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dynamique de l'évolution des populations, appliquée à l'épidémiologie} + % http://sara.etsmtl.ca/fr/guide_redaction_rediger/ + \title{Modélisation de l'évolution des populations, appliquée à l'épidémiologie} \author{Martin \textsc{Lehoux} \and Paul \textsc{Cacheux}} \date{\today} \begin{document} \maketitle +\tableofcontents +\newpage +\section{Abstract} +\begin{itshape} + Research on disease spreading is is likely to lead to a drastic improvement in human health care with any find. + We tried to develop a model of disease spread that could help governments to make the best decisions (in term of vaccination for instance) in + case of epidemic, and to provide a digital tool to use this model. + After we both developped this model, Paul focused on coding the digital tool using Matlab, while Martin extracted data needed by the model from known + statistics of diseases. + We used different models already developped, such as SIR (and SIRS) models, and automated systems with variable dead-time, to build our own model. + This model describe disease dynamics inside a population and between different populations. + Our results lead to understand how a small difference in vaccination can have a huge impact on the survival of a population. The tool provided makes it + posible to simulate different vaccination policy to predict the population evolution. +\end{itshape} + +\textbf{Keywords :} Modeling; Numerical simulation; Disease spreading; Evolution; Demography; + +\newpage +\section{Introduction} +Dans cet article, nous nous intéressons à l'analyse et la modélisation de la dynamique des populations. Nous nous sommes recentrés sur l'étude de la propagation +des virus au sein de populations humaines, car l'impact des études sur la santé humaine peut se réveler très précieux. + +Nous avons décidé de séparer notre travail en deux parties, l'un modélisant la propagation d'un virus au sein d'une population précise, l'autre permettant de +simuler les interactions entre ces différentes populations. Notre objectif est donc de fournir à la fois un modèle de dynamique et un outil numérique pour +exploiter ce modèle. + +Paul s'est principalement occupé de l'implémentation algorithmique sur Matlab des modèles que nous avions élaborés, tandis que Martin a extrait les données des +statistiques sur les virus pour alimenter le modèle. +Nous avons concu notre modèle en respectants certaines hypothèses, nécessaires lors de la modélisation de dynamiques touchant aux êtres vivants, bien trop complexes +dans leur entièreté. + +L'article se décompose en deux parties, l'une traitant de la modélisation de la dynamique d'un "réservoir" de population, à partir du modèle SIR. +La seconde extrapole ces résulats en mettant en concurrence différentes populations. + +\newpage +\section{Revue de littérature} + +\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. On peut représenter les échanges entre différentes catégories dans cette population, par un graphe stochastique, dont les différents coefficients peuvent dépendre @@ -45,12 +84,12 @@ Pour être le plus précis possible, le modèle a besoin de nombreux paramètres $birth$ & Birth rate & 1.85\% per year & $5.0\times10^{-5}$\\ $death_{nat}$ & Natural death rate & 0.78\% per year & $2.1\times10^{-5}$ \\ \hline - $\gamma$ & Infection duration & 24 days & 24\\ + $T$ & Infection duration & 24 days & 24\\ $death_{ill}$ & Disease death rate & 3\% & $1.3\times10^{-3}$\\ $R$ & Recovery rate & 97\% & 0.97\\ - $\theta$ & Incubation duration & 12 days & 12\\ - $p$ & Transmission risk & 60\% & \\ % https://www.cdc.gov/smallpox/clinicians/transmission.html - $\beta$ & Effective contact rate & & \\ + $\theta$ & Incubation duration & 12 days & $12$\\ + $p$ & Transmission risk & 60\cite{smallpox-transmission}\% & $0.6$ \\ + $\gamma$ & Contact rate & 25 per day & 25\\ \hline % v & Vaccination rate among healthy people & 90\%\\ $vaccin_{duration}$ & Efficiency duration of vaccin & 10 years & 3650\\ @@ -69,11 +108,11 @@ Voici comment sont déterminés les paramètres ($||x||$ correspond à la valeur $$ \item $death_{ill}$ est calculé à partir de la durée d'infection ainsi que du taux de mortalité de la maladie: $$ - ||death_{ill}|| = 1-(1-death_{ill})^\frac{1}{\gamma} + ||death_{ill}|| = \frac{death_{ill}}{T} $$ \item $R$ est calculé à partir de la durée d'infection: $$ - R = 1-\death_{ill}^{\frac{1}{\gamma}} + R = \frac{1-death_{ill}}{T} $$ \item $loss$ est calculé à partir de la durée d'efficacité du vaccin: $$ @@ -90,9 +129,9 @@ Voici comment sont déterminés les paramètres ($||x||$ correspond à la valeur $r_k$ & Removed (immune) population\\ $i_k$ & Infected population\\ $s_k$ & Susceptible population\\ - $w_{\theta,k}$ & People with $\theta$ time step incubation\\ + $e_{\theta,k}$ & People with $\theta$ time step incubation\\ ... & ...\\ - $w_{1,k}$ & People with 1 time step incubation\\ + $e_{1,k}$ & People with 1 time step incubation\\ \hline \end{tabular} \caption{Vecteur d'état} @@ -100,39 +139,37 @@ Voici comment sont déterminés les paramètres ($||x||$ correspond à la valeur \end{table} \newpage -\subsection{Modèle d'évolution} -Si on note le vecteur d'état $X_k$ et la matrice de transition $A$, avec -$$ -X_k = -\begin{pmatrix} - r_{k}\\ - i_{k}\\ - s_{k}\\ - w_{\theta,k}\\ - ...\\ - ...\\ - ...\\ - w_{1,k}\\ -\end{pmatrix} -$$ -% $$ -% A= -% \begin{pmatrix} -% 1+nat-\beta & \gamma & v & 0 & & & & \\ -% 0 & 1+ill-\gamma & 0 & 1 & & & & \\ -% \beta & 0 & 1+nat-\alpha-v & 0 & & & & \\ -% & & & 0 & 1+nat & & & \\ -% & & & & 0 & 1+nat & & \\ -% & & & & & 0 & 1+nat & \\ -% & & & & & & 0 & 1+nat\\ -% & & \alpha & & & & & 0\\ -% \end{pmatrix} -% $$ -% 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. +\subsection{Premier modèle d'évolution : SIR} +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. +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. + \item Soit $X$ un individu sain, il a donc $P = \frac{S - \frac{\gamma.p.S}{N}}{S} = 1 - \frac{\gamma.p}{N}$ chances d'être contaminé par un individu $Y$ infecté. + \item Si $X$ veut rester sain faces aux $i_k$ infectés, il n'a donc plus que $P^{i_k}$ chances de le rester. + \item Enfin, on a donc $S.P^{i_k}$ personnes saines qui sont infectées à chaque pas. +\end{itemize} + +\newpage +\subsection{Second modèle d'évolution : SIER} +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 $e_{1,k+1} = p.\frac{i_k}{s_k+i_k+r_k}.s_k$ + \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 $r_{k+1} = vacc_{rate}.s_k + R.i_k + (1-death_{nat}-loss)r_k$ +\end{itemize} + \newpage \section{Modèle mathématique d'interaction entre plusieurs populations} @@ -150,4 +187,40 @@ Les blocs $A_i$, i de 1 à n, sont les blocs correspondant aux évolutions inter Il faut noter que chaque bloc traite plusieurs entrées et sorties, correspondant aux différentes population existantes dans le milieu. 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, + + \emph{Transmission of smallpox}, + + https://www.cdc.gov/smallpox/clinicians/transmission.html, + + Décembre 2016 + \bibitem{sir-model} + Hans Nesse, + + \emph{Global health - SIR model}, + + http://www.public.asu.edu/~hnesse/classes/sir.html + \bibitem{social-contact} + Ruprecht Schmidt-Ott, Markus Schwehm and Martin Eichner, + + \emph{Influence of social contact patterns and demographic factors on influenza simulation results}, + + https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5100331/, + + Novembre 2016 + \bibitem{contact-rate} + 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 +\end{thebibliography} + \end{document} \ No newline at end of file -- GitLab