From 74713bb20473d65c0cdcb793199f7b2a4f467e7c Mon Sep 17 00:00:00 2001
From: Kagamino <martin.lehoux@supelec.fr>
Date: Fri, 15 Jun 2018 01:15:07 +0200
Subject: [PATCH] seiqr

---
 Article/article.pdf | Bin 180566 -> 183056 bytes
 Article/article.tex |  18 +++++++++++++++---
 2 files changed, 15 insertions(+), 3 deletions(-)

diff --git a/Article/article.pdf b/Article/article.pdf
index 3956018d323a13013c258dac73cd8540d158bc15..444506bcf6d0856b5f8d5b122fefb1e3d81ad276 100644
GIT binary patch
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diff --git a/Article/article.tex b/Article/article.tex
index 9eecf30..485c4b3 100644
--- a/Article/article.tex
+++ b/Article/article.tex
@@ -178,13 +178,25 @@ plus concluants dans le modèle SIR simple.
 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}
-    \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 $s_{k+1} = (-death_{nat}-p.\frac{i_k}{s_k+i_k+r_k} -vacc_{rate})s_k + loss.r_k + birth(s_k+i_k+r_k)$
     \item $e_{1,k+1} = p.\frac{i_k}{s_k+i_k+r_k}.s_k$
     \item $\forall l \in [2,\theta], e_{l,k+1} = e_{l-1,k}$
-    \item $i_{k+1} = e_{\theta,k} + (-death_{ill}-R)i_k$
-    \item $r_{k+1} = vacc_{rate}.s_k + R.i_k + (1-death_{nat}-loss)r_k$
+    \item $i_{k+1} = e_{\theta,k} + (-death_{ill}-R-Q).i_k$
+    \item $q_{k+1} = (1-death_{ill}-R).q_k + Q.i_k$
+    \item $r_{k+1} = vacc_{rate}.s_k + R.(i_k+q_k) + (1-death_{nat}-loss)r_k$
 \end{itemize}
 
+La phase d'incubation (\emph{E} pour \emph{exposés}) a un impact assez important, puisqu'elle permet la propagation du virus dans la population sans déclencher de réponse.
+Pendant les premiers jours de l'épidémie, un réservoir très important de personnes exposées va se construire, ce qui va mener dès la fin de l'incubation à une 
+mortalité très élevée. De plus, ces personnes incubées ne guérissent pas, alors qu'elles peuvent tout à fait infecter les autres sains. On voit donc
+que l'ajout de l'incubation permet très clairement à la maladie d'être plus virulente et violente.
+
+D'autre part, la phase de quarantaine joue contre la maladie. En effet, cet état augmente le nombre de personnes perdant le statut d'infecté, responsable en
+grande partie de l'infection de la population saine. Cette population meurt autant et guérit de la même manière que la population infectée, mais ne participe
+pas à la contamination. C'est une phase qui est instaurée par les humains, et, au même titre que le vaccin, le paramètre Q peut être choisi pour augmenter
+l'ampleur de la réaction d'un gouvernement face à une épidémùie importante (surtout quand la vaccination de tout le pays est compromise).
+
+Ce modèle permet de se rapprocher de la réalité : on décrit avec plus de souplesse les phénomènes diverses permettant de modéliser la propagation du virus.
 
 \newpage
 \section{Modèle mathématique d'interaction entre plusieurs populations}
-- 
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