overhaul introduction

chapters/chap01-introduction/chap01-introduction.tex

@@ -8,60 +8,39 @@
 % Version:  01.01.2012
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-\chapter{Introduction   5}
+\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). The pandemic, which originates in
+\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, encompassing the introduction of vaccines and
+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, Germany had recorded over 38,400,000 infection
+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. In light of these
+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 a
+challenging to gain a comprehensive understanding of these factors and develop
-tool that allows for the comparison of disease courses across different diseases
+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 differential equations and other
+on specific factors and modeling these using mathematical tools. These models
-mathematical tools for modeling. These models provide transition rates and
+provide transition rates and parameters that determine the behavior of a disease
-parameters that determine the behavior of a disease within the boundaries of the
+within the boundaries of the model. A fundamental epidemiological model, is the
-model. A fundamental epidemiological model, is the \emph{SIR model}, which was
+\emph{SIR model}, which was first proposed by Kermack and McKendrick~\cite{1927}
-first proposed by Kermack and McKendrick~\cite{1927} in 1927. The SIR model is a
+in 1927. The SIR model is a compartmentalized model that divides the entire
-compartmentalized model that divides the entire population into three distinct
+population into three distinct groups: the \emph{susceptible} compartment, $S$; the
-compartments. The first compartment is the \emph{susceptible} compartment, $S$,
+\emph{infectious} compartment, $I$; and the \emph{removed} compartment, $R$.
-which contains all individuals of the population who are susceptible to
+In the context of the SIR model, the constant parameters of the transmission
-infection. The second group, is the \emph{infectious} compartment, $I$, which
+rate $\beta$ and the recovery rate $\alpha$ serve to quantify and determine the
-comprises all individuals currently infected and capable of infecting
+course of a pandemic. However, pandemic is not a static entity, therefor, Liu
-susceptible individuals. Lastly, the \emph{removed} compartment, $R$, contains
+and Stechlinski~\cite{Liu2012}, and Setianto and Hidayat~\cite{Setianto2023},
-all individuals, who have succumbed to the disease or recovered from it and are
+propose an SIR model with time-dependent transition rates and reproduction number $\Rt$. The SIR model
-therefore no longer susceptible to infection. The model is characterized by two
+is defined by a system of differential equations, that incorporate
-transition rates: the transmission rate $\beta$, which controls the rate of
-individuals becoming infected and consequently transitioning from $S$ to $I$;
-and the recovery rate $\alpha$, which determines the rate at which individuals
-either recover or succumb to the disease, thereby transitioning from $I$ to $R$.
-In the context of the SIR model, the values of $\beta$ and $\alpha$ serve to
-quantify and determine the course of a pandemic.\\
-
-The transition rates of $\beta$ and $\alpha$ serve to quantify a pandemic across
-its entire duration. However, it is important to recognize that a pandemic is
-not a static entity; rather, it evolves, and the infectiousness, deadliness and
-time to recovery associated with it change with each of its numerous variants.
-To address this issue, Liu and Stechlinski, and Setianto and Hidayat~\cite{Liu2012, Setianto2023},
-propose an SIR model with time-dependent transition rates $\beta(t)$ and
-$\alpha(t)$. From these rates, they derive the time-dependent reproductive
-number $\Rt$, which represents the average number of individuals, that are
-infected by one infectious person. A high value for $\Rt$ indicates a rapid
-spread of the disease, while a low value either suggests either an outbreak or
-the disease is declining. This qualifies the time-dependent reproduction number
-$\Rt$ as an indicator of the pandemic's progression.\\
-
-The SIR model is defined by a system of differential equations, that incorporate
 the transition rates, thereby depicting the fluctuation between the three
 compartments. For a given set of data, the transition rate can be identified by
 solving the set of differential systems. Recently, the data-driven approach of
@@ -69,92 +48,98 @@ solving the set of differential systems. Recently, the data-driven approach of
 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 transition rate on synthetic data. Additionally, Millevoi
+able to find the transition rate on data for different diseases. Additionally,
-\etal~\cite{Millevoi2023} were able to identify the reproduction number $\Rt$
+Millevoi \etal~\cite{Millevoi2023} were able to identify the reproduction number
-for both synthetic and Italian COVID-19 data using an approach based on a
+$\Rt$ for both synthetic and Italian COVID-19 data using an approach based on a
 reduced version of the SIR model.\\
 
-The Robert Koch Institute has collected incident and death case data from the
+The objective of this thesis is to identify the transition rates $\beta$ and
-beginning of the outbreak in Germany to the present. This data will be utilitzed
+$alpha$, as well as the reproduction number $\Rt$ of COVID-19 over the first
-in this bachelor thesis to investigate the transition rates and reproduction
+1200 days of recorded data in Germany and its federal states. The Robert Koch
-number for each German state and the country as a whole, employing the
+Institute (RKI) has compiled data on both reported cases and associated
-methodologies proposed by Shaier \etal and Millevoi \etal. Additionally, the
+moralities from the beginning of the outbreak in Germany to the present. We
-findings will be contextualized and correlated with the events of the real
+utilize and preprocess this data according to the required format of our
-world.\\
+approaches. As the raw data lacks information on recovery incidence, we
+introduce the recovery queue that simulates a recovery period. To estimate the
+transition rates 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 transition rates 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
+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.\\
+
 
 % -------------------------------------------------------------------
 
-\section{Related work   2}
+\section{Related work}
 \label{sec:relatedWork}
-In \emph{Forecasting Epidemics Through Nonparametric Estimation of
+In this section, we categorize our work into the context of existing literature
-    Time-Dependent Transmission Rates Using the SEIR Model}~\cite{Smirnova2017},
+on the topic of solving the epidemiological models for real-world data. The
-Smirnova \etal endeavor to identify a stochastic methodology for estimating the
+first work, by Smirnova \etal~\cite{Smirnova2017}, endeavors to identify a
-time-dependent transmission rate $\beta(t)$. This is in response to the
+stochastic methodology for estimating the time-dependent transmission rate
-limitations of earlier parametric estimation methods, which are prone
+$\beta(t)$. They achieve this by projecting the time-dependent transmission rate
-instability due to the difficulty in identifying parameter finding and a low
+onto a finite subspace, that is defined by Legendre polynomials. Subsequently,
-amount of available data. They achieve this by projecting the time-dependent
+they compare the three regularization techniques of variational (Tikhonov’s)
-transmission rate onto a finite subspace, that is defined by Legendre
+regularization, truncated singular value decomposition (TSVD), and modified TSVD
-polynomials. Subsequently, they compare the three regularization techniques of
+to ascertain the most reliable method for forecasting with limited data. Their
-variational (Tikhonov’s) regularization, truncated singular value decomposition
+findings indicate that modified TSVD provides the most stable forecasts on
-(TSVD), and modified TSVD to ascertain the most reliable method for forecasting
+limited data, as demonstrated on both simulated data and real-world data from
-with limited data. Their findings indicate that modified TSVD provides the most
+the 1918 influenza pandemic and the Ebola epidemic. In contrast, we
-stable forecasts on limited data, as demonstrated on both simulated data and
+utilize physics-informed neural networks (PINN) to find the constant transition rates
-real-world data from the 1918 influenza pandemic and the 2014-2015 Ebola
+and the reproduction number for Germany and its states\\
-epidemic.\\
+
-
+Some related works similarly to us apply PINN approaches to COVID-19 and other
-In their publication, entitled \emph{Data-driven approaches for predicting
+real-world disease data such as~\cite{Shaier2021,Berkhahn2022,Olumoyin2021,Millevoi2023}.
-    spread of infectious diseases through DINNs: Disease Informed Neural Networks},
+Specifically in~\cite{Shaier2021}, Shaier \etal put forth a data-driven
-Shaier \etal~\cite{Shaier2021} put forth a data-driven approach for identifying
+approach which they refer to as disease informed neural networks (DINN). In their work,
-the parameters of epidemiological models. The authors apply physics-informed
+they demonstrate the capacity of DINNs to forecast the trajectory of epidemics
-neural networks to the compartmental SIR models, and refer to their method as
+and pandemics. They underpin the efficacy of their approach by applying it to 11
-disease informed neural networks (DINN). In their work, they demonstrate the
+diseases, that have previously been modeled. In their experiments they employ
-capacity of DINNs to forecast the trajectory of epidemics and pandemics. They
+the SIDR (susceptible, infectious, dead, recovered) model. Finally, they present
-underpin the efficacy of their approach by applying it to 11 diseases, that have
+that this method is a robust and effective means of identifying the parameters
-previously been modeled, including examples such as COVID, HIV, Tuberculosis and
+of a SIR model.\\
-Ebola. In their experiments they employ the SIDR (susceptible, infectious, dead,
+
-recovered) model. Finally, they present that this method is a robust and
+Similarly in~\cite{Berkhahn2022}, Berkhahn and Ehrhard employ the susceptible,
-effective means of identifying the parameters of a SIR model.\\
+vaccinated, infectious, hospitalized and removed (SVIHR) model. The proposed
-
+PINN methodology initially estimates the SVIHR model parameters for German
-In their article \emph{A physics-informed neural network to model COVID-19
+COVID-19 data, covering the time span from the inceptions of the outbreak to the
-    infection and hospitalization scenarios}, Berkhahn and Ehrhard~\cite{Berkhahn2022}
+end of 2021. For comparative purposes, Berkhahn and Ehrhard employ the method of
-employ the susceptible, vaccinated, infectious, hospitalized and removed (SVIHR)
+non-standard finite differences (NSFD) as well.  The authors employ both methods
-model. They solve the system of differential equations inherent to the SVIHR
+the two forecasting methods project the trajectory of COVID-19 from mid-April
-model by the means of PINNs. The authors utilize a dataset of German COVID-19
+2023 onwards. Berkhahn and Ehrhard find that the PINN is able to adapt to
-data, covering the time span from the inceptions of the outbreak to the end of
+varying vaccination rates and emerging variants.\\
-2021. The proposed PINN methodology initially estimates the SVIHR model
+
-parameters and subsequently forecasts the data. For comparative purposes,
+Furthermore, Olumoyin \etal~\cite{Olumoyin2021} employ an alternative
-Berkhahn and Ehrhard employ the method of non-standard finite differences (NSFD)
+methodology for identifying the time-dependent transmission rate of an
-as well. In the validation process, the two forecasting methods project the
+asymptomatic-SIR model accounting for unreported infectious cases. The PINN
-trajectory of COVID-19 from mid-April onwards. Berkhahn and Ehrhard find that
+approach they introduce, utilizes the cumulative and daily reported infection
-the PINN is able to adapt to varying vaccination rates and emerging variants.\\
+cases and symptomatic recovered cases, to demonstrate the effect of different
-
+mitigation measures and to ascertain the proportion of non-symptomatic
-In their work, \emph{Data-Driven Deep-Learning Algorithm for Asymptomatic
+individuals and asymptomatic recovered individuals. With this they can
-    COVID-19 Model with Varying Mitigation Measures and Transmission Rate},
-Olumoyin \etal~\cite{Olumoyin2021} employ an alternative methodology for
-identifying the time-dependent transmission rate of an asymptomatic-SIR model.
-On the premise that not all the infectious individuals are reported and included
-in the data available. The algorithm 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 size of the
-part of non-symptomatic individuals in the total number of infective individuals
-and the proportion of 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.\\
 
-In \emph{A Physics-Informed Neural Network approach for compartmental
+Finally, Millevoi \etal~\cite{Millevoi2023} address the issue of the changes in
-    epidemiological models} Millevoi \etal~\cite{Millevoi2023} address the issue
+the transmission rate due to the dynamics of a pandemic.  The authors employ the
-of describing the dynamically changing transmission rate, which is influenced by
+reproduction number to reduce the system of differential equations to a single
-the emergence of new variants or the implementation of non-pharmaceutical
+equation and introduce a reduced-split version of the PINN, which initially
-measures. They employ a PINN to maintain an account of the changes of the
+trains on the data and then trains to minimize the residual of the ODE. They
-transmission rate included in the reproduction number and to approximate the
+test their approach on five synthetic and two real-world scenarios from the
-model state variables. To this end, Millevoi \etal employ the reproduction
+early stages of the COVID-19 pandemic in Italy. This method yields an increase
-number to reduce the system of differential equations to a single equation and
+in both accuracy and training speed. In contrast, to these works, we estimate
-introduce a reduced-split version of the PINN, which initially trains on the
+the rates and the reproduction number for Germany for the entirety of the span
-data and then trains to minimize the residual of the ODE. They test their
+from early March in 2020 to late June in 2023.
-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.
 
 % -------------------------------------------------------------------
 
@@ -164,11 +149,11 @@ 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.
+networks (PINN) and their use in solving ordinary differential equations.~\Cref{chap:methods}
-In~\Cref{chap:methods} outlines the methodology employed in this thesis. First
+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
+we present the PINN approaches, which are inspired by the work of Shaier \etal~\cite{Shaier2021}
-and Millevoi \etal~\cite{Shaier2021,Millevoi2023}.~\Cref{chap:evaluation}
+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 transition rates of $\beta$ and $\alpha$. The subsequent