From 8e0b941d59e163a53da967a4b37e059e21f74eee Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Sat, 24 Jul 2021 12:31:26 +0200 Subject: [PATCH] Expand SIR further --- Mathematik-V-ZF.tex | 34 ++++++++++++++++++++++++++++++---- img/SIR-2.png | Bin 0 -> 114865 bytes 2 files changed, 30 insertions(+), 4 deletions(-) create mode 100644 img/SIR-2.png diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 893330d..2a154f1 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -138,7 +138,7 @@ \begin{center} \Large{ZF Mathematik V} \\ - \small{701-0106-00L Mathematik V, bei M. A. Sprenger} \\ + \small{701-0106-00L Mathematik V} \\ \small{Jannis Portmann \the\year} \\ {\ccbysa} \rule{\linewidth}{0.25pt} @@ -222,9 +222,10 @@ Eigenwerte $\det(\textbf{J} - \lambda \textbf{I}) = 0$ wobei $\lambda \in \mathb SIR: Susceptible-Infected-Recovered \\ +\subsubsection{Single-Strain SIR} \begin{figure}[H] \centering - \includegraphics[width=.3\textwidth]{SIR.png} + \includegraphics[width=.25\textwidth]{SIR.png} \caption{SIR-Modell} \label{fig:sir} \end{figure} @@ -245,9 +246,34 @@ $\beta S I$: Mass-action Infektionsrate \\ $$S_f = \Lambda / \delta_S, I_f=0, R_f=0$$ \item Endemic equilibrium: $$S = \frac{\delta_1 + r}{\beta} , I_e=\frac{\Lambda}{\delta_1} - \frac{\delta_S}{\beta}, R_e = \frac{r}{\delta_R}(\frac{\Lambda}{\delta_1 + r} - \frac{\delta_S}{\beta})$$ - \item \end{itemize} +Für das Disease-free equilibrium ergeben sich die Eigenwerte aus +$$(-\delta_S - \lambda)(\frac{\beta \Lambda}{\delta_S} - \delta_I - r - \lambda)(- \delta R - \lambda) = 0$$ +also +\begin{itemize} + \item $\lambda_1 = -\delta_S$ + \item $\lambda_2 = -\delta_R$ + \item $\lambda_3 = \frac{\beta \Lambda}{\delta_S} - \delta_I - r$ +\end{itemize} + +\subsubsection*{Reproduktionszahl $R_0$} +$$R_0 = \frac{\beta \Lambda}{\delta_S(\delta_I + r)} = \frac{\beta S_f}{\delta_I + r}$$ +\begin{itemize} + \item $R_0 > 1$: Ausbreitung + \item $R_0 < 1$: Aussterben +\end{itemize} + +\subsubsection{Multi-Strain SIR} +\begin{figure}[H] + \centering + \includegraphics[width=.25\textwidth]{SIR-2.png} + \caption{SIR-Modell mit zwei verschiedenen Erregern} + \label{fig:sir-2} +\end{figure} + +Invasion von Strain (2), wenn $R_0^{(1)} < R_0^{(2)}$ + \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ @@ -282,7 +308,7 @@ Jannis Portmann, FS21 \section*{Bildquellen} \begin{itemize} \item Abb. \ref{fig:geo-coordinates}: E\^(nix) \& ttog, \url{https://de.wikipedia.org/wiki/Geographische_Koordinaten#/media/Datei:Geographic_coordinates_sphere.svg} - \item Abb. \ref{fig:sir}: Vorlesungsunterlagen + \item Abb. \ref{fig:sir}, \ref{fig:sir-2}: Vorlesungsunterlagen \end{itemize} \end{multicols*} diff --git a/img/SIR-2.png b/img/SIR-2.png new file mode 100644 index 0000000000000000000000000000000000000000..cee88f1aa428cee5e968bac91dd2334a350ef178 GIT binary patch literal 114865 zcmdSAbzGHO&^8PRN=Sza(jZ+T-6h@K-QC?NDIkrMgmi;+OGtO8q;!ciylbE1IY-~` zd!GNkKR$jN_RZdRtTi)h=9+6}h04i@A|v7=LP0?xi;D>>KtaJtK|#Uf!aoF`B=}9_ zKtVk=GZzw)6BiO9l5?;%F}E^?f)WdjSAC?U*o&L48512n0)rln;enZ>N<~}^1)~6m zM~pxg_Y@-_3}0Kxt|Wv~#YwrdrU9lBr?ZqcpqT+jDb88`iHN*ACdP=fF2^CC>4Def zW=5;==F&5$SJ!nI;`D`Z{ONW4xP$})&oe%BlR3k{5W#RV!bWPPOLc^Whd-2id)37JmQmlvY z_VqBPRLAsiLvpLxft;boRw46Ha@h{`W3*7O4g}KGzxD^Xyy6wuqumVegffc^VE3a~ zKOt7>t zoI#a${R=K@p&TxH*;m4;BbA{EO1NCX1-hY-P~{2SRbF&mbyq04Fgb$vITNDq++MO4 z6YBBr4Q#y?7O;Y~Phw|#q0wb(6xP9x_GL0a<_WeP2Juv|TLnMPhmT{kBx$&1CvT>V 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