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.gitignore
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@ -297,5 +297,3 @@ TSWLatexianTemp*
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*.glstex
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# End of https://www.toptal.com/developers/gitignore/api/latex
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.DS_Store
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@ -138,7 +138,7 @@
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\begin{center}
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\Large{ZF Mathematik V} \\
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\small{701-0106-00L Mathematik V} \\
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\small{701-0106-00L Mathematik V, bei M. A. Sprenger} \\
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\small{Jannis Portmann \the\year} \\
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{\ccbysa}
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\rule{\linewidth}{0.25pt}
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@ -219,66 +219,26 @@ Eigenwerte $\det(\textbf{J} - \lambda \textbf{I}) = 0$ wobei $\lambda \in \mathb
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\subsection{SIR-Modell}
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SIR: Susceptible-Infected-Recovered \\
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\subsubsection{Single-Strain SIR}
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\begin{figure}[H]
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\centering
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\includegraphics[width=.25\textwidth]{SIR.png}
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\caption{SIR-Modell}
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\label{fig:sir}
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\end{figure}
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\begin{align*}
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\frac{dS}{dt} &= \Lambda - \delta_SS - \beta S I \\
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\frac{dI}{dt} &= \beta S I - \delta_I - rI \\
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\frac{dR}{dt} &= rI - \delta_R \\
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\end{align*}
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\vspace{10px}
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$\Lambda$: Geburten- oder Immigrationsrate \\
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$\delta_S, \delta_I, \delta_R$: Sterberaten der jeweiligen (Teil-)populationen \\
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$r$: Erholungsrate von $I$ \\
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$\beta S I$: Mass-action Infektionsrate \\
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$\delta_S, \delta_I, \delta_R$: Sterberaten der jeweiligen (Teil-)populationen
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$r$: Erholungsrate von $I$
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$\beta S I$: Mass-action Infektionsrate
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\begin{itemize}
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\item Disease-free equilibrium:
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$$S_f = \Lambda / \delta_S, I_f=0, R_f=0$$
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\item Endemic equilibrium:
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$$S = \frac{\delta_1 + r}{\beta} , I_e=\frac{\Lambda}{\delta_1} - \frac{\delta_S}{\beta}, R_e = \frac{r}{\delta_R}(\frac{\Lambda}{\delta_1 + r} - \frac{\delta_S}{\beta})$$
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\item
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\end{itemize}
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Für das Disease-free equilibrium ergeben sich die Eigenwerte aus
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$$(-\delta_S - \lambda)(\frac{\beta \Lambda}{\delta_S} - \delta_I - r - \lambda)(- \delta R - \lambda) = 0$$
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also
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\begin{itemize}
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\item $\lambda_1 = -\delta_S$
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\item $\lambda_2 = -\delta_R$
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\item $\lambda_3 = \frac{\beta \Lambda}{\delta_S} - \delta_I - r$
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\end{itemize}
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\subsubsection*{Reproduktionszahl $R_0$}
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$$R_0 = \frac{\beta \Lambda}{\delta_S(\delta_I + r)} = \frac{\beta S_f}{\delta_I + r}$$
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\begin{itemize}
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\item $R_0 > 1$: Ausbreitung
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\item $R_0 < 1$: Aussterben
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\end{itemize}
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\subsubsection{Multi-Strain SIR}
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\begin{figure}[H]
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\centering
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\includegraphics[width=.25\textwidth]{SIR-2.png}
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\caption{SIR-Modell mit zwei verschiedenen Erregern}
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\label{fig:sir-2}
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\end{figure}
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Invasion von Strain (2), wenn $R_0^{(1)} < R_0^{(2)}$
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\section{Taylor-Reihe}
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An der stelle $a$ einer Funtkion $f(x)$
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$$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$
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\section{Operatoren}
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\section{Operators}
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$$\mathrm{div} \, \vec{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}$$
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$$\mathrm{rot} \, \vec{u_{xy}} = \nabla \times \vec{u} = (\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})$$
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@ -308,7 +268,6 @@ Jannis Portmann, FS21
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\section*{Bildquellen}
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\begin{itemize}
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\item Abb. \ref{fig:geo-coordinates}: E\^(nix) \& ttog, \url{https://de.wikipedia.org/wiki/Geographische_Koordinaten#/media/Datei:Geographic_coordinates_sphere.svg}
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\item Abb. \ref{fig:sir}, \ref{fig:sir-2}: Vorlesungsunterlagen
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\end{itemize}
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\end{multicols*}
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