Modeling, analysis and simulation of structured compartmental systems in epidemiology.

français

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 )