Kinetic derivation of Aw-Rascle-Zhang-type traffic models with driver-assist vehicles

G. Dimarco, A. Tosin, M. Zanella

Preprint arXiv, 2021.

In this paper, we derive second order hydrodynamic traffic models from kinetic-controlled equations for driver-assist vehicles. At the vehicle level we take into account two main control strategies synthesising the action of adaptive cruise controls and cooperative adaptive cruise controls. The resulting macroscopic dynamics fulfil the anisotropy condition introduced in the celebrated Aw-Rascle-Zhang model. Unlike other models based on heuristic arguments, our approach unveils the main physical aspects behind frequently used hydrodynamic traffic models and justifies the structure of the resulting macroscopic equations incorporating driver-assist vehicles. Numerical insights show that the presence of driver-assist vehicles produces an aggregate homogenisation of the mean flow speed, hich may also be steered towards a suitable desired speed in such a way that optimal flows and traffic stabilisation are reached


Kinetic models for epidemic dynamics with social heterogeneity

G. Dimarco, B. Perthame, G. Toscani, M. Zanella

Preprint arXiv, 2020.

We introduce a mathematical description of the impact of sociality in the spread of infectious diseases by integrating an epidemiological dynamics with a kinetic modeling of population-based contacts. The kinetic description leads to study the evolution over time of Boltzmann-type equa- tions describing the number densities of social contacts of susceptible, infected and recovered indi- viduals, whose proportions are driven by a classical SIR-type compartmental model in epidemiology. Explicit calculations show that the spread of the disease is closely related to moments of the con- tact distribution. Furthermore, the kinetic model allows to clarify how a selective control can be assumed to achieve a minimal lockdown strategy by only reducing individuals undergoing a very large number of daily contacts. We conduct numerical simulations which confirm the ability of the model to describe different phenomena characteristic of the rapid spread of an epidemic. Motiv- ated by the COVID-19 pandemic, a last part is dedicated to fit numerical solutions of the proposed model with infection data coming from different European countries.


Control of tumour growth distributions through kinetic methods

L. Preziosi, G. Toscani, M. Zanella

Journal of Theoretical Biology, 514: 110579, 2021. (Preprint arXiv)

The mathematical modeling of tumor growth has a long history, and has been mathematically formulated in several different ways. Here we tackle the problem in the case of a continuous distribution using mathematical tools from statistical physics. To this extent, we introduce a novel kinetic model of growth which highlights the role of microscopic transitions in determining a variety of equilibrium distributions. At variance with other approaches, the mesoscopic description in terms of elementary interactions allows to design precise microscopic feedback control therapies, able to influence the natural tumor growth and to mitigate the risk factors involved in big sized tumors. We further show that under a suitable scaling both the free and controlled growth models correspond to Fokker–Planck type equations for the growth distribution with variable coefficients of diffusion and drift, whose steady solutions in the free case are given by a class of generalized Gamma densities which can be characterized by fat tails. In this scaling the feedback control produces an explicit modification of the drift operator, which is shown to strongly modify the emerging distribution for the tumor size. In particular, the size distributions in presence of therapies manifest slim tails in all growth models, which corresponds to a marked mitigation of the risk factors. Numerical results confirming the theoretical analysis are also presented.


Wealth distribution under the spread of infectious diseases

G. Dimarco, L. Pareschi, G. Toscani, M. Zanella

Physical Review E, 102: 022303, 2020. (Preprint arXiv).

We develop a mathematical framework to study the economic impact of infectious diseases by integrating epidemiological dynamics with a kinetic model of wealth exchange.  The multi-agent description leads to study the evolution over time of a system of kinetic equations for the wealth densities of susceptible, infectious and recovered individuals, whose proportions are driven by a classical compartmental model in epidemiology. Explicit calculations show that the spread of the disease seriously affects the distribution of wealth, which, unlike the situation in the absence of epidemics, can converge towards a stationary state with a bimodal form. Furthermore, simulations confirm the ability of the model to describe different phenomena characteristics of economic trends in situations compromised by the rapid spread of an epidemic, such as the unequal impact on the various wealth classes and the risk of a shrinking middle class.


Non-Maxwellian kinetic description of Follow-the-Leader traffic models

Andrea Tosin, Mattia Zanella

Preprint arXiv, 2019.

In this paper we consider a Boltzmann-type kinetic description of Follow-the-Leader traffic dynamics and we study the resulting asymptotic distributions, namely the counterpart of the Maxwellian distribution of the classical kinetic theory. In the Boltzmann-type equation we include a non-Maxwellian, viz. non-constant, collision kernel in order to exclude from the statistical model possibly unphysical interactions. In spite of the increased analytical difficulty caused by this further non-linearity, we show that a careful application of the quasi-invariant limit (an asymptotic procedure reminiscent of the grazing collision limit) successfully leads to a Fokker-Planck approximation of the original Boltzmann-type equation, whence stationary distributions can be explicitly computed. Our analytical results justify, from a genuinely model-based point of view, some empirical results found in the literature by interpolation of experimental data.

Reconstruction of traffic speed distributions from kinetic models with uncertainties

Michael Herty, Andrea Tosin, Giuseppe Visconti, Mattia Zanella

Preprint arXiv, 2019.

In this work we investigate the ability of a kinetic approach for traffic dynamics to predict speed distributions obtained through rough data. The present approach adopts the formalism of uncertainty quantification, since reaction strengths are uncertain and linked to different types of driver behaviour or different classes of vehicles present in the flow. Therefore, the calibration of the expected speed distribution has to face the reconstruction of the distribution of the uncertainty. We adopt experimental microscopic measurements recorded on a German motorway, whose speed distribution shows a multimodal trend. The calibration is performed by extrapolating the uncertainty parameters of the kinetic distribution via a constrained optimisation approach. The results confirm the validity of the theoretical set-up.

Model-based assessment of the impact of driver-assist vehicles using kinetic theory

Benedetto Piccoli, Andrea Tosin, Mattia Zanella

Zeitschrift für Angewandte Mathematik und Physik, 71:152, 2020.

In this paper we consider a kinetic description of follow-the-leader traffic models, which we use to study the effect of vehicle-wise driver-assist control strategies at various scales, from that of the local traffic up to that of the macroscopic stream of vehicles. We provide a theoretical evidence of the fact that some typical control strategies, such as the alignment of the speeds and the optimisation of the time headways, impact on the local traffic features (for instance, the speed and headway dispersion responsible for local traffic instabilities) but have virtually no effect on the observable macroscopic traffic trends (for instance, the flux/throughput of vehicles). This unobvious conclusion, which is in very nice agreement with recent field studies on autonomous vehicles, suggests that the kinetic approach may be a valid tool for an organic multiscale investigation and possibly design of driver-assist algorithms.


Kinetic modelling of multiple interactions in socio-economic systems

Giuseppe Toscani, Andrea Tosin, Mattia Zanella

Network & Heterogeneous Media, 15(3): 519-542, 2020. (Preprint arXiv)

Unlike the classical kinetic theory of rarefied gases, where microscopic interactions among gas molecules are described as binary collisions, the modelling of socio-economic phenomena in a multi-agent system naturally requires to consider, in various situations, multiple interactions among the individuals. In this paper, we collect and discuss some examples related to economic and gambling activities. In particular, we focus on a linearisation strategy of the multiple interactions, which greatly simplifies the kinetic description of such systems while maintaining all their essential aggregate features, including the equilibrium distributions.

Structure preserving schemes for Fokker-Planck equations with nonconstant diffusion matrices


Nadia Loy, Mattia Zanella

Mathematics and Computers in Simulation, to appear. (Preprint arXiv)

In this work we consider an extension of a recently proposed structure preserving numerical scheme for nonlinear Fokker-Planck-type equations to the case of nonconstant full diffusion matrices. While in existing works the schemes are formulated in a one-dimensional setting, here we consider exclusively the two-dimensional case. We prove that the proposed schemes preserve fundamental structural properties like nonnegativity of the solution without restriction on the size of the mesh and entropy dissipation. Moreover, all the methods presented here are at least second order accurate in the transient regimes and arbitrarily high order for large times in the hypothesis in which the flux vanishes at the stationary state. Suitable numerical tests will confirm the theoretical results.

Multiple-interaction kinetic modelling of a virtual-item gambling economy

Giuseppe Toscani, Andrea Tosin, Mattia Zanella

Physical Review E, 100(1): 012308, 2019.

In recent years, there has been a proliferation of online gambling sites, which made gambling more accessible with a consequent rise in related problems, such as addiction. Hence, the analysis of the gambling behaviour at both the individual and the aggregate levels has become the object of several investigations. In this paper, resorting to classical methods of the kinetic theory, we describe the behaviour of a multi-agent system of gamblers participating in lottery-type games on a virtual-item gambling market. The comparison with previous, often empirical, results highlights the ability of the kinetic approach to explain how the simple microscopic rules of a gambling-type game produce complex collective trends, which might be difficult to interpret precisely by looking only at the available data.