2023Activity reportProject-TeamMERGE

From stochastic processes to constrained Hamilton-Jacobi (HJ) equations.

Permanent members: Sylvie Méléard, Gaël Raoul

Most models, for instance for the "normal" bacterial division cycle  70, consider asexual populations with clonal reproduction and vertical inheritance. We want to consider here a more general model with a transfer term, justified by biological considerations in the case of bacterial transfer  74 (see also the application section 4.1). The individual-based population process is given for any $K$ by the jump point-measure Markov process ${\left({\mu }_t^K\right)}_t$ on a trait subset of ${ℝ}^d$ weighted by $1/K$. An individual with trait $x$ gives birth to a new individual with rate $b\left(x\right)$. With probability $1-p_K$, the new individual carries the trait $x$ and with probability $p_K$, it carries a mutant trait $z$ chosen according to the distribution $m(x,z)dz$. An individual with trait $x$ in the population ${\mu }^K$ dies with death rate $d\left(x\right)+C\langle {\mu }^K\left(t\right),1\rangle$. Further an individual with trait $x$ chooses a partner with trait $y$ at rate $h_K(x,y,{\mu }^K)=\tau (x,y)/K\langle {\mu }^K,1\rangle$ and after transfer, the couple $(x,y)$ becomes $(x,x)$. Then for any “good” test function $\phi$, we have

where $M^{K,\phi }$ is a square integrable martingale whose quadratic variation can be easily made explicit. By letting $K$ go to infinity, and $p_K$ go to $p$, we can derive an integro-differential equation with non local non linearities due to both competition and transfer. Uniqueness of its solution is obvious but its long-time behaviour is unknown, as well as the existence of stationary solutions. Formally, applying a limiting procedure for small mutations and time rescaling usually leads to a HJ type equation with constraints, in the formalism introduced in 63, successfully developed in 41, 107, 40, and in many extensions so far. Concentrations in such equations are too fast for realistic evolution  108. Indeed the evolutionary dynamics strongly depend on the positivity of the density although it is exponentially small for some traits. Different papers 108, 84, 100 proposed to slow down the concentration speed by the addition of artificial terms. With Nicolas Champagnat, Sylvie Méléard introduced another point of view. The rare mutation assumption introduced in 50 allowed to obtain a time scale separation between demography and mutation. Under this assumption, they were able to characterize rigorously a general evolutionary jump process describing the successive evolutionary population states 52. This approach was very fruitful and allowed to quantify the complete scheme from individuals to macroscopic behaviors as suggested in 97 and 62. Nevertheless, the assumptions imposed very small mutation rates considered too slow to explain evolution (especially for microorganisms), but also too slow to capture the concentration effects of the HJ equations. At this point one may recall the usual critics by some biologists 125, of unrealistic evolutionary time scale, at least for certain species.

The first task in this study is to integrate fast mutation time scales and to show how stochastic models based on logarithmic scales can capture small populations in large approximations and explain deterministic concentration phenomena. In particular we aim to obtain new singular and constrained HJ equations taking into account the local population extinctions. We hope that these new scales will provide an intermediate approach consistent with biological observations.

The second task is to characterize the different asymptotic behaviors in the Hamilton-Jacobi equation and to understand the role of the trade-off between demography and transfer.

Space-and-trait structured models

Permanent members: Vincent Bansaye, Marie Doumic, Gaël Raoul

Effects of spatial heterogeneity on structured population dynamics need to be studied for many applications, in ecology as well as in microbiology. Here again, relating macroscopic to individual-based models is of key importance for a correct interpretation of macroscopically observed phenomena such as morphogenesis or front propagation. Let us develop two examples: size-and-space structured models and phenotypic trait-and-space structured models.

Non-markovian interactions: from local interaction model to the parabolic-parabolic Keller-Segel system

Permanent members: Milica Tomašević

An important mathematical challenge is to derive mean-field limits for non-Markovian interaction, i.e., when the past also needs to be taken into account. Such models appear for instance in neuroscience  53, 54 and chemotaxis. To model chemotaxis, the parabolic-parabolic Keller-Segel model has been stated phenomenologically, but to interpret it we need to introduce interaction memory, which provides tremendous analysis difficulties since particles are now non-markovian both in time and space. New methods have been proposed by Milica Tomašević, with a stochastic representation of the mild formulation of the equation and a particle approximation  114, 118, 116. The equations obtained have been little studied before Milica Tomašević's Ph.D thesis, so that many questions remain open. Concerning the convergence of the particle systems towards the Keller-Segel model, an important problem is the obtention of explicit convergence rates, when the number of particles tends to infinity, for the propagation of chaos of the particle system in 1D. A possible way is to extend techniques developed by Jabin and Wang  85 for the quantitative study of the mean-field boundaries of particle systems in non-regular Markovian interaction. The aim is to control the relative entropy between the joint law of the particles and the law of $N$ independent copies of the Keller-Segel system. By exploiting the results on the Keller-Segel nonlinearity in 1 dimension and on the Sobolev type estimation on the densities of the system (chapter 4 in  117), a regularization of the interaction kernel of the particles allows to obtain a first convergence rate for the marginal laws in time of an arbitrary particle, explicit but suboptimal (due to the regularization procedure). To obtain the optimal convergence rate, we think to develop an essentially probabilistic approach suggested by the recent works of Veretennikov  122 and Lacker  91 as well as by the partial Girsanov transformations introduced in 86.

Diffusion-growth-fragmentation processes and equations

Permanent members: Marie Doumic, Sylvie Méléard

To model the growth of a bacterial population in a chemostat, a new model of growth and fragmentation, coupled to a differential equation for the resource, was proposed by Josué Tchouanti in his thesis  115. Using a combination of probabilistic and analytical methods, he proved the existence, uniqueness and regularity of solutions, as well as the convergence in large populations of the individual-based model. This model also present similiarities with the proliferation of parasites in dividing cells studied by Vincent Bansaye  38, 37.

One of the very interesting novelties of this model is to consider the growth not as a purely deterministic process, leading to a transport term in the size structured equation $\frac{\partial }{\partial x}\left(\tau \left(x\right)u(t,x)\right)$, but to take into account the intrinsic stochasticity in growth, so that a diffusion-type term $\frac{{\partial }^2}{\partial x^2}\left(D\left(x\right)u(t,x)\right)$ is added, which degenerates at the boundary $x=0.$ We thus want to study further this equation, its long-time dynamics with and without interaction (i.e. in the linear case as well as with nonlinear couplings), how it differs from the much more studied growth-fragmentation equation, and which model seems more relevant in which applicative case. We also want to adapt the model to metabolic heterogeneity cases, i.e. when we model the capacity for bacteria to feed on two distinct nutrients, which leads to distinguish two populations competing for two resources.

Ergodicity analysis and exponential convergence for multi-dimensional growth-fragmentation processes and equations

Permanent members: Vincent Bansaye, Sylvie Méléard, Milica Tomašević

Based on data from the Edinburgh's lab of Meriem El Karoui, Ignacio Madrid Canales introduced during his thesis an adder growth-fragmentation stochastic process modelling the growth of bacteria. He studied its long time behaviour and proved that conveniently renormalized, the associated semigroup converges exponentially to a well defined measure. The aim is now to generalize this result in higher dimensions, motivated by the different growth strategies that bacteria can have under stress. Mathematically the question is also largely open.

Understanding the links between genealogical and population behaviours

Permanent members: Vincent Bansaye, Marie Doumic and Sylvie Méléard

Microfluidic experiments allow one to follow a genealogical lineage of cells, whereas most previous experiments as well as "natural" conditions consist in observing a full population dynamics. The natural question then comes to relate the two models, and to understand how certain phenomena may be observed in one setting but not in the other - for instance, how a few individuals may finally invade the whole population; how survivor cells may emerge from a senescent population; or yet, how to find the "signature" of a phenomenon that happened in the past from the observation of a population at a fixed time.

Estimate growth, division, interaction features in structured populations

The estimation of the division rate in non-interacting populations has been developed in a series of papers over the last decade  19. The question we want to address now is whether growth and division rates are modified by cell-to-cell interaction (or yet by antibiotic resistance or by competition), and reciprocally, how distributed growth and division rates may have an influence on the morphogenesis of the bacterial microcolony. In this task, we aim to provide answers based on more realistic individual-based models. We plan the following steps:

• Develop parametric and non-parametric inference of the interaction function from single individual tracking. A similar study has been carried out by Laetitia Della Maestra and Marc Hoffmann  61 for Mc-Kean-Vlasov equation; we would like to add a size structure and a non-constant number of individuals. We will first assume that the growth and division rates do not depend on the interaction between cells, so that prior to this step we have used the methods already developed to infer these functional parameters. We may also build upon biophysical studies such as in  127.
• Develop statistical hypothesis testing to accept or reject the assumption made in the previous step that division and growth are not influenced by the interaction inferred. Reciprocally, test whether different division or growth rates would give rise to different morphogenesis.
• Generalise the methods and adapt them to new problems, in particular the mycelial networks  64.

Estimate mutation or fragmentation kernel density

The question of estimating the fragmentation kernel in polymer breakage experiments  67 surprisingly rejoins the question of estimating the so-called Distribution of Fitness Effects (DFE) which characterizes the accumulation of mutations in bacteria  111. As shown in  67, these are so-called severely ill-posed inverse problems, for which we aim at developing new approaches, two in particular: rely on short-time instead of long-term behaviour, adapt statistical methods developed for decoumpounding Poisson processes and deconvolution  72.

State estimation and observation inequalities for depolymerisation models

In depolymerisation experiments, prior to parameter estimation, we began to address the question of state estimation, i.e. how to infer the initial condition out of measurements of moments time dynamics. Whereas it is relatively straightforward if we approximate the discrete system by a backward transport equation  29, we address the question of estimating it from the next order approximation, namely a transport-diffusion equation; this new problem is closer to the experimental system but gives rise to a severely ill-posed inverse problem, for which we want to find an observation inequality thanks to Carleman estimates  59, 58.

Calibrating the mycelial network model

The model developed in 119 paves the way to new parametric calibration methods that we wish to confront with the real observations made by mycological colleagues of the LIED laboratory (Paris Diderot University), as well as with their empirical results.

The parametric calibration based on the solutions of the spectral problem can lead to new simple descriptors that characterize the growth of the fungus.

The first objective is to see how values obtained in 64 for the exponential growth rates compare with the one obtained in 119 as a solution of the spectral problem related to the corresponding growth and fragmentation equation. For the latter, there is an interpretation through the main characteristics of the network (ratio between the number of external nodes and the total length of the network at a sufficiently large time $t$).

Then, we could test how these descriptors change in different growth environments. This will allow us to quantify the impact of various forms of stress (nutrient depletion, pH, ...).

From a theoretical point of view, we would have to justify this empirical approach and demonstrate a "many to one" formula to be able to correctly sample our model. It should also be proved that the estimators thus constructed are consistent and converge, when $t\to \infty$, to the quantities they are supposed to approximate.

The bacterial cell cycle

Coordination between cell growth and division is often carried out by ‘size control’ mechanisms, where the cell size has to reach a certain threshold to trigger some event of the cell cycle, such as DNA replication or cell division. Concerning bacteria, recent articles  28, 113 stated the excellent adequacy of the so-called "incremental model", where the structuring variable which triggers division is the size increase of the bacteria since birth, to experimental data. This opens up new questions to refine and analyse this model, test its validity in extreme growth conditions such as antibiotic treatments, and understand its links with intracellular mechanisms. Main research axis: 3, and the CellDiv platform.

Antibiotic response and resistance emergence

To address the emergence of antibiotic-resistant strains of bacteria, it is essential to understand quantitatively the response of bacteria to antibiotic treatments. Under the action of an antibiotic that causes damage to cellular DNA, bacteria change their growth strategy and do not respond homogeneously to this stress. Of particular importance is the so-called SOS response: in response to DNA damage induced by antibiotic treatments, the cell cycle is arrested and DNA repair and mutagenesis are induced (cf. 31). Cells with high SOS response will grow for an abnormal duration, producing long filaments that are impervious to antibiotics. Understanding the distribution of sizes in the population of bacteria will allow a better quanitfication of antibiotics effects. On this subject, we work with Meriem El Karoui who carries out microfluidic experiments in Edinburg university. Main research axis: 2.

Microcolony morphogenesis

When bacterial microcolonies grow, they can aggregate to one another and form a biofilm. How do they interact? How do their growth and division characteristics translate into the shape of the colony? Inside the gut, it has been proved that the immune response acts not by killing bacteria but by making them aggregate after division; how do these aggregates form and break is another question tackled by Claude Loverdo at Lab. Jean Perrin (Sorbonne). Main research axis: 1 (short term in collaboration with Diane Peurichard and Sophie Hecht).

Bacterial growth in a chemostat; the gut as a chemostat

A chemostat is a specific experiment, where the number of bacteria is let constant by a permanent influx and outflux. The functional mechanism of the very gut could be modeled as a chemostat. Main research axis: 2 (mid to long term / only first contacts made).

Mutations

The pace of evolution and possible trajectories depend on the dynamics of mutation incidence and the effects of mutations on fitness. Mutation dynamics has been for the first time analyzed directly by Lydia Robert and co-authors  111, using two different microfluidic experiments which led them to the conclusion of a Poissonnian appearance of bacterial mutations, and to a first parametric estimation of the so-called "distribution of fitness effects" (DFE) of mutations. How to assess better the shape of the DFE, and apply the method not only to deleterious or neutral but also to possibly beneficial mutations, is one of our goals. Main research axis: 3, short term (Guillaume Garnier's ongoing Ph.D).

Horizontal gene transfer

Microorganisms such as bacteria tend to exhibit a relatively large “evolution speed”. They have also the particularity to exchange genes by direct cell-to-cell contact. We are particularly interested in plasmids horizontal gene transfer (HGT): plasmids carry pathogens or genes coding for antibiotics resistances, and plasmid exchange is considered by biologists as the primary reason for antibiotics resistance. Main research axis: 1, both short and long-term research, included in the ERC project SINGER.

Leukemic mutations and hematopoeisis

Hematopoiesis is the process of producing blood cells from stem cells and progenitors. These highly regulated mechanisms keep at equilibrium the number of blood cells such as red blood cells, white blood cells and platelets (mature cells). We want to understand the emergence of leukaemia or resistance to chemotherapy through the mechanisms of erythropoiesis (production of red blood cells) and leukopoiesis (white blood cell formation). This application also rejoins the application 3.1.

Senescence by telomere shortening

Telomeres cap the ends of linear chromosomes, and help maintain genome integrity by preventing the ends being recognized and processed as accidental chromosomal breaks. When telomeres fall below a critical length, cells enter replicative senescence. However, the exact structure(s) of the short or dysfunctional telomeres either triggering permanent replicative senescence or promoting genome instability remains to be defined; this is the main focus of Teresa Teixeira's lab at IBPC, which has developed microfluidic as well as population experiments to follow senescence triggering in yeast cells. Main research axis: 1 and 2. This application is both a long-term goal, in a long-lasting collaboration with Teresa Teixeira and Zhou Xu, and has short and mid-terms objectives, through Anaïs Rat's finishing Ph.D and Jules Olayé forthcoming Ph.D (co-supervised by Milica Tomašević and Marie Doumic).

Ageing in drosophyla

Ageing’s sensitivity to natural selection has long been discussed because of its apparent negative effect on an individual’s fitness. In the recent years, a new 2-phases model of ageing has been proposed by Hervé Tricoire and Michael Rera  60, 121, describing the ageing process not as being continuous but as made of at least 2 consecutive phases separated by a dramatic transition. It was first observed in drosophila, and then shown to be evolutionary conserved; this raises the question of an active selection of the underlying mechanisms throughout evolution. Main research axis: 2 and 3.

Protein polymerisation: amyloid formation and autophagy

Protein polymerisation occurs in many different situations, from functional situations (actin filaments, autophagy) to toxic ones (amyloid diseases). It involves complex reaction networks, making it a challenge to identify the key mechanisms, for instance which mechanisms lead to the initial formation of polymers during the first reaction steps (nucleation), how and where the polymers break, or yet the aggregates formation, out of (at least) two different proteins, in autophagy. With our biologist collaborators, our aim in these applications is to isolate the most meaningful reactions, study their behaviour(Research axis 2), and compare them - qualitatively and, if possible, quantitatively - with experimental data.

Mycelial network

Filamentous fungi are complex expanding organisms that are omnipresent in nature. They form filamentous structures, growing and branching to create huge networks called mycelia. We aim at modelling, understanding and estimating the main mechanisms of mycelial formation. We have already studied a first model without interactions and we will now study the impact of fusion of filaments on the growth of the network. Main research axes: 2 and 3.

Emergence of bacterial resistance in heterogeneous environments

When an antibiotic treatment is applied to a population, bacteria resistant to the treatment have an opportunity to develop. If several treatments are used, life threatening multi-resistant bacteria can appear. Understand the dynamics of bacterial populations in such heterogeneous environments would provide interesting perspectives to improve treatments and keep antibiotic resistance in control. On this topic, we will collaborate with S. Gandon lab at CEFE, that tackles this problem with a combination of theory and experiments. This also rejoins the application domain 3.1., and the main research axis is axis 2.

Dynamics of species submitted to climate change

The impact of climate change on natural species is a complicated matter. An important research effort has been made on the modification of species' niche in coming years, but this is only a partial clue for the future of species. In collaboration with Ophélie Ronce at ISEM, we will investigate how the local adaptation of species will is shook by global changes. With François Massol in CIIL and Nicolas Loeuille in IEES, we will focus on the impact of interspecies effects: predation, parasitism, cooperation, etc. Main research axis: 1.

8.1.1 From the distributions of times of interactions to preys and predators dynamical systems

Participants: Vincent Bansaye, Bertrand Cloez.

In 6, we consider a stochastic individual based model where each predator searches during a random time and then manipulates its prey or rests. The time distributions may be non-exponential. An age structure allows to describe these interactions and get a Markovian setting. The process is characterized by a measure-valued stochastic differential equation. We prove averaging results in this infinite dimensional setting and get the convergence of the slow-fast macroscopic prey predator process to a two dimensional dynamical system. We recover classical functional responses. We also get new forms arising in particular when births and deaths of predators are affected by the lack of food.

8.1.2 Scaling limit of bisexual Galton-Watson process

Participants: Vincent Bansaye, Maria-Emilia Caballero, Jaime San Martin, Sylvie Méléard.

Bisexual Galton-Watson processes are discrete Markov chains where reproduction events are due to mating of males and females. Owing to this interaction, the standard branching property of Galton-Watson processes is lost. In 5, we prove tightness for conveniently rescaled bisexual Galton-Watson processes, based on recent techniques developed by Bansaye, Caballero and Méléard. We also identify the possible limits of these rescaled processes as solutions of a stochastic system, coupling two equations through singular coefficients in Poisson terms added to square roots as coefficients of Brownian motions. Under some additional integrability assumptions, pathwise uniqueness of this limiting system of stochastic differential equations and convergence of the rescaled processes are obtained. Two examples corresponding to mutual fidelity are considered.

8.1.3 From individual-based evolutionary models to Hamilton-Jacobi equations

Participants: Nicolas Champagnat, Sylvie Méléard, Sepideh Mirrahimi, Viet Chi Tran.

In 2, we consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. Traits are vertically inherited unless a mutation occurs, and influence the birth and death rates. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh for the trait values is much smaller than the size of mutation steps. When considering the evolution of the population in a long time scale, the contribution of small sub-populations may strongly influence the dynamics. Our main result quantifies the asymptotic dynamics of sub-population sizes on a logarithmic scale. We establish that under the parameter scaling the logarithm of the stochastic population size process, conveniently normalized, converges to the unique viscosity solution of a Hamilton-Jacobi equation. Such Hamilton-Jacobi equations have already been derived from parabolic integro-differential equations and have been widely developed in the study of adaptation of quantitative traits. Our work provides a justification of this framework directly from a stochastic individual based model, leading to a better understanding of the results obtained within this approach. The proof makes use of almost sure maximum principles and careful controls of the martingale parts. We have thus provided a first answer to a question that has been open for a long time, and we continue to progress in order to generalise these results.

8.1.4 Multispecies cross diffusion

Participants: Marie Doumic, Sophie Hecht, Benoit Perthame, Diane Peurichard.

Systems describing the long-range interaction between individuals have attracted a lot of attention in the last years, in particular in relation with living systems. These systems are quadratic, written under the form of transport equations with a nonlocal self-generated drift. In the article 23, to be published in the Journal of Differential Equations, we establish the localisation limit, that is the convergence of nonlocal to local systems, when the range of interaction tends to 0. These theoretical results are sustained by numerical simulations. The major new feature in our analysis is that we do not need diffusion to gain compactness, at odd with the existing literature. The central compactness result is provided by a full rank assumption on the interaction kernels. In turn, we prove existence of weak solutions for the resulting system, a cross-diffusion system of quadratic type.

8.1.5 Particle approximation of the doubly parabolic Keller-Segel equation

Participants: Nicolas Fournier, Milica Tomašević.

In 3, we study a stochastic system of $N$ particles associated with the parabolic-parabolic Keller-Segel system. This particle system is singular and non Markovian in that its drift term depends on the past of the particles. When the sensitivity parameter is sufficiently small, we show that this particle system indeed exists for any , we show tightness in $N$ of its empirical measure, and that any weak limit point of this empirical measure, as , solves some nonlinear martingale problem, which in particular implies that its family of time-marginals solves the parabolic-parabolic Keller-Segel system in some weak sense. The main argument of the proof consists of a Markovianization of the interaction kernel: We show that, in some loose sense, the two-by-two path-dependant interaction can be controlled by a two-by-two Coulomb interaction, as in the parabolic-elliptic case.

8.2.1 Claire Ecotiere's PhD

Participants: Claire Ecotière, Sylvie Méléard, Régis Ferrière.

Claire Ecotiere's thesis focuses on the stochastic modeling of human behavior in the face of environmental change and the study of related mathematical models. She has mainly developed a stochastic model describing the coupled dynamics of the environment and the population, through the proportion of active individuals facing the environment, in a population composed of active and passive individuals. Individuals can switch from one behavior to another through social interactions or their own assessment of environmental degradation. Active behavior contributes less to environmental degradation, but is more costly to adopt than passive behavior. The stochastic model and its behavior in long time are studied as well as its approximation in large population leading to a deterministic system.

8.2.2 Evolutionary dynamics

Participants: Sirine Boucenna, Vasilis Dakos, Quentin Griette, Sylvie Alfaro, Sylvain Gandon, Gaël Raoul.

The article  80 has been accepted in Evolution Letters. In this article, we have analysed the evolutionary dynamics of pathogens spreading in a heterogeneous host population where selection varies periodically in space. We study both the transient dynamics taking place at the front of the epidemic and the long-term evolution far behind the front. In particular, we identify the conditions where a generalist pathogen carrying multiple adaptations can outrace a coalition of specialist pathogens. We also show that finite host populations promote the spread of generalist pathogens because demographic stochasticity enhances the extinction of locally maladapted pathogens.

The article  112 is in revision for Theoretical Population Biology. Shallow lakes ecosystems may experience abrupt shifts (ie tipping points) from one state to a contrasting degraded alternative state as a result of gradual environmental changes. It is crucial to elucidate how eco-evolutionary feedbacks affect abrupt ecological transitions in shallow lakes. We explore the eco-evolutionary dynamics of submerged and floating macrophytes in a shallow lake ecosystem under asymmetric competition for nutrients and light. We show how rapid trait evolution can result in complex dynamics including evolutionary oscillations, extensive diversification and evolutionary suicide. Overall, this study shows that evolution can have strong effects in the ecological dynamics of bistable ecosystems.

8.2.3 A growth-fragmentation-isolation process on random recursive trees and contact tracing

Participants: Vincent Bansaye, Chenlin Gu, Linglong Yuan.

In 7, we consider a random process on recursive trees, with three types of events. Vertices give birth at a constant rate (growth), each edge may be removed independently (fragmentation of the tree) and clusters (or trees) are frozen with a rate proportional to their sizes (isolation of connected component). A phase transition occurs when the isolation is able to stop the growth fragmentation process and cause extinction. When the process survives, the number of clusters increases exponentially and we prove that the normalized empirical measure of clusters a.s. converges to a limit law on recursive trees. We exploit the branching structure associated with the size of clusters, which is inherited from the splitting property of random recursive trees. This work is motivated by the control of epidemics and contact tracing where clusters correspond to trees of infected individuals that can be identified and isolated. We complement this work by providing results on the Malthusian exponent to describe the effect of control policies on epidemics.

8.2.4 Propagation of chaos for stochastic particle systems with singular mean-field interaction of $L^q-L^p$ type

Participants: Milica Tomašević.

In this work 14, we prove the well-posedness and propagation of chaos for a stochastic particle system in mean-field interaction under the assumption that the interacting kernel belongs to a suitable $L_t^q-L_x^p$ space. Contrary to the large deviation principle approach recently proposed in 83, the main ingredient of the proof here are the Partial Girsanov transformations introduced in 87 and developed in a general setting in this work.

8.2.5 Blow-up for a stochastic model of chemotaxis driven by conservative noise on ${ℝ}^2$

Participants: Avi Mayorcas, Milica Tomašević.

In 13, we establish criteria on the chemotactic sensitivity $\chi$ for the non-existence of global weak solutions (i.e., blow-up in finite time) to a stochastic Keller–Segel model with spatially inhomogeneous, conservative noise on ${ℝ}^2$. We show that if $\chi$ is sufficiently large then blow-up occurs with probability 1. In this regime, our criterion agrees with that of a deterministic Keller–Segel model with increased viscosity. However, for $\chi$ in an intermediate regime, determined by the variance of the initial data and the spatial correlation of the noise, we show that blow-up occurs with positive probability.

8.2.6 Reducing exit-times of diffusions with repulsive interactions

Participants: Paul-Eric Chaudru de Reynal, Pierre Monmarché, Milica Tomašević, Julian Tugaut.

In this work 9, we prove a Kramers’ type law for the low-temperature behavior of the exittimes from a metastable state for a class of self-interacting nonlinear diffusion processes. Contrary to previous works, the interaction is not assumed to be convex, which means that this result covers cases where the exit-time for the interacting process is smaller than the exit-time for the associated non-interacting process. The technique of the proof is based on the fact that, under an appropriate contraction condition, the interacting process is conveniently coupled with a non-interacting (linear) Markov process where the interacting law is replaced by a constant Dirac mass at the fixed point of the deterministic zero-temperature process.

8.3.1 Telomere shortening, a unifying model

Participants: Anaïs Rat, Marie Doumic, Teresa Teixeira, Zhou Xu.

Progressive shortening of telomeres ultimately causes replicative senescence and is linked with aging and tumor suppression. Studying the intricate link between telomere shortening and senescence at the molecular level and its population-scale effects over time is challenging with current approaches but crucial for understanding behavior at the organ or tissue level. In the submitted article 4, we developed a mathematical model for telomere shortening and the onset of replicative senescence using data from Saccharomyces cerevisiae without telomerase. Our model tracks individual cell states, their telomere length dynamics, and lifespan over time, revealing selection forces within a population. We discovered that both cell genealogy and global telomere length distribution are key to determine the population proliferation capacity. We also discovered that cell growth defects unrelated to telomeres also affect subsequent proliferation and may act as confounding variables in replicative senescence assays. Overall, while there is a deterministic limit for the shortest telomere length, the stochastic occurrence of non-terminal arrests drive cells into a totally different regime, which may promote genome instability and senescence escape. Our results offer a comprehensive framework for investigating the implications of telomere length on human diseases.

8.3.2 Calibrating division rates in population dynamics

Participants: Marie Doumic, Marc Hoffmann.

Modelling, analysing and inferring triggering mechanisms in population reproduction is fundamental in many biological applications. It is also an active and growing research domain in mathematical biology. In the book chapter 19, we review the main results developed over the last decade for the estimation of the division rate in growing and dividing populations in a steady environment. These methods combine tools borrowed from PDE's and stochastic processes, with a certain view that emerges from mathematical statistics. A focus on the application to the bacterial cell division cycle provides a concrete presentation, and may help the reader to identify major new challenges in the field.

8.3.3 Fragmentation estimation through short-time dynamics

Participants: Marie Doumic, Miguel Escobedo, Magali Tournus.

Given a phenomenon described by a self-similar fragmentation equation, how to infer the fragmentation kernel from experimental measurements of the solution ? To answer this question at the basis of our work, a formal asymptotic expansion suggested us that using short-time observations and initial data close to a Dirac measure should be a well-adapted strategy. We prove error estimates in Total Variation and Bounded Lipshitz norms; this gives a quantitative meaning to what a ”short” time observation is. Our analysis is complemented by a numerical investigation.

9.2.1 Visits of international scientists

9.2.2 Visits to international teams

9.3.1 Horizon Europe

The ERC Advanced Grant SINGER (Stochastic dynamics of sINgle cells; Growth, Emergence and Resistance) 2022-2027, headed by Sylvie Méléard, involves all members of MERGE and several of our collaborators (Sepideh Mirrahimi in Toulouse, Nicolas Champagnat in BIGS, Viet-Chi Tran in Univ. G. Eiffel). Amount: 2M€.

10.1.1 Scientific events: organisation

10.1.2 Journal

10.1.3 Invited talks

Vincent Bansaye gave invited talks at the Conference on "Branching Processes and applications", Angers, may 23rd; on the Journée Analyse Appliquée Hauts de France, October 17th; at the Evolution Everevol Conference, Grenoble, December 14th.

Marie Doumic gave an invited talk at the 5th International Symposium on Pathomechanisms of Amyloid Diseases, Bordeaux, September 5-7 ; at the workshop "Topics on Neuroscience, Collective Migration and Parameter Estimation", Oxford, July 3-7 ; at the "Journées EDP", Aussois, June 19-23 ; at the READINET conference, Orsay, January 3-6.

Sylvie Méléard gave 12 invited talks, including 7 abroad.

Milica Tomašević gave invited talks at the Mean field interactions with singular kernels and their approximations workshop, IHP Paris 18-22 déc 2023 ; at the Conference Stochastic Flows, thematic program Order and Randomness in Partial Differential Equations, Institut Mittag-Leffler, nov 2023 ; at the Second Berlin-Leipzig Workshop on Fluctuating Hydrodynamics, Freie University, Berlin September 19-20, 2023; at the Conference "A random walk in the land of stochastic processes and numerical probability", CIRM sept 2023 ; at the Conference Mean Field Models, Rennes, 12 juin - 16 juin 2023 ; and 4 seminar talks.

10.1.4 Leadership within the scientific community

Vincent Bansaye has become vice president of the applied mathematics department of Ecole Polytechnique and vice director of the Fondation Mathématiques Jacques Hadamard (FMJH) since September 2023.

Sylvie Méléard leads the Chaire MMB, joint between École Polytechnique, the Muséum national d'Histoire naturelle, the Fondation de l'École Polytechnique and VEOLIA Environnement. It is a major player in bringing together our community of mathematicians, modellers and ecologists, and funds several PhD and post-doctoral grants.

10.1.5 Scientific expertise

Vincent Bansaye is a member of the scientific council of ModCov19.

Marie Doumic has been a member of the CID51 of CNRS (hiring committees for junior and senior research scientists of CNRS), and has been a member for the hiring committee of an associate professor at ENSTA.

10.1.6 Research administration

Vincent Bansaye is a member of the steering committee of the MMB Chaire.

Marie Doumic is a member of the scientific committee of INSMI, of Inria Saclay and of the steering committee of FMJH.

Sylvie Méléard has been a member of the Evaluation Committee of Inria till September 2023, and is a member of the scientific committees of CRM (Montreal), Hausdorff Center (Bonn, Germany) and CMM (Chile), and of the steering committees of FMJH and E4H.

10.2.1 Supervision

Claire Ecotière defended her Ph.D on October 18th, 2023, on the Study and stochastic modelling of human behaviour in the face of environmental change, under the supervision of Sylvie Méléard and Régis Ferrière.

Anaïs Rat defended her Ph.D on May 31st, 2023, on Structured population dynamics: theory, asymptotic and numerical analysis. Application to populations with heterogeneous growth rates and to replicative senescence, under the supervision of Magali Tournus and Marie Doumic.

10.2.2 Juries

Marie Doumic has been a member of the Ph.D thesis committees for Darryl Ondoua (under Yvon Maday's supervision), Léo Meyer (under Magali Ribot and Romain Yvinec's supervision), and for the habilitation thesis of Bertrand Cloez (January 5, 2023).

Sylvie Méléard has been a member for the Marc Yor 2023 prize and the Dargelos 2023 prize committees.

10.3.1 Interventions for undergraduate and highschool students

Marie Doumic gave a talk "Chiche!" at Inria, December 4, 2023, and a talk for high school students at the SMAI/MAM Musée des Arts et Métiers, Paris, March 9.

Sylvie Méléard gave a talk to the Math-Club, Université Paris Cité (bachelor students) and a "Maths en Jeans" conference at Potsdam University in March 2023, organised by the Berlin lycée français and Potsdam University.

10.3.2 General audience talks

Sylvie Méléard gave a talk for her Conférence entry as a foreign correspondent to the Chilean Academy of Sciences, January 2023, and a talk at the actuaries meeting, SCOR Foundation, December 2023.

International journals

• 5 articleV.Vincent Bansaye, M.-E.Maria-Emilia Caballero, S.Sylvie Méléard and J.Jaime San Martín. Scaling limits of bisexual Galton-Watson processes.Stochastics: An International Journal of Probability and Stochastic Processes955February 2023, 749-784HALDOIback to text
• 6 articleV.Vincent Bansaye and B.Bertrand Cloez. From the distributions of times of interactions to preys and predators dynamical systems.Journal of Mathematical Biology871June 2023, 2HALDOIback to text
• 7 articleV.Vincent Bansaye, C.Chenlin Gu and L.Linglong Yuan. A growth-fragmentation-isolation process on random recursive trees and contact tracing.The Annals of Applied Probability336B2023, 5233-5278HALDOIback to text
• 8 articleN.Nicolas Champagnat, S.Sylvie Méléard, S.Sepideh Mirrahimi and V.Viet Chi Tran. Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations.Journal de l'École polytechnique — Mathématiques102023, 1247-1275HALDOI
• 9 articleP.-E.Paul-Eric Chaudru de Raynal, M. H.Manh Hong Duong, P.Pierre Monmarché, M.Milica Tomašević and J.Julian Tugaut. Reducing exit-times of diffusions with repulsive interactions.ESAIM: Probability and Statistics272023, 723-748HALDOIback to text
• 10 articleM.Max Fathi, P.Pierre Le Bris, A.Angeliki Menegaki, P.Pierre Monmarché, J.Julien Reygner and M.Milica Tomašević. Recent progress on limit theorems for large stochastic particle systems.ESAIM: Proceedings and Surveys75December 2023, 2-23HALDOI
• 11 articleN.Nicolas Fournier and M.Milica Tomašević. Particle approximation of the doubly parabolic Keller-Segel equation in the plane.Journal of Functional Analysis28572023, 110064HALDOI
• 12 articleB.Benoît Henry, S.Sylvie Méléard and V.Viet Chi Tran. Time reversal of spinal processes for linear and non-linear branching processes near stationarity.Electronic Journal of Probability28noneJanuary 2023HALDOI
• 13 articleA.Avi Mayorcas and M.Milica Tomašević. Blow-up for a stochastic model of chemotaxis driven by conservative noise on ${}^2$.Journal of Evolution Equations2357September 2023HALDOIback to text
• 14 articleM.Milica Tomašević. Propagation of chaos for stochastic particle systems with singular mean-field interaction of $L^q-L^p$ type.Electronic Communications in Probability28noneAugust 2023, 13HALDOIback to text

Conferences without proceedings

• 15 inproceedingsM.Madeleine Kubasch, V.Vincent Bansaye, F.François Deslandes and E.Elisabeta Vergu. A tribute to Elisabeta Vergu: Multi-level contact models of epidemics.Journées 2023 de l'Action Coordonnée "Modélisation des Maladies Infectieuses"Paris, FranceOctober 2023HAL
• 16 inproceedingsM.Madeleine Kubasch, V.Vincent Bansaye, F.François Deslandes and E.Elisabeta Vergu. Approximations déterministes de modèles stochastiques multi-niveaux en épidémiologie.Colloque Jeunes Probabilistes et StatisticiensSaint Pierre d'Oléron, FranceOctober 2023HAL
• 17 inproceedingsM.Madeleine Kubasch, V.Vincent Bansaye, F.François Deslandes and E.Elisabeta Vergu. Modèles structurés multi-niveaux de dynamiques épidémiques.École de Printemps de la chaire MMBAussois, FranceJune 2023HAL
• 18 inproceedingsM.Madeleine Kubasch. Large population limit for a multilayer SIR model including households and workplaces.Rencontres Mathématiques de RouenRouen, FranceJune 2023HAL

Scientific book chapters

• 19 inbookM.Marie Doumic and M.Marc Hoffmann. Individual and population approaches for calibrating division rates in population dynamics: Application to the bacterial cell cycle.40Modeling and Simulation for Collective DynamicsLecture Notes Series, Institute for Mathematical Sciences, National University of SingaporeWORLD SCIENTIFICFebruary 2023, 1-81HALDOIback to textback to textback to textback to textback to textback to text

Reports & preprints

• 20 miscV.Vincent Bansaye, P.-Y.Pierre-Yves Boëlle, S.Simon Cauchemez, B.Bernard Cazelles, P.Pascal Crepey, L.Lina Cristancho-Fajardo, J.-S.Jean-Stéphane Dhersin, S.Sorin Dumitrescu, P.Pauline Ezanno, A.Antoine Flahault, J.-L.Jean-Louis Golmard, P.Patrick Hoscheit, H.Henri Mermoz Kouye, M.Madeleine Kubasch, C.Catherine Laredo, A.Alain Mallet, L.Lulla Opatowski, C.Clémentine Prieur and A.Amandine Véber. Hommage à Elisabeta Vergu.November 2023HAL
• 21 miscV.Vincent Bansaye, F.François Deslandes, M.Madeleine Kubasch and E.Elisabeta Vergu. The epidemiological footprint of contact structures in models with two levels of mixing.March 2023HAL
• 22 miscV.Vincent Bansaye, X.Xavier Erny and S.Sylvie Méléard. Sharp approximation and hitting times for stochastic invasion processes.December 2023HAL
• 23 miscM.Marie Doumic, S.Sophie Hecht, B.Benoit Perthame and D.Diane Peurichard. Multispecies cross-diffusions: from a nonlocal mean-field to a porous medium system without self-diffusion.May 2023HALback to text
• 24 miscM.Madeleine Kubasch. Large population limit for a multilayer SIR model including households and workplaces.May 2023HAL
• 25 miscA.Anaïs Rat, M.Marie Doumic, M. T.Maria Teresa Teixeira and Z.Zhou Xu. Individual cell fate and population dynamics revealed by a mathematical model linking telomere length and replicative senescence.November 2023HALDOIback to text
• 26 miscM.Milica Tomašević and G.Guillaume Woessner. On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model.September 2022HAL