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% summary of the content in this chapter
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% Version: 20.08.2024
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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-\chapter{Theoretical Background 12}
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+\chapter{Theoretical Background}
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\label{chap:background}
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This chapter introduces the theoretical foundations for the work presented in
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@@ -23,16 +23,16 @@ in~\Cref{sec:pinn}.
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% -------------------------------------------------------------------
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-\section{Mathematical Modelling using Functions 1}
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+\section{Mathematical Modelling using Functions}
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\label{sec:domain}
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-To model a physical problem mathematically, it is necessary to define a set of
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-fundamental numbers\todo{meeting question 1} or quantities upon which the subsequent calculations will be
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-based. These sets may represent, for instance, a specific time interval or a
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-distance. The term \emph{domain} describes these fundamental sets of numbers or
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-quantities~\cite{Rudin2007}. A \emph{variable} is a changing entity living in a
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-certain domain. In this thesis, we will focus on domains of real numbers in
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-$\mathbb{R}$.\\
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+To model a physical problem mathematically, it is necessary to define a
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+fundamental set of numbers or quantities upon which the subsequent calculations
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+will be based. These sets may represent, for instance, a specific time interval
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+or a distance. The term \emph{domain} describes these fundamental sets of
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+numbers or quantities~\cite{Rudin2007}. A \emph{variable} is a changing entity
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+living in a certain domain. In this thesis, we will focus on domains of real
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+numbers in $\mathbb{R}$.\\
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The mapping between variables enables the modeling of a physical process and may
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depict semantics. We use functions in order to facilitate this mapping. Let
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@@ -50,12 +50,12 @@ In this case, time serves as the domain, while the distance is the codomain.
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% -------------------------------------------------------------------
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-\section{Mathematical Modelling using Differential Equations 1}
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+\section{Mathematical Modelling using Differential Equations}
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\label{sec:differentialEq}
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Often, the behavior of a variable or a quantity across a domain is more
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-interesting than its current state. Functions are able to give us the latter, \todo{meeting question 2}
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-but only passively give information about the change of a system. The objective
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+interesting than its current state. Functions are able to give us the latter,
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+but do not contain information about the change of a system. The objective
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is to determine an effective method for calculating the change of a function
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across its domain. Let $f$ be a function and $[a, b]\subset \mathbb{R}$ an
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interval of real numbers. The expression
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@@ -79,11 +79,15 @@ calculates the rate of change of the rate of change and is called the
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$\nicefrac{d^nf}{dx^n}$, the derivative of the $n$'th order. A method for
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obtaining a differential equation is to derive it from the semantics of a
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problem. For example, in physics a differential equation can be derived from the
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-law of the conservation of energy~\cite{Demtroeder2021}. Differential equations \todo{is this good?}
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-find application in several areas such as engineering \eg, the Chua's
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-circuit~\cite{Matsumoto1984}, physics with, \eg, the Schrödinger
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-equation~\cite{Schroedinger1926}, economics, \eg, Black-Scholes
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-equation~\cite{Oksendal2000}, epidemiology, and beyond.\\
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+law of the conservation of energy~\cite{Demtroeder2021}. Differential equations
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+find application in several areas such as engineering \eg, the Kirchhoff's
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+circuit laws~\cite{Kirchhoff1845} to describe the relation between the voltage
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+and current in systems with resistors, inductors, and capacitors; physics with,
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+\eg, the Schrödinger equation, which predicts the probability of finding
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+particles like electrons in specific places or states in a quantum system;
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+economics, \eg, Black-Scholes equation~\cite{Oksendal2000} predicting the price
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+of financial derivatives, such as options, over time; epidemiology with the SIR
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+Model~\cite{1927}; and beyond.\\
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In the context of functions, it is possible to have multiple domains, meaning
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that function has more than one parameter. To illustrate, consider a function
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@@ -127,11 +131,11 @@ models.
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% -------------------------------------------------------------------
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-\section{Epidemiological Models 4}
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+\section{Epidemiological Models}
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\label{sec:epidemModel}
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Pandemics, like \emph{COVID-19}, which have resulted in a significant
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-number of fatalities. Hence, the question arises: How should we analyze a \todo{Better?}
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+number of fatalities. Hence, the question arises: How should we analyze a
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pandemic effectively? It is essential to study whether the employed
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countermeasures are efficacious in combating the pandemic. Given the unfavorable
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public response to measures such as lockdowns, it is imperative to investigate
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@@ -154,7 +158,7 @@ and relations that are pivotal to understanding the problem.
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% -------------------------------------------------------------------
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-\subsection{SIR Model 3}
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+\subsection{SIR Model}
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\label{sec:pandemicModel:sir}
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In 1927, Kermack and McKendrick~\cite{1927} introduced the \emph{SIR Model},
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@@ -252,7 +256,7 @@ emerged.\\
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\begin{picture}(12, 9.5) % Specify the size of the picture environment (width, height)
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% reference
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\put(0, 1.75){
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- \begin{subfigure}{0.4\textwidth}
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+ \begin{subfigure}{0.35\textwidth}
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\centering
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\includegraphics[width=\textwidth]{reference_params_synth.pdf}
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\caption{$\alpha=0.35$, $\beta=0.5$}
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@@ -260,7 +264,7 @@ emerged.\\
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\end{subfigure}
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}
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% 1. row, 1.image (low beta)
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- \put(5.5, 5){
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+ \put(5.25, 5){
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\begin{subfigure}{0.3\textwidth}
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\centering
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\includegraphics[width=\textwidth]{low_beta_synth.pdf}
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@@ -278,7 +282,7 @@ emerged.\\
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\end{subfigure}
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}
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% 2. row, 1.image (low alpha)
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- \put(5.5, 0){
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+ \put(5.25, 0){
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\begin{subfigure}{0.3\textwidth}
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\centering
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\includegraphics[width=\textwidth]{low_alpha_synth.pdf}
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@@ -334,7 +338,7 @@ option in the next~\Cref{sec:pandemicModel:rsir}.
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% -------------------------------------------------------------------
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-\subsection{Reduced SIR Model and the Reproduction Number 1}
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+\subsection{Reduced SIR Model and the Reproduction Number}
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\label{sec:pandemicModel:rsir}
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The~\Cref{sec:pandemicModel:sir} presents the classical SIR model. This model
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contains two scalar parameters $\beta$ and $\alpha$, which describe the course
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@@ -345,10 +349,12 @@ the disease. The reason for this is due to events such as the implementation of
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countermeasures that reduce the contact between the infectious and susceptible
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individuals, the emergence of a new variant of the disease that increases its
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infectivity or deadliness, or the administration of a vaccination that provides
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-previously susceptible individuals with immunity without ever being infected. \todo{sai correction -> is this point not already included?}
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-To address this Millevoi \etal~\cite{Millevoi2023} introduce a model that \todo{are there older sources}
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-simultaneously reduces the size of the system of differential equations and
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-solves the problem of time scaling at hand.\\
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+previously susceptible individuals with immunity without ever being infected.
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+To address this, based on the time-dependent transition rates introduced by Liu
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+and Stechlinski~\cite{Liu2012}, and Setianto and Hidayat~\cite{Setianto2023},
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+Millevoi \etal~\cite{Millevoi2023} present a model that simultaneously reduces
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+the size of the system of differential equations and solves the problem of time
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+scaling at hand.\\
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First, they alter the definition of $\beta$ and $\alpha$ to be dependent on the time interval
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$\mathcal{T} = [t_0, t_f]\subseteq \mathbb{R}_{\geq0}$,
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@@ -413,7 +419,7 @@ situation, due to its fewer input variables.
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% -------------------------------------------------------------------
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-\section{Multilayer Perceptron 2}
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+\section{Multilayer Perceptron}
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\label{sec:mlp}
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In~\Cref{sec:differentialEq}, we demonstrate the significance of differential
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equations in systems, illustrating how they can be utilized to elucidate the
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@@ -532,7 +538,7 @@ solutions to differential systems.
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% -------------------------------------------------------------------
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-\section{Physics Informed Neural Networks 4}
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+\section{Physics Informed Neural Networks}
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\label{sec:pinn}
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In~\Cref{sec:mlp}, we describe the structure and training of MLP's, which are
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@@ -642,7 +648,7 @@ parameter and the observation loss.
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% -------------------------------------------------------------------
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-\subsection{Disease Informed Neural Networks 1}
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+\subsection{Disease Informed Neural Networks}
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\label{sec:pinn:dinn}
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In this section, we describe the capability of MLP's to solve systems of
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differential equations. In~\Cref{sec:pandemicModel:sir}, we describe the SIR
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Phillip Rothenbeck