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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} -- GitLab