3.1.1 Micro-Meso: Graph limits

MUSCLEES permanent members involved: Pierre-Alexandre Bliman, Nastassia Pouradier Duteil

In 2014, Medvedev used techniques from the recent theory of graph limit to derive rigorously the continuum limit of dynamical models on deterministic graphs 133. The limiting equation, so-called “graphon equation” now describes the evolution of the particle's positions $x(t,s)$ as a function of time $t$ and of the “continuous index” $s\in I$ (representing the particle's individual identities, in an infinite population):

In 58, we extended this idea to a collective dynamics model with time-varying weights, adopting the graph point of view described above. We showed that this approach is more general than the mean-field one, and the Graph Limit can be derived for a much greater variety of models.

Our work will involve deriving graph limits for systems of particles that can be structured along a trait that characterizes their interactions, such as volume, mass or phenotype. Among the open problems that we aim to address in collaboration with Nathalie Ayi (LJLL, Sorbonne University), one of them concerns the graph limit for multi-agent systems evolving on weighted random graphs. More specifically, we will consider that the interactions between agents are given by ${\varphi }_{ij}(x_j\left(t\right)-x_i\left(t\right)):={\xi }_{ij}\varphi (x_j\left(t\right)-x_i\left(t\right))$, where ${\left({\xi }_{ij}\right)}_{i,j}$ are random variables whose laws are probability distributions on ${ℝ}_{+}$ that depend on the indices $i,j$. Graphs with random topologies are often used to model systems such as neuronal networks, coupled lasers and communication or power networks. In 133, the continuum limit of collective dynamics on random graphs was derived for graphs whose edge weights ${\xi }_{ij}$ can be either 0 (i.e. there is no edge) or 1. Our aim will be to generalize this results to random weighted graphs, whose weights can be given by any positive real number. Results will then possibly be extended to temporal random graphs, whose edge weights evolve in time as in blinking systems.

In a parallel direction, we will explore the possibilities of the graph-limit formalism in the framework of epidemiological models on graph. A first step was done in 91 by deriving the graph limit of an epidemiological model on graphs, which results in a system of coupled structured PDEs for the susceptible, infected and recovered populations. The graph-limit approach will allow us to ask ourselves fundamental analytical and modeling questions regarding the role of the interaction network in the spread of an epidemic. It will also give us the possibility to address control and optimal control problems aiming to minimize the infected population by controlling the graphon (i.e. the continuous interaction network). Another possibility will be to address inverse problems in order to infer the graph structure based on the epidemic spread. This project will link the research of team members involved in Sections 3.1 and 3.4.

3.1.2 Micro-Meso: Beyond mean-field limits

MUSCLEES permanent members involved: Sophie Hecht, Diane Peurichard, Nastassia Pouradier Duteil

When the interaction between particles is independent of each particle's individual nature, i.e. ${\varphi }_{ij}=\varphi$, the particles are said to be exchangeable, or indistinguishable. In this case, the classical approach to link microscopic and mesoscopic models is a limit process called “mean-field limit”, and consists of approximating the population by a sum of localized point masses, and then of sending the number of agents to infinity, while sending each individual mass to zero 93. In this way, the total mass of the population is conserved throughout the limit process, and everything can be done in the framework of probability measures. The limit PDE is typically a non-linear transport equation of the type

in which $\mu (t,·)\in 𝒫\left({ℝ}^d\right)$ represents the particle distribution at time $t$, and the non-local velocity $V\left[{\mu }_t\right]$ represents the averaged effect of the whole population on each individual. However, this approach has a main drawback: it does not take into account the intrinsic volume of the individuals, since they are approximated by their centers of mass. As a result, in many cases the limiting PDE fails to reproduce the behavior of the microscopic system, in particular when modeling congestion effects due to size constraints.

This is a major modeling limitation, and resolving it is crucial. Several works have highlighted a discrepancy between the microscopic and continuum modeling approaches. For instance, in the context of emergency crowd evacuation, microscopic models are able to reproduce the well-known effect of arch formation in front of exits, resulting in congestion and dramatic slow-down of the crowd's evacuation 131. This effect still eludes all natural continuum limits. Another example can be found in the modeling of cell division: microscopic models capture the fact that the cell population is naturally pushed outwards at the birth of a new daughter cell because of its added volume. This effect is lost in continuum models, as there is no concept of individual size.

The goal of this part of the project is to address this issue. We will first focus on the simple situation of a population of agents whose only interactions are due to “non-overlapping” constraints: if two agents are within a certain distance (representing their diameter), they exert a repulsive force on each other; if their distance is greater than this diameter, there is no interaction. Despite the simplicity of this setting, the micro-macro limit is highly non-trivial due to the role of the agents' size in the dynamics. Indeed, in the continuum description, the information on the agents' size is lost, and the condition on the agent-to-agent distance no longer makes sense, as the concept of individual agents is gone. However, intuitively, one would expect that this distance condition would correspond to a density condition in the continuum setting: interactions take place if and only if the local density is above a critical threshold. We will explore these questions on systems with identical particles (same and fixed sizes), and take a particular interest in how non-overlapping configurations translate into local density constraints at the population level.

In order to gain insights into the role of the individual particle sizes and shapes on the macroscopic structures generated at the population level, we will consider another approach where the particle density distribution for the mean-field limit is structured in space and sizes. In current works (to be submitted), we showed that under reasonable assumptions for the interaction kernel ${\psi }_{r,s}$, the limit PDE describing the particle distribution $\mu (t,x,r)$ (depending on time, space and radius) is of the type:

Proving the convergence of the particle system to the limit PDE with the added radial structure in the density distribution is challenging and is a work in collaboration with Marc Hoffman (Université Paris Dauphine).

3.1.3 Scaling limits

MUSCLEES permanent members involved: Sophie Hecht, Benoît Perthame, Diane Peurichard, Nastassia Pouradier Duteil

In order to link the mesoscopic and the macroscopic model it is common to consider a scaling limit. Depending of the variable of the system the scaling can vary (small particle compared to space, slow division compared to the mechanical interaction, etc).

3.2.1 Regulation Mechanisms of Gene Expression

MUSCLEES permanent members involved: Philippe Robert

The central dogma of molecular biology states that the genetic information flows only in one way, from DNA to RNAs, and to proteins. The production of proteins is a central process of biological cells. It can be described as a two-step process. In the first step, macro-molecules polymerases produce RNAs with genes of the DNA. This is the transcription step. The second step is the production of proteins itself from mRNAs, messenger RNAs, a subset of RNAs, with macro-molecules ribosomes. This is the translation step. An additional feature of this process is that it is consuming an important fraction of energy resources of the cell, to build chains of amino-acids or chains of nucleotides in particular. See  62, 149, 157.

In the context of prokaryotic cells, like bacterial cells or archaeal cells. The cytoplasm of these cells is not as structured as eukaryotic cells, like mammalian cells for example, so that most of the macro-molecules of these cells can potentially collide with each other. This key biological process can be, roughly, described as resulting of multiple encounters/collisions of several types of macro-molecules of the cell: polymerases with DNA, ribosomes with mRNAs, or proteins with DNA, ...

The fact that the cytoplasm of a bacterial cell is a disorganized medium has important implications on the internal dynamics of these organisms. Numerous events are triggered by random events associated to thermal noise. When the external conditions are favorable, these cells can nevertheless multiply via division at a steady pace. A central question is of understanding how the cell adapts to different environments (scarce resources or rich environment).

Important regulation mechanisms of gene expression of bacterial cells are achieved with RNAs. Up to now little is known on the efficiency of this type of regulation from a quantitative point of view. The ambitious goal is of designing and investigating stochastic models integrating the transcription and translation steps as well as the flows of amino-acids within the cell. One of the difficulties is the number of different chemical species involved: genes, RNAs, tRNAs, sRNAs, rRNAs, proteins, Amino-acids, ppGppp, RelA, ...All of them having an important role in this regulation. A scaling approach is investigated to study these multi-dimensional Markov processes. This is a collaboration with Vincent Fromion of the laboratory BioSys "Biology of systems" of Inrae. The main goal of these studies is to evaluate the efficiency of these regulation mechanisms in the cell for the adaptation to changes of environment: switching times, impact of the variation of the flows of amino-acids, ..., and the dependence on the production rates of ppGppp, RelA and sRNAs among others.

3.2.2 Stochastic Chemical Reaction Networks

MUSCLEES permanent members involved: Philippe Robert

The goal of the research project of this section is of investigating a generalization of the law of mass action for biological systems.

For example, if three chemical species $𝒜$, $ℬ$ and $𝒞$ are involved in a chemical reaction of the type,

the classical law of mass action states that the concentration $x_M\left(t\right)$ of the chemical specy $M$ at time $t$ satisfies the relation

The ODE in this case is a quadratic functional of the state vector. In a deterministic context, the famous results by Horn, Johnson and Feinberg give, for some specific topologies, a satisfactory description of the stable states of these networks. See  104 for example. It turns that this description is suitable for systems for which the orders of magnitude of the different chemical species are comparable and that the stochastic components merely vanish. These assumptions are nevertheless not true in some biological settings, when, for example, reactions are driven by a small number of enzymes but with a large reaction rate.

As already mentioned, due to dynamics of binding/unbinding of pairs of macro-molecules within biological cells, it is natural to consider models of chemical reaction networks for which collisions of chemical species occur in a random way. In the above example, it will be assumed that a given couple of $A$ and $B$ particles will collide at rate $k$, so that if $X_M\left(t\right)$ is the number of particles of type $M$ at time $t$, then, at time $t$, a particle of type $C$ is created at rate $k{X}_A\left(t\right)X_B\left(t\right)$. The process $(X_A\left(t\right),X_B\left(t\right),X_c\left(t\right))$ is a Markov process, if we assume that there are external arrivals of $A$ and $B$ particles, it is natural to study the convergence in distribution of this Markov process. There are several conjectures in this domain.

Up to now there are few results in such a random context. The reference  49 shows, by using the results of the deterministic case that the invariant distribution has a product form expression for a specific set of topologies. A challenging question is of extending stability results for networks for which no such product formula holds. New tools, such as scaling techniques, have to be developed to study these important problems.

3.2.3 Neural Networks

MUSCLEES permanent members involved: Benoît Perthame, Philippe Robert

This application domain of this line of research is described in the subsection “Neuroscience” of Section 4.1.

3.3.1 Adaptive phenotype-structured cell population dynamics

MUSCLEES permanent members involved: Jean Clairambault, Benoît Perthame, Nastassia Pouradier Duteil

Initially developed for adaptive dynamics in theoretical ecology and cell population biology models in  86 and in   88, phenotype-structured equations are here studied in the context of cell populations confronted to a changing environment, in particular in the case of cancer and its treatments. Some of these models, developed within the former Inria team, have been reviewed in the survey  77. A more general and extended recent state of the art on phenotype-structured population dynamics is reported in  127.

Our research will focus on the analysis of such phenotype-structured equations, and more particularly, on their long-time behavior, of which little is known. Indeed, the different mathematical terms such as advection (modeling cell differentiation), diffusion (modeling epimutations) and non-local source terms (modeling population growth and phenotype selection) tend to have antagonistic effects. One of the main mathematical challenges consists of understanding the effect of coupling such phenomena on the long-time behavior of the solution.

3.3.2 Around graphon dynamics

MUSCLEES permanent members involved: Nastassia Pouradier Duteil

As introduced in Section 3.1.1, a possible way to describe infinite-dimensional non-exchangeable particle systems is the so-called graphon equation (2). In this equation, the particles' non-exchangeable nature comes from the dependence of the interaction function $\varphi$ on the particles' “continuous index” $s$: often, $\varphi (s,s^{\text{'}},x(t,s^{\text{'}})-x(t,s))=\sigma (s,s^{\text{'}})\tilde{\varphi }(x(t,s^{\text{'}})-x(t,s))$, where the function $\sigma$, known as “graphon”, encodes the graph relation between the continuous particles. Whereas in Section 3.1.1, we focus on deriving the graph limit equation as a mesoscopic limit of particle systems, here we propose to analyse further this graph-limit framework, and to use it to investigate open problems (more specifically, in control theory) that have so far eluded the community in other frameworks.

3.3.3 Analysis of non-local advection-diffusion models for active particles

MUSCLEES permanent members involved: Luca Alasio

Systems of self-propelled interacting particles provide an individual-based description of the motion of agents ranging from bacteria to colloidal surfers 147, 169. Different approaches to the derivation of macroscopic equations from particle dynamics have been considered, and the corresponding limit PDEs exhibit a variety of possible structures and behaviours 68. This work is concerned with the analytical study of some of the above-mentioned PDE models, focusing on regularity and convergence to stationary states. The simplest example is given by the following non-local advection-diffusion equation:

where $\rho (t,x)={\int }_0^{2\pi }f(t,x,\theta )\mathrm{d}\theta$ is the angle-independent density and $𝐞\left(\theta \right)=(cos\theta ,sin\theta )$, with periodic boundary conditions both in the space variable $x\in {(0,2\pi )}^2$ and the angle variable $\theta \in (0,2\pi )$. The constant parameters $\mathrm{Pe}\in ℝ$ and $D_e>0$ are called the Péclet number and spatial diffusion coefficient, respectively. Further details and a preliminary existence theory can be found in 69. In collaboration with Simon Schulz (SNS Pisa) and Jessica Guerand (U. Montpellier), we have proven regularity properties, the Harnack inequality, and exponential convergence to stationary states for weak solutions of equation (6). We apply De Giorgi's method and differentiate the equation with respect to the time variable iteratively to show that weak solutions become smooth away from the initial time. This strategy requires that we obtain improved integrability estimates in order to cater for the presence of the non-local drift. The instantaneous smoothing effect observed for weak solutions is shown to also hold for very weak solutions arising from merely distributional initial data; the proof of this result relies on a uniqueness theorem à la Michel Pierre for low-regularity solutions. The convergence to stationary states is proved using the method of contractive stochastic semigroups (Doeblin–Harris approach), taking advantage of the aforementioned Harnack inequality. This is the first step towards the study of more sophisticated models, for example we are interested in the following:

where the diffusion terms may degenerate to zero. Its microscopic dynamics corresponds to a discrete jump process in position and a continuous Brownian motion in angle. The numerical exploration in 68 shows interesting phase separation effects which connote further analytical challenges.

3.3.4 Analysis of systems with cross-diffusion

MUSCLEES permanent members involved: Luca Alasio

Cross-diffusion systems are related to several models in Mathematical Biology and in Kinetic Theory, for example the SKT model in Population Dynamics 165, tumour growth models 80, and multi-species agent-based models 41. In collaboration with M. Bruna, S. Fagioli and S. Schulz, we have been studying a family of PDE systems with dominant degenerate diffusion, plus cross-diffusion and drift terms. Existence, uniqueness, stability and long-time asymptotics for related systems with standard diffusion have been established in the literature, however the case of degenerate diffusion is considerably harder and requires the development of new techniques. For example, a class of systems with degenerate diffusion has been recently studied taking advantage of their gradient flow structure (in the Wasserstein sense) 113, 72. This structural condition is not always satisfied and we aim to develop alternative approaches under less restrictive assumptions. This is possible thanks to the combination of functional analytic techniques (compactness, lower semi-continuity), Lyapunov functionals, and fixed point results. Study of the long-time asymptotics and stationary states is ongoing. The next steps include further exploration of the connections between degenerate-parabolic and hyperbolic systems. Splitting methods constitute a promising research direction, leading to challenging questions on suitable BV estimates for the solution. We also consider the behaviour of solutions when one species is “frozen”, i.e. it does not evolve in time. Such species acts as a spatially heterogeneous obstacle to the evolution of the other components. Finally, efficient model comparison requires new continuous dependence results allowing the study of non-local terms such as interaction potentials describing collective behaviour (in the absence of strong parabolicity).

3.4.1 Vector-borne diseases

MUSCLEES permanent members involved: Pierre-Alexandre Bliman, Benoît Perthame

3.4.2 Infectious diseases

MUSCLEES permanent members involved: Pierre-Alexandre Bliman

3.5.1 Individual-based models for micro-colony growth

MUSCLEES permanent members involved: Sophie Hecht, Diane Peurichard

Individual-based models allow the description of a population at the microscopic level. These models consider each particle as autonomous entities and define their dynamics according to their local environments. For this reason it is an ideal tool to confront mathematical models and experimental data. In a previous work 96, we have developed a model to study growth of micro-colonies of elongated bacteria such as E. coli. In this paper, bacteria are represented by sphero-cylinders characterized by their length, their orientation and the position of their center of mass. The motion of bacteria is supposed to be only due to steric interaction with their close neighbors to prevent the overlapping of cells during growth and division (passive motion). This repulsion is realized via a potential based on Hertzian theory. Fragmentation occurs when the increment of length of a bacteria reaches a given threshold, distributed according to an experimental law. A key aspect of the paper is to propose a model taking into account asymmetric friction and a non-uniform distribution of mass along the length of bacteria, which impact the movement of particles. These two mechanisms were shown to improve significantly the comparison between experimental data and numerical simulations, yet we failed to reproduce one of the primordial characteristics such as the high density of bacteria in the microcolony (where all the space within the convex envelope of the colony seems occupied). This property is not reproduced to date in the models proposed in the literature 96, 99.

A discussion with the experimenter Nicolas Desprat (ABCD biophysics Lab - ENS) highlighted the possible impact of the deformation of bacteria in a micro-colony. After observation, it appears that at the point of inflexion in the colony, bacteria are often curved. The small deformation observed could be the key to the dense character of the colonies and modify their global organisations. It is therefore interesting to consider the deformable character of bacteria in order to best reproduce the organization observed experimentally. To do this, many approaches are possible 112, 134. We will consider an individual-based model where each bacterium is modeled by a string of spheres linked with spring and angular spring. This description will allow local bending for the bacterium. We will then test different modelling assumptions in order to reproduced as close as possible observed phenomena during the micro-colony growth.

After deriving the new model, we will study the influence of the different parameters and compare numerical simulations with experimental data. This work will be a collaboration with the biophysics laboratory of Nicolas Desprat, giving us access to datasets of micro-colony of strains of Escherichia coli and Pseudomonas aeruginous growing between glass and agarose. On these datasets, segmentation has been previously performed to track individual bacteria as spherocylinder. However, the purpose of this study requires to identify bacteria as deformable solids. Thus, a first step to compare experimental data to numerical simulations will be to develop new segmentation process, adapting techniques existing for clustered nuclei. In a second part, the comparison will require the development of new tools to better quantify the evolution of the colony. Among the quantifiers we found to study the growth of bacterial structure, we found the one related to the shape of the colony. In the literature, the quantifiers used to characterize the shape often consist in comparing the colony to an ellipse. However, the colonies, although elongated, have shapes that are not necessarily ellipsoidal. To develop a new sophisticated quantifier an idea is to consider the modes of the elliptical Fourier transform of the envelope of a colony in order to characterize its shape 178. Similar work will be done on other quantifiers characterising the local organisation, bending, four cell array arrangement, etc...

3.5.2 Energy-driven models of tissue organisation and architecture

MUSCLEES permanent members involved: Sophie Hecht, Diane Peurichard

This research axis is in the frame of a long standing collaboration with a team of biologists from RESTORE (Toulouse), which led to the ANR grant ENERGENCE (2023-2026) recently awarded to D. Peurichard. The goal here is to propose a general framework to understand the combined role of mechanics and energy exchanges in tissue development, repair and decline. To our knowledge, very few mathematical models have been proposed for tissue organization combining both energetical and mechanical interactions, while numerous evidences suggest that energy exchanges and mechanical forces can feedback on each other at different stages of tissue life, and that large perturbations of one or the other are associated with degeneration and diseases. Therefore, we propose to build a complete framework to theoretically and numerically model the complex interplay between energy and mechanics at different spatiotemporal scales. We will focus on adipose tissue (AT) as a relevant biological model because its architecture is relatively simple and largely dependent on energy exchanges (food supplies), and as a target with the world-wide development of obesity’s epidemic.

This project will rely on a synthetic approach based on a dual use of mathematical modelling and in-vitro/in-vivo experiments. We will propose a new view of biological tissues as complex ecological/social systems whose architecture emergence is driven by few key determinants, interacting together mechanically and constantly exchanging energy/matter with their environment. We will aim to first develop individual-based models (IBM), which promises exciting theoretical and experimental challenges such as the determination of complex feedback loops between energy intakes and local growth laws, and the study of metastable states and phase transitions applied to changes in energy fluxes, modelling cafeteria diet and food deprivation. The biological calibration of the IBM via in vitro and in vivo experiments (performed at the RESTORE lab) will go through determining how energy is distributed among the different agents and their interactions. A user-friendly interface will also be developed based on the IBM and will be used to resolve some unsolved questions such as how the AT architecture is modified by the amplitude, frequency and length of energy intake modifications.

In a second aspect of the ENERGENCE project, we will tackle the important challenges contained in the derivation of a Continuum Model (CM) from our IBM, in order to obtain a computationally efficient CM containing as much as possible the mechanisms of the microscale. Numerous technical and conceptual barriers will have to be lifted in this more theoretical part of the project, due to the nature of our IBM, the presence of correlations between the agents at the microscale and the complex mechanical and energetical feedback loops. If successful, this model will be the first continuum description of two immiscible fluids composed of cells and (anisotropic) fiber elements obtained from an agent-based description, and promises exciting new and invaluable insights into how specific microscopic effects translate at the macroscale. Our CM will rely on the complete and valid IBM and, if successful, will enable to study the interplay between energy balance and whole tissue architecture during a lifespan and at the organ scale (long-term and large-scale effects).

The impacts of the highly interdisciplinary ANR project ENERGENCE are twofold. On the biological viewpoint, the energy/mechanics coupling view of tissue emergence and changes will provide a new understanding of aging at different spatio-temporal scales that will pave the way for new rejuvenative therapies to treat age-related dysfunctions, and also impact the tissue engineering field in which metabolism remains often overlooked. On the mathematical viewpoint, the ENERGENCE project will provide involved numerical treatments and innovative sensitivity analysis methods for IBM, and tackle important theoretical challenges related to the derivation of continuous biphasic fluid models from IBM, promising exciting new understanding of the micro- macro- link. Although focused on adipose tissue, the theory and the mathematical modelling developed in this project will be general enough to apply to other biological systems such as muscle tissues and, if successful, will constitute the basis for collaborations with other European research teams through the building of an ERC Synergy.

The ENERGENCE project involves several members of our project team MUSCLEES and will be completely integrated in the team activities: the development and parametric analysis of Agent-Based Models will rely on the expertise of S. Hecht together with D. Peurichard, the challenges of deriving PDE models from IBM will be completely integrated in Axis 1 of the team (together with S. Hecht, N. Pouradier-Duteil, B. Perthame), and the analysis of the resulting PDE models will be enriched by the results of the team in Axes 2 and 3. By combining biological experiments and mathematical modelling to study the multi-scale and temporal effects of metabolism and mechanics, the ENERGENCE project will be one of the most applicative activities of MUSCLEES, and, if successful, will represent a significant step forward to understand the emergence of metastable organized structures in living matter.

3.5.3 A traffic model for the interkinetic nuclear migration (IKNM)

MUSCLEES permanent members involved: Sophie Hecht

In the past years, members of MUSCLEES have studied the cell cycle with age structured transport equations 75, 63. These models considered the transition between the different phases of the cell cycle depending of the cell age. However, recent works 110 have highlighted that the transition between these phases are likely to be impacted by the moving positions of the nuclei. Thus, we will introduce a space structured model in order to consider the influence of the movement of nuclei on the cell cycle and its transition.

As mentioned in section 4.2, in pseudo-stratified epithelium, nuclei undergo IKNM during the cell cycle. Namely, nuclei in the phase G2 move toward the apical membrane to divide while nuclei in G1 move in the opposite direction to return in the depth of the tissue. The nuclei in S do not have a clear direction in their motion. This phenomenon can be viewed as a one-dimensional traffic problem. Therefore we will model this system with a 3 species, bidirectional PDE system. The transition between the phases will be modeled by reaction terms and boundary conditions. We will study the new system of equations and answer the classical question of existence and uniqueness. Additionally we will focus on the long time behaviour, understanding the range of parameters leading to a slowdown of growth with realistic distributions of the nuclei in the different cell phases.

The model will be compared to experimental data provided by Jean-Paul Vincent's laboratory in the Francis Crick Institute (Epithelial Cell Interactions Laboratory). Existing data of the distribution of the nuclei in the different phases in the apical/basal axis at different times of development will allow to tune the different parameters of the model. The model will then allow us to test hypothesis proposed in a previous work 110 where we developed a microscopic model. In this paper, we conjectured a mechanism to explain the transition between G1 and S phase but were limited in the test due to the small number of nuclei we could consider due to computational cost. The new model we proposed would allow a further study of the influence of this mechanism.

3.5.4 Models for collective behavior in gregarious fish

MUSCLEES permanent members involved: Nastassia Pouradier Duteil

Many living systems exhibit fascinating dynamics of collective behavior during locomotion, from bacterial colonies to human crowds. The emergence of such complex spatio-temporal patterns can be described using local, short-range interactions between nearest neighbours. Fish schools are a typical example of this kind of self-organization: in order to perceive the position or kinematics of close neighbors, fish rely essentially on vision and sensing of hydrodynamic disturbances. However, the role of each of these senses is not clearly elucidated today. Our objective is to model the visual interaction within a group of animals experiencing a dynamic visual disturbance (temporal variation of the ambient light intensity). Previous experiments have revealed a correlation between illumination and group cohesion, measured in terms of geometric parameters (polarization, rotational moment, nearest-neighbour distance).

In collaboration with a team of experimental physicists of the PMMH laboratory of ESPCI and Sorbonne University, we aim to study this behaviour using mathematical models of collective motion. Numerical simulations could elucidate the influence of illumination on the field of view of the fish (distance or angle of the cone of vision), and the role of density in the emergence or not of strong rotational motion when increasing light intensity. The model used will be a variation of the Persistant Turning Walker model, a system of coupled ordinary differential equations in which each fish's angular velocity evolves in time due to alignement with its closest neighbors, attraction towards the group, and random perturbations.

3.5.5 Mathematical models of retinal biochemistry

MUSCLEES permanent members involved: Luca Alasio, Benoît Perthame, Philippe Robert

3.5.6 Modelling the Retinal Pigment Epithelium in Age-Related Macular Degeneration

MUSCLEES permanent members involved: Luca Alasio, Benoît Perthame

Biological and social networks

Collective phenomena can emerge from local interactions in biological and social networks. Social animals tend to organize themselves into highly coherent groups, such as schools of fish, bird flocks, swarms of insects, herds of sheep, or even human crowds. Much research is currently undertaken in various scientific communities (including biologists, sociologists, computer scientists and mathematicians) to understand how and why certain types of collective behavior (such as flocking 81, alignment 173, or consensus 111) are observed. Despite this surge of interest, many questions remain open and our research aims to address some of them. In particular, can the emergence of global behavior such as consensus be predicted from initial conditions? Are there sufficient or necessary conditions on the interaction network ensuring convergence to a coherent asymptotic state?

Bacterium colony growth

Bacteria are unicellular organisms, whose biomass exceeds that of all other living organisms, and on which our survival is dependent. In the human body, the number of bacteria almost equals the one of cells. Despite the fact that most of the bacteria are harmless, some pathogenic strains are the cause of infectious diseases such as tuberculosis, cholera, bacterial meningitis, and salmonella among others. It makes it essential to understand in which way bacteria multiply and disrupt the normal functions of our bodies. Numerous studies have been done to grasp how a bacterium, from a single organism, develops into organized micro-colonies and biofilm structures 96, 99. Still, some phenomena are not explained. At early stages of the development, going from one bacterium to a structured micro-colony, we will investigate mechanisms leading to poorly understood properties, such as the elongated shape of the colonies, the four cell arrays arrangement and the high density 163. At latter stages of development, we will question the impact of these microscopic phenomena on macroscopic structures.

Cell population dynamics: the classic homogeneous case

Self-organization is often observed in cell population dynamics, both within a single cell population or between two or more distinct populations. Interestingly, the forward and backward epithelial-mesenchymal cellular transitions (EMT-MET), which play a crucial role in embryonic development, tissue repair and cancer metastasis, can be modeled either as a transition between three homogeneous cell populations (epithelial, mesenchymal and hybrid), or as the evolution of a single heterogeneous cell population, structured by an epithelial-to-mesenchymal phenotype. In order to achieve self-organization, cell populations often display local communication strategies, whether it be within a cell population or between different cell types. For instance, chemotaxis refers to the directed movement of cells in response to a chemical gradient produced by neighbouring cells (Keller-Segel-type models). Mechanosensing is another well-established cell-cell communication strategy, that relies simply on mechanical constraints. Communication between cells can also be driven by the secretion and subsequent binding of ligands, as in the case of the EMT-MET 172.

When considering interactions between several cell populations, interactions may be mutualistic as in the case of cancer cell populations and trophic healthy cell populations (breast cancer and adipocytes  155, or leukaemic cells and supporting somatic cells  145 for instance), or cells can be in competition (in particular tumour-immune interactions  45, 126). This latter aspect will continue to be one of our present objectives in modelling cancer cell populations. We will address it in the sequel in the adapted framework of heterogeneous cell populations.

Cell population dynamics: heterogeneous cell populations and trait-structured models

One of the main challenges when modeling a single cell population is to take into account the biological variability, aka intrinsic heterogeneity, of the population. A now classic way of modelling, introduced in adaptive dynamics, firstly in theoretical ecology, then in cell population dynamics, is to use continuous trait (or phenotype)-structured population dynamics settings.

How to deal with them depends on the heterogeneity question at stake and on the choice of traits used to structure an adaptive cell population: should they be well-identified biological molecules or gene expression determinants, (e.g., specific to a given drug and a given population under drug exposure  156)? Or should they be hidden, but general and linked to cell fates, in other words potentials to develop such and such a trait or phenotype  48, 76, 78, 120, 164, as in theoretical ecology models (viability, fecundity, plasticity of individuals)?

Due to the lack of measurable markers of relevant biological variability (i.e., heterogeneity) recorded in continuous time from experimental teams, we are often bound to stick to their more hidden and abstract version. However, this will certainly never free us from keeping watch over incoming biological developments amenable to at least partly identify possible molecular markers of such a priori abstract phenotypes.

Of note, in the framework of adaptive structured cell population dynamics, emergence of phenotypes is always reversible. Which means that, according to changes in the cell population environment, new phenotypes may appear, and they can equally disappear if the environment changes. In other words, we address the question of cell differentiations, not mutations, recalling that cell differentiations occur in an isogenic cell population, not modifying its genome, only gene expressions due to the action of epigenetic enzymes, whereas mutations change the genome by modifying its constituting base pairs in the sequences ATGC.

Some of the questions that we aim to address by means of mathematical modelling by structured population models, in particular in the context of the EMT-MET (reversible phenotype transition) and phenotype divergence (reversible evolution between phenotype monomorphism and dimorphism) are the following: Can different cell phenotypes co-exist at the same time in a population, and if only some of them persist, which are they? What effect do growth and death of the population have on the phenotype distribution of the population? What effect do growth and environmental changes have on transient phenomena, such as the hysteretic behaviour observed in the Epithelial-Mesenchymal Transition, and on asymptotic behaviour of the cell populations? What role can be attributed to phenotype plasticity in such transient or established phenomena?

Neuroscience

In neuroscience, learning and memory are usually associated with long-term changes of connection strength between neurons. In this context, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. A Spike-Timing Dependent Plasticity (STDP) rule is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons.

There is a rich mathematical literature on biological neural networks but mainly when the connectivity of the network is fixed, i.e. when the synaptic weights are constant. In a series of articles  159, 160, 158, 174, a new, general, mathematical framework to study the phenomenon of synaptic plasticity associated to STDP rules has been introduced and analyzed for a system composed of two neuronal cells connected by a single synapse whose weight is time-varying.

Experiments show that long-term synaptic plasticity evolves on a much slower timescale than the cellular mechanisms driving the activity of neuronal cells. A scaling model has been introduced and limiting results have been proved. The central result obtained is an averaging principle for the stochastic process associated to the synaptic weight.

We plan to investigate mathematical models of plastic synapticity in a more general network. The question is of determining under which conditions the coordinates of the matrix of synaptic weights of a given subset $S$ of cells grow without bound or not. This property can be expressed by the fact that the cells of $S$ exhibit a collective behavior.

A difficult modelling problem in this context is of having a priori two scaling parameters with two different types of convergence: Averaging principles or mean-field approximations.

1. The factor of the time-scale of fast cellular processes;

   The main assumption is that the timescale of the time evolution of the synaptic weights is slow. This is the framework of 158. This scaling leads to a possible averaging principle.

2. The number of nodes of the network.

   A given neural cell receives an input from a large number of cells and to each of them is associated a synaptic weight. This scaling, with appropriate symmetry properties of the topology, may give a mean-field approximation of the network.

Both of these parameters should be large, and are a priori uncorrelated. A central question is to determine how possible scaling results can give an insight on the plastic synapticity at the level of such a network.

Pseudo-stratified epithelial tissue development

Understanding how tissue growth and development is regulated is crucial in biology. Both proliferation and regulation of cells' growth are fundamental for the development of healthy tissues in animals and plants. Pseudo-stratified epithelium tissues are composed of narrow and elongated cells arranged in a packed one-layer tissue. The positions of the nuclei are variable along the depth of the tissue. Each cell is connected to the so-called basal and apical surface. During development, each cell follows a series of events leading to cell division. This process, known as the cell cycle, is composed of four steps: G1, where the cell prepares for DNA replication; S, where the DNA is replicated; G2, where the cell prepares to divide; and mitosis M, where the cell divides. In pseudostratified epithelia, the nuclei move along the apical/basal axis during the inter-kinetic phases G1 and G2 35. This motion is called inter-kinetic nuclear migration (IKNM). The IKNM has become a point of interest in the past years with numerous studies being published 118. Some of the questions we will aim to answer with the development and analysis of mathematical models are the following. Are the motions in G1 and G2 active or passive motions? How is the IKNM impacted by the increase of crowding during the tissue development? Which mechanism allows the transition of the cell in the different phases of the cell cycle?

Energy driven development of tissue architecture

One of the main socio-economic challenges in the twenty-first century is to ensure that increasing lifespan is accompanied by the prevention of decline to achieve similar or greater increases in health. Organized architecture that supports organ function emerges rapidly and locally during the first period of life (during development), where the extracellular matrix (ECM) plays a key role by giving rise to the mechanical macrostructure. This 3D architecture is then globally maintained during the maturity period, before progressively declining corresponding to degeneration and loss of functions. Throughout all these steps, the evolving architecture and its constant turn-over is powered by energy exchanges through metabolism. Numerous evidences suggest that energy exchanges and mechanical forces can feedback on each other and that large perturbations of one or the other are associated with degeneration and diseases. Therefore, understanding the dynamics of biological tissues at different spatiotemporal scales requires to account simultaneously for energy exchanges and mechanical considerations, a view that is currently lacking. We will aim to bridge this gap by taking a particular focus on the complex interplay between metabolism and mechanics in tissue development and ageing via the dual use of mathematical modelling and in vitro/in vivo experiments.

Living tissues as multiphase flows

At the continuum (macroscopic) level, a living tissue might be seen as a multiphase flow (different types of cells, liquid, molecules) through a porous media (extra-cellular matrix), a view encompassed in the so-called mixture theory (see 82). In mathematical terms, this leads to strongly nonlinear degenerate parabolic Cahn-Hilliard (PDE) equations  151. Although widely used in the literature to describe the mechanical properties of living tissues, it remains unclear how these continuum models (at the population level) can be obtained from a mechanical description at the cell level. We will take an interest in the derivation of such models from mesoscopic (kinetic) models, in order to understand the relation between compressible and incompressible porous-medium models.

Tumour-immune cell interactions and immunotherapies

In a model of tumour-immune cell interactions under development, the behaviour of interacting heterogeneous cell populations is described by a set of coupled PDEs of the nonlocal Lotka-Volterra type. The cell population densities are structured by a continuous trait (aka phenotype) standing for malignancy identified to a potential of de-differentiation (so-called `stemness'), in tumour cells, and, similarly, a continuous trait representing anti-tumour aggressiveness in immune cells. As modern immunotherapeutic drugs, in particular Immune Checkpoint Inhibitors, have recently been introduced as boosters of such aggressiveness, i.e., of cancer cell kill by T-lymphocytes, and even more recently also by NK-lymphocytes, their impact on tumour-immune interactions is represented in the present model under development by a target in the effector lymphocyte population. Questions at stake are: Can we model in a relevant way and mathematically analyse these interactions between cell populations, so as to obtain a qualitative description of the so-called immunoediting, that is known to yield extinction, equilibrium or escape in the tumour cell population? Can we show `proof of concept' situations in which the impact of immunotherapies can reverse tumour escape towards extinction, or at least equilibrium? Can we design theoretical optimised strategies to deliver time-scheduled immunotherapies to attain this goal? Can we analyse these interactions and their therapeutic control by immunotherapies in terms of concentration (or not) of the traits?

Phenotypic divergence in cancer and in the emergence of multicellularity

The question of understanding the cancer disease from an integrative physiology and long-time evolution point of view has stimulated many authors for quite a long time. In this respect, the atavistic theory of cancer, presented in  83, 176, proposes that tumours represent, roughly speaking, a reverse evolution to a previous, incoherent, disorganised and very plastic state of multicellularity in animals, which the authors call Metazoa 1.0. This theory involves a billion year-long evolutionary perspective of the emergence of multicellularity from collections of unicellular beings to the first organised animals, so-called Urmetazoa  140. Phenotypic divergence under environmental constraints is involved in both evolutionary/developmental and cancer biology. In the former, it is the fundamental phenomenon by which cell differentiation yields new cell types with emerging functions, leading in particular to multicellular beings such as animals (aka metazoa). In the latter, the process of bet hedging in cancer is a response to cellular stress to describe the multiple fates of a plastic cancer cell population as a fail-safe strategy to face deadly insults, e.g., due to anticancer drugs. The question of phenotypic divergence in an isogenic cell population is thus crucial. We will address it by phenotype-structured PDEs of the reaction-advection-diffusion type  48, 76, 78, 120, 164, and explore what mechanisms (mutations, differentiation, selection) are responsible for concentration of the population around a unique phenotype (a singleton in phenotypic space); or, on the contrary, for continuous or discrete heterogeneity of the population, the discrete cases being represented by discrete sets of phenotypes, cases among which divergence stricto sensu, leading to a doubleton (phenotypic dimorphism), is the simplest one.

Elastic description of the Retinal Pigment Epithelium (RPE).

A further modelling effort is necessary in order to capture both biological and mechanical features of the RPE monolayer, with specific attention to topological changes such as lesion formation, closure and fusion. So far, hybrid models combining elastic deformations and motion by mean curvature seem very promising in terms of analysis, simulation, and qualitative adherence to experimental data.

Vector-borne epidemics

Every year, around 700,000 deaths are due to diseases transmitted by (female) mosquitoes, like malaria, yellow fever, dengue, Zika, chikungunya, Nile virus... They are indeed the most dangerous animals for humankind. For many of these diseases, no efficient remedy or vaccine presently exists, and an essential strategy to control vector-borne disease outbreaks consists in the control of mosquito vector populations that transmit these diseases (Aedes species for the diseases previously cited).

The insecticides, which have non-specific actions and strongly affect biodiversity, are now recognized as a highly unsatisfying solution, and innovative methods of biological control are being searched for and tested. Among these, the sterile or incompatible insect techniques (SIT/IIT) and replacement strategies (Wolbachia) attract strong attention. SIT is based on the release of male insects after their sterilization (traditionally by means of irradiation): sterile males will mate with wild females without producing any offspring, reducing or suppressing the wild population. The sterile insects are not self-replicating and, therefore, cannot become established in the environment. On the other hand, Wolbachia is a natural intracellular bacterial symbiont, maternally transmitted to offspring. Some of its strains cause a drastic decrease in the capacity to transmit dengue, zika or chikungunya of the mosquitoes, directly (by interfering with their vector competence) or indirectly (by shortening lifespan, etc.). Contrary to SIT, this offers theoretically a permanent protection against the outbreaks.

The application in the field of these promising techniques to control mosquitoes is not easy, and models are a useful tool to study the various issues at stake, and to propose and scale control strategies. In particular, it is important to take into account the spatial extension (and possible heterogeneities) of the operation and other aspects like the seasonality, migration from outside the treated domain, release of mosquitoes imperfectly treated, effects of the treatment on the epidemic risk and so on. The uncertainties on the biological processes and the imprecision of the measures make the whole issue quite intricate, and we intend to see what control science has to say to solve the related problems.

Infectious diseases

The progress of the pandemic of Covid-19 has highlighted on a scale never seen before the complexity and intricateness of the factors that shape the spread of an epidemic, from the biological aspects at various scales (from virus to world population), to the economic, social and politic aspects, without forgetting the many feedback loops binding them2. Our interest is to participate to the understanding and disentanglement of the important factors, to the design and analysis of relevant mathematical models, and to their use to shape adequate control strategies.

For the accomplishment of this task, we plan to take advantage of a reservoir of tools and ideas from control theory, in addition to the more classical techniques developed in mathematical epidemiology. This is a point in common with our other topic of interest previously mentioned, the vector-borne diseases. First, we will routinely consider control issues — not only in the sense of controlling a disease, but using the term as in “control theory”. The control inputs we will encounter represent the available “means of action” on the epidemic, typically vaccination campaigns or social distancing measures (or sterile mosquito releases in the case of vector-borne diseases previously mentioned). Constraints on the intensity of the input variables like the duration of lockdown periods are pertinent (total number of released mosquitoes for the control of vector-borne diseases), but also on the state variables, e.g. on a maximal room occupancy rate in Intensive Treatment Units (maximal number of female mosquitoes, to limit both nuisance and epidemiological risk in the vector-borne diseases context). Optimal control involves non-conventional cost functions, such as the peak of infectious people (peak of female mosquito population...) or the time spent above a given value, which do not lead to Bolza problem. Robustness issues are also important in this context where the reality is imperfectly described by approximate models.

Second, we will pay particular attention to the models, the data and their cross-relations. Contrary to the engineering sciences, where models come from a combination of general principles and empirical laws, there is no such situation in mathematical epidemiology. In fact, it is not fully clear what are the key phenomena and quantities that influence decisively such complex situations, and thus deserve to be included in a model. On the other hand, the data themselves are imprecise and questionable, due to reasons that range from the evolving biological reality and our imperfect knowledge, to the characteristics of the data collection process by the Health system. In this context, we will be specially interested in questions of observability and identifiability (“given a model of the system and specific input-output experiments supposed error free, is it possible to determine uniquely the actual system state value and the parameters of the model ?"), and of observation and identification, their realization counterparts (“given a model observable or identifiable, how to practically estimate the state or parameter values ?").

7.1.1 Large-population limits

Participants: Nastassia Pouradier Duteil.

7.1.2 Scaling limits

Participants: Sophie Hecht, Benoit Perthame, Diane Peurichard.

A Scaling Approach to Stochastic Chemical Reaction Networks.

In 123, we have investigated the asymptotic properties of Markov processes associated to stochastic chemical reaction networks (CRNs) driven by the kinetics of the law of mass action. Their transition rates exhibit a polynomial dependence on the state variable, with possible discontinuities of the dynamics along the boundary of the state space. As a natural choice to study stability properties of CRNs, the scaling parameter considered in this paper is the norm of the initial state. Compared to existing scalings of the literature, this scaling does not change neither the topology of a CRN, nor its reactions constants. Functional limit theorems with this scaling parameter can be used to prove positive recurrence of the Markov process. This scaling approach also gives interesting insights on the transient behavior of these networks, to describe how multiple time scales drive the time evolution of their sample paths for example. General stability criteria are presented as well as a possible framework for scaling analyses. Several simple examples of CRNs are investigated with this approach. A detailed stability and scaling analyses of a CRN with slow and fast timescales is worked out.

Analysis of Stochastic Chemical Reaction Networks with a Hierarchy of Timescales

In124 we investigate a class of stochastic chemical reaction networks with $n\ge 1$ chemical species $S_1$, ..., $S_n$, and whose complexes are only of the form $k_i{S}_i$, $i=1$,..., $n$, where $\left(k_i\right)$ are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter $N$. A natural hierarchy of fast processes, a subset of the coordinates of $\left(X_i\left(t\right)\right)$, is determined by the values of the mapping $i\mapsto k_i$. We show that the scaled vector of coordinates $i$ such that $k_i=1$ and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as $N$ gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions.

Stochastic Chemical Reaction Networks with Discontinuous Limits and AIMD processes

In 125 we study a class of stochastic chemical reaction networks (CRNs) for which chemical species are created by a sequence of chain reactions. We prove that under some convenient conditions on the initial state, some of these networks exhibit a discrete-induced transitions (DIT) property: isolated, random, events have a direct impact on the macroscopic state of the process. If this phenomenon has already been noticed in several CRNs, in auto-catalytic networks in the literature of physics in particular, there are up to now few rigorous studies in this domain. A scaling analysis of several cases of such CRNs with several classes of initial states is achieved. The DIT property is investigated for the case of a CRN with four nodes. We show that on the normal timescale and for a subset of (large) initial states and for convenient Skorohod topologies, the scaled process converges in distribution to a Markov process with jumps, an Additive Increase/Multiplicative Decrease (AIMD) process. This asymptotically discontinuous limiting behavior is a consequence of a DIT property due to random, local, blowups of jumps occurring during small time intervals. With an explicit representation of invariant measures of AIMD processes and time-change arguments, we show that, with a speed-up of the timescale, the scaled process is converging in distribution to a continuous deterministic function. The DIT property analyzed in this paper is connected to a simple chain reaction between three chemical species and is therefore likely to be a quite generic phenomenon for a large class of CRNs.

7.3.1 Modelling phenotypic divergence in cancer and in the emergence of multicellularity by phenotype-structured equations of cell population dynamics

Participants: Jean Clairambault, Lia Sela.

7.3.2 Analysis of non-local advection-diffusion models for active particles

Participants: Luca Alasio.

7.4.1 Biological control of vectors

Participants: Pierre-Alexandre Bliman, Nga Nguyen, Manon de la Tousche.

7.4.2 Control of infectious diseases

Participants: Pierre-Alexandre Bliman, Marcel Fang, Assane Savadogo.

7.5.1 Modeling of milling and schooling in gregarious fish.

Participants: Nastassia Pouradier Duteil.

The study of collective behavior has attracted much attention in the last twenty years, both among mathematicians and experimentalists, with the aim of explaining how local interactions between the group members lead to the emergence of global patterns, a phenomenon referred to as self-organization. Importantly, a group's capacity to transition between different global configurations is related to its survival capacity. For example, fish schools adopt different collective behaviors when foraging for food or facing predators.

However, the mechanisms provoking phase transition in a group of interacting agents are often difficult to identify experimentally. We have initiated a collaboration with R. Godoy-Diana and B. Thiria of the laboratory PMMH of ESPCI, in order to focus on exploring the effect of two main mechanisms in the collective behavior of gragarious fish (Hemmigramus rhodostomus): (i) the individuals' fields of vision; and (ii) the population's heterogeneity. Both mechanisms are particularly challenging to study exclusively through numerical experiments, which justifies the tight collaboration between the two teams. First results were obtained during the internship of A. Savalle (May-July 2024). We proved that a simple agent-based model with just two forces (self-propulsion and alignment dynamics) is able to reproduce the milling behavior of the group when the interactions are non-symmetric, due to the directed field of vision of each individual.

7.5.2 3D Modeling of biological tissue emergence and repair

Participants: Diane Peurichard.

7.5.3 Modelling the Retinal Pigment Epithelium in Age-Related Macular Degeneration

Participants: Luca Alasio.

7.5.4 Mathematical models of retinal biochemistry

Participants: Luca Alasio.

8.1.1 Inria associate team not involved in an IIL or an international program

8.1.2 STIC/MATH/CLIMAT AmSud projects

9.1.1 Scientific events: selection

9.1.2 Journal

9.1.3 Invited talks

Lucie Laurence has given a talk at the SPT-CRN conference in Pulla, Italia, 10/06-14/06.

Philippe Robert has given a seminar at Toulouse at the IRIT on September 13, in Tolede, for the ECMTB conference, 22/07-26/07 and at the online seminar Morn Seminar 23/05. Philippe Robert has given lectures at the Journées Math Bio Santé 2024, Nantes, 24/06-28/06.

Marcel Fang presented a contribution at the 15th Conference on Dynamical Systems Applied to Biology and Natural Sciences, DSABNS 2024, in February.

Manon De La Tousche presented a contribution at the 15th Conference on Dynamical Systems Applied to Biology and Natural Sciences, DSABNS 2024, in February.

Pierre-Alexandre Bliman gave seminars at Université Mohamed 6 Polytechnique, Maroc in July; and at Department of Mathematics and Applications, University of Naples Federico II in September. He presented contributions at the conference “Numerical Analysis and Modelling in Applied Sciences (NAMAS)”, Gaeta, Italy, in September, and (online)in the Conférence Maths and Decision: Mini-Symposium Biologiacal Complex Systems, Rabat, Marocco (online) in December.

Nastassia Pouradier Duteil gave presentations at the workshops “Alhambra PDE days” (University of Granada), “Collective models for networked particle systems” (University of Pavia), “Variational Analysis, Models and Methods in Measure Spaces” (CIRM). She gave a course at the Kinmat Summer School “Kinetic description of collective dynamics and related emergent phenomena” (Bedlewo, Poland). She also gave a seminar at the “Laurent Schwarz seminars” at Ecole Polytechnique.

Diane Peurichard gave presentations at the conferences GIMC SIMAI Young (Naples, Italy), ECMTB2024 (Toledo, Spain), 'International Summer school on mathematical biology' (Shanghai, China), workshop MATIDY (IHP, Paris), Plant Biology Workshop (Lyon), GDR Mecanobiology (Metz).

Jean Clairambault has given an in-presence talk to the on-invitation workshop “Modelling Complexity in Mechanics and Applied Mathematics: Theory, Experiments, and Simulations” in Ortigia, Syracuse, Italy, September 2024, and another one to the online “Mathematical Modelling in Biomedicine” scientific seminar, RUDN University, Moscow, February 2024.

Sophie Hecht gave a presentation at the GdT Normand on biomathematics in May 2024.

9.1.4 Scientific expertise

Diane Peurichard is member of the commission d'évaluation (CE) Inria, of the commission des emplois scientifiques (CES) Inria Paris, of the commité de suivi doctoral (CSD) Inria Paris, and of the pole ecoute, LJLL, Sorbonne Université.

9.1.5 Research administration

Diane Peurichard is coordinator of the ANR project ENERGENCE.

Nastassia Pouradier Duteil is coordinator of the ANR project FISH.

Pierre-Alexandre Bliman is coordinator of the ANR project NOCIME and of the STIC AMSUd project BIO-CIVIP.

9.2.1 Supervision

Pierre-Alexandre Bliman has been PhD supervisor of Marcel Fang , Assane Savadogo and Manon De La Tousche .

Nastassia Pouradier Duteil has supervised the M2-level internship of A. Savalle.

Diane Peurichard has supervised the post-doctorate of S. Toste (co-supervised with RESTORE, Toulouse) and the post-doctorate of H. Horii (together with Sophie Hecht).

Sophie Hecht has supervised the post-doctorate of H. Horii (together with Diane Peurichard).

9.2.2 Juries

Philippe Robert has been a reviewer of the PhD document “Stochastic analysis of unreliable large-scale storage systems” by Soukaina El Masmari, University of Casablanca, Morocco. He has been a member of the jury for the PhD defense of Lucie Laurence in December

Diane Peurichard has been in the CRCN/ISFP juries for Inria Saclay and Inria Nancy as a member of the commission d'évaluation (CE), campaign 2024. She was also member of the jury for the Prix Junior Maryam Mirzakhani 2024 of the fondation Hadamar, rewarding three young students (license, master) for a first research work.

Sophie Hecht has been in the jury for the recruitment of a MdC at Université Paris Nord.

9.3.1 Productions (articles, videos, podcasts, serious games, ...)

Nastassia Pouradier Duteil was the first guest in Nathalie Ayi's podcast “Tête-à-tête chercheuse(s)”, seeking to promote a diversified image of mathematical researchers. She is now one of the permanent hosts of the podcast.

9.3.2 Participation in Live events

Nastassia Pouradier Duteil gave an outreach presentation for high school and university students at the Classes Préparatoires “Les Lazaristes”.

International journals

• 2 articleP.-A.Pierre-Alexandre Bliman, D.Daiver Cardona-Salgado, Y.Yves Dumont and O.Olga Vasilieva. Optimal Control Approach for Implementation of Sterile Insect Techniques.Journal of Mathematical Sciences2795March 2024, 607-622HALDOIback to text
• 3 articleP.-A.Pierre-Alexandre Bliman, N.Nga Nguyen and N.Nicolas Vauchelet. Efficacy of the Sterile Insect Technique in the presence of inaccessible areas: A study using two-patch models.Mathematical BiosciencesNovember 2024, 109290In press. HALDOIback to text
• 4 articleP.-A.Pierre-Alexandre Bliman, M.Manon de la Tousche and Y.Yves Dumont. Feasibility and optimization results for elimination by mass-trapping in a metapopulation model.Applied Mathematical Modelling144March 2025, 116047HALDOIback to text
• 5 articleV.Valeria Caliaro, D.Diane Peurichard and O.Osvaldo Chara. How a reaction-diffusion signal can control spinal cord regeneration in axolotls: A modeling study.iScience277July 2024, 110197HALDOIback to text
• 6 articleP.Pauline Chassonnery, J.Jenny Paupert, A.Anne Lorsignol, C.Childérick Séverac, M.Marielle Ousset, P.Pierre Degond, L.Louis Casteilla and D.Diane Peurichard. Fibre crosslinking drives the emergence of order in a three-dimensional dynamical network model.Royal Society Open Science1112024, 231456HALDOIback to text
• 7 articleG.Giorgia Ciavolella, N.Nathalie Ferrand, M.Michéle Sabbah, B.Benoît Perthame and R.Roberto Natalini. A model for membrane degradation using a gelatin invadopodia assay.Bulletin of Mathematical Biology863January 2024, 30HALDOI
• 8 articleM.Marie Doumic, S.Sophie Hecht, B.Benoît Perthame and D.Diane Peurichard. Multispecies cross-diffusions: from a nonlocal mean-field to a porous medium system without self-diffusion.Journal of Differential Equations389April 2024, 228-256In press. HALDOIback to text
• 9 articleC.Charles Elbar, B.Benoît Perthame, A.Andrea Poiatti and J.Jakub Skrzeczkowski. Nonlocal Cahn-Hilliard equation with degenerate mobility: Incompressible limit and convergence to stationary states.Archive for Rational Mechanics and Analysis2483May 2024, 41HALDOIback to text
• 10 articleF.Frank Ernesto Alvarez and J.Jean Clairambault. Phenotype divergence and cooperation in isogenic multicellularity and in cancer.Mathematical Medicine and BiologyJuly 2024HALDOIback to text
• 11 articleV.Vincent Fromion, P.Philippe Robert and J.Jana Zaherddine. A Stochastic Analysis of Particle Systems with Pairing.Stochastic Processes and their Applications178September 2024, 104480HALDOI
• 12 articleM.Maxime Herda, P.Pierre Monmarché and B.Benoît Perthame. Wasserstein contraction for the stochastic Morris-Lecar neuron model.Kinetic and Related Models 181February 2025, 1-18HALDOI
• 13 articleL.Laura Kanzler, B.Benoît Perthame and B.Benoît Sarels. Structured Model Conserving Biomass for the Size-spectrum Evolution in Aquatic Ecosystems.Journal of Mathematical Biology883February 2024, 26HALDOI
• 14 articleN.Nadia Loy and B.Benoît Perthame. A Hamilton-Jacobi approach to nonlocal kinetic equations.Nonlinearity3710September 2024, 105019HALDOIback to text
• 15 articleA.Anastasia Pacary, D.Diane Peurichard, L.Laurence Vaysse, P.Paul Monsarrat, C.Clémence Bolut, A.Adeline Girel, C.Christophe Guissard, A.Anne Lorsignol, V.Valérie Planat-Benard, J.Jenny Paupert, M.Marielle Ousset and L.Louis Casteilla. A computational model reveals an early transient decrease in fiber cross-linking that unlocks adult regeneration.NPJ Regenerative medicine91October 2024, 29HALDOIback to text
• 16 articleP.Philippe Robert and G.Gaëtan Vignoud. A Palm Space Approach to Non-Linear Hawkes Processes.Electronic Journal of Probability29noneJanuary 2024, 1-37HALDOI

International peer-reviewed conferences

• 17 inproceedingsP.-A.Pierre-Alexandre Bliman and M.Marcel Fang. A two-stage seirs reinfection model with multiple endemic equilibria.15th Conference on Dynamical Systems Applied to Biology and Natural Sciences, DSABNS 2024Lisbonne, Portugal2024HAL
• 18 inproceedingsM.Manon de la Tousche, P.-A.Pierre-Alexandre Bliman and Y.Yves Dumont. Modeling population dynamics and control strategies for a unique species evolving in heterogenous landscape.15 th Conference on Dynamical Systems Applied to Biology and Natural Sciences (DSABNS 2024)Lisbonne, Portugal2024, 194HAL

Doctoral dissertations and habilitation theses

• 19 thesisM.Marcel Fang. Mathematical modelling, observation and identification of epidemiological models with reinfection.Sorbonne UniversiteDecember 2024HAL
• 20 thesisA.Assane Savadogo. Modeling and mathematical study of an epidemiological infection in an ecological community and within a commuting population.Sorbonne Université; Université Nazi Boni (Bobo-Dioulasso, Burkina Faso)December 2024HAL

Reports & preprints

• 21 miscL.Luca Alasio, J.Jessica Guerand and S. M.Simon M Schulz. Regularity and trend to equilibrium for a non-local advection-diffusion model of active particles.March 2024HALback to text
• 22 miscN.Nathalie Ayi and N.Nastassia Pouradier Duteil. Large-population limits of non-exchangeable particle systems.January 2024HALback to text
• 23 miscN.Nathalie Ayi, N.Nastassia Pouradier Duteil and D.David Poyato. Mean-field limit of non-exchangeable multi-agent systems over hypergraphs with unbounded rank.October 2024HALDOIback to text
• 24 miscP.-A.Pierre-Alexandre Bliman. Basic offspring number and robust feedback design for the biological control of vectors by sterile insect release technique.October 2024HALback to text
• 25 miscP.-A.Pierre-Alexandre Bliman, B.Boureima Sangaré and A.Assane Savadogo. A framework for the modelling and the analysis of epidemiological spread in commuting populations.August 2024HALback to text
• 26 miscX.Xu'An Dou, B.Benoît Perthame, D.Delphine Salort and Z.Zhennan Zhou. Noisy integrate-and-fire equation: continuation after blow-up.September 2024HAL
• 27 miscM.Marie Doumic, S.Sophie Hecht, M.Marc Hoffmann and D.Diane Peurichard. Scaling limits for a population model with growth, division and cross-diffusion.October 2024HALback to text
• 28 miscM.Marcel Fang, P.-A.Pierre-Alexandre Bliman, D.Denis Efimov and R.Rosane Ushirobira. Nonlinear adaptive observers for SIS system with primary infections.October 2024HAL
• 29 miscM. A.Marco A. Fontelos, N.Nastassia Pouradier Duteil and F.Francesco Salvarani. Self-similar solutions, regularity and time asymptotics for a nonlinear diffusion equation arising in game theory.July 2024HAL
• 30 miscB.Benoît Perthame, D.Delphine Salort and C.Clément Rieutord. Strongly nonlinear age structured equation, time-elapsed model and large delays.March 2025HAL
• 31 miscB.Benoît Perthame and C.Chiara Villa. Regularity and stability in a strongly degenerate nonlinear diffusion and haptotaxis model of cancer invasion.December 2024HAL
• 32 miscB.Benoît Perthame and M.Mingyue Zhang. Patterns in the Keller-Segel system with density cut-off.December 2024HAL

Scientific popularization

• 33 articleJ.Jean Clairambault. Cancer as a localised disorganisation of the body plan.Revue Française de Psychosomatique66November 2024, 83-98HALback to text