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index 5569c91cff4d04c6f5b9048111f96479be2f0e47..e4c31a5df30f6e0fa8b60214f0c182e42c7583e4 100644
--- a/Article/article.aux
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@@ -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 4906691412d21591d934c0754253c0faae03846e..db1a7a71ff6d098d90c6ef4738512c9f1efdbcb8 100644
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index c3eb4ad91f2bfbd9523d1c83e588b12949facd67..1596e47adffdbc2bc230b4a2eb2582d71b9d01e6 100644
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index ce66872a9cc7cdf3836e1cc1cd5e879b59300316..36494c2a3fbdfe7bcdbbd6e81bd52ccc203a8953 100644
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diff --git a/Article/article.pdf b/Article/article.pdf
index 2cf992debc3a025cd92a00c4b58741c5bf5de78e..a804f9f89bd7550574f768222a2d3464e819920a 100644
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diff --git a/Article/article.synctex.gz b/Article/article.synctex.gz
index 05190910748c17de54985f1f857cb479db538c04..b0b60d8c87e9a8dc707316131a8277e309003ffb 100644
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diff --git a/Article/article.tex b/Article/article.tex
index 38d61b2c9c503402ea9060301cad580443e64f58..69afd3d08895f1a9a261ca2c1fb25d0c2b56d4d4 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 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}