Par ordre alphabétique :
Henri Berestycki - EHESS
The stability-compactness method and qualitative properties of nonlinear elliptic equations
Résumé : In this talk I report on a series of works with Cole Graham on semi-linear elliptic equations with positive non-linearities, and Dirichlet boundary conditions. Solutions represent stationary states of reaction-diffusion equations. We focus in particular on uniqueness. The main motivation is to study these equations in general unbounded domains, which exhibit remarkably rich behavior. Our method rests on decomposing the problem into a compact part and one for which a stability result can be derived and then to combine the two. This approach has proved to be unexpectedly versatile and in fact encompasses past works on the subject such as the general moving plane method.
Céline Bonnet - ENS Lyon
A piecewise deterministic and Markovian approach to study the role of quiescence dynamics in blood cancers.
Résumé : We will see an approach to study the impact of a small microscopic population of cancer cells on a macroscopic population of healthy cells, with an example inspired by pathological hematopoiesis. Hematopoiesis is the biological phenomenon of blood cells production by differentiation of cells called hematopoietic stem cells (HSCs). Cancer HSCs produce a large number of cancer blood cells but randomly stop to produce them (during such a period, the cancer HSC is called quiescent). We will study the impact of such a quiescent state on the production of cancer blood cells and on healthy cells through regulation. We will describe the evolution over time of the number of healthy and cancer cells using a multi-type Markov process. A single cancer HSC is considered while other populations are in large numbers. We show the convergence in law of this process towards a piecewise deterministic Markov process (PDMP). We then study the long time behavior of this limit process. We show the existence and uniqueness of an invariant probability measure using the works of Benaim and co-authors. We finally identify this measure using the solution of a stationary system of partial differential equations describing the impact of cancer HSC quiescent phases and regulation on the cell density of the hematopoietic system studied.
Lucilla Corrias - Univ. Evry
Rigorous derivation of cross-diffusion dynamics from fast reaction PDE systems
Résumé : In this talk, we will consider a class of cross diffusion systems obtained as the equilibrium limit of reaction diffusion systems with fast reaction terms. The fast reaction diffusion systems model the dynamic of two species in competition, one of which switch between two different states. We will show how the limit, as the conversion rate goes to zero, is rigorously obtained through a family of energy functionals. The existence of weak solutions of the corresponding cross diffusion system is thus obtained. A weak-strong stability/uniqueness result will be also presented. This talk is based on joint works with E. Brocchieri, L. Desvillettes and H. Dietert.
Fabien Crauste - Univ. Paris Cité
Multiscale models of the CD8 T cell Immune response: Mathematical and Computational Approaches
Résumé : Immune cells allow a priori fast and efficient responses against non-self agents. They rely upon the ability of the organism to identify threats and trigger the most appropriate reactions. Cytotoxic immune responses aim in particular at inducing infected cell death, and to do so they integrate early on information about the nature of the infection in order to perform an appropriate differentiation program. This leads to an important inter-individual variability in terms of cell counts and temporal dynamics among individuals of a given population (for instance, miceor humans). Most theoretical models of immune responses, either mathematical or computational models, usually consider only population-aggregated values such as mean and standard deviation. I will discuss complex, multiscale models of the T cell immune response, accounting for coupled descriptions of both molecular and cellular dynamics, and I will show that it represents an original tool for investigating the influence of early molecular events on the long-term cellular dynamics in silico.
Léonard Dekens - Univ. Paris
Sharp habitat shifts, evolutionary tipping points and rescue: modelling the perilous path of a specialist species toward a refugium in a changing environment
Résumé : Over the last decades, numerous studies have been documenting the impacts of climate deregulation on species ranges. Among them, specialists, which thrive under specific environmental conditions, typically in narrow geographic ranges, are widely recognised as one of the most threatened categories. Many might rely on both their potential to adapt and to disperse toward an environmental refugium to avoid extinction. It is thus crucial to understand the influence of environmental conditions on the unfolding process of adaptation.
Here, I study the eco-evolutionary dynamics of a sexually reproducing specialist species in a two-patch quantitative genetic model with moving optima. The analysis relies on a separation of time scales shown to occur in an analogous model under stable environment, and the subsequent study of phase lines. Aligning with previous theoretical studies, I derive the critical environmental speed beyond which the environment changes too fast for the species persistance. This quantity reflects how much the existence of a refugium can delay extinction. Moreover, my analysis provides key quantitative insights about the sharp dynamics that arise on the path towards this refugium. I show that after an initial increase of population size, there exists a lower critical environmental speed for which the species crosses a tipping point, resulting into an abrupt habitat switch. Besides, when selection for local adaptation is relatively strong, this habitat switch passes through an evolutionary “death valley”, from where a small portion of thepopulation can get rescued by adapting fast enough to the local conditions of the refugium.
Marie Doumic - INRIA & Sorbonne Univ.
Analysis and calibration of depolymerising systems
Résumé : Shrinkage of large particles, either through depolymerisation (i.e. progressive shortening) or through fragmentation (breakage into smaller pieces) may be modelled by discrete equations, of Becker-Döring type, or by continuous ones. In many applications, the dynamic nature of the experiments, as well as their nanoscale, makes it challenging to estimate their features. In this talk, we review some inverse problems linked to the estimation of the initial size-distribution and of the fragmentation characteristics.
Departing from a model of discrete depolymerisation, we first evaluate the impact of using continuous approximations to solve the initial-state estimation problem. At second order, the asymptotic model becomes an advection-diffusion equation, where the diffusion is a corrective term. This approximation is much more accurate, but we face a classical accuracy versus stability trade-off: the inverse reconstruction reveals to be severely ill-posed. Thanks to Carleman inequalities and to log-convexity estimates, we prove observability results and error estimates for a Tikhonov regularisation. This is a joint work with P. Moireau, inspired by experiments by H. Rezaei.
To estimate the fragmentation kernel in experiments of polymer breakage, we propose several approaches based on the continuous fragmentation equation, studying and making use either of the long-term, the transient or the short-term dynamics. Error estimates in Bounded Lipshitz norm are obtained. This is a joint work with M. Escobedo and M. Tournus, based on biological questions and experiments of W.F. Xue.
François Hamel - Univ. Aix-Marseille
Large-time dynamics of solutions of reaction-diffusion equations in RN with general initial support
Résumé : The talk will focus on the large-time dynamics of bounded solutions of reaction-diffusion equations in RN with general bounded or unbounded initial support. I will discuss the existence of spreading speeds and spreading sets of the solutions in any direction, in connection with the existence of planar traveling fronts. I will also explain some results on the asymptotic one-dimensional symmetry of the elements of the Ω-limit sets of the solutions. Lastly, I will discuss the influence of the fragmentation of the initial support on the large-time dynamics. The talk is based on joint works with Matthieu Alfaro, Lionel Roques, and Luca Rossi.
Florence Hubert - Univ. Aix-Marseille
Some mathematical models for cell migration
Résumé : Cell migration is a complex biological phenomenon playing an important
role in many processes such as embryogenesis, but also in the development
of pathologies such as cancer. The main driver of the motility is the actin
network, the dynamics of which is regulated by many proteins. Mathematical
models have been developped in the last decades to better understand this
complexity. One of the difficulty leads on the representation of this moving
domain. Several approaches have been proposed: Lagrangian Markers Cells
[Edelstein-Kechet et al.(2011)], Level-set methods [Tesson et al.(2020)]
or phase fields models [Ziebert-Aronson (2011)]. We propose in this talk
to illustrate these methods with two biological issues. In the first one, we
will model the impact of the microtubules on the process using the level set
method. In the second one, we will use phase field models to explain atypical
cases of adhesive haptotaxis [Luo et al (2020)], [Seveau Phd (2022)].
Madeleine Kubasch - Ecole Polytechnique
Large population limit for a multilayer SIR model including households and workplaces
Résumé : We study a multilayer SIR model with two levels of mixing, namely a global level which is uniformly mixing, and a local level with two layers distinguishing household and workplace contacts, respectively. We establish the large population convergence of the corresponding stochastic process. In a general setting where a non-exponential distribution of infectious periods may be considered, convergence to the unique deterministic solution of a measure-valued equation is obtained. In the particular case of exponentially distributed infectious periods, we show that it is possible to further reduce the obtained deterministic limit, leading to a closed, finite dimensional dynamical system capturing the epidemic dynamics.
Sepideh Mirrahimi - Univ. Montpellier
A moment-based approach for the analysis of the infinitesimal model in the regime of small variance
Résumé : We provide an asymptotic analysis of a nonlinear integro-differential equation which describes the evolutionary dynamics of a population which reproduces sexually and which is subject to selection and competition. The sexual reproduction is modeled via a nonlinear integral term, known as the ’infinitesimal model’. We consider a regime of small segregational variance, where a parameter in the infinitesimal operator which measures the deviation between the trait of the offspring and the mean parental trait, is small. We prove that in this regime the phenotypic distribution remains close to a Gaussian profile with a fixed small variance and we characterize the dynamics of the mean phenotypic trait via an ordinary differential equation. While similar properties were already proved for a closely related model using a Hopf-Cole transformation and perturbative analysis techniques, we provide an alternative proof which relies on a direct study of the dynamics of the moments of the phenotypic distribution and a contraction property of the Wasserstein distance. This is a joint work with Jessica Guerand and Matthieu Hillairet.
Grégoire Nadin - Sorbonne Univ.
Optimization of the total size of a population
Résumé : In this talk I will present several results obtained with I. Mazari and Y. Privat. This work investigates the maximization of $\int_\Omega u$ with respect to $m$, where $u$ is the solution of the Fisher-KPP equation $-\mu \Delta u = u\big(m(x)-u\big)$ in a bounded domain. Here, $m$ is assumed to be bounded, with a prescribed integral. I will present results about the bang-bang nature of optimizers, their behaviour when the dispersion parameter is either large or small. Lastly, I will present some recent results characterization the fragmentation and the periodic nature of the optimizers when $\mu$ is small.
Clair Poignard - INRIA Bordeaux
Mathematical challenges for tumor ablation by electroporation
Résumé : Electroporation is a significant increase in the electrical conductivity and permeability of the cell membrane that occurs when pulses of large amplitude (a few hundred volts per centimeter) are applied to the cells: due to the electric field, the cell membrane is permeabilized. If the pulse duration is sufficiently short (a few milliseconds or a few microseconds, depending on the pulse amplitude), the cell membrane reseals within several tens of minutes: reversible electroporation, preserves the cell viability and is used in electrochemotherapy to vectorize the drugs until the cell inside. If the pulses are too long, too numerous or if their amplitude is too high, the cell membrane is irreversibly destroyed and the cells are killed. Irreversible electroporation provides thus a novel non thermal and minimally invasive ablation therapy.
In this talk I will present some results on the mathematical modeling of electroporation, and its impact of pulsed electric field on the growth of microtumors. I will eventually present some mathematical challenges to address in order to help interventional radiologists in their practice of percutaneous liver tumor ablation by irreversible electroporation.
Ariane Trescases - Univ. Toulouse
Cross-diffusions systems, chemotaxis and local sensing
Résumé : Chemotaxis is a process that drives cellular motility, which is ubiquitous in Biology. We present a class of systems with cross-diffusion that model cell aggregation by chemotaxis. We will focus on local sensing, that is, the cells respond to a certain concentration of chemoattractant perceived locally (as opposed to gradient sensing, when the cells are able to perceive a gradient of concentration). We study the well- posedness and long-time behaviour of these models.
Samuel Tréton - Univ. Rouen Normandie
Diffusive Dialogues: Mathematical Models of Individual Exchanging Across Spatial Domains
Résumé : In this talk, we examine mathematical models that describe the diffusion and exchange of individuals across spatial domains.
We begin with the field-road model, emphasizing its biological foundations and its importance in understanding fast diffusion channels in population dynamics and ecology. Following this, we explain how to derive the explicit solutions for the field-road model and provide estimates on the asymptotic decay rate of these solutions.
This analytical framework paves the way for exploring non-linear issues, including Fujita-type blow-up phenomena, which we explain by outlining the key concepts involved.
We then turn to the Heat-exchanger model, which serves as a tractable first approach for deriving some coupled-by-diffusion Fujita-type systems. We proceed to characterize the purely diffusive version of this model. With this foundation, we tackle the question of blow-up vs. global existence that arises when incorporating super-linear Fujita-type reaction terms.
The presentation concludes with insights into a stochastic simple exclusion process for the field-road diffusion model. We provide a brief overview of the mechanics of this particle system, drawing parallels with the canonical example of the Heat equation.
Hatem Zaag - Univ. Sorbonne Paris Nord
How can mathematics help with Inflammatory Bowel Diseases (IBD)?
Résumé : We consider two IBD: Crohn’s Disease (CD) and Ulcerative Colitis (UC). While the symptoms are the same, with a severe degradation of life’s quality, these diseases differently affect the interior of the digestive track. While inflammation is deep and discontinuous for CD, it is superficial and continuous for UC, affecting only one part of the colon. In this talk, we give a PDE system presenting a Turing mechanism, as a model for CD. We also present some image processing techniques to help standardize the severity assessment for UC. We also combine the image processing technique with a Fisher-KPP model, in order to measure the spreading speed of the inflammation for different patients with UC.
