Category Archives: Optimization and Control

Breaking consensus in kinetic opinion formation models on graphons

by Mattia Zanella

B. Düring, J. Franceschi, M.-T. Wolfram, M. Zanella

Preprint arXiv, 2024

In this work we propose and investigate a strategy to prevent consensus in kinetic models for opinion formation. We consider a large interacting agent system, and assume that agent interactions are driven by compromise as well as self-thinking dynamics and also modulated by an underlying static social network.

This network structure is included using so-called graphons, which modulate the interaction frequency in the corresponding kinetic formulation. We then derive the corresponding limiting Fokker Planck equation, and analyze its large time behavior. This microscopic setting serves as a starting point for the proposed control strategy, which steers agents away from mean opinion and is characterised by a suitable penalization depending on the properties of the graphon. We show that this minimalist approach is very effective by analyzing the quasi-stationary solutions mean-field model in a plurality of graphon structures. Several numerical experiments are also provided the show the effectiveness of the approach in preventing the formation of consensus steering the system towards a declustered state.

On the optimal control of kinetic epidemic models with uncertain social features

by Mattia Zanella

J. Franceschi, A. Medaglia, M. Zanella.

Optimal Control, Applications and Methods, online first. (Preprint arXiv)

It is recognized that social heterogeneities in terms of the contact distribution have a strong influence on the spread of infectious diseases. Nevertheless, few data are available and their statistical description does not possess universal patterns and may vary spatially and temporally. It is therefore essential to design optimal control strategies, mimicking the effects of non-pharmaceutical interventions, to limit efficiently the number of infected cases.

In this work, starting from a recently introduced kinetic model for epidemiological dynamics that takes into account the impact of social contacts of individuals, we consider an uncertain contact formation dynamics leading to slim-tailed as well as fat-tailed distributions of contacts. Hence, we analyse the effects of an optimal control strategy of the system of agents. Thanks to classical methods of kinetic theory, we couple uncertainty quantification methods with the introduced mathematical model to assess the effects of social limitations. Finally, using the proposed modelling approach and starting from available data, we show the effectiveness of the proposed selective measures to dampen uncertainties together with the epidemic trends.

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

by Mattia Zanella

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.

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

by Mattia Zanella

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.

Optimal control of epidemic spreading in the presence of social heterogeneity

by Mattia Zanella

G. Dimarco, G. Toscani, M. Zanella

Philosophical Transactions of the Royal Society A, 380:20210160, 2022. (Preprint arXiv)

The spread of COVID-19 has been thwarted in most countries through non-pharmaceutical interventions. In particular, the most effective measures in this direction have been the stay-at-home and closure strategies of businesses and schools.

However, population-wide lockdowns are far from being optimal carrying  heavy economic consequences. Therefore, there is nowadays a strong interest in designing more efficient restrictions. In this work, starting from a recent  kinetic-type model which takes into account the heterogeneity described by the social contact of individuals, we analyze the effects of introducing an optimal control strategy into the system, to limit selectively the mean number of contacts and reduce consequently the number of infected cases. Thanks to a data-driven approach, we show that this new mathematical model permits to assess the effects of the social limitations.  Finally, using the model introduced here and starting from the available data, we show the effectivity of the proposed selective measures to dampen the epidemic trends.

Kinetic-controlled hydrodynamics for multilane traffic models

by Mattia Zanella

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.

Control with uncertain data of socially structured compartmental epidemic models

by Mattia Zanella

G. Albi, L. Pareschi, M. Zanella

Journal of Mathematical Biology, 82, 63, 2021. (Preprint arXiv)

The adoption of containment measures to reduce the amplitude of the epidemic peak is a key aspect in tackling the rapid spread of an epidemic. Classical compartmental models must be modified and studied to correctly describe the effects of forced external actions to reduce the impact of the disease. In addition, data are often incomplete and heterogeneous, so a high degree of uncertainty must naturally be incorporated into the models. In this work we address both these aspects, through an  optimal control formulation of the epidemiological model in presence of uncertain data. After the introduction of the optimal control problem, we formulate an instantaneous approximation of the control that allows us to derive new feedback controlled compartmental models capable of describing the epidemic peak reduction. The need for long-term interventions shows that alternative actions based on the social structure of the system can be as effective as the more expensive global strategy. The importance of the timing and intensity of interventions is particularly relevant in the case of uncertain parameters on the actual number of infected people. Simulations related to data from the recent COVID-19 outbreak in Italy are presented and discussed.

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

by Mattia Zanella

B. Piccoli, A. Tosin, M. 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.

 

Uncertainty damping in kinetic traffic models by driver-assist controls

by Mattia Zanella

Andrea Tosin, Mattia Zanella

Mathematical Control and Related Fields, 11(3): 681-713, 2021. (Preprint arXiv)

In this paper, we propose a kinetic model of traffic flow with uncertain binary interactions, which explains the scattering of the fundamental diagram in terms of the macroscopic variability of aggregate quantities, such as the mean speed and the flux of the vehicles, produced by the microscopic uncertainty. Moreover, we design control strategies at the level of the microscopic interactions among the vehicles, by which we prove that it is possible to dampen the propagation of such an uncertainty across the scales. Our analytical and numerical results suggest that the aggregate traffic flow may be made more ordered, hence predictable, by implementing such control protocols in driver-assist vehicles. Remarkably, they also provide a precise relationship between a measure of the macroscopic damping of the uncertainty and the penetration rate of the driver-assist technology in the traffic stream.

 

Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles

by Mattia Zanella

Andrea Tosin, Mattia Zanella

Multiscale Modeling and Simulation, 17(2): 716-749, 2019. (Preprint arXiv)

We develop a hierarchical description of traffic flow control by means of driver-assist vehicles aimed at the mitigation of speed-dependent road risk factors. Microscopic feedback control strategies are designed at the level of vehicle-to-vehicle interactions and then upscaled to the global flow via a kinetic approach based on a Boltzmann-type equation. Then first and second order hydrodynamic traffic models, which naturally embed the microscopic control strategies, are consistently derived from the kinetic-controlled framework via suitable closure methods. Several numerical examples illustrate the effectiveness of such a hierarchical approach at the various scales.