Modeling, analysis and simulation of structured compartmental systems in epidemiology.
Speciality : Mathématiques Appliquées
24/10/2024 - 13:30 Robin VAUDRY (Université Grenoble Alpes) Bâtiment IMAG, salle 306
Keywords :
- Modèles compartimentaux en épidémiologie
- Analyse des EDP
- Structure en âge
- Fonction à base radiale
- Méthode de différences finies.
Humans are the animal species most affected by infectious diseases. The development of an epidemic is fraught with consequences for populations : increased mortality, saturation of hospitals, economic repercussions... The epidemiology of infectious diseases is the science that studies the frequency and distribution of these diseases in time and space, their impact on populations and the parameters that determine them. The recent COVID-19 pandemic that affected the world between December 2019 and May 2023 has reinforced the need to create high-performance mathematical models to limit the health impact of such epidemics. The aim of this thesis is to propose a realistic model of the evolution of the health status of a population during an epidemic. In this research work, we are building a compartmental SEHIRVD model structured according to age of illness and age of immunity. This model takes into account the progressive loss of immunity of individuals, as well as vaccination. The equations governing this model are a system of coupled partial differential equations with integral terms. For this model, we propose a numerical scheme based on the Finite Difference method supported by Radial Basis Functions (RBF-FD) for age operator approximation, and semi-implicit for discretization in time. The integrals are approximated by a Quasi-Monte Carlo method. We then establish theoretical properties of our SEHIRVD model. We prove its well-posedness : the system of equations admits a unique solution that depends continuously on the parameters and initial conditions of the problem. We also demonstrate that the size of each compartment in the model remains positive over time, and does not exceed that of the total population. Finally, we present numerical simulations of our model performed in Python. These simulations explore the impact of age structures on the evolution of the system, as well as the effect of vaccination.
Directors:
- PR Clémentine PRIEUR (Université Grenoble Alpes )
- PR Didier GEORGES (Grenoble INP / Université Grenoble Alpes )
Raporteurs:
- DR Alain RAPAPORT (INRAE )
- DR Amandine VEBER (CNRS )
Examinators:
- PR Joseph WINKIN (Université de Namur )
- CR Paolo FRASCA (CNRS )