From a82b986aa7c215dde2b3e905a1a60f37c8512fcc Mon Sep 17 00:00:00 2001 From: Kagamino <martin.lehoux@supelec.fr> Date: Wed, 18 Apr 2018 18:49:48 +0200 Subject: [PATCH] update article --- Article/article.aux | 5 ++ Article/article.fdb_latexmk | 43 ++++++--- Article/article.fls | 50 +++++++++-- Article/article.log | 174 ++++++++++++++++++++++++++---------- Article/article.pdf | Bin 83535 -> 126048 bytes Article/article.synctex.gz | Bin 4360 -> 14457 bytes Article/article.tex | 84 +++++++++++++++-- Article/article.toc | 2 - 8 files changed, 291 insertions(+), 67 deletions(-) delete mode 100644 Article/article.toc diff --git a/Article/article.aux b/Article/article.aux index 5569c91..e4c31a5 100644 --- a/Article/article.aux +++ b/Article/article.aux @@ -8,3 +8,8 @@ \@writefile{lof}{\select@language{french}} \@writefile{lot}{\select@language{french}} \@writefile{toc}{\contentsline {section}{\numberline {1}Mod\IeC {\`e}le math\IeC {\'e}matique de la dynamique d'un r\IeC {\'e}servoir}{1}} +\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Param\IeC {\`e}tres du mod\IeC {\`e}le}}{1}} +\newlabel{parameters}{{1}{1}} +\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Vecteur d'\IeC {\'e}tat}}{2}} +\newlabel{variables}{{2}{2}} +\@writefile{toc}{\contentsline {section}{\numberline {2}Mod\IeC {\`e}le math\IeC {\'e}matique d'interaction entre plusieurs populations}{3}} diff --git a/Article/article.fdb_latexmk b/Article/article.fdb_latexmk index 4906691..db1a7a7 100644 --- a/Article/article.fdb_latexmk +++ b/Article/article.fdb_latexmk @@ -1,15 +1,24 @@ # Fdb version 3 -["pdflatex"] 1524062144 "/home/kagamino/Documents/Cours/CentraleSupelec 1A/dynamique-populations/Article/article.tex" "article.pdf" "article" 1524062144 +["pdflatex"] 1524069504 "/home/kagamino/Documents/Cours/CentraleSupelec 1A/dynamique-populations/Article/article.tex" "article.pdf" "article" 1524069505 "/etc/texmf/web2c/texmf.cnf" 1522149631 475 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z>-cj6_MVXY-_KR9#!=~fzh}=Xg`YcH3*SF6Oy}#}F@td$_a@)XC+10iJalnFZuk9= zzEkN(dk=c?X?!<1b>{Ju85UOWT$jeBUwiEAWpz?$&${_yW<^R_Cfd>)4c4-66<n1r za^_KG*CyuHXBR7`n)V-a{T2LQ+4z_Hv59K8p4M7e>3pBNCu5Szq`Z0Am65!MZn-6z z#Z)ce>6iW4rZmy^XZcTYzc=gu`_EbPU$-Iuz}d<6ci!_qpMUTEyK{aIzPwXNeEURN drBA71>;8%z|7t4!l$3pvU;l=qx5Jfz0RR>^uc-h4 diff --git a/Article/article.tex b/Article/article.tex index 38d61b2..69afd3d 100644 --- a/Article/article.tex +++ b/Article/article.tex @@ -1,7 +1,10 @@ -\documentclass[twocolumn]{article} +\documentclass[onecolumn]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[french]{babel} + \usepackage{fixltx2e} + \usepackage{amsmath} + \usepackage{amssymb} \title{Rapport sur la dynamique de l'évolution des populations, appliquée à l'épidémiologie} \author{Martin \textsc{Lehoux} \and Paul \textsc{Cacheux}} @@ -9,10 +12,6 @@ \begin{document} \maketitle -\newpage - -\tableofcontents -\newpage \section{Modèle mathématique de la dynamique d'un réservoir} On s'intérèsse à l'évolution d'une population sans échanges avec l'extérieur. Cette population est composée de catégories, que nous expliciterons dans la suite. @@ -24,4 +23,79 @@ et on ne peut le représenter par une chaîne de Markov. Deux options se présen \item Pour Python, on peut tout simplement recalculer la matrice de transition à chaque pas de calcul. \end{itemize} +\begin{table} + \centering + \begin{tabular}{|c|l|r|} + \hline + Paramètre & Description & Valeur\\ + \hline + nat & Natural population balance rate & 1\%\\ + ill & Ill population balance rate & ...\\ + v & Vaccination rate among healthy people & 90\%\\ + $\alpha$ & Contamination rate & ...\\ + n & Natality rate & 1.5\%\\ + $\gamma$ & Recovery rate among ill people & ...\\ + $\beta$ & Unimmunty rate among healed people & ...\\ + $\theta$ & Pure delay for incubation of the disease & ...\\ + \hline + \end{tabular} + \caption{Paramètres du modèle} + \label{parameters} +\end{table} + +\begin{table} + \centering + \begin{tabular}{|c|l|} + \hline + Variable & Description\\ + \hline + $h_k$ & Healthy people\\ + $s_k$ & Sick people\\ + $c_k$ & Cured people\\ + $i_{\theta,k}$ & People with $\theta$ time step incubation\\ + ... & ...\\ + $i_{1,k}$ & People with 1 time step incubation\\ + \hline + \end{tabular} + \caption{Vecteur d'état} + \label{variables} +\end{table} + +Si on note le vecteur d'état $X_k$ et la matrice de transition $A$, avec +$$ +X_k = +\begin{pmatrix} + c_{k}\\ + s_{k}\\ + h_{k}\\ + i_{\theta,k}\\ + ...\\ + ...\\ + ...\\ + i_{1,k}\\ +\end{pmatrix} +$$ +$$ +A= +\begin{pmatrix} + 1+nat-\beta & \gamma & v & 0 & & & & \\ + 0 & 1+ill-\gamma & 0 & 1 & & & & \\ + \beta & 0 & 1+nat-\alpha-v & 0 & & & & \\ + & & & 0 & 1+nat & & & \\ + & & & & 0 & 1+nat & & \\ + & & & & & 0 & 1+nat & \\ + & & & & & & 0 & 1+nat\\ + & & \alpha & & & & & 0\\ +\end{pmatrix} +$$ +Alors l'équation aux différences du système se réécrit de manière matricielle : +$$ +X_{k+1}=AX_k +$$ +Le problème de non-linéarité se trouve au niveau du coefficient $\alpha$, qui dépend en réalité du nombre total de personnes infectées ou sous incubation. + +\newpage +\section{Modèle mathématique d'interaction entre plusieurs populations} +Après avoir étudier le fonctionnement d'un réservoir en tant que système fermé, on peut si'intéresser à l'interaction entre plusieurs réservoirs. +Pour ne pas perdre les résulats de l'étude précédente, on peut scinder le nouveau fonctionnement à l'aide d'un schéma-bloc. \end{document} \ No newline at end of file diff --git a/Article/article.toc b/Article/article.toc deleted file mode 100644 index fbd2329..0000000 --- a/Article/article.toc +++ /dev/null @@ -1,2 +0,0 @@ -\select@language {french} -\contentsline {section}{\numberline {1}Mod\IeC {\`e}le math\IeC {\'e}matique de la dynamique d'un r\IeC {\'e}servoir}{1} -- GitLab