From a82b986aa7c215dde2b3e905a1a60f37c8512fcc Mon Sep 17 00:00:00 2001
From: Kagamino <martin.lehoux@supelec.fr>
Date: Wed, 18 Apr 2018 18:49:48 +0200
Subject: [PATCH] update article

---
 Article/article.aux         |   5 ++
 Article/article.fdb_latexmk |  43 ++++++---
 Article/article.fls         |  50 +++++++++--
 Article/article.log         | 174 ++++++++++++++++++++++++++----------
 Article/article.pdf         | Bin 83535 -> 126048 bytes
 Article/article.synctex.gz  | Bin 4360 -> 14457 bytes
 Article/article.tex         |  84 +++++++++++++++--
 Article/article.toc         |   2 -
 8 files changed, 291 insertions(+), 67 deletions(-)
 delete mode 100644 Article/article.toc

diff --git a/Article/article.aux b/Article/article.aux
index 5569c91..e4c31a5 100644
--- a/Article/article.aux
+++ b/Article/article.aux
@@ -8,3 +8,8 @@
 \@writefile{lof}{\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{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}}
diff --git a/Article/article.fdb_latexmk b/Article/article.fdb_latexmk
index 4906691..db1a7a7 100644
--- a/Article/article.fdb_latexmk
+++ b/Article/article.fdb_latexmk
@@ -1,15 +1,24 @@
 # Fdb version 3
-["pdflatex"] 1524062144 "/home/kagamino/Documents/Cours/CentraleSupelec 1A/dynamique-populations/Article/article.tex" "article.pdf" "article" 1524062144
+["pdflatex"] 1524069504 "/home/kagamino/Documents/Cours/CentraleSupelec 1A/dynamique-populations/Article/article.tex" "article.pdf" "article" 1524069505
   "/etc/texmf/web2c/texmf.cnf" 1522149631 475 c0e671620eb5563b2130f56340a5fde8 ""
-  "/home/kagamino/Documents/Cours/CentraleSupelec 1A/dynamique-populations/Article/article.tex" 1524062143 1299 05844c8b3cb30e9feb28bb387bee95b0 ""
+  "/home/kagamino/Documents/Cours/CentraleSupelec 1A/dynamique-populations/Article/article.tex" 1524069503 3667 590cb9bae66a2f578cd97fdae424579b ""
   "/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1480098670 3287 e6b82fe08f5336d4d5ebc73fb1152e87 ""
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   "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx1440.tfm" 1480098696 3584 13049b61b922a28b158a38aeff75ee9b ""
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   "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/eccc1200.tfm" 1480098696 3072 1e3dcc56ea731d0f3645bbefe8c8608c ""
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   "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1728.tfm" 1480098696 3584 3c76ccb63eda935a68ba65ba9da29f1a ""
+  "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1480098698 1004 54797486969f23fa377b128694d548df ""
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+  "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1480098698 924 9904cf1d39e9767e7a3622f2a125a565 ""
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+  "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm" 1480098698 940 75ac932a52f80982a9f8ea75d03a34cf ""
+  "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1480098698 940 228d6584342e91276bf566bcf9716b83 ""
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   "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm" 1480098701 1520 eccf95517727cb11801f4f1aee3a21b4 ""
@@ -19,12 +28,28 @@
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   "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1480098701 1120 8b7d695260f3cff42e636090a8002094 ""
+  "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1480098733 30251 6afa5cb1d0204815a708a080681d4674 ""
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   "/usr/share/texlive/texmf-dist/tex/generic/babel/switch.def" 1498771725 12650 fb4c49ceb13037455c02ab8dd235f9ec ""
+  "/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1480098820 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c ""
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+  "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty" 1480098820 2210 5c54ab129b848a5071554186d0168766 ""
+  "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty" 1480098820 4160 c115536cf8d4ff25aa8c1c9bc4ecb79a ""
+  "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty" 1485129665 84329 81aa65c5042562f79cb421feff9b8bdc ""
+  "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty" 1480098820 4115 318a66090112f3aa3f415aeb6fe8540f ""
+  "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty" 1480098820 2431 fe3078ec12fc30287f568596f8e0b948 ""
   "/usr/share/texlive/texmf-dist/tex/latex/base/article.cls" 1480098821 19821 310da678527a7dfe2a02c88af38079b7 ""
+  "/usr/share/texlive/texmf-dist/tex/latex/base/fixltx2e.sty" 1492297155 15510 1c4e54b5314ada40f53d6cb0c6970568 ""
   "/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty" 1492297155 4571 13977df0eda144b93597fc709035ad1f ""
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@@ -37,8 +62,8 @@
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   "/usr/share/texmf/fonts/type1/public/cm-super/sfbx1440.pfb" 1522838260 135942 859a90cad7494a1e79c94baf546d7de5 ""
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@@ -46,11 +71,9 @@
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-  "article.toc" 1524062144 147 ae1521e80ddf4aba51e60bd7de7e9bc1 ""
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+  "article.tex" 1524069503 3667 590cb9bae66a2f578cd97fdae424579b ""
   (generated)
-  "article.log"
-  "article.aux"
   "article.pdf"
-  "article.toc"
+  "article.aux"
+  "article.log"
diff --git a/Article/article.fls b/Article/article.fls
index c3eb4ad..1596e47 100644
--- a/Article/article.fls
+++ b/Article/article.fls
@@ -37,6 +37,22 @@ INPUT /usr/share/texlive/texmf-dist/tex/latex/carlisle/scalefnt.sty
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 INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty
 INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty
+INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fixltx2e.sty
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+INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
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+INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty
 INPUT article.aux
 INPUT article.aux
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 INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm
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-INPUT article.toc
-INPUT article.toc
-INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx1000.tfm
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 INPUT /var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map
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 INPUT /usr/share/texmf/fonts/enc/dvips/cm-super/cm-super-t1.enc
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diff --git a/Article/article.log b/Article/article.log
index ce66872..36494c2 100644
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diff --git a/Article/article.tex b/Article/article.tex
index 38d61b2..69afd3d 100644
--- a/Article/article.tex
+++ b/Article/article.tex
@@ -1,7 +1,10 @@
-\documentclass[twocolumn]{article}
+\documentclass[onecolumn]{article}
     \usepackage[utf8]{inputenc}
     \usepackage[T1]{fontenc}
     \usepackage[french]{babel}
+    \usepackage{fixltx2e}
+    \usepackage{amsmath}
+    \usepackage{amssymb}
 
     \title{Rapport sur la dynamique de l'évolution des populations, appliquée à l'épidémiologie}
     \author{Martin \textsc{Lehoux} \and Paul \textsc{Cacheux}}
@@ -9,10 +12,6 @@
 
 \begin{document}
 \maketitle
-\newpage
-
-\tableofcontents
-\newpage
 
 \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.
@@ -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 fbd2329..0000000
--- 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}
-- 
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