Second order traffic flow models on road networks and real data applications

This thesis concerns macroscopic traffic models and data-driven models. In the first part we deal with the extension of Generic Second Order Models (GSOM) for traffic flow to road networks. We define a Riemann Solver at the junction based on a priority rule, providing an iterative algorithm able to build the solution to junctions with n incoming and m outgoing roads. The logic underlying our solver is the following: the flow is maximised respecting the priority rule, but the latter can be modified if the outgoing road supply exceeds the demand of the road with higher priority. We provide bounds on the total variation of waves interacting with the junction, giving explicit computations for intersections with two incoming and two outgoing roads. These estimates are fundamental to prove the existence of weak solutions to Cauchy problems on networks via Wave-Front-Tracking. GSOM are used to analyse traffic dynamics and their effects on the production of pollutant emissions. First we apply the proposed Riemann Solver to simulate traffic dynamics on diverge and merge junctions and on roundabouts obtained by combining these two types of intersection. Then, we propose a methodology to estimate the pollutant emissions deriving from traffic dynamics. The emission model is calibrated and validated using the NGSIM dataset of real trajectory data. Furthermore, we set up a minimisation problem aimed at finding the optimal priority rule for our Riemann Solver that reduces the emission rates due to the traffic dynamic. Finally, we analyse some chemical reactions which lead to the production of ozone, focusing on the effects on pollution of the presence of traffic lights on the road. Next, we introduce a macroscopic two-dimensional multi-class traffic model on a single road, aimed at including lane-changes and different types of vehicles. The multi-class model consists of a coupled system of conservation laws in two space dimensions. Besides the study of the Riemann problems, we present a Lax-Friedrichs type discretisation scheme and we recover the theoretical results by means of numerical tests. We then calibrate and validate the multi-class model with real trajectory data and we test its ability of simulating vehicles overtaking. Finally, we present a new methodology to recover mass movements from snapshots of its distribution. To this end we put in place an algorithm based on the combination of two methods: first, we use the dynamic mode decomposition to create a system of equations describing the mass transfer; second, we use the Wasserstein distance to reconstruct the underlying velocity field that is responsible for the displacement. We conclude this part with a real-life application: the algorithm is employed to study the travel flows of people in large populated areas using, as input, presence data of people in given region domains derived from the mobile phone network, at different time instants

Fractional SIS epidemic models

In this paper we consider the fractional SIS epidemic model ($\alpha$-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order $\alpha \uparrow 1$ (the SIS model) is also given. We analyse the effects of the fractional derivatives by comparing the SIS and the $\alpha$-SIS models.Comment: 17 pages, 6 figure

Understanding Mass Transfer Directions via Data-Driven Models with Application to Mobile Phone Data

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 partial differential equations (PDEs), 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 instances. This kind of data are provided by the Italian telecommunication company TIM and are derived by mobile phone usage.Comment: 19 pages, 10 figure

Understanding Human Mobility Flows from Aggregated Mobile Phone Data

In this paper we deal with the study of travel flows and patterns of people in large populated areas. Information about the movements of people is extracted from coarse-grained aggregated cellular network data without tracking mobile devices individually. Mobile phone data are provided by the Italian telecommunication company TIM and consist of density profiles (i.e. the spatial distribution) of people in a given area at various instants of time. By computing a suitable approximation of the Wasserstein distance between two consecutive density profiles, we are able to extract the main directions followed by people, i.e. to understand how the mass of people distribute in space and time. The main applications of the proposed technique are the monitoring of daily flows of commuters, the organization of large events, and, more in general, the traffic management and control.Comment: 6 pages, 14 figure

Forecasting Visitors’ behaviour in Crowded Museums

In this paper, we tackle the issue of measuring and understanding the visitors’ dynamics in a crowded museum in order to create and calibrate a predictive mathematical model. The model is then used as a tool to manage, control and optimize the fruition of the museum. Our contribution comes with one successful use case, the Galleria Borghese in Rome, Italy

A Reduced Order Model formulation for left atrium flow: an Atrial Fibrillation case

A data-driven Reduced Order Model (ROM) based on a Proper Orthogonal Decomposition - Radial Basis Function (POD-RBF) approach is adopted in this paper for the analysis of blood flow dynamics in a patient-specific case of Atrial Fibrillation (AF). The Full Order Model (FOM) is represented by incompressible Navier-Stokes equations, discretized with a Finite Volume (FV) approach. Both the Newtonian and the Casson's constitutive laws are employed. The aim is to build a computational tool able to efficiently and accurately reconstruct the patterns of relevant hemodynamics indices related to the stasis of the blood in a physical parametrization framework including the cardiac output in the Newtonian case and also the plasma viscosity and the hematocrit in the non-Newtonian one. Many FOM-ROM comparisons are shown to analyze the performance of our approach as regards errors and computational speed-up.Comment: 21 pages, 14 figure

A two-dimensional multi-class traffic flow model

The aim of this work is to introduce a two-dimensional macroscopic traffic model for multiple populations of vehicles. Starting from the paper [20], where a two-dimensional model for a single class of vehicles is proposed, we extend the dynamics to a multi-class model leading to a coupled system of conservation laws in two space dimensions. Besides the study of the Riemann problems we also present a Lax-Friedrichs type discretization scheme recovering the theoretical results by means of numerical tests. We calibrate the multi-class model with real data and compare the fitted model to the real trajectories. Finally, we test the ability of the model to simulate the overtaking of vehicles.Comment: 21 pages, 14 figure