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% Author:   Phillip Rothenbeck
% Title:    Investigating the Evolution of the COVID-19 Pandemic in Germany Using Physics-Informed Neural Networks
% File:     chap02/chap02.tex
% Part:     theoretical background
% Description:
%         summary of the content in this chapter
% Version:  05.08.2024
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\chapter{Theoretical Background}
\label{chap:background}

This chapter introduces the theoretical knowledge that forms the foundation of
the work presented in this thesis. In~\Cref{sec:domain}
and~\Cref{sec:differentialEq}, we talk about differential equations and the
underlying theory. In these Sections both the explanations and the approach are
strongly based on the book on analysis by Rudin~\cite{Rudin2007} and the 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
in~\Cref{sec:pinn}.

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\section{Mathematical Modelling using Functions}
\label{sec:domain}

To model a physical problem using mathematical tools, it is necessary to define
a set of fundamental numbers or quantities upon which the subsequent calculations
will be based. These sets may represent, for instance, a specific time interval
or a distance. The term \emph{domain} describes these fundamental sets of
numbers or quantities~\cite{Rudin2007}. A \emph{variable} is a changing entity
living in a certain domain. In this thesis, we will focus on domains of real
numbers in $\mathbb{R}$.\\

The mapping between variables enables the modeling of the process and depicts
the semantics. We use functions in order to facilitate this mapping. Let
$A, B\subset\mathbb{R}$ be to subsets of the real numbers, then we define a
function as the mapping
\begin{equation}
  f: A\rightarrow B.
\end{equation}
In other words, the function $f$ maps elements $x\in A$ to values
$f(x)\in B$. $A$ is the \emph{domain} of $f$, while $B$ is the \emph{codomain}
of $f$. Functions are capable of representing the state of a system as a value
based on an input value from their domain. One illustrative example is a
function that maps a time point to the distance covered since a starting point.
In this case, time serves as the domain, while the distance is the codomain.

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\section{Basics of Differential Equations}
\label{sec:differentialEq}

Often, the change of a system is more interesting than its current state.
Functions are able to give us the latter, but only passively give information
about the change of a system. The objective is to determine an effective method
for calculating the change of a function across its domain. Let $f$ be a
function and $[a, b]\subset \mathbb{R}$ an interval of real numbers, the
expression
\begin{equation}
  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,
we need to narrow 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}
if it exists. $\frac{df}{dx}$ is the \emph{derivative}, or
\emph{differential equation}, it returns the momentary rate of change of $f$ for
each value $x$ of $f$'s domain. Repeating this process on $\frac{df}{dx}$ yields
$\frac{d^2f}{dx^2}$, which is the function that calculates the rate of change of
the rate of change and is called the second order derivative. Iterating this $n$
times results in $\frac{d^nf}{dx^n}$, the derivative of the $n$'th order.
Another method for obtaining a differential equation is to create it from the
semantics of a problem. This method is useful if no basic function exists for a
system. Differential equations find application in several areas such as
engineering, physics, economics, epidemiology, 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
operating in two-dimensional space, wherein each parameter represents one axis
or one that, employs with time and locations as inputs. The term
\emph{partial differential equations} (\emph{PDE}'s) describes differential
equations of such functions, which require a derivative for each of their
domains. In contrast, \emph{ordinary differential equations} (\emph{ODE}'s) are
the single derivatives for a function having only one domain. In this thesis, we
only need ODE's.\\

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.\\

Tenenbaum and Pollard~\cite{Tenenbaum1985} provide many examples for ODE's,
including the \emph{Motion of a Particle Along a Straight Line}. Further,
Newton's second law states that ``the rate of change of the momentum of a body
($momentum = mass \cdot velocity$) is proportional to the resultant external
force $F$ acted upon it''~\cite{Tenenbaum1985}. Let $m$ be the mass of the body
in kilograms, $v$ its velocity in meters per second and $t$ the time in seconds.
Then, Newton's second law translates mathematically to
\begin{equation} \label{eq:newtonSecLaw}
  F = m\frac{dv}{dt}.
\end{equation}
It is evident that the acceleration, $a=\frac{dv}{dt}$, as the rate of change of
the velocity is part of the equation. Additionally, the velocity of a body is
the derivative of the distance traveled by that body. Based on these findings,
we can rewrite the~\Cref{eq:newtonSecLaw} to
\begin{equation}
  F=ma=m\frac{d^2s}{dt^2}.
\end{equation}\\
This explanation of differential equations focuses on the aspects deemed crucial
for this thesis and is not intended to be a complete explanation of the subject.
To gain a better understanding of it, we recommend the books mentioned
above~\cite{Rudin2007,Tenenbaum1985}. In the following section we
describe the application of these principles in epidemiological models.

% -------------------------------------------------------------------

\section{Epidemiological Models}
\label{sec:epidemModel}

Pandemics, like \emph{COVID-19}, which has resulted in a significant
number of fatalities. The question arises: How should we fight a pandemic
correctly? Also, it is essential to study whether the employed countermeasures
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 aforementioned events we need to
develop a method to quantize the pandemic along with its course of
progression.\\

The real world is a highly complex system, which presents a significant
challenge attempting to describe it fully in a model. Therefore, the model must
reduce the complexity while retaining the essential information. Furthermore, it
must address the issue of limited data availability. For instance, during
COVID-19 institutions such as the Robert Koch Institute
(RKI)\footnote[1]{\url{https://www.rki.de/EN/Home/homepage_node.html}} were only
able to collect data on infections and mortality cases. Consequently, we require
a model that employs an abstraction of the real world to illustrate the events
and relations that are pivotal to understanding the problem.

% -------------------------------------------------------------------

\subsection{SIR Model}
\label{sec:pandemicModel:sir}

In 1927, Kermack and McKendrick~\cite{1927} introduced the \emph{SIR Model},
which subsequently became one of the most influential epidemiological models.
This model enables the modeling of infections during epidemiological events such as pandemics.
The book \emph{Mathematical Models in Biology}~\cite{EdelsteinKeshet2005}
reiterates the model and serves as the foundation for the following explanation
of SIR models.\\

The SIR model is capable of illustrating diseases, which are transferred through
contact or proximity of an individual carrying the illness and a healthy
individual. This is possible due to the distinction between infected beings
who are carriers of the disease and the part of the population, which is
susceptible to infection. In the model, the mentioned groups are capable to
change, e.g.,  healthy individuals becoming infected.  The model assumes the
size $N$ of the population remains constant throughout the duration of the
pandemic. The population $N$ comprises three distinct groups: the
\emph{susceptible} group $S$, the \emph{infectious} group $I$ and the
\emph{removed} group $R$ (hence SIR model). Let $\mathcal{T} = [t_0, t_f]\subseteq
  \mathbb{R}_{\geq0}$ be the time span of the pandemic, then,
\begin{equation} \label{eq:N_char}
  S: \mathcal{T}\rightarrow\mathbb{N}, \quad I: \mathcal{T}\rightarrow\mathbb{N}, \quad R: \mathcal{T}\rightarrow\mathbb{N},
\end{equation}
give the values of $S$, $I$ and $R$ at a certain point of time
$t\in\mathcal{T}$. For $S$, $I$, $R$ and $N$ applies:
\begin{equation} \label{eq:N_char}
  N = S + I + R.
\end{equation}
The model makes another assumption by stating that recovered people are immune
to the illness and infectious individual 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]
  \centering
  \includegraphics[scale=0.3]{sir_graph.png}
  \caption{A visualization of the SIR model, illustrating $N$ being split in the
    three groups $S$, $I$ and $R$.}
  \label{fig:sir_model}
\end{figure}
As visualized in the~\Cref{fig:sir_model} the
individuals may transition between groups based on transition rates. The
transmission rate $\beta$ is responsible for individuals becoming infected,
while the rate of removal or recovery rate $\alpha$ (also referred to as
$\delta$ or $\nu$, e.g.,~\cite{EdelsteinKeshet2005,Millevoi2023}) moves
individuals from $I$ to $R$.\\

We can describe this problem mathematically using a system of differential
equations (see ~\Cref{sec:differentialEq}). Thus, Kermack and
McKendrick~\cite{1927} propose the following set of differential equations:
\begin{equation}\label{eq:sir}
  \begin{split}
    \frac{dS}{dt} &= -\beta S I,\\
    \frac{dI}{dt} &= \beta S I - \alpha I,\\
    \frac{dR}{dt} &= \alpha I.
  \end{split}
\end{equation}
This, according to Edelstein-Keshet, is based on the following assumption:
``The rate of transmission of a microparasitic disease is proportional to the
rate of encounter of susceptible and infective individuals modelled by the
product ($\beta S I$)''~\cite{EdelsteinKeshet2005}. The system shows the change
in size of the groups per time unit due to infections, recoveries, and deaths.\\

The term $\beta SI$ describes the rate of encounters of susceptible and infected
individuals. This term is dependent on the size of $S$ and $I$, thus Anderson
and May~\cite{Anderson1991} propose a modified model:
\begin{equation}\label{eq:modSIR}
  \begin{split}
    \frac{dS}{dt} &= -\beta \frac{SI}{N},\\
    \frac{dI}{dt} &= \beta \frac{SI}{N} - \alpha I,\\
    \frac{dR}{dt} &= \alpha I.
  \end{split}
\end{equation}
In which $\beta SI$ gets normalized by $N$, which 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.
The infectious group has not yet infected any individuals thus
neither recovery nor mortality is possible. Let $I_0\in\mathbb{N}$ be
the number of infected individuals at the beginning of the disease. Then,
\begin{equation}
  \begin{split}
    S(0) &= N - I_{0},\\
    I(0) &= I_{0},\\
    R(0) &= 0,
  \end{split}
\end{equation}
describes the initial configuration of a system in which a disease has just
emerged.\\

\begin{figure}[h]
  %\centering
  \setlength{\unitlength}{1cm} % Set the unit length for coordinates
  \begin{picture}(12, 9.5) % Specify the size of the picture environment (width, height)
    % reference
    \put(0, 2.5){
      \begin{subfigure}{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{reference_params_synth.png}
        \caption{$\alpha=0.35$, $\beta=0.5$}
        \label{fig:synth_norm}
      \end{subfigure}
    }
    % 1. row, 1.image (low beta)
    \put(5, 5){
      \begin{subfigure}{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{low_beta_synth.png}
        \caption{$\alpha=0.25$, $\beta=0.5$}
        \label{fig:synth_low_beta}
      \end{subfigure}
    }
    % 1. row, 2.image (high beta)
    \put(9, 5){
      \begin{subfigure}{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{high_beta_synth.png}
        \caption{$\alpha=0.45$, $\beta=0.5$}
        \label{fig:synth_high_beta}
      \end{subfigure}
    }
    % 2. row, 1.image (low alpha)
    \put(5, 0){
      \begin{subfigure}{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{low_alpha_synth.png}
        \caption{$\alpha=0.35$, $\beta=0.4$}
        \label{fig:synth_low_alpha}
      \end{subfigure}
    }
    % 2. row, 2.image (high alpha)
    \put(9, 0){
      \begin{subfigure}{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{high_alpha_synth.png}
        \caption{$\alpha=0.35$, $\beta=0.6$}
        \label{fig:synth_high_alpha}
      \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.}
  \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
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
contrast, an early and high peak means that the disease is rapidly transmitted
through the population, with a significant proportion of individuals having been
infected.~\Cref{fig:sir_model} illustrates the impact of modifying either
$\beta$ or $\alpha$ while simulating  a pandemic using a model such
as~\Cref{eq:modSIR}. It is evident that both the transmission rate $\beta$
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. This
means, that it is crucial to know both $\beta$ and $\alpha$ to be able to
simulate a pandemic using the SIR model.\\

The SIR model makes a number of assumptions that are intended to reduce the
model's overall complexity while simultaneously increasing its divergence from
actual reality. One such assumption is that the size of the population, $N$,
remains constant. This depiction is not an accurate representation of the actual
relations observed in the real world, as the size of a population is subject to
a number of factors that can contribute to change. The population is increased
by the occurrence of births and decreased by the occurrence of deaths. There are
different reasons for mortality, including the natural aging process or the
development of other diseases. Other examples are the absence of the possibility
for individuals to be susceptible again, after having recovered, or the
possibility for the transition rates to change due to new variants or the
implementation of new countermeasures. We address this latter option in the
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. The model
comprises two parameters $\beta$ and $\alpha$, 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 at each time point. 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 previously
susceptible individuals with immunity without ever being infectious. To address
this Millevoi et al.~\cite{Millevoi2023} introduce a model that simultaneously
reduces the size of the system of differential equations and solves the problem
of time scaling at hand.\\

First, they alter the definition of $\beta$ and $\alpha$ to be dependent on the time interval
$\mathcal{T} = [t_0, t_f]\subseteq \mathbb{R}_{\geq0}$,
\begin{equation}
  \beta: \mathcal{T}\rightarrow\mathbb{R}_{\geq0}, \quad\alpha: \mathcal{T}\rightarrow\mathbb{R}_{\geq0}.
\end{equation}
Another crucial element is $D(t) = \frac{1}{\alpha(t)}$, which represents the initial time
span an infected individual requires to recuperate. Subsequently, at the initial time point
$t_0$, the \emph{reproduction number},
\begin{equation}
  \RO = \beta(t_0)D(t_0) = \frac{\beta(t_0)}{\alpha(t_0)},
\end{equation}
represents the number of susceptible individuals, that one infectious individual
infects at the onset of the pandemic.In light of the effects of $\beta$ and
$\alpha$ (see~\Cref{sec:pandemicModel:sir}), $\RO > 1$ indicates that the
pandemic is emerging. In this scenario $\alpha$ is relatively low due to the
limited number of infections resulting from $I(t_0) << S(t_0)$. When $\RO < 1$,
the disease is spreading rapidly across the population, with an increase in $I$
occurring at a high rate. Nevertheless, $\RO$ does not cover the entire time
span. For this reason, Millevoi et al.~\cite{Millevoi2023} introduce $\Rt$
which has the same interpretation as $\RO$, with the exception that $\Rt$ is
dependent on time. The definition of the time-dependent reproduction number on
the time interval $\mathcal{T}$ with the population size $N$,
\begin{equation}\label{eq:repr_num}
  \Rt=\frac{\beta(t)}{\alpha(t)}\cdot\frac{S(t)}{N}
\end{equation}
includes the rates of change 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
a state variable in the model and enabling the following reduction of the SIR
model.\\

\Cref{eq:N_char} allows for the calculation of the value of the group $R$ using
$S$ and $I$, with the term $R(t)=N-S(t)-I(t)$. Thus,
\begin{equation}
  \begin{split}
    \frac{dS}{dt} &= \alpha(\Rt-1)I(t),\\
    \frac{dI}{dt} &= -\alpha\Rt I(t),
  \end{split}
\end{equation}
is the reduction of~\Cref{eq:sir} on the time interval $\mathcal{T}$ using this
characteristic and the reproduction number \Rt (see ~\Cref{eq:repr_num}).
Another issue that Millevoi et al.~\cite{Millevoi2023} seek to address is the
extensive range of values that the SIR groups can assume, spanning from $0$ to
$10^7$. Accordingly, they initially scale the time interval $\mathcal{T}$ using
its borders to calculate the scaled time $t_s = \frac{t - t_0}{t_f - t_0}\in
  [0, 1]$. Subsequently, they calculate the scaled groups,
\begin{equation}
  S_s(t_s) = \frac{S(t)}{C},\quad I_s(t_s) = \frac{I(t)}{C},\quad R_s(t_s) = \frac{R(t)}{C},
\end{equation}
using a large constant scaling factor $C\in\mathbb{N}$. Applying this to the
variable $I$, results in,
\begin{equation}
  \frac{dI_s}{dt_s} = \alpha(t_f - t_0)(\Rt - 1)I_s(t_s),
\end{equation}
a further reduced version of~\Cref{eq:sir} results in a more streamlined and
efficient process, as it entails the elimination of a parameter($\beta$) and two
state variables ($S$ and $R$), while adding the state variable $\Rt$. This is a
crucial aspect for the automated resolution of such differential equation
systems, as we describe in~\Cref{sec:mlp}.

% -------------------------------------------------------------------

\section{Multilayer Perceptron}
\label{sec:mlp}
In~\Cref{sec:differentialEq} we show the importance of differential equations to
systems, being able to show the change of it dependent on a certain parameter of
the parameter. In~\Cref{sec:epidemModel} we show specific applications for
differential equations in an epidemiological context. Now, the last point is to
solve these equations. For this problem, there are multiple methods to reach
this goal one of them is the \emph{Multilayer Perceptron}
(MLP)~\cite{Hornik1989}. In the following we briefly tackle the structure,
training and usage of these neural networks.

% -------------------------------------------------------------------

\section{Physics Informed Neural Networks}
\label{sec:pinn}

% -------------------------------------------------------------------

\subsection{Disease Informed Neural Networks}
\label{sec:pinn:dinn}

% -------------------------------------------------------------------
