2023Activity reportProject-TeamMAMBA

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 145. The limiting equation, so-called “graphon equation” now describes the evolution of the particle's positions $x(t,s)$ as a function of time $t$ and of the “continuous index” $s\in I$ (representing the particle's individual identities, in an infinite population):

In 73, we extended this idea to a collective dynamics model with time-varying weights, adopting the graph point of view described above. We showed that this approach is more general than the mean-field one, and the Graph Limit can be derived for a much greater variety of models.

Our work will involve deriving graph limits for systems of particles that can be structured along a trait that characterizes their interactions, such as volume, mass or phenotype. Among the open problems that we aim to address in collaboration with Nathalie Ayi (LJLL, Sorbonne University), one of them concerns the graph limit for multi-agent systems evolving on weighted random graphs. More specifically, we will consider that the interactions between agents are given by ${\varphi }_{ij}(x_j\left(t\right)-x_i\left(t\right)):={\xi }_{ij}\varphi (x_j\left(t\right)-x_i\left(t\right))$, where ${\left({\xi }_{ij}\right)}_{i,j}$ are random variables whose laws are probability distributions on ${ℝ}_{+}$ that depend on the indices$i,j$. Graphs with random topologies are often used to model systems such as neuronal networks, coupled lasers and communication or power networks. In 145, 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 106 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 108. In this way, the total mass of the population is conserved throughout the limit process, and everything can be done in the framework of probability measures. The limit PDE is typically a non-linear transport equation of the type

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

This is a major modeling limitation, and resolving it is crucial. Several works have highlighted a discrepancy between the microscopic and continuum modeling approaches. For instance, in the context of emergency crowd evacuation, microscopic models are able to reproduce the well-known effect of arch formation in front of exits, resulting in congestion and dramatic slow-down of the crowd's evacuation 143. This effect still eludes all natural continuum limits. Another example can be found in the modeling of cell division: microscopic models capture the fact that the cell population is naturally pushed outwards at the birth of a new daughter cell because of its added volume. This effect is lost in continuum models, as there is no concept of individual size.

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

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

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

3.1.3 Scaling limits

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

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

3.2.1 Regulation Mechanisms of Gene Expression

MUSCLEES permanent members involved: Philippe Robert

The central dogma of molecular biology states that the genetic information flows only in one way, from DNA to RNAs, and to proteins. The production of proteins is a central process of biological cells. It can be described as a two-step process. In the first step, macro-molecules polymerases produce RNAs with genes of the DNA. This is the transcription step. The second step is the production of proteins itself from mRNAs, messenger RNAs, a subset of RNAs, with macro-molecules ribosomes. This is the translation step. An additional feature of this process is that it is consuming an important fraction of energy resources of the cell, to build chains of amino-acids or chains of nucleotides in particular. See  77, 161, 168.

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

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

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

3.2.2 Stochastic Chemical Reaction Networks

MUSCLEES permanent members involved: Philippe Robert

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

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

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

The ODE in this case is a quadratic functional of the state vector. In a deterministic context, the famous results by Horn, Johnson and Feinberg give, for some specific topologies, a satisfactory description of the stable states of these networks. See  119 for example. It turns that this description is suitable for systems for which the orders of magnitude of the different chemical species are comparable and that the stochastic components merely vanish. These assumptions are nevertheless not true in some biological settings, when, for example, reactions are driven by a small number of enzymes but with a large reaction rate.

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

Up to now there are few results in such a random context. The reference  64 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  101 and in   103, 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  92. A more general and extended recent state of the art on phenotype-structured population dynamics is reported in  139.

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

3.3.2 Around graphon dynamics

MUSCLEES permanent members involved: Nastassia Pouradier Duteil

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

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

MUSCLEES permanent members involved: Luca Alasio

Systems of self-propelled interacting particles provide an individual-based description of the motion of agents ranging from bacteria to colloidal surfers 159, 180. 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 83. This work is concerned with the analytical study of some of the above-mentioned PDE models, focusing on regularity and convergence to stationary states. The simplest example is given by the following non-local advection-diffusion equation:

where $\rho (t,x)={\int }_0^{2\pi }f(t,x,\theta )\mathrm{d}\theta$ is the angle-independent density and $𝐞\left(\theta \right)=(cos\theta ,sin\theta )$, with periodic boundary conditions both in the space variable $x\in {(0,2\pi )}^2$ and the angle variable $\theta \in (0,2\pi )$. The constant parameters $\mathrm{Pe}\in ℝ$ and $D_e>0$ are called the Péclet number and spatial diffusion coefficient , respectively. Further details and a preliminary existence theory can be found in 84. In collaboration with Simon Schulz (SNS Pisa) and Jessica Guerand (U. Montpellier), we are working on 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 83 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 176, tumour growth models 95, and multi-species agent-based models 57. 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) 128, 87. 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 will 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 111, 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 111, 114.

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 127, 146. 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 189. 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 90, 78. These models considered the transition between the different phases of the cell cycle depending of the cell age. However, recent works 125 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 125 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 96, alignment 184, or consensus 126) 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 111, 114. 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 174. 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 183.

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  166, or leukaemic cells and supporting somatic cells  157 for instance), or cells can be in competition (in particular tumour-immune interactions  7, 138). 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  167)? 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  63, 91, 93, 135, 175, 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  170, 171, 169, 185, a new, general, mathematical framework to study the phenomenon of synaptic plasticity associated to STDP rules has been introduced and analyzed for a system composed of two neuronal cells connected by a single synapse whose weight is time-varying.

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

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

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

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

   The main assumption is that the timescale of the time evolution of the synaptic weights is slow. This is the framework of 169. 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 51. 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 133. 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 97). In mathematical terms, this leads to strongly nonlinear degenerate parabolic Cahn-Hilliard (PDE) equations  163. 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  98, 187, 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   152. 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  63, 91, 93, 135, 175, 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.

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 Graph Limit for Interacting Particle Systems on Weighted Random Graphs

In 28, we studied the large-population limit of interacting particle systems posed on weighted random graphs. In that aim, we introduced a general framework for the construction of weighted random graphs, generalizing the concept of graphons. We proved quantitative convergence results in probability, as the number of particles tends to infinity, of the finite-dimensional system towards the solution of a deterministic graph-limit equation. In this limit equation, the graphon prescribing the interaction is given by the first moment of the weighted random graph law. We also studied interacting particle systems posed on switching weighted random graphs, which are obtained by resetting the weighted random graph at regular time intervals. We revealed the interplay between the large-population limit and the switching time. In particular, we show that for a fixed switching time, these systems converge as the number of particles tends to infinity to the same graph-limit equation, in which the interaction is prescribed by the constant-in-time first moment of the weighted random graph law.

7.1.2 Localization limit for multispecies cross-diffusions models

In 34, we studied the localization limit of models describing the long-range interaction between individuals, of particular interest for living systems. The starting point models are quadratic, written under the form of transport equations with a nonlocal self-generated drift. We established the localisation limit, that is the convergence of nonlocal to local systems, when the range of interaction tends to 0. These theoretical results are sustained by numerical simulations. The major new feature in our analysis is that we do not need diffusion to gain compactness, at odd with the existing literature. The central compactness result is provided by a full rank assumption on the interaction kernels. In turn, we prove existence of weak solutions for the resulting system, a cross-diffusion system of quadratic type.

7.1.3 Asymptotic preserving schemes for nonlinear kinetic equations

In 40, we studied the diffusion limit of nonlinear kinetic equations modeling the pattern formation in bacterial populations moving by a chemotaxis process. We first proved, by formal arguments, that the diffusion limit of these equations, when both the transport term and the turning operator are density-dependent, lead to volume-exclusion chemotactic equations. We generalise an asymptotic preserving scheme for such nonlinear kinetic equations based on a micromacro decomposition. By properly discretizing the nonlinear term implicitly-explicitly in an upwind manner, the scheme produces accurate approximations also in the case of strong chemosensitivity. We showed, via detailed calculations, that the scheme presents the following properties: asymptotic preserving, positivity preserving and energy dissipation, which are essential for practical applications. We extend this scheme to two dimensional kinetic models and we validate its efficiency by means of 1D and 2D numerical experiments of pattern formation in biological systems.

7.1.4 Derivation of the degenerate Cahn-Hilliard equation

The degenerate Cahn-Hilliard equation can be derived from interacting particle systems with short and long range interactions. These scales lead to consider non-local versions of the equation as well as the local limit, 16. The degenerate Cahn-Hilliard equation describes surface tension between two types of cells and this produces a pressure jump at the interface in the compressible limit which is computed in 15.

7.2.1 Hawkes Processes

Participants: Philippe Robert, Gaëtan Vignoud.

In 19 a general study of Hawkes processes on $ℝ$ has been achieved. A Hawkes process is a point process whose intensity function at time $t$ is a functional of its past activity before time $t$. It is defined by its activation function $\Phi$ and its memory function $h$. In this work, the Hawkes property is expressed as an operator on the sub-space of non-negative sequences associated to distances between its points. By using the classical correspondence between a stationary point process and its Palm measure, we establish a characterization of the corresponding Palm measure as an invariant distribution of a Markovian kernel. We prove that if $\Phi$ is continuous and its growth rate is at most linear with a rate below some constant, then there exists a stationary Hawkes point process. The classical Lipschitz condition of the literature for an unbounded function $\Phi$ is relaxed. Our proofs rely on a combination of coupling methods, monotonicity properties of linear Hawkes processes and classical results on Palm distributions. An investigation of the Hawkes process starting from the null measure, the empty state, on ${ℝ}_{-}$ plays also an important role. The linear case of Hawkes and Oakes is revisited at this occasion.

If the memory function $h$ is an exponential function, under a weak condition it is shown that there exists a unique stationary Hawkes point process. In this case, its Palm measure is expressed in terms of the invariant distribution of a one-dimensional Harris ergodic Markov chain. When the activation function is a polynomial $\Phi$ with degree $>1$, there does not exist a stationary Hawkes process and if the Hawkes process starts from the empty state, a scaling result for the accumulation of its points is obtained.

7.2.2 Stochastic Chemical Networks

Participants: Lucie Laurence, Philippe Robert.

The general goal of this PhD work, started in September 2020, is of developing a scaling approach to analyze stochastic models of chemical networks.

7.2.3 Allocation of Resources in Prokaryotic Cells

Participants: Vincent Fromion, Philippe Robert, Jana Zaherddine.

The information flow from DNA genes to proteins is a fundamental process that is common to all living organisms. We analyze it in the context of a bacterial cell. The production of proteins is the most important process that takes place inside a bacterium consuming around 90% of its resources.

Bacterium adapts very quickly to changes in the environment. It succeeds in making a balance between supply allocating the resources available in the environment for its growth. All of this is achieved via regulatory mechanisms in the bacterium.

7.3.1 Applications to cancer-immune interactions and to evolutionary biology in multicellularity and in cancer.

Mentioned in the Section 'Adaptive phenotype-structured cell population dynamics' of the research program, the PhD theses of Zineb Kaid, on interactions of a tumour cell population with populations of immune cells, NK- and T-lymphocytes, and of Frank Ernesto Alvarez Borges, on phenotype divergence in early development of multicellular animals and in tumours, have been defended, respectively at Tlemcen University (Algeria) in October, and at Paris-Dauphine University, in December, both under the mentoring of Jean Clairambault. It is expected that the collaborative research work with the latter will be pursued, developing more in-depth the idea, proposed in 27, that specialisation and cooperation between cell lines, and later between tissues, is the result of an optimisation process that leads to division of work in the development of organised multicellularity in animals, a process which is partially, nevertheless in an instable and reversible way, conserved in tumours, in which cooperation has indeed been evidenced between subpopulations of cells.

7.3.2 Long-time behaviour of an advection-selection equation

In 43 we studied the long-time behaviour of the advection-selection equation

with $\rho \left(t\right)={\int }_{{ℝ}^d}n(t,x)dx$ and with an initial condition $n(0,·)=n^0$. In the field of adaptive dynamics, this equation typically describes the evolution of a phenotype-structured population over time. In this case, $x\mapsto n(t,x)$ represents the density of the population characterised by a phenotypic trait $x$, the advection term `$\nabla ·\left(f\left(x\right)n(t,x)\right)$' a cell differentiation phenomenon driving the individuals toward specific regions, and the selection term `$\left(r\left(x\right)-\rho \left(t\right)\right)n(t,x)$' the growth of the population, which is of logistic type through the total population size $\rho \left(t\right)={\int }_{{ℝ}^d}n(t,x)dx$.

In the one-dimensional case $x\in ℝ$, we prove that the solution to this equation can either converge to a weighted Dirac mass or to a function in $L^1$. Depending on the parameters $n^0$, $f$ and $r$, we determine which of these two regimes of convergence occurs, and we specify the weight and the point where the Dirac mass is supported, or the expression of the $L^1$-function which is reached. This work was part of the PhD thesis of Jules Guilberteau, co-directed with C. Pouchol.

7.3.3 Infinite times renewal equation

In neuroscience, the time elapsed since the last discharge has been used to predict the probability of the next discharge. Such predictions can be improved taking into account the last discharge times. Such multi-time processes arise in many other areas and there is no universal limitation on the number of times to be used. This observation leads us to study the infinite-times renewal equation as a simple model to understand the meaning and properties of such partial differential equations depending on an infinite number of variables. We define notions of solutions, prove existence and uniqueness of solutions and prove the long time convergence, with exponential rate, to the steady state in different, strong or weak, topologies depending on the coefficients. See 32.

Work with Xuan Dou, Chenjiayue Qi, Delphine Salort and Zhennan Zhou.

7.3.4 Wasserstein contraction for the stochastic Morris-Lecar neuron model

Neuron models have attracted a lot of attention recently, both in mathematics and neuroscience. In 45 we study long-time and large-population emerging properties in a simplified toy model. From a mathematical perspective, this amounts to study the long-time behaviour of a degenerate reflected diffusion process. Using coupling arguments, the flow is proven to be a contraction of the Wasserstein distance for long times, which implies the exponential relaxation toward a (non-explicit) unique globally attractive equilibrium distribution. The result is extended to a McKean-Vlasov type non-linear variation of the model, when the mean-field interaction is sufficiently small. The ergodicity of the process results from a combination of deterministic contraction properties and local diffusion, the noise being sufficient to drive the system away from non-contractive domains.

Work with Maxime Herda and Pierre Monmarché.

7.4.1 Biological control of vectors

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

7.4.2 Control of infectious diseases

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

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

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). In 42, we propose 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 demonstrate 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 reveal 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. This work was part of the PhD thesis of Jules Guilberteau, and was conducted in collaboration with Mohit Kumar Jolly and Paras Jain (Indian Institute of Science, Bangalore) and Camille Pouchol (Université Paris Cité).

7.5.2 Mathematical modeling of axolotl spinal cord regeneration

Axolotls are uniquely able to completely regenerate the spinal cord after amputation. The underlying governing mechanisms of this regenerative response have not yet been fully elucidated. We previously found that spinal cord regeneration is mainly driven by cell cycle acceleration of ependymal cells, recruited by a hypothetical signal propagating from the injury. However, the nature of the signal and its propagation remain unknown. In 30, we took a step forward and proposed a theoretical study in which we investigated whether the regeneration-inducing signal could follow a reaction-diffusion process. We developed a computational model, validated it with experimental data and showed that the signal dynamics can be understood in terms of reaction-diffusion mechanism. By developing a theory of the regenerating outgrowth in the limit of fast reaction-diffusion, we demonstrated that control of regenerative response solely relies on cell-to-signal sensitivity and the signal reaction-diffusion characteristic length. This study lays foundations for further identification of the signal controlling regeneration of the spinal cord.

7.5.3 Development and analaysis of 3D dynamical network models

The Extra-Cellular-Matrix (ECM) is a complex interconnected 3D network that provides structural support for the cells and tissues and defines organ architecture key for their healthy functioning. However, the intimate mechanisms by which ECM acquire their 3D architecture are still largely unknown. In 12, we addressed this question by means of a 3D individual based model of interacting fibers able to spontaneously crosslink or unlink to each other and align at the crosslinks. We showed that such systems are able to spontaneously generate different types of architectures, and performed a thorough analysis of the emerging structures by an exhaustive parametric analysis and the use of appropriate visualization tools and quantifiers in 3D. The most striking result is that the emergence of ordered structures can be fully explained by a single emerging variable : the proportion of crosslinks in the network. This simple variable becomes an important putative target to control and predict the structuring of biological tissues, to suggest possible new therapeutic strategies to restore tissue functions after disruption, and to help in the development of collagen-based scaffolds for tissue engineering. Moreover, the model revealed that the emergence of architecture is a spatially homogeneous process following a unique evolutionary path, and highlights the essential role of dynamical crosslinking in tissue structuring.

Contract with TreeFrog Pharmaceuticals

Simulation of growth efficiency and cell yield in multiple in vitro experimental settings to better understand the impact of the chosen culturing method and to guide potential improvements of the outcome.

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

9.1.2 STIC/MATH/CLIMAT AmSud projects

9.2.1 Visits of international scientists

Eva Loeser (UCSD San Diego) visited the team for a week in February.

Osvaldo Chara (University of Nottingham and Sysbio Group, University of la Plata, Argentina) visited the team for a week in July 2023.

Alberto Ceccarelli (PhD Student under the supervision of O. Chara University of la Plata, Argentina) visited the team for two weeks in July 2023.

9.2.2 Visits to international teams

9.3.1 ANR

Emergence Grant POMATH(Sorbonne University) 2023-2024

Nastassia Pouradier Duteil POMATH: POpulation-based MAthematical models for Tumor Heterogeneity.

10.1.1 Scientific events: organisation

Nastassia Pouradier Duteil is a member of the organization committee of the “Journée de modélisation mathématique en sciences de la vie et santé”, common to the three laboratories LAGA-LJLL-MAP5. The first two editions took place in March and November 2023.

10.1.2 Scientific events: selection

10.1.3 Journal

10.1.4 Invited talks

Lucie Laurence has given talks

• April 2023: online “Stochastic system seminar” organized by Ruth Williams (UCSD San Diego)
• June 2023: INFORMS Applied Probability Society Conference (Nancy)
• November 2023: online MoRN seminar
• November 2023: Probability Seminar, University of Pisa (visit for a week of Andrea Agazzi).

Nastassia Pouradier Duteil has given the following talks

• Sept. 2023: French-Korean Webinar Kinetic and fluid equations for collective behavior
• April 2023: PEIPS Seminar, CMAP, Ecole Polytechnique (Saclay)
• June 2023: Workshop “Round Meanfield II” (Rome, Italy)
• July 2023: Seminar at Gipsa Lab (Grenoble)
• July 2023: Seminar at Universidad de Cantabria, (Santander, Spain)
• Oct. 2023: PDE Seminar at the Mathematical Institute, Oxford University (UK)
• Nov. 2023: Groupe de Travail Biomath Normand (Rouen)
• Nov. 2023: Presentation for the HCERES Laboratory Evaluation at Laboratoire Jacques-Louis Lions (Paris)
• Dec. 2023: Demi-heure de sciences at Inria (Paris)
• Dec. 2023: Séminaire Laurent Schwartz, Ecole Polytechnique (Saclay)

Diane Peurichard has given the following talks:

• Dec 5 2023 Talk in the frame of the MMB chair at Natural History Museum, Paris, http://www.cmap.polytechnique.fr/chaire-mmb/
• Sept 11-15 IMACS2023, Rome, Italy https://www.imacs2023.eu/index.php/IMACS2023/IMACS2023
• July 3-7 2023 Workshop ‘Topics on Neuroscience, Collective Migration and Parameter Estimation’, Oxford https://sites.google.com/view/neurocollectiveestimation/home?authuser=0
• June 13 2023 Seminar ÉDP, Modélisation et Calcul Scientifique, Lyon-Saint Etienne http://math.univ-lyon1.fr/wikis/seminaire-mmcs/doku.php?id=accueil
• June 8 2023 ‘Calcul Scientifique et Modélisation Mathématique’, laboratoire de Mathématiques d'Amiens

Pierre-Alexandre Bliman presented seminaries at

• EPI BIOCORE, Centre Inria d'Université Côte d'Azur;
• Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil;
• COPPE, Universidade Federal do Rio de Janeiro, Brazil
• Instituto de Biociência, Universidade Estadual Paulista “Júlio de Mesquita Filho”, Câmpus de Botucatu, Brazil;
• Department of Mathematical Sciences, Aoyama Gakuin University, Tokyo, Japan.

10.1.5 Leadership within the scientific community

Philippe Robert has organized two sessions at the INFORMS Applied Probability Society Conference (Nancy) in June.

Pierre-Alexandre Bliman co-organized an invited session devoted to “Control theory perspectives on mathematical epidemiology" together with Alain Rapaport, Inrae.

10.1.6 Scientific expertise

Pierre-Alexandre Bliman is Inria representative in the Scientific committee of MATH AmSud program.

Pierre-Alexandre Bliman is expert for the Belgian Fonds National de Recherche Scientifique.

10.2.1 Teaching

Lucie Laurence and Jana Zaherddine are teaching assistants in the course “Mathematics for scientists”, in L1 at Jussieu.

Nastassia Pouradier Duteil Duteil is teaching a course on Mathematics in the Sorbonne University program RESPE “Retour aux Etudes Supérieures des Personnes Exilées”. She also supervised an L3 project on Evolutionary Games. She also taught a 30-hour course on Differential Equations at master's level (M1) at the Institut de Mathématiques et de Sciences Physiques at Porto-Novo (Benin).

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

Diane Peurichard gave a 4h course for the “Master de physique cellulaire", Strasbourg University on january 26th, 2023 entitled 'Mathematical modelling in biological systems'

Guillaume Garnier gave exercise classes in L1 (linear algebra and probabilities) and L3 (measure theory).

Marcel Fang and Manon De La Tousche gave exercise classes at licence level in Sorbonne Université.

Thi Nguyen gave exercise classes at licence level in Université Sorbonne Paris-Nord.

10.2.2 Supervision

Pierre-Alexandre Bliman is PhD co-advisor of Assane Savadogo , jointly with Prof. Boureima Sangaré, Université Nazi Boni, Bobo Dioulasso, Burkina Faso.

Pierre-Alexandre Bliman is PhD co-advisor of Thi Nguyen , jointly with Prof. Nicolas Vauchelet, Sorbonne Paris-Nord.

Pierre-Alexandre Bliman is PhD advisor of Marcel Fang .

Pierre-Alexandre Bliman is PhD co-advisor for Manon De La Tousche , jointly with Yves Dumont, CIRAD.

Nastassia Pouradier Duteil co-advised the PhD of Jules Guilberteau , together with Camille Pouchol, Université Paris Cité.

Diane Peurichard co-advised the PhD of Pauline Chassonnery , together with L. Casteilla, RESTORE, Toulouse. The PhD was defended on december 2023 at Université de Toulouse III.

Diane Peurichard co-advised the PhD of Valeria Caliaro together with O. Chara, Nottingham University and University of La Plata. The PhD was abandonned in december 2023.

10.2.3 Juries

Philippe Robert has been a member of the jury for the PhD defense of Jana Zaherddine in December.

Philippe Robert has been a reviewer of the PhD document of Suney Toste (ENS-PSL) defended in September.

Pierre-Alexandre Bliman has been member of the jury for the HDR defense of Frédéric Grognard (Inria Sophia) in February.

Diane Peurichard has been a member of the jury for the PhD defense of Pauline Chassonnery in December.

Nastassia Pouradier Duteil has been a member of the jury for the PhD defense of Pauline Chassonnery in December.

Diane Peurichard has been member of the jury CRCN Inria Saclay 2023

10.3.1 Internal or external Inria responsibilities

Diane Peurichard is member of the following comittees/commissions:

• From Sept 2023: Member of the Evaluation Committee (CE) https://www.inria.fr/en/inria-evaluation-committee
• Since 2019: Member of the Comité de Suivi Doctoral (CSD) Inria
• Since 2018: Member of the Comission des Emplois Scientifiques (CES) Inria
• Since 2020 Member of the Pôle Ecoute, LJLL Sorbonne Université https://www.ljll.math.upmc.fr/fr/presentation/article/parite-pole-ecoute-et-prevention?lang=fr

Pierre-Alexandre Bliman is Member of the Pôle Ecoute, LJLL Sorbonne Université since 2023.

10.3.2 Interventions

Nastassia Pouradier Duteil took part in Speed Meetings with high-school students at the event “Master-class lycénnes” at Institut Henri Poincaré in April 2023. She was also interviewed by Nathalie Ayi for the podcast “Tête à tête chercheuses”. She also took part in organizing a one-day course for high-school math teachers at the Jacques-Louis Lions Laboratory.

Diane Peurichard participated in Speed Meetings with high-school students online in the frame of Femmes et Mathématiques association https://femmes-et-maths.fr/de-lecole-au-lycee/speed-meetings/speed-meetings-reguliers/

International journals

• 7 articleL.Luís Almeida, C.Chloe Audebert, E.Emma Leschiera and T.Tommaso Lorenzi. Discrete and continuum models for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes and tumour cells.Mathematical Medicine and BiologyJanuary 2023HALDOIback to text
• 8 articleL.Luis Almeida, A.Alexis Léculier and N.Nicolas Vauchelet. Analysis of the "Rolling carpet" strategy to eradicate an invasive species.SIAM Journal on Mathematical Analysis551February 2023, 275-309HALDOI
• 9 articleC.Carles Barril, P.-A.Pierre-Alexandre Bliman and S.Sílvia Cuadrado. Final Size for Epidemic Models with Asymptomatic Transmission.Bulletin of Mathematical Biology856June 2023, 52HALDOIback to text
• 10 articleP.-A.Pierre-Alexandre Bliman, A.Alessio Carrozzo-Magli, A.Alberto D’onofrio and P.Piero Manfredi. Tiered social distancing policies and epidemic control.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences4782268January 2023HALDOIback to text
• 11 articleP.-A.Pierre-Alexandre Bliman, Y.Yves Dumont, O. E.Oscar Eduardo Escobar-Lasso, H.Hector Martinez-Romero and O.Olga Vasilieva. A sex-structured model of Wolbachia invasion to design sex-biased release strategies in Aedes spp mosquitoes populations.Applied Mathematical Modelling119March 2023, 391-412HALDOIback to text
• 12 articleP.Pauline Chassonnery, J.Jenny Paupert, A.Anne Lorsignol, C.Childérick Séverac, M.Marielle Ousset, P.Pierre Degond, L.Louis Casteilla and D.Diane Peurichard. Fiber crosslinking drives the emergence of order in a 3D dynamical network model.Royal Society Open Science2024HALback to text
• 13 articleJ.Jules Dichamp, G.Geraldine Cellière, A.Ahmed Ghallab, R.Reham Hassan, N.Noemie Boissier, U.Ute Hofmann, J.Joerg Reinders, S.Selahaddin Sezgin, S.Sebastian Zühlke, J.Jan Hengstler and D.Dirk Drasdo. In-vitro to in-vivo acetaminophen hepatotoxicity extrapolation using classical schemes, pharmaco-dynamic models and a multiscale spatial-temporal liver twin.Frontiers in Bioengineering and Biotechnology112023HALDOI
• 14 articleC.Charles Elbar, M.Marco Mason, B.Benoît Perthame and J.Jakub Skrzeczkowski. From Vlasov equation to degenerate nonlocal Cahn-Hilliard equation.Communications in Mathematical Physics4011February 2023, 1033-1057HALDOI
• 15 articleC.Charles Elbar, B.Benoît Perthame and J.Jakub Skrzeczkowski. Pressure jump and radial stationary solutions of the degenerate Cahn-Hilliard equation (Extended version).Comptes Rendus. Mécanique2023HALback to text
• 16 articleC.Charles Elbar and J.Jakub Skrzeczkowski. Degenerate Cahn-Hilliard equation: From nonlocal to local.Journal of Differential Equations364August 2023, 576-611HALDOIback to text
• 17 articleV.Vincent Fromion, P.Philippe Robert and J.Jana Zaherddine. Stochastic Models of Regulation of Transcription in Biological Cells.Journal of Mathematical Biology8752023, 65HALDOIback to text
• 18 articleZ.Zineb Kaid, C.Camille Pouchol and J.Jean Clairambault. A phenotype-structured model for the tumour-immune response.Mathematical Modelling of Natural PhenomenaAugust 2023HAL
• 19 articleP.Philippe Robert and G.Gaëtan Vignoud. A Palm Space Approach to Non-Linear Hawkes Processes.Electronic Journal of Probability29noneJanuary 2024, 1-37HALDOIback to text

International peer-reviewed conferences

• 20 inproceedingsP.-A.Pierre-Alexandre Bliman and A.Alain Rapaport. Control theory perspectives on mathematical epidemiology.IFAC World Congress 2023 - 22nd World Congress of the International Federation of Automatic ControlThe 22nd World Congress of the International Federation of Automatic ControlYokohama, JapanJuly 2023, 3p.HALback to text
• 21 inproceedingsP.-A.Pierre-Alexandre Bliman and A.Alain Rapaport. On the problem of minimizing the epidemic final size for SIR model by social distancing.IFAC World Congress 2023 - 22nd World Congress of the International Federation of Automatic ControlYokohama, JapanJuly 2023HALback to text
• 22 inproceedingsJ.Jean Clairambault. Mathematical modelling of cancer growth and drug treatments: taking into account cell population heterogeneity and plasticity.ECC 23 - European Control ConferenceBucarest, RomaniaJune 2023HAL
• 23 inproceedingsM.Marcel Fang, P.-A.Pierre-Alexandre Bliman, D.Denis Efimov and R.Rosane Ushirobira. Nonlinear Adaptive Observers for an SIS System Counting Primo-infections.22nd World Congress of the International Federation of Automatic ControlYokohama, JapanJuly 2023HALback to text

Scientific book chapters

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

Reports & preprints

• 25 miscL.Luis Almeida, J.Jesús Bellver Arnau, Y.Yannick Privat and C.Carlota Rebelo. Vector-borne disease outbreak control via instant releases.January 2024HALback to text
• 26 miscL.Luís Almeida, P.-A.Pierre-Alexandre Bliman, N.Nga Nguyen and N.Nicolas Vauchelet. Steady-state solutions for a reaction-diffusion equation with Robin boundary conditions: Application to the control of dengue vectors.September 2023HALDOIback to text
• 27 miscF.Frank Alvarez and J.Jean Clairambault. Phenotype divergence and cooperation in isogenic multicellularity and in cancer.June 2023HALback to text
• 28 miscN.Nathalie Ayi and N.Nastassia Pouradier Duteil. Graph Limit for Interacting Particle Systems on Weighted Random Graphs.July 2023HALback to text
• 29 miscN.Nathalie Ayi and N.Nastassia Pouradier Duteil. Large-population limits of non-exchangeable particle systems.January 2024HAL
• 30 miscV.Valeria Caliaro, D.Diane Peurichard and O.Osvaldo Chara. How a reaction-diffusion signal can control spinal cord regeneration in axolotls: A modelling study.August 2023HALDOIback to text
• 31 miscJ.Jean Clairambault. Mathematical modelling of cancer growth and drug treatments: taking into account cell population heterogeneity and plasticity.April 2023HAL
• 32 miscX.Xu'An Dou, B.Benoît Perthame, C.Chenjiayue Qi, D.Delphine Salort and Z.Zhennan Zhou. An infinite-times renewal equation.April 2023HALback to text
• 33 miscM.Marie Doumic, M.Miguel Escobedo and M.Magali Tournus. An inverse problem: recovering the fragmentation kernel from the short-time behaviour of the fragmentation equation.February 2023HAL
• 34 miscM.Marie Doumic, S.Sophie Hecht, B.Benoit Perthame and D.Diane Peurichard. Multispecies cross-diffusions: from a nonlocal mean-field to a porous medium system without self-diffusion.May 2023HALback to text
• 35 miscC.Charles Elbar, P.Piotr Gwiazda, J.Jakub Skrzeczkowski and A.Agnieszká Swierczewska-Gwiazda. From nonlocal Euler-Korteweg to local Cahn-Hilliard via the high-friction limit.May 2023HAL
• 36 miscC.Charles Elbar, B.Benoît Perthame, A.Andrea Poiatti and J.Jakub Skrzeczkowski. Nonlocal Cahn-Hilliard equation with degenerate mobility: Incompressible limit and convergence to stationary states.May 2023HAL
• 37 miscC.Charles Elbar, B.Benoît Perthame and J.Jakub Skrzeczkowski. On the limit problem arising in the kinetic derivation of the Cahn-Hilliard equation.June 2023HAL
• 38 miscC.Charles Elbar and A.Alexandre Poulain. Analysis and numerical simulation of a generalized compressible Cahn-Hilliard-Navier-Stokes model with friction effects.May 2023HAL
• 39 miscC.Charles Elbar and J.Jakub Skrzeczkowski. On the inviscid limit connecting Brinkman's and Darcy's models of tissue growth with nonlinear pressure.June 2023HAL
• 40 miscG.Gissell Estrada-Rodriguez, D.Diane Peurichard and X.Xinran Ruan. Asymptotic preserving schemes for nonlinear kinetic equations leading to volume-exclusion chemotaxis in the diffusive limit.November 2022HALback to text
• 41 miscV.Vincent Fromion, P.Philippe Robert and J.Jana Zaherddine. A Stochastic Analysis of Particle Systems with Pairing.October 2023HALback to text
• 42 miscJ.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.September 2023HALback to text
• 43 miscJ.Jules Guilberteau, C.Camille Pouchol and N. P.Nastassia Pouradier Duteil. Long-time behaviour of an advection-selection equation.January 2023HALback to text
• 44 miscQ.Qingyou He, H.-L.Hai-Liang Li and B.Benoît Perthame. Incompressible limit of porous media equation with chemotaxis and growth.November 2023HAL
• 45 miscM.Maxime Herda, P.Pierre Monmarché and B.Benoît Perthame. Wasserstein contraction for the stochastic Morris-Lecar neuron model.July 2023HALback to text
• 46 miscL.Laura Kanzler, B.Benoît Perthame and B.Benoît Sarels. Structured Model Conserving Biomass for the Size-spectrum Evolution in Aquatic Ecosystems.October 2023HAL
• 47 miscL.Lucie Laurence and P.Philippe Robert. A Scaling Approach to Stochastic Chemical Reaction Networks.October 2023HALback to text
• 48 miscA.Alexis Léculier and N.Nga Nguyen. A control strategy for Sterile Insect Techniques using exponentially decreasing releases to avoid the hair-trigger effect.April 2023HALback to text
• 49 miscA.Anaïs Rat, M.Marie Doumic, M. T.Maria Teresa Teixeira and Z.Zhou Xu. Individual cell fate and population dynamics revealed by a mathematical model linking telomere length and replicative senescence.November 2023HALDOI

Cited publications

• 50 articleA.A.G .McKendrick. Applications of mathematics to medical problems.Proc. Edinburgh Math. Soc.541926, 98--130back to text
• 51 articleC.C .Norden. Pseudostratified epithelia \textendash} cell biology, diversity and roles in organ formation at a glance.}.Journal of Cell Science1302017, 1859--1863back to text
• 52 articleB.B .Perthame, F.F .Quirós and J. ..J .L. Vázquez. The Hele--Shaw asymptotics for mechanical models of tumor growth.Arch. Ration. Mech. Anal.2122014, 93--127back to text
• 53 articleB.B .Perthame and N.N .Vauchelet.. Incompressible limit of a mechanical model of tumour growth with viscosity..Philos. Trans. Roy. Soc. A, Mathematical, physical, and engineering sciences3732015back to text
• 54 articleN. S.N. Salehi A. Bressan. On the Optimal Control of Propagation Fronts.preprint2022back to text
• 55 articleL. J.Laith J. Abu-Raddad and N. M.Neil M. Ferguson. Characterizing the symmetric equilibrium of multi-strain host-pathogen systems in the presence of cross immunity..Journal of Mathematical Biology5055 2005DOIback to text
• 56 articleA.Agustina Alaimo, G. G.Guadalupe García Liñares, J. M.Juan Marco Bujjamer, R. M.Roxana Mayra Gorojod, S. P.Soledad Porte Alcon, J. H.Jimena Hebe Martínez, A.Alicia Baldessari, H. E.Hernán Edgardo Grecco and M. L.Mónica Lidia Kotler. Toxicity of blue led light and A2E is associated to mitochondrial dynamics impairment in ARPE-19 cells: implications for age-related macular degeneration.Archives of Toxicology932019, 1401--1415back to text
• 57 articleL.Luca Alasio, M.Maria Bruna, S.Simone Fagioli and S.Simon Schulz. Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts.Nonlinear Analysis2232022, 113064back to text
• 58 articleL.Luca Alasio. Towards a new mathematical model of the visual cycle.2022back to text
• 59 articleN. D.Nicholas D. Alikakos, P. W.Peter W. Bates and X.Xinfu Chen. Periodic traveling waves and locating oscillating patterns in multidimensional domains.Trans. Amer. Math. Soc.35171999, 2777--2805URL: https://doi.org/10.1090/S0002-9947-99-02134-0DOIback to text
• 60 unpublishedL.Luís Almeida, C.Chloe Audebert, E.Emma Leschiera and T.Tommaso Lorenzi. Discrete and continuum models for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes and tumour cells.September 2021, working paper or preprintHALback to text
• 61 unpublishedL.Luis Almeida, A.Alexis Leculier and N.Nicolas Vauchelet. Analysis of the ''Rolling carpet'' strategy to eradicate an invasive species.June 2021, working paper or preprintHALback to textback to text
• 62 articleL.L. Alphey, M.M. Benedict, R.R. Bellini, G.G.G. Clark, D.D.A. Dame, M.M.W. Service and S.S.L. Dobson. Sterile-insect methods for control of mosquito-borne diseases: an analysis.Vector Borne Zoonotic Dis.2010back to text
• 63 articleF. E.Frank Ernesto Alvarez, J. A.José Antonio Carrillo and J.Jean Clairambault. Evolution of a structured cell population endowed with plasticity of traits under constraints on and between the traits.Journal of Mathematical Biologyon line, September 2022September 2022HALback to textback to textback to text
• 64 articleD. F.David F Anderson, G.Gheorghe Craciun and T. G.Thomas G Kurtz. Product-form stationary distributions for deficiency zero chemical reaction networks.Bulletin of mathematical biology7282010, 1947--1970back to text
• 65 bookR. M.Roy M Anderson and R. M.Robert M May. Infectious diseases of humans: dynamics and control.Oxford university press1992back to text
• 66 articleV.Viggo Andreasen, J.Juan Lin and S. A.Simon A. Levin. The dynamics of cocirculating influenza strains conferring partial cross-immunity.Journal of Mathematical Biology3578 1997DOIback to text
• 67 incollectionJ.Julien Arino. Diseases in metapopulations.Modeling and dynamics of infectious diseasesWorld Scientific2009, 64--122back to textback to text
• 68 articleJ.Julien Arino, C. C.C Connell McCluskey and P.Pauline van den Driessche. Global results for an epidemic model with vaccination that exhibits backward bifurcation.SIAM Journal on Applied Mathematics6412003, 260--276back to textback to text
• 69 articleJ.Julien Arino and P.P Van den Driessche. Disease spread in metapopulations.Fields Institute Communications4812006, 1--13back to text
• 70 articleE. F.Edilson F Arruda, S. S.Shyam S Das, C. M.Claudia M Dias and D. H.Dayse H Pastore. Modelling and optimal control of multi strain epidemics, with application to COVID-19.PLoS One1692021, e0257512back to text
• 71 articleP.Pierre Auger, E.Etienne Kouokam, G.Gauthier Sallet, M.Maurice Tchuente and B.Berge Tsanou. The Ross--Macdonald model in a patchy environment.Mathematical biosciences21622008, 123--131back to text
• 72 inbookA.Aylin Aydoğdu, M.Marco Caponigro, S.Sean McQuade, B.Benedetto Piccoli, N.Nastassia Pouradier Duteil, F.Francesco Rossi and E.Emmanuel Trélat. Interaction Network, State Space, and Control in Social Dynamics.Active Particles, Volume 1 : Advances in Theory, Models, and ApplicationsN.Nicola BellomoP.Pierre DegondE.Eitan TadmorChamSpringer International Publishing2017, 99--140URL: https://doi.org/10.1007/978-3-319-49996-3_3DOIback to text
• 73 articleN.Nathalie Ayi and N. P.Nastassia Pouradier Duteil. Mean-field and graph limits for collective dynamics models with time-varying weights.Journal of Differential Equations2992021, 65-110URL: https://www.sciencedirect.com/science/article/pii/S0022039621004472DOIback to text
• 74 articleI. A.Isa Abdullahi Baba, B.Bilgen Kaymakamzade and E.Evren Hincal. Two-strain epidemic model with two vaccinations.Chaos, Solitons & Fractals1062018, 342--348back to text
• 75 articleN. H.N. H. Barton and M.M. Turelli. Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of Allee effects.The American Naturalist1782011, E48-E75back to textback to text
• 76 articleN. G.Nicolas G Bazan. Cell survival matters: docosahexaenoic acid signaling, neuroprotection and photoreceptors.Trends in neurosciences2952006, 263--271back to text
• 77 articleO. G.O. G. Berg. A model for the statistical fluctuations of protein numbers in a microbial population.Journal of theoretical biology714apr 1978, 587--603back to text
• 78 articleF.Frédérique Billy, J.Jean Clairambault, O.Olivier Fercoq, S.Stéphane Gaubert, T.Thomas Lepoutre, T.Thomas Ouillon and S.Shoko Saito. Synchronisation and control of proliferation in cycling cell population models with age structure..Mathematics and Computers in Simulation962014, 66-94back to text
• 79 articleP.-A.Pierre-Alexandre Bliman. A feedback control perspective on biological control of dengue vectors by Wolbachia infection.European Journal of Control592021, 188--206back to text
• 80 articleS.Samuel Bowong, Y.Yves Dumont and J. J.Jean Jules Tewa. A patchy model for chikungunya-like diseases.Biomath212013, 1307237back to text
• 81 articleF.Fred Brauer. Backward bifurcations in simple vaccination models.Journal of Mathematical Analysis and Applications29822004, 418--431back to text
• 82 bookF.Fred Brauer and C.Carlos Castillo-Chavez. Mathematical models for communicable diseases.SIAM2012back to text
• 83 articleM.Maria Bruna, M.Martin Burger, A.Antonio Esposito and S. M.Simon M Schulz. Phase separation in systems of interacting active Brownian particles.SIAM Journal on Applied Mathematics8242022, 1635--1660back to textback to text
• 84 articleM.Maria Bruna, M.Martin Burger, A.Antonio Esposito and S.Simon Schulz. Well-posedness of an integro-differential model for active Brownian particles.SIAM Journal on Mathematical Analysis5452022, 5662--5697back to text
• 85 articleF.Federica Bubba, B.Benoît Perthame, C.Camille Pouchol and M.Markus. Schmidtchen. Hele--Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues.Archive for Rational Mechanics and Analysis2362020back to text
• 86 articleC.Christian Caprara and C.Christian Grimm. From oxygen to erythropoietin: relevance of hypoxia for retinal development, health and disease.Progress in retinal and eye research3112012, 89--119back to text
• 87 articleJ. A.José A Carrillo, S.Simone Fagioli, F.Filippo Santambrogio and M.Markus Schmidtchen. Splitting schemes and segregation in reaction cross-diffusion systems.SIAM Journal on Mathematical Analysis5052018, 5695--5718back to text
• 88 articleN.Nicolas Champagnat and P.-E.Pierre-Emmanuel Jabin. The evolutionary limit for models of populations interacting competitively via several resources.J. Differential Equations25112011, 176--195URL: https://doi-org.accesdistant.sorbonne-universite.fr/10.1016/j.jde.2011.03.007DOIback to text
• 89 articleC.C. Christiansen-Jucht, K.K. Erguler, C.C.Y. Shek, M.M.G. Basánez and P.P.E. Parham. Modelling Anopheles gambiae s.s. Population Dynamics with Temperature- and Age-Dependent Survival.Int J Environ Res Public Health1262015, 5975-6005back to text
• 90 articleJ.Jean Clairambault, P.Philippe Michel and B.Benôit Perthame. Circadian rhythm and tumour growth.C. R. Acad. Sci. (Paris) Ser. I Mathématique3422006, 17-22back to text
• 91 incollectionJ.Jean Clairambault. Plasticity in Cancer Cell Populations: Biology, Mathematics and Philosophy of Cancer.``Mathematical and Computational Oncology'', Proceedings of the Second International Symposium, ISMCO 2020, San Diego, CA, USA, October 8-10, 2020, G. Bebis, M. Alekseyev, H. Cho, J. Gevertz, M. Rodriguez Martinez (Eds.), Springer LNBI 12508, pp. 3-9, October 2020.12508LNBI - Lecture Notes in BioinformaticsSpringerDecember 2020, 3-9HALDOIback to textback to text
• 92 articleJ.Jean Clairambault and C.Camille Pouchol. A survey of adaptive cell population dynamics models of emergence of drug resistance in cancer, and open questions about evolution and cancer.BIOMATH81Copyright 2019 Clairambault et al. This article is distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.May 2019, 23HALDOIback to text
• 93 articleJ.Jean Clairambault. Stepping From Modeling Cancer Plasticity to the Philosophy of Cancer.Frontiers in Genetics2020, 11:579738back to textback to text
• 94 articleM.Marcello Colombino and R. S.Roy S. Smith. A Convex Characterization of Robust Stability for Positive and Positively Dominated Linear Systems.IEEE Transactions on Automatic Control612016, 1965-1971back to text
• 95 articleC.Chloé Colson, F.Faustino Sánchez-Garduño, H. M.Helen M Byrne, P. K.Philip K Maini and T.Tommaso Lorenzi. Travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion.Proceedings of the Royal Society A47722562021, 20210593back to text
• 96 articleF.F. Cucker and S.S. Smale. Emergent behavior in flocks.IEEE Transactions on Automatic Control522007, 852--862back to textback to textback to text
• 97 articleN.Noemi David and B.Benoît Perthame. Free boundary limit of tumor growth model with nutrient.Journal de Mathématiques Pures et Appliquées155https://arxiv.org/abs/2003.107312021, 62-82HALDOIback to text
• 98 articleP. C.Paul C.W. Davies and C. H.Charles H. Lineweaver. Cancer tumors as Metazoa 1.0: tapping genes of ancient ancestors.Physical Biology81feb 2011, 015001URL: https://doi.org/10.1088/1478-3975/8/1/015001DOIback to text
• 99 inbookP.Pierre Degond. MATHEMATICAL MODELS OF COLLECTIVE DYNAMICS AND SELF-ORGANIZATION.Proceedings of the International Congress of Mathematicians (ICM 2018)3925-3946URL: https://www.worldscientific.com/doi/abs/10.1142/9789813272880_0206DOIback to text
• 100 articleL. V.Lucian V Del Priore, Y.-H.Ya-Hui Kuo and T. H.Tongalp H Tezel. Age-related changes in human RPE cell density and apoptosis proportion in situ.Investigative ophthalmology & visual science43102002, 3312--3318back to text
• 101 incollectionO.Odo Diekmann. A BEGINNER'S GUIDE TO ADAPTIVE DYNAMICS.63MATHEMATICAL MODELLING OF POPULATION DYNAMICSBANACH CENTER PUBLICATIONS, INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA2004, 47--86back to text
• 102 bookO.Odo Diekmann and J. A.Johan Andre Peter Heesterbeek. Mathematical epidemiology of infectious diseases: model building, analysis and interpretation.5John Wiley & Sons2000back to text
• 103 articleO.Odo Diekmann, P.-E.Pierre-Emmanuel Jabin, S.Stéphane Mischler and B.Benoît Perthame. The dynamics of adaptation: An illuminating example and a Hamilton--Jacobi approach.Theoretical Population Biology672005, 257--271back to text
• 104 articleW.Weiwei Ding and H.Hiroshi Matano. Dynamics of time-periodic reaction-diffusion equations with compact initial support on $\mathbb{R}$.J. Math. Pures Appl. (9)1312019, 326--371URL: https://doi.org/10.1016/j.matpur.2019.09.010DOIback to text
• 105 articleW.Weiwei Ding and H.Hiroshi Matano. Dynamics of time-periodic reaction-diffusion equations with front-like initial data on $\mathbb{R}$.SIAM J. Math. Anal.5232020, 2411--2462URL: https://doi.org/10.1137/19M1268987DOIback to text
• 106 miscB. A.Blanca Ayuso de Dios, S.Simone Dovetta and L. V.Laura V. Spinolo. On the continuum limit of epidemiological models on graphs: convergence results, approximation and numerical simulations.2022, URL: https://arxiv.org/abs/2211.01932DOIback to text
• 107 articleR.Ramsès Djidjou-Demasse, C.Christian Selinger and M. T.Mircea T Sofonea. Épidémiologie mathématique et modélisation de la pandémie de Covid-19: enjeux et diversité.Revue Francophone des Laboratoires20205262020, 63--69back to text
• 108 articleR. L.R. L. Dobrushin. Vlasov equations.Functional Analysis and Its Applications1321979, 115--123URL: https://doi.org/10.1007/BF01077243DOIback to text
• 109 articleJ.Jean Dolbeault and G.Gabriel Turinici. Heterogeneous social interactions and the COVID-19 lockdown outcome in a multi-group SEIR model.Mathematical Modelling of Natural Phenomena152020, 36back to text
• 110 articleM.Marie Doumic, K.Klemens Fellner, M.Mathieu Mezache and H.Human Rezaei. A bi-monomeric, nonlinear Becker-Döring-type system to capture oscillatory aggregation kinetics in prion dynamics.Journal of Theoretical Biology48011 2019DOIback to text
• 111 articleM.Marie Doumic, S.Sophie Hecht and D.Diane Peurichard. A purely mechanical model with asymmetric features for early morphogenesis of rod-shaped bacteria micro-colony.Mathematical Biosciences and Engineering176October 2020, http://aimspress.com/article/doi/10.3934/mbe.2020356HALDOIback to textback to textback to text
• 112 articleC.C. Dufourd and Y.Y. Dumont. Impact of environmental factors on mosquito dispersal in the prospect of sterile insect technique control.Comput. Math. Appl.6692013, 1695--1715back to text
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• 114 articleM.-C.Marie-Cécilia Duvernoy, T.Thierry Mora, M.Maxime Ardré, V.Vincent Croquette, D.David Bensimon, C.Catherine Quilliet, J.-M.Jean-Marc Ghigo, M.Martial Balland, C.Christophe Beloin, S.Sigolène Lecuyer and others. Asymmetric adhesion of rod-shaped bacteria controls microcolony morphogenesis.Nature communications912018, 1--10back to textback to text
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