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Deux options se présen \item Pour Matlab, il est nécessaire de linéariser le système autour d'un point d'équilibre afin d'effectuer des analyses de stabilité. \item Pour Python, on peut tout simplement recalculer la matrice de transition à chaque pas de calcul. \end{itemize} + +\subsection{Hypothèses} Pour notre premier modèle, nous avons besoin des hypothèses suivantes: \begin{itemize} \item Homogénéité: les individus de chaque catégorie se mélangent parfaitement dans le milieu. + \item Distribution des ages: les individus ne sont pas différents selon leur age. + \item Mis à part l'incubation, tous les phénomènes suivent des lois géométriques. + \item Les enfants des malades et des immunisés sont sains. \end{itemize} -\begin{table} + +\newpage +\subsection{Détermination des paramètres} +Pour être le plus précis possible, le modèle a besoin de nombreux paramètres. +\begin{table}[!ht] \centering - \begin{tabular}{|c|l|r|} + \begin{tabular}{|c|l|r|r|} \hline - Paramètre & Description & Valeur\\ + Paramètre & Description & Value & Normalized value (per day)\\ \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 & ...\\ + $birth$ & Birth rate & 1.85\% per year & $5.0\times10^{-5}$\\ + $death_{nat}$ & Natural death rate & 0.78\% per year & $2.1\times10^{-5}$ \\ + \hline + $\gamma$ & Infection duration & 24 days & 24\\ + $death_{ill}$ & Disease death rate & 3\% & $1.3\times10^{-3}$\\ + $R$ & Recovery rate & 97\% & 0.97\\ + $\theta$ & Incubation duration & 12 days & 12\\ + $p$ & Transmission risk & 60\% & \\ % https://www.cdc.gov/smallpox/clinicians/transmission.html + $\beta$ & Effective contact rate & & \\ + \hline + % v & Vaccination rate among healthy people & 90\%\\ + $vaccin_{duration}$ & Efficiency duration of vaccin & 10 years & 3650\\ + $loss$ & Loss of immunity & & $2.7\times10^{-4}$\\ \hline \end{tabular} - \caption{Paramètres du modèle} + \caption{Paramètres du modèle et valeurs moyennes, exemple de la smallpox} \label{parameters} \end{table} +Voici comment sont déterminés les paramètres ($||x||$ correspond à la valeur normalisée du paramètre $x$): +\begin{itemize} + \item $birth$, $death_{nat}$ sont déterminés à partir de leur valeur annuelle: + $$ + ||x|| = (1+x)^{\frac{1}{365}}-1 + $$ + \item $death_{ill}$ est calculé à partir de la durée d'infection ainsi que du taux de mortalité de la maladie: + $$ + ||death_{ill}|| = 1-(1-death_{ill})^\frac{1}{\gamma} + $$ + \item $R$ est calculé à partir de la durée d'infection: + $$ + R = (1-\death_{ill}^{\gamma})^{\frac{1}{\gamma}} + $$ + \item $loss$ est calculé à partir de la durée d'efficacité du vaccin: + $$ + loss = \frac{1}{vaccin_{duration}} + $$ +\end{itemize} + \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\\ + $r_k$ & Removed (immune) population\\ + $i_k$ & Infected population\\ + $s_k$ & Susceptible population\\ + $w_{\theta,k}$ & People with $\theta$ time step incubation\\ ... & ...\\ - $i_{1,k}$ & People with 1 time step incubation\\ + $w_{1,k}$ & People with 1 time step incubation\\ \hline \end{tabular} \caption{Vecteur d'état} \label{variables} \end{table} +\newpage +\subsection{Modèle d'évolution} Si on note le vecteur d'état $X_k$ et la matrice de transition $A$, avec $$ X_k = \begin{pmatrix} - c_{k}\\ + r_{k}\\ + i_{k}\\ s_{k}\\ - h_{k}\\ - i_{\theta,k}\\ + w_{\theta,k}\\ ...\\ ...\\ ...\\ - i_{1,k}\\ -\end{pmatrix} -$$ -$$ -A= -\begin{pmatrix} - 1+nat-\beta & \gamma & v & 0 & & & & \\ - 0 & 1+ill-\gamma & 0 & 1 & & & & \\ - \beta & 0 & 1+nat-\alpha-v & 0 & & & & \\ - & & & 0 & 1+nat & & & \\ - & & & & 0 & 1+nat & & \\ - & & & & & 0 & 1+nat & \\ - & & & & & & 0 & 1+nat\\ - & & \alpha & & & & & 0\\ + w_{1,k}\\ \end{pmatrix} $$ -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. +% $$ +% A= +% \begin{pmatrix} +% 1+nat-\beta & \gamma & v & 0 & & & & \\ +% 0 & 1+ill-\gamma & 0 & 1 & & & & \\ +% \beta & 0 & 1+nat-\alpha-v & 0 & & & & \\ +% & & & 0 & 1+nat & & & \\ +% & & & & 0 & 1+nat & & \\ +% & & & & & 0 & 1+nat & \\ +% & & & & & & 0 & 1+nat\\ +% & & \alpha & & & & & 0\\ +% \end{pmatrix} +% $$ +% 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} -- GitLab