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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 fbd2329dc168f9b157b2d4ab7c72cd1eee4b4531..0000000000000000000000000000000000000000 --- 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}