DETERMINISTIC MATHEMATICAL MODELS FOR BIOLOGICAL PROCESSES

Many phenomena in mathematical biology can be described by using the mathematical properties of the transport equation and more generally a kinetic description of a large population of individuals. One main example is the pattern formation that occurs naturally while observing a flock of birds, a school of fish or a swarm of bees. A possible mathematical description of this behaviour uses the kinetic gas description of statistical mechanics to describe the interaction of a large number of individuals. With this perspective, swarming would be a consequence of an equilibrium between large-range attraction between moving individuals of the same species and a long-range repulsion intended to avoid collisions. The mathematical framework of these models involves the use of the transport equation and the associated collision and alignment alignment models, based on the Boltzmann equation. The goal of this course is to give an introduction to these tools, in particular, renewal equations and their wellposedness, transport equations (classical, weak and renormalized solutions) and time asymptotics.

This course is designed in two parts. The first part is dedicated to an introduction of dynamical models used in epidemiology, based on the SIR Susceptible-Infected-Removed model. Building on the basic ODE model, various disease-related specificities will be included, such as vertical transmission or vector-borne diseases. The course will then address a classical metric in mathematical epidemiology, the basic reproduction number R0, which can be defined as the number of secondary cases generated by an average infectious individual in a susceptible population. The next-generation method will be used for its computation. Finally, there will be a focus on plant epidemiology, which is fairly specific compared to human and animal epidemiology. (Introductory course - 3 hours: half lecture, half exercise session)
The second part of the course is dedicated to the design of optimal control strategies in plant epidemiological models. An introduction to optimal control theory will be given. Then, based on several examples in crop protection, the choice of the optimisation criterion will be examined, which will in some cases lead to “standard” optimisation problems rather than optimal control problems. The BOCOP software will be used to solve the optimal control problems. (Advanced course - 3 hours: half lecture, half practical session).

This course aims to present the incompressible Navier-Stokes equations that describe the behavior of incompressible viscous flows. The aim of the course is to give an introduction and provide standard mathematical results such as the existence of solutions.
Two applications will be also presented: one to oncology and the other one to the ventilation process.

The aim of the lecture is to present a hierarchy of models of ventilation and gas diffusion in the lung. First a coupled system of ODEs describing the evolution of global quantities shall be presented, then systems of PDEs describing the transport and diffusion of air as a gas mixture along the bronchial tree. These later models are based on 1D advection-diffusion-reaction equations and they enable the recovery of standard breathing quantities and scenarios. These models can be further used to control breathing.

Title. New methods for solving inverse problems of determining the order of fractional derivatives of partial differential equations

Title. Recent developments and ideas on metapopulations, metacommunities and opinion models

Title. Discrete-time dynamical systems of quadratic stochastic operators

Title. Growth-fragmentation equation to model the metastatic emission by clusters