Numerical methods in the context of compartmental models in epidemiology
Résumé
We consider compartmental models in epidemiology. For the study of the divergence of the stochastic model from its corresponding deterministic limit (i.e., the solution of an ODE) for long time horizon, a large deviations principle suggests a thorough numerical analysis of the two models. The aim of this paper is to present three such motivated numerical works. We first compute the solution of the ODE model by means of a non-standard finite difference scheme. Next we solve a constraint optimization problem via discrete-time dynamic programming: this enables us to compute the leading term in the large deviations principle of the time of extinction of a given disease. Finally, we apply the {\tau}-leaping algorithm to the stochastic model in order to simulate its solution efficiently. We illustrate these numerical methods by applying them to two examples.
On considere des modeles comportementaux en epidemiologie. An detudier lecart en temps long entre le modele stochastique et sa limite loi des grands nombres (qui est la solution dune EDO), onse base sur un principe des grandes daviations, qui nous conduit a mener une etude numerique des deux modeles, sur trois aspects di erents. Tout dabord, nous calculons une solution approchee de lEDO a laide dune methode numerique dite non standard . Ensuite une resolvons numeriquement un probleme de controle sous contrainte, a n de calculer le terme principal des grandes deviations du temps de sortie dune situation endemique. Enn nous mettons en oeuvre lalgorithme du leaping pour simuler e cacement la solution du systeme stochastique. Nous illustrons ces simulations numeriques en les appliquant a deux exemples
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