finish SIR and start MLP

chapters/chap02/chap02.tex

@@ -163,16 +163,17 @@ 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. 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:
+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}
@@ -201,7 +202,7 @@ McKendrick~\cite{1927} propose the following set of differential equations:
   \begin{split}
     \frac{dS}{dt} &= -\beta S I,\\
     \frac{dI}{dt} &= \beta S I - \alpha I,\\
-    \frac{dR}{dt} &= \alpha I,
+    \frac{dR}{dt} &= \alpha I.
   \end{split}
 \end{equation}
 This, according to Edelstein-Keshet, is based on the following assumption:
@@ -221,7 +222,7 @@ and May~\cite{Anderson1991} propose a modified model:
   \end{split}
 \end{equation}
 In which $\beta SI$ gets normalized by $N$, which is more correct in a
-real world aspect.\\
+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.
@@ -239,42 +240,56 @@ describes the initial configuration of a system in which a disease has just
 emerged.\\
 
 \begin{figure}[h]
-  \centering
-  \begin{subfigure}[h]{0.3\textwidth}
-    \centering
-    \includegraphics[width=\textwidth]{reference_params_synth.png}
-    \caption{Basic configuration, $\alpha=0.35$, $\beta=0.5$}
-    \label{fig:synth_norm}
-  \end{subfigure}
-  \hfill
-  \begin{subfigure}[h]{0.3\textwidth}
-    \centering
-    \includegraphics[width=\textwidth]{high_beta_synth.png}
-    \caption{High $\alpha$ configuration, $\alpha=0.45$, $\beta=0.5$}
-    \label{fig:synth_high_beta}
-  \end{subfigure}
-  \hfill
-  \begin{subfigure}[h]{0.3\textwidth}
-    \centering
-    \includegraphics[width=\textwidth]{low_beta_synth.png}
-    \caption{Low $\alpha$ configuration, $\alpha=0.25$, $\beta=0.5$}
-    \label{fig:synth_low_beta}
-  \end{subfigure}
-  \hfill
-  \begin{subfigure}[b]{0.3\textwidth}
-    \centering
-    \includegraphics[width=\textwidth]{high_alpha_synth.png}
-    \caption{High $\beta$ configuration, $\alpha=0.35$, $\beta=0.6$}
-    \label{fig:synth_high_alpha}
-  \end{subfigure}
-  \begin{subfigure}[b]{0.3\textwidth}
-    \centering
-    \includegraphics[width=\textwidth]{low_alpha_synth.png}
-    \caption{Low $\beta$ configuration, $\alpha=0.35$, $\beta=0.3$}
-    \label{fig:synth_low_alpha}
-  \end{subfigure}
-  \caption{Synthetic data, using~\Cref{eq:modSIR} and $N=7.9\cdot 10^6$,
-    $I_0=10$ with different sets of parameters.}
+  %\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}
 
@@ -293,7 +308,21 @@ 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.
+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}.
 
 % -------------------------------------------------------------------
 
@@ -377,6 +406,14 @@ 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.
 
 % -------------------------------------------------------------------