Sensitivity analysis approaches for stochastic models. Application to compartmental models in epidemiology.

Keywords :

• Indices de Sobol’
• Méthodes de Monte-Carlo
• Processus stochastiques
• Modèles épidémiologiques
• Algorithmes de simulation exacte.

This thesis focuses on the sensitivity analysis of stochastic models. These models include uncertainties that originate mainly from two sources: the parametric uncertainty due to the lack of knowledge of parameters and the intrinsic randomness that represents the noise inherent to the model coming from the way chance intervenes in the description of the modeled phenomenon. The presence of intrinsic randomness is a challenge in sensitivity analysis because, on the one hand, it is generally hidden and therefore cannot be characterized and, on the other hand, it acts as noise when evaluating the impact of the parameters on the model output However, in epidemiology, the issues associated with the sensitivity of a model can be important in the control of epidemics because they impact the decisions made on the basis of this model. This thesis studies approaches for sensitivity analysis of stochastic models such as epidemiological models based on stochastic processes, in the framework of variance-based analysis. In a general context, we introduce a method for estimating sensitivity indices that optimizes the trade-off between the number of input parameter values of the model and the number of replications of model evaluation in each of these values. For this method, we consider the class of quantities of interest of stochastic model outputs that are in the form of conditional expectations with respect to uncertain parameters. In the context of estimation of sensitivity indices by the Monte Carlo method, we control the quadratic risk of the estimators, show its convergence and find a trade-off between the bias related to the presence of the intrinsic randomness and the variance. In the specific context of stochastic compartmental models in epidemiology, we characterize the intrinsic randomness of the stochastic processes on which these models are based. These stochastic processes can be Markovian or non-Markovian. For Markovian processes, we use Gillespie algorithms to make explicit the intrinsic randomness and to separate it from uncertain parameters. Regarding non-Markovian processes, we extend to a large class of compartmental models the Sellke construction, which was originally introduced to describe epidemic dynamics of the SIR model in a framework that is not necessarily Markovian. This extension has allowed us to develop an algorithm that generates exact trajectories in a non-Markovian framework for a large class of compartmental models but also to be able to separate intrinsic randomness from parameter uncertainty in the output of these models. Thus, for both types of processes, Markovian and non-Markovian, the separation of the two sources of uncertainty has been obtained and it allows to represent model outputs as a deterministic function of the uncertain parameters and the variables representing the intrinsic randomness. When the uncertainty on the parameters is assumed to be independent of the intrinsic randomness, this representation allows to assess the contributions of the intrinsic randomness on the model outputs, in addition to the contributions of the parameters. It is also possible to characterize different interactions. This thesis has contributed to develop an approach to estimate sensitivity indices and to evaluate the contribution of intrinsic randomness in compartmental models in epidemiology based on stochastic processes.

Directors:

• DR Elisabeta VERGU (INRAE (Université Paris-Saclay) )
• PR Clémentine PRIEUR (Université Grenoble Alpes )
• CR Gildas MAZO (INRAE (Université Paris-Saclay) )

Raporteurs:

• MCF Christophette BLANCHET (Ecole Centrale Lyon )
• PR Olivier ROUSTANT (INSA Toulouse )

Examinators:

• PR Pierre BARBILLON (AgroParisTech (Université Paris-Saclay) )
• DR Robert FAIVRE (INRAE Toulouse )
• DR Olivier LE MAITRE (CNRS (Ecole Polytechnique) )