Generalized Wasserstein distances allow us to quantitatively compare two continuous or atomic mass distributions with equal or different total masses. In this paper, we propose four numerical methods for the approximation of three different generalized Wasserstein distances introduced in the past few years, giving some insights into their physical meaning. After that, we explore their usage in the context of a sensitivity analysis of differential models for traffic flow. The quantification of the models’ sensitivity is obtained by computing the generalized Wasserstein distances between two (numerical) solutions corresponding to different inputs, including different boundary conditions.

Releasing capsules are widely employed in biomedical applications as smart carriers of therapeutic agents, including drugs and bioactive compounds. Such delivery vehicles typically consist of a loaded core, enclosed by one or multiple concentric coating strata. In this work, we extended existing mechanistic models to account for such multi-strata structures, including possible concurrent erosion of the capsule itself, and we characterized the release kinetics of the active substance into the surrounding medium. We presented a computational study of drug release from a spherical microcapsule, modeled through a non-linear diffusion equation incorporating radial asymmetric diffusion and space- and time-discontinuous coefficients, as suggested by the experimental data specifically collected for this study. The problem was solved numerically using a finite volume scheme on a grid with adaptive spatial and temporal resolution. Analytical expressions for concentration and cumulative release were derived for all strata, enabling the exploration of parameter sensitivity—such as coating permeability and internal diffusivity—on the overall release profile. The resulting release curves provide mechanistic insight into the transport processes and offer design criteria for achieving controlled release. Model predictions were benchmarked against in vitro experimental data obtained under physiologically relevant conditions, showing good agreement and validating the key features of the model. The proposed model thus serves as a practical tool for predicting the behavior of composite coated particles, supporting performance evaluation and the rational design of next-generation drug delivery systems with reduced experimental effort.

The increasing availability of traffic data from sensor networks has created new opportunities for understanding vehicular dynamics and identifying anomalies. In this study, we employ clustering techniques to analyse traffic flow data with the dual objective of uncovering meaningful traffic patterns and detecting anomalies, including sensor failures and irregular congestion events. We explore multiple clustering approaches, i.e. partitioning and hierarchical methods, combined with various time series representations and similarity measures. Our methodology is applied to real-world data from highway sensors, enabling us to assess the impact of different clustering frameworks on traffic pattern recognition. We also introduce a clustering-driven anomaly detection methodology that identifies deviations from expected traffic behaviour based on distance-based anomaly scores. Results indicate that hierarchical clustering with symbolic representations provides robust segmentation of traffic patterns, while partitioning methods such as k-means and fuzzy c-means yield meaningful results when paired with Dynamic Time Warping. The proposed anomaly detection strategy successfully identifies sensor malfunctions and abnormal traffic conditions with minimal false positives, demonstrating its practical utility for real-time monitoring. Real-world vehicular traffic data are provided by Autostrade Alto Adriatico S.p.A.

The study at the Peggy Guggenheim Collection in Venice highlights critical interactions between indoor air quality, visitor dynamics, and microclimatic conditions, offering insights into preventive conservation of modern artworks. By analyzing pollutants such as ammonia, formaldehyde, and organic acids, alongside visitor density and environmental data, the research identified key patterns and risks. Through three seasonal monitoring campaigns, the concentrations of SO2 (sulphur dioxide), NO (nitric oxide), NO2 (nitrogen dioxide), NOx (nitrogen oxides), HONO (nitrous acid), HNO3 (nitric acid), O3 (ozone), NH3 (ammonia), CH3COOH (acetic acid), HCOOH (formic acid), and HCHO (formaldehyde) were determined using passive samplers, as well as temperature and relative humidity data loggers. In addition, two specific short-term monitoring campaigns focused on NH3 were performed to evaluate the influence of visitor presence on indoor concentrations of the above compounds and environmental parameters. NH3 and HCHO concentrations spiked during high visitor occupancy, with NH3 levels doubling in crowded periods. Short-term NH3 campaigns confirmed a direct correlation between visitor numbers and the above indoor concentrations, likely due to human emissions (e.g., sweat, breath) and off-gassing from materials. The indoor/outdoor ratios indicated that several pollutants originated from indoor sources, with ammonia and acetic acid showing the highest indoor concentrations. By measuring the number of visitors and microclimate parameters (temperature and humidity) every 3 s, we were able to precisely estimate the causality and the temporal shift between these quantities, both at small time scale (a few minute delay between peaks) and at medium time scale (daily average conditions due to the continuous inflow and outflow of visitors).

In this paper we propose a novel macroscopic (fluid dynamics) model for describing pedestrian flow in low and high density regimes. The model is characterized by the fact that the maximal density reachable by the crowd – usually a fixed model parameter – is instead a state variable. To do that, the model couples a conservation law, devised as usual for tracking the evolution of the crowd density, with a Burgers-like PDE with a nonlocal term describing the evolution of the maximal density. The variable maximal density is used here to describe the effects of the psychological/physical pushing forces which are observed in crowds during competitive or emergency situations. Specific attention is also dedicated to the fundamental diagram, i.e., the function which expresses the relationship between crowd density and flux. Although the model needs a well defined fundamental diagram as known input parameter, it is not evident a priori which relationship between density and flux will be actually observed, due to the time-varying maximal density. An a posteriori analysis shows that the observed fundamental diagram has an elongated “tail” in the congested region, thus resulting similar to the concave/concave fundamental diagram with a “double hump” observed in real crowds. The main features of the model are investigated through 1D and 2D numerical simulations. The numerical code for the 1D simulation is freely available on this Gitlab repository.

In this work, we deal with a mathematical model describing the dissolution process of irregularly shaped particles. In particular, we consider a complete dissolution model accounting for surface kinetics, convective diffusion, and relative velocity between fluid and dissolving particles, for three drugs with different solubility and wettability: theophylline, griseofulvin, and nimesulide. The possible subsequent recrystallization process in the bulk fluid is also considered. The governing differential equations are revisited in the context of the level-set method and Hamilton-Jacobi equations, then they are solved numerically. This choice allows us to deal with the simultaneous dissolution of hundreds of different polydisperse particles. We show the results of many computer simulations which investigate the impact of the particle size, shape, area/volume ratio, and the dependence of the interfacial mass transport coefficient on the surface curvature.

In this paper, we derive a kinetic description of swarming particle dynamics in an interacting multi-agent system featuring emerging leaders and followers. Agents are classically characterized by their position and velocity plus a continuous parameter quantifying their degree of leadership. The microscopic processes ruling the change of velocity and degree of leadership are independent, non-conservative and non-local in the physical space, so as to account for long-range interactions. Out of the kinetic description, we obtain then a macroscopic model under a hydrodynamic limit reminiscent of that used to tackle the hydrodynamics of weakly dissipative granular gases, thus relying in particular on a regime of small non-conservative and short-range interactions. Numerical simulations in one- and two-dimensional domains show that the limiting macroscopic model is consistent with the original particle dynamics and furthermore can reproduce classical emerging patterns typically observed in swarms.

In this paper, we develop new methods to join machine learning techniques and macroscopic differential models, aimed at estimate and forecast vehicular traffic. This is done to complement respective advantages of data-driven and model-driven approaches. We consider here a dataset with flux and velocity data of vehicles moving on a highway, collected by fixed sensors and classified by lane and by class of vehicle. By means of a machine learning model based on an LSTM recursive neural network, we extrapolate two important pieces of information: (1) if congestion is appearing under the sensor, and (2) the total amount of vehicles which is going to pass under the sensor in the next future (30 min). These pieces of information are then used to improve the accuracy of an LWR-based first-order multi-class model describing the dynamics of traffic flow between sensors. The first piece of information is used to invert the (concave) fundamental diagram, thus recovering the density of vehicles from the flux data, and then inject directly the density datum in the model. This allows one to better approximate the dynamics between sensors, especially if an accident/bottleneck happens in a not monitored stretch of the road. The second piece of information is used instead as boundary conditions for the equations underlying the traffic model, to better predict the total amount of vehicles on the road at any future time. Some examples motivated by real scenarios will be discussed. Real data are provided by the Italian motorway company Autovie Venete S.p.A.

This paper investigates the model for pedestrian flow firstly proposed in [Cristiani, Priuli, and Tosin, SIAM J. Appl. Math., 75:605-629, 2015]. The model assumes that each individual in the crowd moves in a known domain, aiming at minimizing a given cost functional. Both the pedestrian dynamics and the cost functional itself depend on the position of the whole crowd. In addition, pedestrians are assumed to have predictive abilities, but limited in time.

In this paper we deal with pedestrian modeling, aiming at simulating crowd behavior in normal and emergency scenarios, including highly congested mass events. We are specifically concerned with a new agent-based, continuous-in-space, discrete-in-time, nondifferential model, where pedestrians have finite size and are compressible to a certain extent. The model also takes into account the pushing behavior appearing at extremely high densities. The main novelty is that pedestrians are not assumed to generate any kind of "field" which governs the dynamics of the others in the space around them. Instead, the behavior of each pedestrian solely relies on its knowledge of the environment and the evaluation of interpersonal distances between it and the others. The model is able to reproduce the concave/concave fundamental diagram with a "double hump" (i.e. with a second peak) which shows up when body forces come into play. We present several numerical tests (some of them being inspired by the recent ISO 20414 standard), which show how the model can reproduce classical self-organizing patterns.

Computer Vision and 3D printing have rapidly evolved in the last 10 years but interactions among them have been very limited so far, despite the fact that they share several mathematical techniques. We try to fill the gap presenting an overview of some techniques for Shape-from-Shading problems as well as for 3D printing with an emphasis on the approaches based on nonlinear partial differential equations and optimization. We also sketch possible couplings to complete the process of object manufacturing starting from one or more images of the object and ending with its final 3D print. We will give some practical examples of this procedure.

Presents several mathematical problems for 3D and 3D printing plus a survey that gives the links between the two areas Contains contributions from highly reputed academic and industrial researchers with a long experience Shows several techniques, applications and benchmarks that can be useful for young researchers approaching the field

In this paper we devise a microscopic (agent-based) mathematical model for reproducing crowd behavior in a specific scenario: a number of pedestrians, consisting of numerous social groups, flow along a corridor until a gate located at the end of the corridor closes. People are not informed about the closure of the gate and perceive the blockage observing dynamically the local crowd conditions. Once people become aware of the new conditions, they stop and then decide either to stay, waiting for reopening, or to go back and leave the corridor forever. People going back hit against newly incoming people creating a dangerous counter-flow. We run several numerical simulations varying parameters which control the crowd behavior, in order to understand the factors which have the greatest impact on the system dynamics. We also study the optimal way to inform people about the blockage in order to prevent the counter-flow. We conclude with some useful suggestions directed to the organizers of mass events.

Starting from recent experimental observations of starlings and jackdaws, we propose a minimal agent-based mathematical model for bird flocks based on a system of second-order delayed stochastic differential equations with discontinuous (both in space and time) right-hand side. The model is specifically designed to reproduce self-organized spontaneous sudden changes of direction, not caused by external stimuli like predator's attacks. The main novelty of the model is that every bird is a potential turn initiator, thus leadership is formed in a group of indistinguishable agents. We investigate some theoretical properties of the model and we show the numerical results. Biological insights are also discussed.

We present an all-around study of the visitors flow in crowded museums: a combination of Lagrangian field measurements and statistical analyses enable us to create stochastic digital-twins of the guest dynamics, unlocking comfort- and safety-driven optimizations. Our case study is the Galleria Borghese museum in Rome (Italy), in which we performed a real-life data acquisition campaign. We specifically employ a Lagrangian IoT-based visitor tracking system based on Raspberry Pi receivers, displaced in fixed positions throughout the museum rooms, and on portable Bluetooth Low Energy beacons handed over to the visitors. Thanks to two algorithms: a sliding window-based statistical analysis and an MLP neural network, we filter the beacons RSSI and accurately reconstruct visitor trajectories at room-scale. Via a clustering analysis, hinged on an original Wasserstein-like trajectory-space metric, we analyze the visitors paths to get behavioral insights, including the most common flow patterns. On these bases, we build the transition matrix describing, in probability, the room-scale visitor flows. Such a matrix is the cornerstone of a stochastic model capable of generating visitor trajectories in silico. We conclude by employing the simulator to enhance the museum fruition while respecting numerous logistic and safety constraints. This is possible thanks to optimized ticketing and new entrance/exit management.

In this paper, we propose two models describing the dynamics of heavy and light vehicles on a road network, taking into account the interactions between the two classes. The models are tailored for two-lane highways where heavy vehicles cannot overtake. This means that heavy vehicles cannot saturate the whole road space, while light vehicles can. In these conditions, the creeping phenomenon can appear, i.e., one class of vehicles can proceed even if the other class has reached the maximal density. The first model we propose couples two first-order macroscopic LWR models, while the second model couples a second-order microscopic follow-the-leader model with a first-order macroscopic LWR model. Numerical results show that both models are able to catch some second-order (inertial) phenomena such as stop and go waves. Models are calibrated by means of real data measured by fixed sensors placed along the A4 Italian highway Trieste-Venice and its branches, provided by Autovie Venete S.p.A.

The idea that individuals tend to choose a romantic partner following similarities on personality traits has always attracted much attention in the psychological literature, although results were controversial. We conducted a new data analysis approach to personality traits of 235 newlywed couples. Univariate analysis revealed that a neurotic husband is usually paired with a lesser extrovert and open wife. To figure out if this mating selection pattern may be translated in a mathematical predictive model a twofold approach was employed by using Partial Least Squares regression and machine learning algorithm. The experimental results demonstrate that marital assortment for personality is a multi-trait complementarity process but these data are unable to predict human mating.

In this survey we consider mathematical models and methods recently developed to control crowd dynamics, with particular emphasis on egressing pedestrians. We focus on two control strategies: the first one consists in using special agents, called leaders, to steer the crowd towards the desired direction. Leaders can be either hidden in the crowd or recognizable as such. This strategy heavily relies on the power of the social influence (herding effect), namely the natural tendency of people to follow group mates in situations of emergency or doubt. The second one consists in modify the surrounding environment by adding in the walking area multiple obstacles optimally placed and shaped. The aim of the obstacles is to naturally force people to behave as desired. Both control strategies discussed in this paper aim at reducing as much as possible the intervention on the crowd. Ideally the natural behavior of people is kept, and people do not even realize they are being led by an external intelligence. Mathematical models are discussed at different scales of observation, showing how macroscopic (fluid-dynamic) models can be derived by mesoscopic (kinetic) models which, in turn, can be derived by microscopic (agent-based) models.

The aim of this paper is to solve an inverse problem which regards a mass moving in a bounded domain. We assume that the mass moves following an unknown velocity field and that the evolution of the mass density can be described by a partial differential equation, which is also unknown. The input data of the problems are given by some snapshots of the mass distribution at certain times, while the sought output is the velocity field that drives the mass along its displacement. To this aim, we put in place an algorithm based on the combination of two methods: first, we use the dynamic mode decomposition to create a mathematical model describing the mass transfer; second, we use the notion of Wasserstein distance (also known as earth mover's distance) to reconstruct the underlying velocity field that is responsible for the displacement. Finally, we consider a real-life application: the algorithm is employed to study the travel flows of people in large populated areas using, as input data, density profiles (i.e., the spatial distribution) of people in given areas at different time instants. These kinds of data are provided by the Italian telecommunication company TIM and are derived by mobile phone usage.

We tackle the issue of measuring and analyzing the visitors’ dynamics in crowded museums. We propose an IoT-based system – supported by artificial intelligence models – to reconstruct the visitors’ trajectories throughout the museum spaces. Thanks to this tool, we are able to gather wide ensembles of visitors’ trajectories, allowing useful insights for the facility management and the preservation of the art pieces. Our contribution comes with one successful use case: the Galleria Borghese in Rome, Italy.