A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow

In this paper we present a Semi-Lagrangian scheme for a regularized version of the Hughes model for pedestrian flow. Hughes originally proposed a coupled nonlinear PDE system describing the evolution of a large pedestrian group trying to exit a domain as fast as possible. The original model corresponds to a system of a conservation law for the pedestrian density and an Eikonal equation to determine the weighted distance to the exit. We consider this model in presence of small diffusion and discuss the numerical analysis of the proposed Semi-Lagrangian scheme. Furthermore we illustrate the effect of small diffusion on the exit time with various numerical experiments

Modeling Edar expression reveals the hidden dynamics of tooth signaling center patterning.

When patterns are set during embryogenesis, it is expected that they are straightly established rather than subsequently modified. The patterning of the three mouse molars is, however, far from straight, likely as a result of mouse evolutionary history. The first-formed tooth signaling centers, called MS and R2, disappear before driving tooth formation and are thought to be vestiges of the premolars found in mouse ancestors. Moreover, the mature signaling center of the first molar (M1) is formed from the fusion of two signaling centers (R2 and early M1). Here, we report that broad activation of Edar expression precedes its spatial restriction to tooth signaling centers. This reveals a hidden two-step patterning process for tooth signaling centers, which was modeled with a single activator-inhibitor pair subject to reaction-diffusion (RD). The study of Edar expression also unveiled successive phases of signaling center formation, erasing, recovering, and fusion. Our model, in which R2 signaling center is not intrinsically defective but erased by the broad activation preceding M1 signaling center formation, predicted the surprising rescue of R2 in Edar mutant mice, where activation is reduced. The importance of this R2-M1 interaction was confirmed by ex vivo cultures showing that R2 is capable of forming a tooth. Finally, by introducing chemotaxis as a secondary process to RD, we recapitulated in silico different conditions in which R2 and M1 centers fuse or not. In conclusion, pattern formation in the mouse molar field relies on basic mechanisms whose dynamics produce embryonic patterns that are plastic objects rather than fixed end points

Numerical study of macroscopic pedestrian flow models

We analyze numerically macroscopic models of crowd dynamics: classical Hughes model and the second order model being an extension to pedestrian motion of the Payne-Whitham vehicular traffic model. The desired direction of motion is determined by solving an eikonal equation with density dependent running cost function standing for minimization of the travel time and avoidance of congested areas. We apply a mixed finite volume-element method to solve the problems and present error analysis in case of the eikonal solver, gradient computation and the second order model yielding a first order convergence. We show that Hughes' model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit. In particular, we analyze the effect on the total evacuation time of the level of compression, the desired speed and the presence of obstacles.On analyse plusieurs modèles numériques pour la dynamique des foules : le modèle classique de Hughes et le modèle du second ordre, qui est une extension au mouvement de piétons du modèle de Payne-Whitham pour le trafic routier. La direction du mouvement est obtenue par résolution d'une équation Eikonale avec une fonction de coût dépendant de la densité, de manière à minimiser le temps de déplacement et éviter les endroits congestionnés. On utilise une méthode mixte éléments / volumes finis pour résoudre le problème et on présente une analyse d'erreur pour la résolution de l'équation Eikonale, le calcul de gradient et le modèle du second ordre, aboutissant à une précision du premier ordre. On montre que le modèle de Hughes est incapable de reproduire la dynamique des foules complexes, comme les ondes de type "stop-and-go" et le phénomène de colmatage apparaissant aux goulots d'étranglement. Enfin, avec le modèle du second ordre, on étudie numériquement l'évacuation de piétons d'une salle avec une sortie étroite. An particulier, on analyse l'effet du niveau de compression, de la vitesse de déplacement et la présence d'obstacle sur le temps total d'évacuation

Mathematical model for transport of DNA plasmids from the external medium up to the nucleus by electroporation

We propose a mathematical model for the transport of DNA plasmids from the extracellular matrix up to the cell nucleus. The model couples two phenomena: the electroporation process, describing the cell membrane permeabilization to plasmids and the intracellular transport enhanced by the presence of microtubules. Numerical simulations of cells with arbitrary geometry, in 2D and 3D, and a network of microtubules show numerically the importance of the microtubules and the electroporation on the effectiveness of the DNA transfection, as observed by previous biological data. The paper proposes efficient numerical tools for forthcoming optimized procedures of cell transfection.Initiative d'excellence de l'Université de Bordeau

Mathematical models and methods for crowd dynamics control

In this survey we consider mathematical models and methods recently developedto control crowd dynamics, with particular emphasis on egressing pedestrians.We focus on two control strategies: The first one consists in using specialagents, called leaders, to steer the crowd towards the desired direction.Leaders can be either hidden in the crowd or recognizable as such. Thisstrategy heavily relies on the power of the social influence (herding effect),namely the natural tendency of people to follow group mates in situations ofemergency or doubt. The second one consists in modify the surroundingenvironment by adding in the walking area multiple obstacles optimally placedand shaped. The aim of the obstacles is to naturally force people to behave asdesired. Both control strategies discussed in this paper aim at reducing asmuch as possible the intervention on the crowd. Ideally the natural behavior ofpeople is kept, and people do not even realize they are being led by anexternal intelligence. Mathematical models are discussed at different scales ofobservation, showing how macroscopic (fluid-dynamic) models can be derived bymesoscopic (kinetic) models which, in turn, can be derived by microscopic(agent-based) models