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Commit a82b986a authored by Martin Lehoux's avatar Martin Lehoux
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update article

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Article/article.aux
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View file @ a82b986a
...@@ -8,3 +8,8 @@ ...@@ -8,3 +8,8 @@
\@writefile{lof}{\select@language{french}} \@writefile{lof}{\select@language{french}}
\@writefile{lot}{\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{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}}
Article/article.fdb_latexmk
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View file @ a82b986a
# Fdb version 3 # Fdb version 3
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Article/article.fls
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\documentclass[twocolumn]{article} \documentclass[onecolumn]{article}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc} \usepackage[T1]{fontenc}
\usepackage[french]{babel} \usepackage[french]{babel}
\usepackage{fixltx2e}
\usepackage{amsmath}
\usepackage{amssymb}
\title{Rapport sur la dynamique de l'évolution des populations, appliquée à l'épidémiologie} \title{Rapport sur la dynamique de l'évolution des populations, appliquée à l'épidémiologie}
\author{Martin \textsc{Lehoux} \and Paul \textsc{Cacheux}} \author{Martin \textsc{Lehoux} \and Paul \textsc{Cacheux}}
...@@ -9,10 +12,6 @@ ...@@ -9,10 +12,6 @@
\begin{document} \begin{document}
\maketitle \maketitle
\newpage
\tableofcontents
\newpage
\section{Modèle mathématique de la dynamique d'un réservoir} \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. 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 ...@@ -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. \item Pour Python, on peut tout simplement recalculer la matrice de transition à chaque pas de calcul.
\end{itemize} \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} \end{document}
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\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}
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