finish sir section

chapters/chap02/chap02.tex

@@ -191,14 +191,14 @@ $\beta$ 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
+In order to model the problem mathematically using a system of differential
+equations as we describe in Section~\ref{sec:differentialEq}, it is necessary to
+make an assumption serving as the foundation for the model. In their book,
+Edelstein-Keshet makes 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}. Kermack and McKendrick~\cite{1927}
+thus propose the following set of differential equations:
 \begin{equation}
   \begin{split}
     \frac{dS}{dt} &= -\beta S I,\\
@@ -206,49 +206,94 @@ by Kermack and McKendrick is thus
     \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.
+The system shows the change of size of the groups per time unit due to
+infections, recoveries, and deaths.\\
 
-\begin{figure}
+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.\\
+
+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 $I_0\in\mathbb{N}_{\geq0}$ 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
-  \begin{subfigure}[b]{0.3\textwidth}
+  \begin{subfigure}[h]{0.3\textwidth}
     \centering
     \includegraphics[width=\textwidth]{synth_alpha_beta}
     \caption{Basic configuration, $\alpha=0.35$, $\beta=0.2$}
     \label{fig:synth_norm}
   \end{subfigure}
   \hfill
-  \begin{subfigure}[b]{0.3\textwidth}
+  \begin{subfigure}[h]{0.3\textwidth}
     \centering
     \includegraphics[width=\textwidth]{synth_alpha_high_beta}
     \caption{High $\alpha$ configuration, $\alpha=0.45$, $\beta=0.2$}
     \label{fig:synth_high_beta}
   \end{subfigure}
   \hfill
-  \begin{subfigure}[b]{0.3\textwidth}
+  \begin{subfigure}[h]{0.3\textwidth}
     \centering
     \includegraphics[width=\textwidth]{synth_alpha_low_beta}
     \caption{Low $\alpha$ configuration, $\alpha=0.25$, $\beta=0.2$}
     \label{fig:synth_low_beta}
   \end{subfigure}
   \hfill
-  \begin{subfigure}[h]{0.3\textwidth}
+  \begin{subfigure}[b]{0.3\textwidth}
     \centering
     \includegraphics[width=\textwidth]{synth_high_alpha_beta}
     \caption{High $\beta$ configuration, $\alpha=0.35$, $\beta=0.3$}
     \label{fig:synth_high_alpha}
   \end{subfigure}
-  \hfill
-  \begin{subfigure}[h]{0.3\textwidth}
+  \begin{subfigure}[b]{0.3\textwidth}
     \centering
     \includegraphics[width=\textwidth]{synth_low_alpha_beta}
     \caption{Low $\beta$ configuration, $\alpha=0.35$, $\beta=0.1$}
     \label{fig:synth_low_alpha}
   \end{subfigure}
-  \caption{Synthetic data with different sets of parameters.}
+  \caption{Synthetic data, using the Equations~\ref{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 the pace with the rate of recovery, resulting
+in its demise before it can exert a significant influence on the population. In
+contrast, an early, high peak means that the disease is rapidly transmitted
+through the population, with a significant proportion of individuals having been
+infected. Figure~\ref{fig:sir_model} illustrates the impact of modifying either
+$\beta$ or $\alpha$ while simulating  a pandemic using a model
+such as~\ref{eq:modSIR}. It is evident that both the transmission rate $\beta$
+and the recovery rate $\alpha$ influence the height and time of occurrence 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 quantize a pandemic using the SIR model.
+
 % -------------------------------------------------------------------
 
 \subsection{reduced SIR Model}

thesis.bbl

@@ -1,4 +1,4 @@
-\begin{thebibliography}{Rud07}
+\begin{thebibliography}{And91}
 
 % this bibliography is generated by alphadin.bst [8.2] from 2005-12-21
 
@@ -7,6 +7,11 @@
   \providecommand{\doi}[1]{doi: #1}\else
   \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi
 
+\bibitem[And91]{Anderson1991}
+\textsc{Anderson}, Robert~M. Roy Malcolm;~May~M. Roy Malcolm;~May:
+\newblock \emph{Infectious diseases of humans : dynamics and control}.
+\newblock Oxford University Press, 1991
+
 \bibitem[EK05]{EdelsteinKeshet2005}
 \textsc{Edelstein-Keshet}, Leah:
 \newblock \emph{Mathematical Models in Biology}.

thesis.bib

@@ -84,4 +84,11 @@
   year      = {2005},
 }
 
+@Book{Anderson1991,
+  author    = {Anderson, Roy Malcolm; May, Robert M.},
+  publisher = {Oxford University Press},
+  title     = {Infectious diseases of humans : dynamics and control},
+  year      = {1991},
+}
+
 @Comment{jabref-meta: databaseType:bibtex;}