The pandemic, which originates in
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+Wuhan, China, had a profound impact on the global community, paralyzing it for
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+over two years. In response to the pandemic, the German government employed a
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+multifaceted approach, encompassing the introduction of vaccines and
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+non-pharmaceutical mitigation policies such as lockdowns. Between mitigation
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+policies and varying strains of COVID-19, which have exhibited varying degrees
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+of infectiousness and lethality, Germany had recorded over 38,400,000 infection
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+cases and 174,000 deaths, as of the end of June in 2023. In light of these
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+figures the need for an analysis arises.\\
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+
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+The dynamics of the spread of disease transmission in the real-world are
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+complex. A multitude of factors influence the course of a disease, and it is
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+challenging to gain a comprehensive understanding of these factors and develop a
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+tool that allows for the comparison of disease courses across different diseases
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+and time points. The common approach in epidemiology to address this is the
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+utilization of epidemiological models that approximate the dynamics by focusing
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+on specific factors and modeling these using differential equations and other
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+mathematical tools for modeling. These models provide transition rates and
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+parameters that determine the behavior of a disease within the boundaries of the
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+model. A fundamental epidemiological model, is the \emph{SIR model}, which was
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+first proposed by Kermack and McKendrick~\cite{1927} in 1927. The SIR model is a
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+compartmentalized model that divides the entire population into three distinct
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+compartments. The first compartment is the \emph{susceptible} compartment, $S$,
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+which contains all individuals of the population who are susceptible to
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+infection. The second group, is the \emph{infectious} compartment, $I$, which
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+comprises all individuals currently infected and capable of infecting
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+susceptible individuals. Lastly, the \emph{removed} compartment, $R$, contains
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+all individuals, who have succumbed to the disease or recovered from it and are
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+therefore no longer susceptible to infection. The model is characterized by two
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+transition rates: the transmission rate $\beta$, which controls the rate of
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+individuals becoming infected and consequently transitioning from $S$ to $I$;
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+and the recovery rate $\alpha$, which determines the rate at which individuals
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+either recover or succumb to the disease, thereby transitioning from $I$ to $R$.
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+In the context of the SIR model, the values of $\beta$ and $\alpha$ serve to
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+quantify and determine the course of a pandemic.\\
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+
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+The transition rates of $\beta$ and $\alpha$ serve to quantify a pandemic across
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+its entire duration. However, it is important to recognize that a pandemic is
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+not a static entity; rather, it evolves, and the infectiousness, deadliness and
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+time to recovery associated with it change with each of its numerous variants.
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+To address this issue, Liu and Stechlinski, and Setianto and Hidayat~\cite{Liu2012, Setianto2023},
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+propose an SIR model with time-dependent transition rates $\beta(t)$ and
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+$\alpha(t)$. From these rates, they derive the time-dependent reproductive
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+number $\Rt$, which represents the average number of individuals, that are
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+infected by one infectious person. A high value for $\Rt$ indicates a rapid
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+spread of the disease, while a low value either suggests either an outbreak or
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+the disease is declining. This qualifies the time-dependent reproduction number
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+$\Rt$ as an indicator of the pandemic's progression.\\
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+
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+The SIR model is defined by a system of differential equations, that incorporate
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+the transition rates, thereby depicting the fluctuation between the three
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+compartments. For a given set of data, the transition rate can be identified by
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+solving the set of differential systems. Recently, the data-driven approach of
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+\emph{physics-informed neural networks} (PINN) has gained attention due to its
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+capability of finding solutions to differential equations by fitting its
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+predictions to both given data and the governing system of differential
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+equations. By employing this methodology, Shaier \etal~\cite{Shaier2021} were
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+able to find the transition rate on synthetic data. Additionally, Millevoi
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+\etal~\cite{Millevoi2023} were able to identify the reproduction number $\Rt$
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+for both synthetic and Italian COVID-19 data using an approach based on a
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+reduced version of the SIR model.\\
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+
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+The Robert Koch Institute has collected incident and death case data from the
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+beginning of the outbreak in Germany to the present. This data will be utilitzed
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+in this bachelor thesis to investigate the transition rates and reproduction
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+number for each German state and the country as a whole, employing the
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+methodologies proposed by Shaier \etal and Millevoi \etal. Additionally, the
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+findings will be contextualized and correlated with the events of the real
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+world.\\
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+
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% -------------------------------------------------------------------
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\section{Related work 2}
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@@ -85,8 +156,24 @@ approach on five synthetic and two real-world scenarios from the early stages of
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the COVID-19 pandemic in Italy. This method yields an increase in both accuracy
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and training speed.
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+% -------------------------------------------------------------------
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+\section{Overview}
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+This thesis is comprised of four chapters. \Cref{chap:background}
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+presents with the theoretical overview of mathematical modeling in epidemiology,
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+with a particular focus on the SIR model. Subsequently, it shifts its focus to
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+neural networks, specifically on the background of physics-informed neural
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+networks (PINN) and their use in solving ordinary differential equations.
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+In~\Cref{chap:methods} outlines the methodology employed in this thesis. First
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+we present the data, that was collected by the Robert Koch Institute (RKI). Then
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+we present the PINN approaches, which are inspired by the work of Shaier \etal
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+and Millevoi \etal~\cite{Shaier2021,Millevoi2023}.~\Cref{chap:evaluation}
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+presents the setups and results of the experiments that we conduct. This chapter
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+is divided into two sections. The first section presents and discusses the
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+results concerning the transition rates of $\beta$ and $\alpha$. The subsequent
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+section presents the results concerning the reproduction value $\Rt$. Finally,
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+in \Cref{chap:conclusions}, we connect our results with the events of the
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+real-world and give an overview of potential further work.
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-
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-% -------------------------------------------------------------------
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+% -------------------------------------------------------------------
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Phillip Rothenbeck