include notes + add sir diff eq

chapters/chap02/chap02.tex

@@ -27,13 +27,13 @@ Section~\ref{sec:pinn}.
 \section{Mathematical Modelling using Functions}
 \label{sec:domain}
 
-In order to model a mathematical problem, 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
-$\mathbb{R}$.\\
+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
@@ -78,7 +78,8 @@ 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.\\
+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
@@ -90,91 +91,123 @@ 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{differential system} is the name for a collective of differential
-equations. The derivatives in a differential system each have their own
-codomain, which is part of the problem, while they all share the same domain.\\
+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} provides many examples for ODE's,
-including the \emph{Motion of a Particle Along a Straight Line}. Newton's second
-law states that ``the rate of change of the momentum of a body
+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 seconds per meter and $t$ the time in seconds.
+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, is the velocity of a body
+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 equation~\ref{eq:newtonSecLaw} to
 \begin{equation}
   F=ma=m\frac{d^2s}{dt^2}.
-\end{equation}
+\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}
 
-After a pandemic like \emph{COVID-19}, which has resulted in a significant
-number of fatalities, the question remains: How should we fight a pandemic
-correctly. Also, it is necessary to study whether the employed countermeasures
-efficacious in combating the pandemic. In the light of the unfavorable public
-responce 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 conduct the aforementioned investigations
-we need to develop a method to quantize the pandemic and its course of progression.
+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. The model must therefor
+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) 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.
+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}
 
-One of the most influential epidemiological models is the \emph{SIR Model}
-introduced by Kermack and McKendrick~\cite{1927} in 1927. The book
-\emph{Mathematical Models in Biology}~\cite{EdelsteinKeshet2005} re-iterates the
-model and the following explanation will be based on it.\\
-
-The SIR Model is able to depict diseases, which are transferred by contact or
-close proximity of an individual carrying the illness and a healthy one. This is
-possible due to the distinction between infected beings carrying the disease and
-the group if people that are healthy but can be infected. In the model the
-mentioned are able to change, by healthy individuals getting infected. In the
-real world the size of a population has many causes to change. Births increase
-the population, while deaths make it decease. There are different reasons for
-people dying, for instance old age, or another disease. To omit this factor of
-complexity, the model assumes the size $N$ of the population is constant across
-the duration of the epidemic. Three groups make up the population $N$: the
-\emph{susceptible} group $S$, the \emph{infectious} group $I$ and the
-\emph{removed} group $R$ (hence SIR Model). For $S$, $I$, $R$ and $N$ applies:
+In 1927, Kermack and McKendrick~\cite{1927} introduced the \emph{SIR Model},
+which subsequently became one of the most influential epidemiological models.
+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, by healthy individuals becoming infected. In the real world 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. To omit this factor
+of complexity, the model assumes the size $N$ of the population remains constant
+throughout the duration of the epidemic. The population $N$ is comprised of
+three distinct groups: the \emph{susceptible} group $S$, the \emph{infectious}
+group $I$ and the \emph{removed} group $R$ (hence SIR Model). For $S$, $I$, $R$
+and $N$ applies:
 \begin{equation}
   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 and dead,
-which both cannot carry the disease anymore. As visualized in the
-Figure~\ref{fig:sir_model} the individuals can traverse from one group to
-another on the bases of rates. The transmission rate $\beta$ is responsible for
-people being infected, while the rate of removal or recovery rate $\alpha$
-(in literature also $\delta$ or $\nu$) moves people from $I$ to $R$.
+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{SIR Model}
   \label{fig:sir_model}
 \end{figure}
+As visualized in the Figure~\ref{fig:sir_model} the
+individuals may transition between groups based on 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 $\nu$ in the
+literature) moves individuals from $I$ to $R$.\\
+
+Having established all components of the model, all that is left is to describe
+the relations using mathematical modelling specifically employing a system of
+differential equations as mentioned in Section~\ref{sec:differentialEq}. To be
+capable to create this system another assumption is made: ``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 of differential equations
+by Kermack and McKendrick is thus
+\begin{equation}
+  \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}
+The system shows the change of size of the groups per day due to infections,
+recoveries, and deaths.
 
 \begin{figure}
   \centering

thesis.tex

@@ -15,7 +15,7 @@
 \input{macros.tex} % define your own abbreviations,commands, etc. in file macros.tex
 
 % widow and club penalty
-\widowpenalty = 1000
+\widowpenalty = 10000
 \clubpenalty = 10000
 \displaywidowpenalty = 10000