2025​​​‌Activity reportProject-TeamMUSCLEES‌

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 128.​‌ 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” $\mathrm{s‌}\in I$ (representing the​​ particle's individual identities, in​​​‌ an infinite population):

In 50,​​ 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 ${\mathrm{\varphi }}_{ij}({\mathrm{x‌}}_j(t)-x_i(‌t)):={\xi }_{i\mathrm{j‌}}\varphi (x_{\mathrm{j}}\left(t\right)-‌x_i\left(\mathrm{t}\right))$, where​‌ ${\left({\xi }_{i\mathrm{j}}\right)}_{i,\mathrm{j‌}}$ 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 128,​​ 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 84​​​‌ 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 86.‌ 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 $\mu (t,·)‌\in 𝒫\left({\mathrm{ℝ‌}}^d\right)$ represents the‌​‌ particle distribution at time​​ $t$, and the​​​‌ non-local velocity $V[‌{\mu }_t]$ 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 126. 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 $\mu (\mathrm{t},x,\mathrm{r‌})$ (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 54, 146‌, 155.

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 $𝒜$, $\mathrm{ℬ}$ and $𝒞$ are involved​​​‌ in a chemical reaction‌ of the type,

the classical​​​‌ law of mass action‌ states that the concentration‌​‌ $x_M\left(\mathrm{t}\right)$ 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 98 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 $\mathrm{A‌}$ and $B$ particles will​​ collide at rate $\mathrm{k‌}$, so that if​ $X_M\left(\mathrm{t‌}\right)$ is the number​​ of particles of type​​​‌ $M$ at time $\mathrm{t}$, then, at time​‌ $t$, a particle​​ of type $C$ is​​​‌ created at rate $\mathrm{k}{X}_A\left(\mathrm{t‌}\right)X_B(t)$. The​​​‌ process $(X_{\mathrm{A}}\left(t\right),‌X_B\left(\mathrm{t}\right),X_{\mathrm{c‌}}\left(t\right))$ is a Markov process,​‌ if we assume that​​ there are external arrivals​​​‌ of $A$ and $\mathrm{B}$ 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 41​‌ 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 79​ and in  81,​‌ 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 70.​ A more general and​‌ extended recent state of​​ the art on phenotype-structured​​​‌ population dynamics is reported​ in 120.

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, $\varphi (\mathrm{s},s^{\text{'}},‌x(t,‌s^{\text{'}})-‌‌x(t,s))=‌\sigma (s,‌s^{\text{'}})\widetilde{\mathrm{\varphi ‌‌}}(x(t,s^{\text{'}‌})-x(‌t,s)‌‌)$, 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​‌ 143, 167.​​ 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 60‌​‌. 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​‌ $\rho (t,x)={\int ‌}_0^{2\pi }\mathrm{f‌}(t,\mathrm{x‌‌},\theta )\mathrm{d}\theta$ is the angle-independent​​​‌ density and $𝐞(‌\theta )=(‌‌cos\theta ,sin\theta )$, with​​​‌ periodic boundary conditions both‌ in the space variable‌​‌ $x\in {(\mathrm{0},2\pi )‌}^2$ and the angle‌ variable $\theta \in (‌‌0,2\mathrm{\pi })$. The constant​​​‌ parameters $\mathrm{Pe}\in \mathrm{ℝ‌}$ 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 61​​​‌. 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 60 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 163​​, tumour growth models​​​‌ 73, and multi-species​ agent-based models 33.​‌ 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) 107,​​​‌ 64. 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 89, 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 89, 92​​.

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 106, 129​​. 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 176​. 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 68, 55​. These models considered​‌ the transition between the​​ different phases of the​​​‌ cell cycle depending of​ the cell age. However,​‌ recent works 104 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​‌ 104 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 74,​​ alignment 171, or​​​‌ consensus 105) 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 89,​​​‌ 92. 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​​​‌ 161. 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 170.‌​‌

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 153, or leukaemic‌​‌ cells and supporting somatic​​ cells 140 for instance),​​​‌ or cells can be‌ in competition (in particular‌​‌ tumour-immune interactions 37,​​ 119). 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 154)?​​​‌ 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 40, 69​​, 71, 114​​​‌, 162, 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​ 157, 158,​‌ 156, 172,​​ 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 $\mathrm{S‌}$ of cells grow without‌​‌ bound or not. This​​ property can be expressed​​​‌ by the fact that‌ the cells of $\mathrm{S‌‌}$ 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 156. 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​‌ 27. 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 112‌. 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 75​​). In mathematical terms,​​​‌ this leads to strongly​ nonlinear degenerate parabolic Cahn-Hilliard​‌ (PDE) equations 148.​​ 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​‌ 76, 174,​​ 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 135.‌ 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 40​​​‌, 69, 71‌, 114, 162‌​‌, 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.

Visco-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. A​​​‌ visco-elastic description of the‌ tissue has been developed‌​‌ in collaboration with the​​ group of Prof. M.​​​‌ Paques at Hôpital National‌ des Quinze-Vingts (Paris Eye‌​‌ Imaging). We are confidently​​ working towards new results​​​‌ in terms of analysis,‌ simulation, and qualitative adherence‌​‌ to experimental data. The​​ main goal is to​​​‌ support and integrate ophthalmological‌ research, contributing to the‌​‌ prediction of the evolution​​ of atrophic lesions during​​​‌ the progression of Age-Related‌ Macular Degeneration. Recent advances‌​‌ have been made in​​ the study of the​​​‌ connection between microscopic and‌ macroscopic models for the‌​‌ RPE tissue, but this​​ point of view is​​​‌ still in an early‌ phase. In the spirit‌​‌ of describing tissue ageing,​​​‌ new models structured with​ radius or senescence variables​‌ are being constructed and​​ analysed.

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 tissueMORPH

• Keywords:​​​‌
  Systems Biology, Computational biology,​ Physiology, Mechanistic modeling
• Functional​‌ Description:
  tissueMORPH is a​​ software that permits 2D​​​‌ agent-based simulations of tissue​ morphogenesis and regeneration/wound healing.​‌ Simulation platform enabling to​​ perform simulations of systems​​​‌ composed of individual cells​ (disks) growing and interacting​‌ in a dynamical fiber​​ network (composed of cross-linked​​​‌ fibers - segments). The​ platform enables to explore​‌ the mechanisms of tissue​​ construction (morphogenesis) and tissue​​​‌ reparation after injury (regeneration/wound​ healing) . The software​‌ includes many modules specifically​​ tailored to support the​​​‌ simulation and analysis of​ virtual tissues including 2D​‌ visualization and image processing​​ tools. Cell and fiber​​​‌ network parameters can be​ independently varied which facilitates​‌ specific simulations and allows​​ for detailed analyses of​​​‌ growth dynamics and links​ between matrix mechanical properties​‌ and tissue construction/reconstruction. Applications:​​ adipose tissues morphogenesis, tissue​​​‌ reparation, wound healing, muscles,​ dynamical fiber networks.
• Publications:​‌
  hal-01576486, version 1,​​ hal-02345773, version 1
• Contact:​​​‌
  Diane Peurichard
• Participant:
  6​ anonymous participants

8.1.1​‌ Scaling limits

Participants: Sophie​​ Hecht, Benoit Perthame​​​‌, Diane Peurichard.​

8.1.2 Collective motion​​​‌ of non-exchangeable particle systems‌

Participants: Nastassia Pouradier Duteil‌​‌, Angelina Jammart.​​

Many living systems exhibit​​​‌ fascinating dynamics of collective‌ behavior during locomotion, from‌​‌ bacterial colonies to human​​ crowds. The celebrated Cucker-Smale​​​‌ model describes the dynamics‌ of a group of‌​‌ interacting particles, whose velocities​​​‌ evolve in time according​ to alignment dynamics. The​‌ particles are said to​​ be exchangeable (or identical)​​​‌ if the dynamics does​ not depend explicitely on​‌ their labels. In the​​ opposite exchangeable case, the​​​‌ Cucker-Smale system is known​ to exhibit a flocking​‌ behaviour, that is the​​ asymptotic alignment of all​​​‌ the individual agent velocities,​ under a “fat-tail” condition​‌ on the interaction kernel,​​ see for instance the​​​‌ surveys. These results were​ extended to the non-exchangeable​‌ case in several works,​​ under some additional conditions​​​‌ on the communication weights.​ The internship of Angelina​‌ Jammart , co-supervised by​​ Nastassia Pouradier Duteil and​​​‌ Benoît Bonnet-Weill, consisted in​ extending the existing results​‌ of convergence to flocking​​ for the microscopic system,​​​‌ in particular for time-dependent​ coefficients with positive scrambling​‌, with some time-integral​​ condition.

8.1.3 Multiscale analysis​​​‌ of a kinetic equation​ for mechanotaxis

Participants: Benoit​‌ Perthame.

In 24​​, we present a​​​‌ new kinetic equation for​ cell migration driven by​‌ mechanical interactions with the​​ substrate, an effect not​​​‌ previously captured in kinetic​ models, and essential for​‌ explaining observed collective behaviors​​ such as those in​​​‌ bacterial colonies. The model​ introduces an acceleration term​‌ that accounts for the​​ dynamics of motile cells​​​‌ undergoing mechanotaxis, where extracellular​ signals modulate the forces​‌ arising from cell-substrate interactions.​​ From this formulation, we​​​‌ derive a family of​ macroscopic limit equations and​‌ analyze their principal properties.​​ In particular, we examine​​​‌ linear stability and pattern​ formation ability through theoretical​‌ analysis, supported by numerical​​ simulations.

8.1.4 Incompressible limit​​​‌ of porous media equation​ with chemotaxis and growth​‌

Participants: Benoit Perthame.​​

In 15, we​​​‌ revisit the problem of​ proving the incompressible limit​‌ for the compressible porous​​ media equation with Newtonian​​​‌ drift and growth. The​ question is motivated by​‌ models of living tissues​​ development including chemotaxis. We​​​‌ extend the problem, already​ treated by the authors​‌ and several other contributions,​​ in using a simplified​​​‌ approach, in treating dimensions​ two or higher, and​‌ in incorporating the pressure​​ driven growth term. We​​​‌ also complete the analysis​ with stronger estimates on​‌ the pressure gradient. The​​ major difficulty is to​​​‌ prove the strong convergence​ of the pressure gradient​‌ which is obtained here​​ by a new observation​​​‌ on an algebraic relation​ involving the pressure gradient​‌ for weak limits.

8.1.5​​ Deriving sub-diffusion equations

Participants:​​​‌ Benoit Perthame.

Sub-diffusion​ equations are used in​‌ a large range of​​ applications including fluids, plasma​​​‌ physics and biology. Their​ mathematical analysis is advanced​‌ even if a much​​ larger literature addresses super-diffusions.​​​‌ In 25, we​ provide the microscopic mechanism​‌ and rigorous derivation of​​ sub-diffusions when the waiting​​​‌ time distribution of particles​ follows an age-structured equation​‌ and jumps occur at​​ each renewal. The major​​​‌ difficulty to recover sub-diffusions,​ unlike normal diffusions, is​‌ that the assumption of​​ long waiting time implies​​​‌ lack of integrability for​ the age equilibrium. This​‌ prevents to establish strong​​ a priori estimates. Here,​​​‌ the Laplace transform plays​ the role that Fourier​‌ transform plays for the​​ more traditional case of​​ fast diffusions.

Stochastic Chemical Reaction​​​‌ Networks with Discontinuous Limits‌ and AIMD processes

In‌​‌ 118 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. Joint work‌ with Lucie Laurence (University‌​‌ of Berne).

Analysis of​​ Stochastic Chemical Reaction Networks​​​‌ with a Hierarchy of‌ Timescales

In 117 we‌​‌ investigate a class of​​ stochastic chemical reaction networks​​​‌ with $n\ge \mathrm{1‌}$ chemical species $S_{\mathrm{1‌‌}}$, ..., $S_{\mathrm{n}}$, and whose complexes​​​‌ are only of the‌ form $k_i{\mathrm{S‌‌}}_i$, $i=1$,..., $n$,​​​‌ where $\left(k_{\mathrm{i‌}}\right)$ 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 offast processes,‌ a subset of the‌​‌ coordinates of $\left({\mathrm{X}}_i(t)‌\right)$, is determined‌ by the values of‌​‌ the mapping $i\mapsto k_i$. We​​​‌ show that the scaled‌ vector of coordinates $\mathrm{i‌‌}$ such that $k_{\mathrm{i}}=1$ 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.​‌ Joint work with Lucie​​ Laurence (University of Berne).​​​‌

Stochastic Models of Resource​ Allocation in Chemical Reaction​‌ Networks

In 22 we​​ investigate a stochastic model​​​‌ of a chemical reaction​ network with three types​‌ of chemical species $\mathrm{ℛ}$, $ℳ$ and $\mathrm{𝒰‌}$ that interact to transform​ a flow of external​‌ resources, the chemical species​​ $𝒬$, to produce​​​‌ a product, the chemical​ species ${𝒫}_r$.​‌ A regulation mechanism involving​​ the sequestration of the​​​‌ chemical species $ℛ$ when​ the flow of resources​‌ is too low is​​ investigated. The original motivation​​​‌ of the study is​ of analyzing the qualitative​‌ properties of a key​​ regulation mechanism of gene​​​‌ expression in biological cells,​ the stringent response.​‌

A scaling analysis of​​ a Markov process in​​​‌ ${\mathbb{N}}^5$ representing the​ state of the chemical​‌ reaction network is achieved.​​ It is shown that,​​​‌ depending on the parameters​ of the model, there​‌ are, quite surprisingly, three​​ possible asymptotic regimes. To​​​‌ each of them corresponds​ a stochastic averaging principle​‌ with a fast process​​ expressed in terms of​​​‌ a network of $\mathrm{M}/M/\mathrm{\infty ‌}$ queues. One of these​​ regimes, the optimal sequestration​​​‌ regime, does not seem​ to have been identified​‌ up to now. Under​​ this regime, the input​​​‌ flow of resources is​ low but the state​‌ of the network is​​ still acceptable in terms​​​‌ of unused macro-molecules, showing​ the remarkable efficiency of​‌ this regulation mechanism. The​​ technical proofs of the​​​‌ main convergence results rely​ on a combination of​‌ coupling arguments, technical estimates​​ of the solutions of​​​‌ SDEs, of sample paths​ of fast processes in​‌ particular, and the stability​​ properties of some non-linear​​​‌ dynamical systems in ${\mathrm{ℝ}}^2$. Joint work​‌ with Vincent Fromion (INRAE)​​ and Jana Zaherddine (INRIA-Paris).​​​‌

8.3.1​​​‌ Structured Continuity Equations in​ Fibred Wasserstein Spaces

Participants:​‌ Nastassia Pouradier Duteil.​​

In 19, in​​​‌ collaboration with Benoît Bonnet-Weill,​ we developed a comprehensive​‌ ODE-theory for structured continuity​​ equations in fibred measure​​​‌ spaces, which refer to​ a class of heterogeneous​‌ continuity equations arising as​​ the macroscopic approximation of​​​‌ nonexchangeable particle systems. After​ investigating in depth the​‌ topologies induced by the​​ so-called fibred and classical​​​‌ Wasserstein metrics over such​ probability spaces, we studied​‌ both local and nonlocal​​ structured continuity equations over​​​‌ fibred Wasserstein spaces. We​ notably established quantitative Cauchy-Lipschitz​‌ and qualitative Carathéodory-Peano well-posedness​​ results for the latter,​​​‌ along with precise correspondences​ between the latter and​‌ classical Lagrangian dynamics and​​ continuity equations. In keeping​​​‌ with what has long​ been known for exchangeable​‌ particle systems, we provided​​ a general meanfield approximation​​​‌ result for solutions of​ structured continuity equations, along​‌ with a quantitative variant​​ thereof under practically reasonable​​​‌ regularity assumptions on the​ driving field and initial​‌ data.

8.3.2 An integrative​​ phenotype-structured partial differential equation​​ model for the population​​​‌ dynamics of epithelial-mesenchymal transition‌

Participants: Nastassia Pouradier Duteil‌​‌.

In 9,​​ in the framework of​​​‌ Jules Guilberteau's PhD thesis‌ (defended in 2023), we‌​‌ developed a collaboration with​​ a team of system's​​​‌ biologists at the Indian‌ Institute of Science of‌​‌ Bangalore to study epithelial-mesenchymal​​ cell transition using a​​​‌ structured PDE model. Phenotypic‌ heterogeneity along the epithelial-mesenchymal‌​‌ (E-M) axis contributes to​​ cancer metastasis and drug​​​‌ resistance. Recent experimental efforts‌ have collated detailed time-course‌​‌ data on the emergence​​ and dynamics of E-M​​​‌ heterogeneity in a cell‌ population. However, it remains‌​‌ unclear how different possible​​ processes interplay in shaping​​​‌ the dynamics of E-M‌ heterogeneity: a) intracellular regulatory‌​‌ interaction among biomolecules, b)​​ cell division and death,​​​‌ and c) stochastic cell-state‌ transition (biochemical reaction noise‌​‌ and asymmetric cell division).​​ Here, we proposed a​​​‌ Cell Population Balance (Partial‌ Differential Equation (PDE)) based‌​‌ model that captures the​​ dynamics of cell population​​​‌ density along the E-M‌ phenotypic axis due to‌​‌ abovementioned multi-scale cellular processes.​​ We demonstrated how population​​​‌ distribution resulting from intracellular‌ regulatory networks driving cell-state‌​‌ transition gets impacted by​​ stochastic fluctuations in E-M​​​‌ regulatory biomolecules, differences in‌ growth rates among cell‌​‌ subpopulations, and initial population​​ distribution. Further, we revealed​​​‌ that a linear dependence‌ of the cell growth‌​‌ rate on the population​​ heterogeneity is sufficient to​​​‌ recapitulate the faster in‌ vivo growth of orthotopic‌​‌ injected heterogeneous E-M subclones​​ reported before experimentally. Overall,​​​‌ our model contributes to‌ the combined understanding of‌​‌ intracellular and cell-population levels​​ dynamics in the emergence​​​‌ of E-M heterogeneity in‌ a cell population.

8.3.3‌​‌ Modelling phenotypic divergence in​​ cancer and in the​​​‌ emergence of multicellularity by‌ phenotype-structured equations of cell‌​‌ population dynamics

Participants: Jean​​ Clairambault, Lia Sela​​​‌.

8.3.4​​ Analysis of non-local advection-diffusion​​​‌ models for active particles​

Participants: Luca Alasio.​‌

8.3.5 On a‌ relaxed Cahn-Hilliard tumour growth‌​‌ model with single-well potential​​ and degenerate mobility

Participants:​​​‌ Benoit Perthame.

In‌ 20, we consider‌​‌ a phase-field system modelling​​ solid tumour growth. This​​​‌ system consists of a‌ Cahn-Hilliard equation coupled with‌​‌ a nutrient equation. The​​ former is characterised by​​​‌ a degenerate mobility and‌ a singular potential. Both‌​‌ equations are subject to​​ suitable reaction terms which​​​‌ model proliferation and nutrient‌ consumption. Chemotactic effects are‌​‌ also taken into account.​​ Adding an elliptic regularisation,​​​‌ depending on a relaxation‌ parameter , in the‌​‌ equation for the chemical​​ potential, we prove the​​​‌ existence of a weak‌ solution to an initial‌​‌ and boundary value problem​​ for the relaxed system.​​​‌ Then, we let go‌ to zero, and we‌​‌ recover the existence of​​ a weak solution to​​​‌ the original system.

8.3.6‌ Mathematical analysis of macroscopic‌​‌ models for neurons

Participants:​​ Benoit Perthame.

This​​​‌ section summarizes the recent‌ results obtained in the‌​‌ analysis of various macroscopic​​ models for neuronal dynamics.​​​‌

8.4.1 Biological control of​​​‌ vectors

Participants: Pierre-Alexandre Bliman​, Manon de la​‌ Tousche, Morgane Doukhan​​.

8.4.2 Control of infectious​ diseases

Participants: Pierre-Alexandre Bliman​‌, Marcel Fang,​​ Bernard Cazelles.

8.5.1 Modeling​​​‌ of milling and schooling‌ in gregarious fish

Participants:‌​‌ Nastassia Pouradier Duteil,​​ Sam Gaborieau.

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.‌ Together with Benjamin Thiria‌​‌ and Laurent Boudin, we​​ are currently co-supervising a​​​‌ PhD student, Sam Gaborieau,‌ who won an INLIFE‌​‌ fellowship (Initiative sciences aux​​ interfaces du vivent) to​​​‌ study numerically and experimentally‌ the phase transition in‌​‌ fish collective behavior.

8.5.2​​ Modeling of biological tissue​​​‌ emergence and repair

Participants:‌ Diane Peurichard, Sophie‌​‌ Hecht, Pierre-Alexandre Grott​​.

This subsection presents​​​‌ our previous and current‌ activities towards biological tissue‌​‌ modeling using an agent-based​​ formalism. This section is​​​‌ based on a long‌ standing collaboration with the‌​‌ RESTORE laboratory (biological lab​​ in Toulouse), with whom​​​‌ we developed a 2D‌ computational agent based-model (ABM,‌​‌ cf 150), enabling​​ to recapitulate the emergence​​​‌ of complex adipose tissue‌ architectures in 2D via‌​‌ simple physical principles between​​ their core components (cells​​​‌ and fibers). When extended‌ to account for the‌​‌ mechanisms of repair after​​ injury 149, 142​​​‌ (combined in-silico/in-vivo study at‌ the core of the‌​‌ PhD of A. Pacary,​​ 2021-2024, defended december 2024),​​​‌ the model enabled to‌ suggest a new in-vivo‌​‌ validated therapeutic target (ECM​​ cross-linking) and a temporal​​​‌ window of modulation of‌ this parameter to induce‌​‌ regeneration in adult mammals​​ adipose tissues. These studies​​​‌ opened new therapeutic approaches‌ targeting ECM cross-linking to‌​‌ induce tissue regeneration in​​ adult mammals, and positioned​​​‌ our computational model as‌ a solid candidate for‌​‌ digital twinning. We hereby​​ present the current works​​​‌ developed with the RESTORE‌ laboratory around this modeling‌​‌ framework.

8.5.3 Modelling‌​‌ the Retinal Pigment Epithelium​​ in Age-Related Macular Degeneration​​​‌

Participants: Luca Alasio,‌ Sophie Hecht, Diane‌​‌ Peurichard, Naoufel Cresson​​, Clara Choukroun.​​​‌

8.5.4​ Mathematical models of retinal​‌ biochemistry

Participants: Luca Alasio​​.

8.5.5​​​‌ Modelling bacterial micro-colony growth​

Participants: Sophie Hecht,​‌ Diane Peurichard.

This​​ project is a collaboration​​​‌ between N Desprat (ABCD​ biophysics Lab - ENS),​‌ S. Hecht, D. Peurichard​​ and the post-doctorate H.​​​‌ Horii (end of contract​ October 2025) funded by​‌ IMPT. We developed an​​ individual-based model where each​​​‌ bacterium is modeled by​ a string of spheres​‌ linked with linear and​​ angular springs. This description​​​‌ allows local bending of​ the individual bacteria. Numerical​‌ simulations have showed that​​ the curvature of the​​​‌ bacteria modify the global​ organisations of the micro-colony​‌ and reproduce the dense​​ organisation we observe experimentally.​​ A characteristic we were​​​‌ focused on, since it‌ controls the capacity of‌​‌ the colony to create​​ a barrier with its​​​‌ environment. The next part‌ of the project is‌​‌ a rigorous comparaison with​​ experimental data on different​​​‌ strains of bacteria.

8.5.6‌ Modelling ovocyte deformation

Participants:‌​‌ Sophie Hecht, Diane​​ Peurichard.

The CIRB​​​‌ lab (Centre interdisciplinaire de‌ recherche en biologie) of‌​‌ the College de France​​ is working on developing​​​‌ a micro-constriction setup to‌ facilitate ovocyte selection for‌​‌ FIV. Together with D.​​ Peurichard, S. Hecht, and​​​‌ L. Barbier (CIRB) we‌ are developing a model‌​‌ to reproduce the deformation​​ of mice ovocyte in​​​‌ AFM and micro-constriction experiments.‌ The objective is to‌​‌ optimise the transition of​​ the experimental setup to​​​‌ human ovocyte study.

9.1.1 STIC/MATH/CLIMAT​​ AmSud projects

9.2.1 Visits​​ to international teams

10.1.1​‌ Journal

Philippe Robert is​​ an associate editor of​​​‌ the journal "Stochastic Models".​

10.1.2 Invited talks

Pierre-Alexandre​‌ Bliman presented contributions at​​ the conferences Biomath (Sofia,​​​‌ Bulgaria, June) and European​ Control Conference (Thessaloniki, Greece,​‌ July). He also presented​​ seminars at COPPE, Universidade​​​‌ Federal de Rio de​ Janeiro (November)and GT Contrôle​‌ at Laboratoire Jacques-Louis Lions​​ (December).

Jean Clairambault was​​​‌ invited to give a​ talk at the conference​‌ "Second Workshop on Multiscale​​ and Nonlocal Problems in​​​‌ PDEs" on June 19​ and 20, 2025 in​‌ Bari at the Politecnico​​ to celebrate the conclusion​​​‌ of the PRIN 2022​ "Evolution problem involving interacting​‌ scales". He was also​​ invited to give a​​​‌ virtual seminar at the​ "Online conference on Mathematical​‌ modelling in biology and​​ medicine, May 19-23, 2025"​​​‌ organised by Vitaly Volpert,​ and to give a​‌ talk at the "Premières​​ journées de la biologie​​​‌ théorique" (J-BIOT 2025), Nov.​ 24-25, 2025, Grenoble.

Nastassia​‌ Pouradier Duteil was invited​​ to present at the​​​‌ Gazteak Spanish Conference of​ Young Researchers in Mathematics​‌ (Bilbao, Spain), at the​​ workshop “Cabo de Gata​​​‌ PDE days”, (Almeria, Spain),​ at the INdAM workshop​‌ “Differential equations and nonlinear​​ models”, (Rome, Italy), and​​​‌ at the conference “Population​ dynamics: model design, optimization​‌ & control” (Nice, France).​​ She was also invited​​​‌ to give a seminar​ at the Partial Differential​‌ Equations Seminar, University of​​ Granada (Spain).

Diane Peurichard​​​‌ was invited to the​ Workshop for Young Women​‌ in Math Biology (Bonn,​​ germany), the conférence 'Round​​​‌ Meanfield IV: N-body sul​ Canal Grande' (Venise, Italy),​‌ the Theoretical Biology days​​ (JBIOT Grenoble, France), and​​​‌ working seminars in France​ (Nice, Institut Biomécanique, Paris,​‌ Math-bio days, Toulouse)

Luca​​ Alasio gave a presentation​​​‌ in the following events:​ Workshop on Multiscale modeling​‌ of ocular and cardiovascular​​ systems, September 29 to​​ October 3, 2025, at​​​‌ the American Institute of‌ Mathematics, Pasadena, California. Workshop‌​‌ Mathematical Biology: Applications and​​ Analysis, at the Faculty​​​‌ of Mathematics, Informatics and‌ Mechanics of the University‌​‌ of Warsaw, from July​​ 28 to July 31,​​​‌ 2025. Mathematical Biology and‌ Ecology Seminar, Mathematuical Institute,‌​‌ Oxford, June 06 2025.​​ Workshop Recent advances in​​​‌ mathematical modelling for medicine‌ and biology at Laboratoire‌​‌ des Mathematiques Raphael Salem​​ in Rouen, 22nd to​​​‌ 24th of January 2025.‌

Sophie Hecht was invited‌​‌ to the workshop Mathematical​​ Biology: Analysis and Application​​​‌ (Warsaw, Poland), Round Meanfield‌ IV: N-body sul Canal‌​‌ Grande (Venise, Italy) and​​ to the CIMPA summer​​​‌ school Mathematical models in‌ biology and related applications‌​‌ of partial differential equations​​ (La Havana, Cuba)

Philippe​​​‌ Robert has been invited‌ at the University of‌​‌ Casablanca (Morocco) From April​​ 27 to April 29,​​​‌ at the “Stochastic Reaction‌ Networks Workshop” (Politecnico di‌​‌ Torino) from June 16​​ to June 18, and​​​‌ at the University of‌ Berne from November 4‌​‌ to November 7. He​​ gave a summer school​​​‌ at Torgnon (Italy) on‌ stochastic chemical reaction networks‌​‌ from June 9 to​​ June 14.

10.1.3 Scientific​​​‌ expertise

Pierre-Alexandre Bliman has‌ reviewed proposals submitted to‌​‌ the Belgian Fonds national​​ de la recherche scientifique​​​‌ (FNRS).

Diane Peurichard is‌ member of the Commission‌​‌ d’évaluation (CE) Inria, of​​ the Commission des emplois​​​‌ scientifiques (CES) Inria Paris,‌ and of the Comité‌​‌ de suivi doctoral (CSD)​​ Inria Paris.

10.1.4 Research​​​‌ administration

Diane Peurichard is‌ coordinator of the ANR‌​‌ project ENERGENCE. She is​​ also member of the​​​‌ Pôle écoute at LJLL,‌ Sorbonne Université.

Nastassia Pouradier‌​‌ Duteil is coordinator of​​ the ANR project FISH.​​​‌

Luca Alasio is coordinator‌ of the Action Exploratoire‌​‌ RADIOS.

Pierre-Alexandre Bliman is​​ coordinator of the ANR​​​‌ project NOCIME and of‌ the STIC AMSUd project‌​‌ BIO-CIVIP. He is also​​ member of the Pôle​​​‌ écoute at LJLL, Sorbonne‌ Université.

10.2.1 Teaching

Luca Alasio‌ and Naoufel Cresson gave‌​‌ the course Méthodes Variationnelles​​ et EDP (Introduction to​​​‌ PDEs and finite differences)‌ in the context of‌​‌ M1 Mathématiques, Modélisation, Apprentissage,​​ Université Paris Cité (MAP5).​​​‌

Diane Peurichard made a‌ 4h intervention in the‌​‌ M2 program CARe, Toulouse,​​ entitled 'Mathematical modeling of​​​‌ biological systems', aimed at‌ promoting mathematical modeling and‌​‌ simulation to biology students.​​

Diane Peurichard animated a​​​‌ 4h M2 course in‌ the Cell physics M2‌​‌ program at Université de​​ Strasbourg, entitled 'Mathematical modeling​​​‌ of biological systems' aimed‌ at forming physics students‌​‌ to mathematical modelling for​​ biology.

Manon de la​​​‌ Tousche has been teaching‌ assistant in Licence at‌​‌ Sorbonne Université.

Philippe Robert​​ is teaching the master​​​‌ M2 course `Modèles Stochastiques‌ de la Biologie Moléculaire`”‌​‌ at Sorbonne Université.

10.2.2​​ Supervision

Pierre-Alexandre Bliman is​​​‌ PhD co-supervisor of Manon‌ De La Tousche and‌​‌ Morgane Doukhan , together​​ with Yves Dumont (CIRAD).​​​‌

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).

Jean Clairambault​‌ is currently co-supervising with​​ Emmanuel Trélat (CAGE Inria​​​‌ team) at LJLL and​ Jean-Philippe Foy at CRSA,​‌ Saint-Antoine Hospital, the PhD​​ thesis of Lia Sela,​​​‌ funded since October 2024​ by a SU PDIC​‌ (Programme Doctoral Interdisciplinaire en​​ Cancérologie) grant.

Nastassia Pouradier​​​‌ Duteil co-supervised the M2​ internship of Angelina Jammart​‌ , and is currently​​ co-supervising her PhD thesis​​​‌ (funded by the ANR​ JCJC “FISH” obtained in​‌ 2024), together with Mario​​ Sigalotti (CAGE Inria team)​​​‌ and Benoît Bonnet-Weill. She​ also co-supervised the M2​‌ internship and is currently​​ co-supervising the PhD thesis​​​‌ of Sam Gaborieau, in​ collaboration with Laurent Boudin​‌ (LJLL, Sorbonne University) and​​ Benjamin Thiria (PMMH, ESPCI).​​​‌

Diane Peurichard supervised the​ post-doctorate of Suney Toste​‌ (2024-2025, ANR ENERGENCE), the​​ post-doctorate of H. Horii​​​‌ (2024-2025, co-supervised with Sophie​ Hecht ), and co-supervised​‌ with Sophie Hecht the​​ M2 internship of V.​​​‌ Brulard (summer 2025). Diane​ Peurichard is currently co-supervising​‌ the PhD of P-A​​ Grott (2025-2028 with J.​​​‌ Paupert, RESTORE), the PhD​ of N. Martinez Tomas​‌ (2025-2028 with Sophie Hecht​​ and A. Trescases (IMT​​​‌ Toulouse)), the post-doctorate of​ Louis Fostier (feb 2026-​‌ feb 2027, ANR ENERGENCE)​​ and will co-supervise the​​​‌ M2 internship of Aurélien​ Astesiano (starting may 2026,​‌ co-supervised with A. Manhart,​​ Vienna Austria).

Luca Alasio​​​‌ is supervising the PhD​ project of Naoufel Cresson,​‌ jointly with Prof. Marcela​​ Szopos (Université Paris Cité)​​​‌ and Prof. Michel Paques​ (Hôpital National des Quinze-Vingts​‌ and Sorbonne Université). Luca​​ Alasio is supervising an​​​‌ M2 project of the​ student Eloi Bedek in​‌ the programme Mathématiques pour​​ les sciences du vivant,​​​‌ at Université Paris Saclay.​

10.2.3 Juries

Nastassia Pouradier​‌ Duteil was a jury​​ member for the thesis​​​‌ of Carlos Pulido, who​ defended in January 2025​‌ at the University of​​ Granada. She was also​​​‌ a reviewer for the​ thesis of Alexis Bejar,​‌ who will defend his​​ thesis in March 2026​​​‌ at the University of​ Granada.

Diane Peurichard (member​‌ of the CE) was​​ member of the instruction​​​‌ jury for the team​ BIOTIC, and was member​‌ of the following selection​​ committees for the 2025​​​‌ campaign:

• Inria Nice CRCN/ISFP​ 2025 campaign
• Inria Saclay​‌ CRCN/ISFP 2025 campaign
• Inria​​ Grenoble CRCN/ISFP 2025 campaign​​​‌
• Maitre de conférences position​ 2025 at Université de​‌ Bordeaux

Diane Peurichard (member​​ of the commission des​​​‌ emplois scientifiques) was member​ of the jury for​‌ the delegations Inria.

Diane​​ Peurichard was member of​​​‌ the jury for the​ FSMP junior price Maryam​‌ Mirzakhani (link),​​ awarding three womens (in​​​‌ their final year of​ a bachelor's or master's​‌ degree) for their first​​ research work or bibliographical​​​‌ study in mathematics.

Sophie​ Hecht was member of​‌ the selection committee for​​ MCF positions in Brest​​​‌ Université and Université Paris​ Nord.

Pierre-Alexandre Bliman has​‌ been a reviewer of​​ the PhD manuscript `Mathematics​​​‌ and Infectious Diseases: Analysis​ and Application of Compartmental​‌ Models' by Alicja Kubik​​ (Universidad Complutense de Madrid,​​​‌ Facultad de Ciencias Matemáticas,​ March 2025). He was​‌ also reviewer of the​​ PhD manuscript `Modélisation de​​ la Technique de l’Insecte​​​‌ Stérile dans un contexte‌ agricole : examen des‌​‌ facteurs biologiques et techniques​​ susceptibles d’atténuer son efficacité'​​​‌ by Marine Courtois (Université‌ Côte d'Azur, November 2025)‌​‌ and member of the​​ board.

Philippe Robert was​​​‌ a member of the‌ jury of the HDR‌​‌ of Hanène Mohamed (Université​​ de Nanterre) on May​​​‌ 22.

10.2.4 Educational and‌ pedagogical outreach

Sophie Hecht‌​‌ and Diane Peurichard were​​ invited professors at the​​​‌ CIMPA school 'Mathematical models‌ in biology and related‌​‌ applications of partial differential​​ equations', La Havana, Cuba,​​​‌ taking place from 9‌ to 20 june 2025‌​‌ (cf link). In​​ this event, Sophie Hecht​​​‌ presented a 6h master‌ course entitled 'Singular limits‌​‌ arising in mechanical models​​ of tissue growth' and​​​‌ Diane Peurichard did a‌ 6hours Master course entitled‌​‌ 'Simulation and numerical treatment​​ of PDEs in Mathematical​​​‌ Biology'.

10.3.1‌ Productions (articles, videos, podcasts,‌​‌ serious games, ...)

Nastassia​​ Pouradier Duteil is a​​​‌ recurrent host in Nathalie‌ Ayi's podcast “Tête-à-tête chercheuse(s)”,‌​‌ seeking to promote a​​ diversified image of mathematical​​​‌ researchers.

International journals

• 2 article‌​‌L.Luca Alasio and​​ S.Simon Schulz.​​​‌ Regularity and uniqueness for‌ a model of active‌​‌ particles with angle-averaged diffusions​​.Nonlinear Differential Equations​​​‌ and ApplicationsJanuary 2025‌HALDOI
• 3 article‌​‌A.Amit Bhaya and​​ P.-A.Pierre-Alexandre Bliman.​​​‌ Feedback design for biological‌ control by the sterile‌​‌ insect release technique exploiting​​ monotone system theory.​​​‌European Journal of Control‌86July 2025,‌​‌ 101292In press. HAL​​DOIback to text​​​‌
• 4 articleP.-A.Pierre-Alexandre‌ Bliman, A.Anas‌​‌ Bouali, P.Patrice​​ Loisel, A.Alain​​​‌ Rapaport and A.Arnaud‌ Virelizier. On the‌​‌ problem of minimizing the​​ epidemic final size for​​​‌ SIR model by social‌ distancing.Mathematical Biosciences‌​‌ and Engineering233​​January 2026, 567–593​​​‌HALDOIback to‌ text
• 5 articleP.-A.‌​‌Pierre-Alexandre Bliman and M.​​Marcel Fang. Reinfection​​​‌ induced multistability in an‌ epidemic model.Nonlinear‌​‌ ScienceDecember 2025HAL​​back to text
• 6​​​‌ 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 Modelling144​​​‌March 2025, 116047‌HALDOIback to‌​‌ text
• 7 articleM.​​Marie Doumic, S.​​​‌Sophie Hecht, M.‌Marc Hoffmann and D.‌​‌Diane Peurichard. Scaling​​ limits for a population​​​‌ model with growth, division‌ and cross-diffusion.Mathematical‌​‌ Models and Methods in​​ Applied Sciences3512​​​‌November 2025, 2611-2660‌HALDOIback to‌​‌ textback to text​​
• 8 articleG.Gissell​​​‌ Estrada-Rodriguez, D.Diane‌ Peurichard and X.Xinran‌​‌ Ruan. From a​​​‌ nonlinear kinetic equation to​ a volume-exclusion chemotaxis model​‌ via asymptotic preserving methods​​.Kinetic and Related​​​‌ Models 186May​ 2025, 989-1015HAL​‌DOIback to text​​
• 9 articleJ.Jules​​​‌ Guilberteau, P.Paras​ Jain, M. K.​‌Mohit Kumar Jolly,​​ C.Camille Pouchol and​​​‌ N.Nastassia Pouradier Duteil​. An integrative phenotype-structured​‌ partial differential equation model​​ for the population dynamics​​​‌ of epithelial-mesenchymal transition.​npj Systems Biology and​‌ Applications111March​​ 2025, 24HAL​​​‌DOIback to text​
• 10 articleM.Maxime​‌ Herda, P.Pierre​​ Monmarché and B.Benoît​​​‌ Perthame. Wasserstein contraction​ for the stochastic Morris-Lecar​‌ neuron model.Kinetic​​ and Related Models 18​​​‌1February 2025,​ 1-18HALDOIback​‌ to text
• 11 article​​B.Benoît Perthame,​​​‌ C.Clément Rieutord and​ D.Delphine Salort.​‌ Strongly nonlinear age structured​​ equation, time-elapsed model and​​​‌ large delays.Journal​ of Mathematical Biology91​‌5March 2025,​​ 65HALDOIback​​​‌ to text

Conferences without​ proceedings

• 12 inproceedingsP.-A.​‌Pierre-Alexandre Bliman. Basic​​ offspring number and robust​​​‌ feedback design for the​ biological control of vectors​‌ by sterile insect release​​ technique.23rd European​​​‌ Control ConferenceThessaloniki, Greece​2025HALback to​‌ text
• 13 inproceedingsP.-A.​​Pierre-Alexandre Bliman and M.​​​‌Marcel Fang. Reinfection​ induced multistability in an​‌ epidemic model.Biomath​​ 2025 - international conference​​​‌ on Mathematical Methods and​ Models in BiosciencesSofia,​‌ BulgariaJune 2025HAL​​
• 14 inproceedingsP.-A.Pierre-Alexandre​​​‌ Bliman, M.Manon​ de la Tousche and​‌ Y.Yves Dumont.​​ Feasibility and optimisation of​​​‌ fly elimination by adult​ mass trapping and larval​‌ treatment : a stage-structured​​ metapopulation approach.BIOMATH2025​​​‌ - International Conference on​ Mathematical Methods and Models​‌ in BiosciencesSofia, Bulgaria​​June 2025HAL

Scientific​​​‌ book chapters

• 15 inbook​Q.Qingyou He,​‌ H.-L.Hai-Liang Li and​​ B.Benoît Perthame.​​​‌ Incompressible limit of porous​ media equation with chemotaxis​‌ and growth.18​​Partial Differential Equations: Waves,​​​‌ Nonlinearities and NonlocalitiesAbel​ SymposiaSpringer Nature Switzerland​‌August 2025, 111-128​​HALDOIback to​​​‌ text

Reports & preprints​

• 16 reportS.Soraya​‌ Arias, M.Michel​​ Bergmann, F.Fabien​​​‌ Campillo, M.-A.Marie-Agnès​ Enard, C.Christian​‌ Fabre, F.Frédérick​​ Garcia, B.Benjamin​​​‌ Guedj, E.Emmanuel​ Jeannot, G.Giovanni​‌ Neglia, D.Diane​​ Peurichard, D.Daniel​​​‌ Racoceanu, B.Benoît​ Sagot and G.Gaëlle​‌ Tworkowski. Reflections on​​ the Use of Generative​​​‌ AI for Research Professions​.InriaJuly 2025​‌HAL
• 17 reportS.​​Soraya Arias, M.​​​‌Michel Bergmann, F.​Fabien Campillo, M.-A.​‌Marie-Agnès Enard, C.​​Christian Fabre, F.​​​‌Frédérick Garcia, B.​Benjamin Guedj, E.​‌Emmanuel Jeannot, G.​​Giovanni Neglia, D.​​​‌Diane Peurichard, D.​Daniel Racoceanu, B.​‌Benoît Sagot and G.​​Gaëlle Tworkowski. Réflexions​​​‌ sur l'usage de l'IA​ générative pour les métiers​‌ de la recherche.​​Inria2025, 1-10​​HAL
• 18 miscP.-A.​​​‌Pierre-Alexandre Bliman, M.‌Manon de la Tousche‌​‌ and Y.Yves Dumont​​. Sterile Insect Technique​​​‌ in a n-patch system‌ with Allee effect and‌​‌ mass trapping: modeling, analysis​​ and simulations.December​​​‌ 2025HALback to‌ text
• 19 miscB.‌​‌Benoît Bonnet-Weill and N.​​Nastassia Pouradier Duteil.​​​‌ Structured Continuity Equations in‌ Fibred Wasserstein Spaces.‌​‌November 2025HALback​​ to text
• 20 misc​​​‌C.Cecilia Cavaterra,‌ M.Matteo Fornoni,‌​‌ M.Maurizio Grasselli and​​ B.Benoît Perthame.​​​‌ On a relaxed Cahn-Hilliard‌ tumour growth model with‌​‌ single-well potential and degenerate​​ mobility.December 2025​​​‌HALback to text‌
• 21 miscJ.Jean‌​‌ Clairambault and L.Lia​​ Sela. The animal​​​‌ body plan revisited in‌ light of the failure‌​‌ of local tissue cohesion​​ in cancer.November​​​‌ 2025HALback to‌ text
• 22 miscV.‌​‌Vincent Fromion, P.​​Philippe Robert and J.​​​‌Jana Zaherddine. Stochastic‌ Models of Resource Allocation‌​‌ in Chemical Reaction Networks​​.November 2025HAL​​​‌back to text
• 23‌ miscB.Benoît Perthame‌​‌, C.Clément Rieutord​​ and D.Delphine Salort​​​‌. A Fokker-Planck equation‌ with superlinear drift at‌​‌ infinity for Integrate-and-Fire model​​.January 2026HAL​​​‌back to text
• 24‌ miscB.Benoît Perthame‌​‌, F.Francesco Salvarani​​ and S.Shugo Yasuda​​​‌. Multiscale analysis of‌ a kinetic equation for‌​‌ mechanotaxis.January 2026​​HALback to text​​​‌
• 25 miscB.Benoît‌ Perthame and M.Min‌​‌ Tang. Deriving sub-diffusion​​ equations.July 2025​​​‌HALback to text‌