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retard pur pour le modèle SEIQR} +Afin de complexifier le modèle, on pense tout d'abord à instaurer un état d'incubation. Cet état diffère des autres puisqu'il est déféni par un retard pur +pour la maladie, avant que celle ci affecte réellement le malade \cite{pure-delay}. Dans cet article, le modèle du retard pur est implémenté à l'aide de +plusieurs réservoirs, en réalité un réservoir par pas de calcul, pour la durée totale de la période d'incubation. +Ainsi, à chaque pas de calcul, le contenu de chaque réservoir est déversé dans le suivant, tandis que le dernier s'ajoute à la population malade. \newpage \section{Modèle mathématique de la dynamique d'un réservoir} @@ -140,20 +145,19 @@ Voici comment sont déterminés les paramètres ($||x||$ correspond à la valeur \newpage \subsection{Premier modèle d'évolution : SIR} -Le modéle SIR est le modéle fondamental dans l'étude compartimental en épidémiologie\cite{sir-model}. -Voici les différentes équations régissant l'évolution pas à pas du système (tout d'abord sans incubation): +Voici les différentes équations régissant l'évolution pas à pas du système (tout d'abord sans l'incubation): \begin{itemize} \item $s_{k+1} = (-death_{nat}-p.\frac{i_k}{s_k+i_k+r_k})s_k + loss.r_k + birth(s_k+i_k+r_k)$ \item $i_{k+1} = p.\frac{i_k}{s_k+i_k+r_k}.s_k + (-death_{ill}-R)i_k$ \item $r_{k+1} = vacc_{rate}.s_k + R.i_k + (1-death_{nat}-loss)r_k$ \end{itemize} - En particulier, le terme $p.\frac{i_k}{s_k+i_k+r_k}.s_k$ correspond à la transmission de la maladie. Il est important de noter qu'il dépend à la fois du nombre de personnes susceptibles, mais aussi de la proportion de personnes infectées parmis la population. Cela pose un problème car ce terme rend le système non-linéaire. -Dans la recherche d'une modélisation plus proche de la réalité, nous avons pensé à une autre formule concernant la transmission du virus. +On se propose donc d'améliorer cette formule en utilisant un modèle combinatoire. -On procéde ainsi : +Le terme $s_k.(1-\frac{\gamma.p}{N})^I$ représente la transmission du virus. +Voici comment obtenir cette formule : \begin{itemize} \item Un individu infecté effectue $\gamma$ contacts par jour, donc seule une proportion $p$ est susceptible de transùettre la maladie. \item Parmis ces contacts, seuls $\frac{S}{N}$ atteignent des personnes saines. On a donc $\frac{\gamma.p.S}{N}$ transmissions par individu infecté. @@ -162,14 +166,15 @@ On procéde ainsi : \item Enfin, on a donc $S.P^{i_k}$ personnes saines qui sont infectées à chaque pas. \end{itemize} -Le terme $s_k.(1-\frac{\gamma.p}{N})^I$ représente donc la transmission du virus. -On a alors : -\begin{itemize} - \item $s_{k+1} = (-death_{nat}-(1-\frac{\gamma.p}{N})^I)s_k + loss.r_k + birth(s_k+i_k+r_k)$ -\end{itemize} +Cette formule est plus intéressante, car elle fait intervenir des données plus facilement récupérables : les contacts par personne et par jour \cite{social-contact} \cite{contact-rate} ainsi le taux +d'infectiosité \cite{smallpox-transmission} sont aiséments trouvables dans des travaux de recherche. + +Néanmoins, l'utilisation de cette formule plus représentative du fonctionnement réel de la propagation du virus, n'est pas compatible avec les évolutions +spécifiées dans le modèle suivant. Par conséquent, nous avons décidé de conserver la première expression. La nouvelle formule donne toutefois des résulats +plus concluants dans le modèle SIR simple. \newpage -\subsection{Second modèle d'évolution : SEIR} +\subsection{Second modèle d'évolution : SEIQR} On peut ensuite introduire dans notre modèle le phénomène d'incubation. Pour cela, on doit utiliser des variables stockant les personnes à différents stades de l'incubation, comme vu à la figure \ref{variables}. Cela change légèrement les équations : \begin{itemize} @@ -199,9 +204,6 @@ Il faut noter que chaque bloc traite plusieurs entrées et sorties, correspondan Le bloc $M$ modélise dans le cas présent les échanges entre les différents milieux. Il est donc appliqué à chaque pas après les évolutions internes. \newpage -% \bibliography{biblio.lib}{} -% \bibliographystyle{plain} - \begin{thebibliography}{9} \bibitem{smallpox-transmission} Center for disease control and prevention, @@ -226,11 +228,18 @@ Le bloc $M$ modélise dans le cas présent les échanges entre les différents m Novembre 2016 \bibitem{contact-rate} - Hao Hu, Karima Nigmatulina and Philip Eckhoff, + Hao Hu, Karima Nigmatulina and Philip Eckhoff, + + \emph{The scaling of contact rates with population density for the infectious disease models}, + + https://www.sciencedirect.com/science/article/pii/S0025556413001235 + + \bibitem{pure-delay} + C. Stoica, M.R. Arahal, D.E. Rivera, P. Rodriguez-Ayerbe and D. Dumur, + + \emph{Application of Robustified Model Predictive Control to a Production-Inventory System} - \emph{The scaling of contact rates with population density for the infectious disease models}, - - https://www.sciencedirect.com/science/article/pii/S0025556413001235 + 22 August 2017 \end{thebibliography} \end{document} \ No newline at end of file -- GitLab