Multiphase modeling and qualitative analysis of the growth of tumor cords
Andrea Tosin
Networks & Heterogeneous Media 2008, 3(1): 43-83 doi: 10.3934/nhm.2008.3.43
In this paper a macroscopic model of tumor cord growth is developed, relying on the mathematical theory of deformable porous media. Tumor is modeled as a saturated mixture of proliferating cells, extracellular fluid and extracellular matrix, that occupies a spatial region close to a blood vessel whence cells get the nutrient needed for their vital functions. Growth of tumor cells takes place within a healthy host tissue, which is in turn modeled as a saturated mixture of non-proliferating cells. Interactions between these two regions are accounted for as an essential mechanism for the growth of the tumor mass. By weakening the role of the extracellular matrix, which is regarded as a rigid non-remodeling scaffold, a system of two partial differential equations is derived, describing the evolution of the cell volume ratio coupled to the dynamics of the nutrient, whose higher and lower concentration levels determine proliferation or death of tumor cells, respectively. Numerical simulations of a reference two-dimensional problem are shown and commented, and a qualitative mathematical analysis of some of its key issues is proposed.
keywords: free boundary problems. theory of mixtures mathematical modeling Tumor growth
Existence and approximation of probability measure solutions to models of collective behaviors
Andrea Tosin Paolo Frasca
Networks & Heterogeneous Media 2011, 6(3): 561-596 doi: 10.3934/nhm.2011.6.561
In this paper we consider first order differential models of collective behaviors of groups of agents, based on the mass conservation equation. Models are formulated taking the spatial distribution of the agents as the main unknown, expressed in terms of a probability measure evolving in time. We develop an existence and approximation theory of the solutions to such models and we show that some recently proposed models of crowd and swarm dynamics fit our theoretic paradigm.
keywords: nonlocal flux. Systems of interacting agents probability distribution continuity equation
Kinetic models for traffic flow resulting in a reduced space of microscopic velocities
Gabriella Puppo Matteo Semplice Andrea Tosin Giuseppe Visconti
Kinetic & Related Models 2017, 10(3): 823-854 doi: 10.3934/krm.2017033

The purpose of this paper is to study the properties of kinetic models for traffic flow described by a Boltzmann-type approach and based on a continuous space of microscopic velocities. In our models, the particular structure of the collision kernel allows one to find the analytical expression of a class of steady-state distributions, which are characterized by being supported on a quantized space of microscopic speeds. The number of these velocities is determined by a physical parameter describing the typical acceleration of a vehicle and the uniqueness of this class of solutions is supported by numerical investigations. This shows that it is possible to have the full richness of a kinetic approach with the simplicity of a space of microscopic velocities characterized by a small number of modes. Moreover, the explicit expression of the asymptotic distribution paves the way to deriving new macroscopic equations using the closure provided by the kinetic model.

keywords: Kinetic models traffic flow Boltzmann equation equilibrium distributions discrete velocity models
Fundamental diagrams for kinetic equations of traffic flow
Luisa Fermo Andrea Tosin
Discrete & Continuous Dynamical Systems - S 2014, 7(3): 449-462 doi: 10.3934/dcdss.2014.7.449
In this paper we investigate the ability of some recently introduced discrete kinetic models of vehicular traffic to catch, in their large time behavior, typical features of theoretical fundamental diagrams. Specifically, we address the so-called ``spatially homogeneous problem'' and, in the representative case of an exploratory model, we study the qualitative properties of its solutions for a generic number of discrete microscopic states. This includes, in particular, asymptotic trends and equilibria, whence fundamental diagrams originate.
keywords: discrete kinetic models Traffic flow stochastic games fundamental diagrams. asymptotic trends
On the dynamics of social conflicts: Looking for the black swan
Nicola Bellomo Miguel A. Herrero Andrea Tosin
Kinetic & Related Models 2013, 6(3): 459-479 doi: 10.3934/krm.2013.6.459
This paper deals with the modeling of social competition, possibly resulting in the onset of extreme conflicts. More precisely, we discuss models describing the interplay between individual competition for wealth distribution that, when coupled with political stances coming from support or opposition to a Government, may give rise to strongly self-enhanced effects. The latter may be thought of as the early stages of massive unpredictable events known as Black Swans, although no analysis of any fully-developed Black Swan is provided here. Our approach makes use of the framework of the kinetic theory for active particles, where nonlinear interactions among subjects are modeled according to game-theoretical principles.
keywords: generalized kinetic theory Social systems stochastic games. complexity active particles
Kinetic description of collision avoidance in pedestrian crowds by sidestepping
Adriano Festa Andrea Tosin Marie-Therese Wolfram
Kinetic & Related Models 2018, 11(3): 491-520 doi: 10.3934/krm.2018022

In this paper we study a kinetic model for pedestrians, who are assumed to adapt their motion towards a desired direction while avoiding collisions with others by stepping aside. These minimal microscopic interaction rules lead to complex emergent macroscopic phenomena, such as velocity alignment in unidirectional flows and lane or stripe formation in bidirectional flows. We start by discussing collision avoidance mechanisms at the microscopic scale, then we study the corresponding Boltzmann-type kinetic description and its hydrodynamic mean-field approximation in the grazing collision limit. In the spatially homogeneous case we prove directional alignment under specific conditions on the sidestepping rules for both the collisional and the mean-field model. In the spatially inhomogeneous case we illustrate, by means of various numerical experiments, the rich dynamics that the proposed model is able to reproduce.

keywords: Crowd dynamics collision avoidance Boltzmann-type kinetic model mean-field approximation
Transport of measures on networks
Fabio Camilli Raul De Maio Andrea Tosin
Networks & Heterogeneous Media 2017, 12(2): 191-215 doi: 10.3934/nhm.2017008

In this paper we formulate a theory of measure-valued linear transport equations on networks. The building block of our approach is the initial and boundary-value problem for the measure-valued linear transport equation on a bounded interval, which is the prototype of an arc of the network. For this problem we give an explicit representation formula of the solution, which also considers the total mass flowing out of the interval. Then we construct the global solution on the network by gluing all the measure-valued solutions on the arcs by means of appropriate distribution rules at the vertexes. The measure-valued approach makes our framework suitable to deal with multiscale flows on networks, with the microscopic and macroscopic phases represented by Lebesgue-singular and Lebesgue-absolutely continuous measures, respectively, in time and space.

keywords: Network transport equation measure-valued solutions distribution conditions

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