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Following some general ideas on the discrete kinetic and stochastic game theory proposed by one of the authors in a previous work, this paper develops a discrete velocity mathematical model for vehicular traffic along a one-way road. The kinetic scale is chosen because, unlike the macroscopic one, it allows to capture the probabilistic essence of the interactions among the vehicles, and offers at the same time, unlike the microscopic one, the opportunity of a profitable analytical investigation of the relevant global features of the system. The discretization of the velocity variable, rather than being a pure mathematical technicality, plays a role in including the intrinsic granular nature of the flow of vehicles in the mathematical theory of traffic. Other important characteristics of the model concern the gain and loss terms of the kinetic equations, namely the construction of a density-dependent table of games to model velocity transitions and the introduction of a visibility length to account for nonlocal interactions among the vehicles.

This review reports the existing literature on traffic flow modelling in the framework of a critical overview which aims to indicate research perspectives. The contents mainly refer to modelling by fluid dynamic and kinetic equations and are arranged in three parts. The first part refers to methodological aspects of mathematical modelling and to the interpretation of experimental results. The second part is devoted to modelling and deals both with methodological aspects and with the description of some specific models. The third part reports about an overview on applications and research perspectives.

This paper deals with some methodological aspects related to the discretization of a class of integro-differential equations modelling the evolution of the probability distribution over the microscopic state of a large system of interacting individuals. The microscopic state includes both mechanical and socio-biological variables. The discretization of the microscopic state generates a class of dynamical systems defining the evolution of the densities of the discretized state. In general, this yields a system of partial differential equations replacing the continuous integro-differential equation. As an example, a specific application is discussed, which refers to modelling in the field of social dynamics. The derivation of the evolution equation needs the development of a stochastic game theory.

This paper deals with a critical analysis and some developments related to the mathematical literature on multiscale modelling of multicellular systems involving tumor immune cells competition at the cellular level. The analysis is focused on the development of mathematical methods of the classical kinetic theory to model the above physical system and to recover macroscopic equation from the microscopic description. Various hints are given toward research perspectives, with special attention on the modelling of the interplay of microscopic (at the cellular level) biological and mechanical variables on the overall evolution of the system. Indeed the final aim of this paper consists of organizing the various contributions available in the literature into a mathematical framework suitable to generate a mathematical theory for complex biological systems.

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This paper develops a variety of mathematical tools to model the dynamics of large systems of interacting cells. Interactions are ruled not only by laws of classical mechanics, but also by some biological functions. The mathematical approach is the one of kinetic theory and non-equilibrium statistical mechanics. The paper deals both with the derivation of evolution equations and with the design of specific models consistent with the above-mentioned mathematical framework. Various hints for research perspectives are proposed in the last part of the paper.