Monte Carlo stochastic Galerkin methods for non-Maxwellian kinetic models of multiagent systems with uncertainties

A. Medaglia, A. Tosin, M. Zanella

Partial Differential Equations and Applications, 3, 51, 2022. (Preprint arXiv)

In this paper, we focus on the construction of a hybrid scheme for the approximation of non- Maxwellian kinetic models with uncertainties. In the context of multiagent systems, the introduction of a kernel at the kinetic level is useful to avoid unphysical interactions.

The methods here proposed, combine a direct simulation Monte Carlo (DSMC) in the phase space together with stochastic Galerkin (sG) methods in the random space. The developed schemes preserve the main physical properties of the solution together with accuracy in the random space. The consistency of the methods is tested with respect to surrogate Fokker-Planck models that can be obtained in the quasi-invariant regime of parameters. Several applications of the schemes to non-Maxwellian models of multiagent systems are reported.

Effects of vaccination efficacy on wealth distribution in kinetic epidemic models

E. Bernardi, L. Pareschi, G. Toscani, M. Zanella

Entropy, 22:216, 2022. (Preprint arXiv)

The spreading of Covid-19 pandemic has highlighted the close link between economics and health in the context of emergency management. A widespread vaccination campaign is considered the main tool to contain the economic consequences. This paper will focus, at the level of wealth distribution modelling, on the economic improvements induced by the vaccination campaign in terms of its effectiveness rate. The economic trend during the pandemic is evaluated resorting to a mathematical model joining a classical compartmental model including vaccinated individuals with a kinetic model of wealth distribution based on binary wealth exchanges. The interplay between wealth exchanges and the progress of the infectious disease is realized by assuming on the one hand that individuals in different compartments act differently in the economic process and on the other hand that the epidemic affects risk in economic transactions. Using the mathematical tools of kinetic theory, it is possible to identify the equilibrium states of the system and the formation of inequalities due to the pandemic in the wealth distribution of the population. Numerical experiments highlight the importance of the vaccination campaign and its positive effects in reducing economic inequalities in the multi-agent society

Uncertainty quantification and control of kinetic models for tumour growth under clinical uncertainties

A. Medaglia, G. Colelli, L. Farina, A. Bacila, P. Bini, E. Marchioni, S. Figini, A. Pichiecchio, M. Zanella

International Journal of Non-Linear Mechanics, 141:103933. (Preprint arXiv)

In this work, we develop a kinetic model for tumour growth taking into account the effects of clinical uncertainties characterising the tumours’ progression.

The action of therapeutic protocols trying to steer the tumours’ volume towards a target size is then investigated by means of suitable selective-type controls acting at the level of cellular dynamics. By means of classical tools of statistical mechanics for many-agent systems, we are able to prove that it is possible to dampen clinical uncertainties across the scales. To take into account the scarcity of clinical data and the possible source of error in the image segmentation of tumours’ evolution, we estimated empirical distributions of relevant parameters that are considered to calibrate the resulting model obtained from real cases of primary glioblastoma. Suitable numerical methods for uncertainty quantification of the resulting kinetic equations are discussed and, in the last part of the paper, we compare the effectiveness of the introduced control approaches in reducing the variability in tumours’ size due to the presence of uncertain quantities.

A multi-agent description of the influence of higher education on social stratification

G. Dimarco, G. Toscani, M. Zanella

Journal of Economic Interaction and Coordination, in press. (Preprint arXiv)

We introduce and discuss a system of one-dimensional kinetic equations describing the influence of higher education in the social stratification of a multi-agent society.

The system is obtained by coupling a model for knowledge formation with a kinetic description of the social climbing in which the parameters characterizing the elementary interactions leading to the formation of a social elite are assumed to depend on the degree of knowledge/education of the agents. In addition, we discuss the case in which the education level of an individual is function of the position occupied in the social ranking. With this last assumption we obtain a fully coupled model in which knowledge and social status influence each other. In the last part, we provide several numerical experiments highlighting the role of education in reducing social inequalities and in promoting social mobility.

Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

G. Albi, G. Bertaglia, W. Boscheri, G. Dimarco, L. Pareschi, G. Toscani, M. Zanella.

Predicting Pandemics in a Globally Connected World Vol.1, 2022. (Preprint arXiv)

In this survey we report some recent results in the mathematical modeling of epidemic phenomena through the use of kinetic equations.

We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker’s perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed. Finally, the kinetic modeling is extended to spatially dependent settings using multiscale transport models that can characterize the impact of movement dynamics on epidemic advancement on both one-dimensional networks and realistic two-dimensional geographic settings.

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

G. Dimarco, A. Tosin, M. Zanella

Journal of Statistical Physics, 186, 17, 2022. (Preprint arXiv)

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-controlled hydrodynamics for multilane traffic models

R. Borsche, A. Klar, M. Zanella

Physica A: Statistical Mechanics and its Applications, 587: 126486, 2022. (Preprint arXiv)

We study the application of a recently introduced hierarchical description of traffic flow control by driver-assist vehicles to include lane changing dynamics. Lane-dependent feedback control strategies are implemented at the level of vehicles and the aggregate trends are studied by means of Boltzmann-type equations determining three different hydrodynamics based on the lane switching frequency. System of first order macroscopic equations describing the evolution of densities along the lanes are then consistently determined through a suitable closure strategy. Numerical examples are then presented to illustrate the features of the proposed hierarchical approach.

Kinetic models for epidemic dynamics with social heterogeneity

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

Journal of Mathematical Biology, 83, 4, 2021. (Preprint arXiv)

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.