Identifiabilité de paramètres pour des systèmes décrits par des équations aux dérivées partielles. Application à la dynamique des populations

Abstract : This thesis aims at studying the identifiability of an epidemiological model described by semilinear integro-differential partial differential equations (PDE) of reaction-transport type. To achieve this goal, we start with a literature survey of inverse problems devoted to parameter identifiability. We study the mathematical bases of the existing techniques, outlining the systems to which they are applied or could be extended. In finite dimension, the three main methods for systems of ordinary differential equations are based on: Taylor series expansion, algebro-differential elimination, and the state isomorphism theorem. In infinite dimension, for PDE systems, two methods are generally applied in the linear case: a spectral approach, and another one based on Carleman estimates. The latter is also applied to some semilinear PDE systems in particular situations where the identifiability problem amounts to studying a linear system. However, this method cannot be, or can hardly be applied to our system owing to the complexity of its nonlinearity. We then perform the identifiability analysis of the epidemiological model. We first build a formal identifiability framework adapted to semilinear PDE systems. This framework requires that a solution space is defined for the PDE problem. Therefore, we determine a functional framework compatible with the biological conditions imposed by the model and prove the existence and uniqueness of the solution. Second, we perform the identifiability analysis of the model by adapting the algebro-differential elimination method. We obtain sufficient identifiability conditions for given parameter classes. We finally discuss and interpret the results we obtain, and provide numerical simulations.
Document type :
Theses
Complete list of metadatas

Cited literature [79 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-00464272
Contributor : Antoine Perasso <>
Submitted on : Tuesday, March 16, 2010 - 4:14:16 PM
Last modification on : Thursday, January 11, 2018 - 6:21:22 AM
Long-term archiving on : Friday, June 18, 2010 - 9:27:07 PM

Identifiers

  • HAL Id : tel-00464272, version 1

Collections

Citation

Antoine Perasso. Identifiabilité de paramètres pour des systèmes décrits par des équations aux dérivées partielles. Application à la dynamique des populations. Mathématiques [math]. Université Paris Sud - Paris XI, 2009. Français. ⟨tel-00464272⟩

Share

Metrics

Record views

661

Files downloads

5572