\documentclass[a4paper,12pt]{article}
%
% graphic files used:
%	pandemic_spreading, pandemic_damping, pandemic_SIR
%
\usepackage{beamerarticle}
% 
%\usepackage{harvard} 
%
\usepackage[latin1]{inputenc}  % Erlaubt mehr als nur ASCII-Zeichen (z.B. Umlaute) im Text  
\usepackage[T1]{fontenc}       % Sorgt fuer richtige Trennung auch bei Umlauten im Wort
\usepackage{graphicx}   
\usepackage[english]{babel}
%
\usepackage{amsmath}           % more symbols
%
%\usepackage{tikz}                  % trees
%\usetikzlibrary{fit}               % for trees, too
%\usetikzlibrary{shapes.geometric}  % for trees, too
\definecolor{myblue}{HTML}{92dcec} % colors for trees
\colorlet{myfillgray}{blue!5}
\colorlet{mydrawgray}{blue!20}
\colorlet{mydarkgray}{blue!80}
%
%
\definecolor{lightgray}{gray}{.8}
%
\usepackage{textcomp}
%\usepackage{bera} %????
%
\newcommand{\nc}[1]{
\begin{frame}
\begin{center}
\textcolor{structure}{\huge #1}
\end{center}
\end{frame}
}
%
\renewcommand{\labelenumi}{(\arabic{enumi})}
%
\input{../../layout/TeXLayout}
%
\begin{document}
\thispagestyle{empty}
%
\input{../lecture/title_growth}
\begin{center}
{\Large \bf Economic Growth} \\[2ex]
{\large \bf Problems Sets} \\[2ex]
{\bf Winter Semester 2025
}\\[2ex]
{\large \bf Karl-Josef Koch}\\[5ex]
\end{center}
\subsection*{Problem Set 1}
\subsubsection*{Exercise 1}
%%%%%%   \input{growth_exercise_01_SIR_with_solution}
%
\lecture{Economic Growth}{w02}
\only<article>{ \subsection*{Digression: The SIR-model of Mathematical Epidemiology} }
\begin{frame}
\begin{block}{\only<presentation>{Digression: Mathematical Epidemiology: The SIR Model}}
Three real variables $S,I,N$ are relevant measures in the context of a pandemic 
\begin{itemize}
\item [] $S$ is the number of susceptible individuals, 
\item [] $I$ the number of infectious individuals, 
\item [] $N$ the size of the population
\end{itemize}
Moreover, $c$ is the infection rate, $\omega$ the recovery rate.%
\only<article>{\footnote{The inverse of these rates is the average time it takes to get infected or to recover.}}
\pause
\only<presentation>{\vspace{-4ex}

}
\begin{eqnarray*}
\dot I &=& c \, I \, S - \omega \, I\\ \pause
\dot S &=& -c \, I \, S + \omega \, I\\ \pause
N      &=& I + S
\end{eqnarray*}
\pause
Verify: 
\begin{enumerate}
\item $N$ is constant over time!
\item The differential equations above are interdependent. Although the differential equation for $I$ looks linear in $I$, it is actually quadratic.
\item The system of differential equations $(\dot I,\; \dot S)$ can be transformed into two very similar unconnected, quadratic differential equations. Using a short hand $A = c \, N - \omega$ one of them is
\[ \dot I = (A - c \, I) \, I \] 
\item If $I$ takes the value $I_0$ at time zero the following function is the solution for $I(t)$.
\[ I(t) = \frac{I_0 e^{At}}{1+\frac{c}{A} I_0 \left(e^{At} -1\right)} \]
\item Find the solution for $S(t)$.
\item For $t \rightarrow \infty$ the number of infected $I$ will converge to $N - \omega / c$.
\end{enumerate}
\pause
%\[ \dot I = c \, I \, (N-I) - \omega \, I \]
\end{block}
\end{frame} 
%
\begin{frame}
\begin{block}{\only<presentation>{Digression: Mathematical Epidemiology}}
Let us put some numbers to the model get a feeling for what can happen in such a system.
\[ c = 2 \cdot 10^{-9} ,\; \quad \omega = 0,03 \]
The right hand side of the differential equation %
\only<presentation>{ \[ \dot I = (A - c \, I) \, I \] }
decomposes into two effects. 
\begin{itemize}
\item The positive linear part $A \, I$ of $\dot I$ causes exponential growth due to spreading of the virus. This makes up the numerator of the solution.
\item This linear part drives the infecion process if there are no limiting obstacles for the virus to spread. Various reaactions may cause a damping of spreading of the virus if the share of infected people is high.
\item The negative quadratic part $- c \, I^2$ causes damping of the exponential growth due to saturation making up the denominator.
\item The result may be almost exponential spreading at the beginning of the pandemic whereas spreading may come to a halt before the share of infected people reaches 100\%.
\end{itemize}
\end{block}
\end{frame} 
%
\begin{frame}
\begin{block}{\only<presentation>{Digression: Mathematical Epidemiology}}
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{../../graphics/pandemic_spreading}
\end{center}
\caption{Exponential spreading}
\end{figure}
\end{block}
\end{frame} 
%
\begin{frame}
\begin{block}{\only<presentation>{Digression: Mathematical Epidemiology}}
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{../../graphics/pandemic_damping}
\end{center}
\caption{Damping of the exponential spreading}
\end{figure}
\end{block}
\end{frame}  
%
\begin{frame}
\begin{block}{\only<presentation>{Digression: Mathematical Epidemiology}}
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{../../graphics/pandemic_SIR}
\end{center}
\caption{The SIR-model of mathematical epidemiology}
\end{figure}
\end{block}
\end{frame} 
%
\end{document}