Boltzmann-type models with uncertain binary interactions

In this paper we study binary interaction schemes with uncertain parameters for a general class of Boltzmann-type equations with applications in classical gas and aggregation dynamics. We consider deterministic (i.e., a priori averaged) and stochastic kinetic models, corresponding to different ways of understanding the role of uncertainty in the system dynamics, and compare some thermodynamic quantities of interest, such as the mean and the energy, which characterise the asymptotic trends. Furthermore, via suitable scaling techniques we derive the corresponding deterministic and stochastic Fokker-Planck equations in order to gain more detailed insights into the respective asymptotic distributions. We also provide numerical evidences of the trends estimated theoretically by resorting to recently introduced structure preserving uncertainty quantification methods

A fully-discrete-state kinetic theory approach to modeling vehicular traffic

This paper presents a new mathematical model of vehicular traffic, based on the methods of the generalized kinetic theory, in which the space of microscopic states (position and velocity) of the vehicles is genuinely discrete. While in the recent literature discrete-velocity kinetic models of car traffic have already been successfully proposed, this is, to our knowledge, the first attempt to account for all aspects of the physical granularity of car flow within the formalism of the aforesaid mathematical theory. Thanks to a rich but handy structure, the resulting model allows one to easily implement and simulate various realistic scenarios giving rise to characteristic traffic phenomena of practical interest (e.g., queue formation due to roadworks or to a traffic light). Moreover, it is analytically tractable under quite general assumptions, whereby fundamental properties of the solutions can be rigorously proved.Comment: 22 pages, 3 figure

Reducing complexity of multiagent systems with symmetry breaking: an application to opinion dynamics with polls

In this paper we investigate the possibility of reducing the complexity of a system composed of a large number of interacting agents, whose dynamics feature a symmetry breaking. We consider first order stochastic differential equations describing the behavior of the system at the particle (i.e., Lagrangian) level and we get its continuous (i.e., Eulerian) counterpart via a kinetic description. However, the resulting continuous model alone fails to describe adequately the evolution of the system, due to the loss of granularity which prevents it from reproducing the symmetry breaking of the particle system. By suitably coupling the two models we are able to reduce considerably the necessary number of particles while still keeping the symmetry breaking and some of its large-scale statistical properties. We describe such a multiscale technique in the context of opinion dynamics, where the symmetry breaking is induced by the results of some opinion polls reported by the media

Fundamental diagrams for kinetic equations of traffic flow

In this paper we investigate the ability of some recently introduced discrete kinetic models of vehicular traffic to catch, in their large time behavior, typical features of theoretical fundamental diagrams. Specifically, we address the so-called "spatially homogeneous problem" and, in the representative case of an exploratory model, we study the qualitative properties of its solutions for a generic number of discrete microstates. This includes, in particular, asymptotic trends and equilibria, whence fundamental diagrams originate.Comment: 14 page

The Changing Face of the Unaccompanied Alien Child: A Portrait of Foreign-Born Children in Federal Foster Care and How to Best Meet Their Needs

In June 2011, the United States Conference of Catholic Bishops/Migration and Refugee Services (USCCB/MRS) began an analysis of children placed in the Unaccompanied Alien Children (UAC) and Unaccompanied Refugee Minor (URM) foster care programs. To explore the changing face of the children coming into care, USCCB/MRS considered children referred for foster care services from the Department of Health and Human Services (DHHS)/Office of Refugee Resettlement (ORR) between October 1, 2007, and June 1, 2011. Of the 279 children referred to USCCB/MRS, the sample size for this paper included 98 children from across the study years. The goal of this paper is to inform ORR and other stakeholders about the profile of unaccompanied children entering foster care and how to better serve them and their needs. Through a greater understanding of the changing face of the UAC/URM population, all stakeholders can better shape their organizational capacity development to meet the increasingly complex needs of these children.This paper provides an in-depth analysis of the profile of children coming into federal foster care and how that profile has changed over the years. Across all study years, male referrals dominated, accounting for 65 percent, while 35 percent of the referrals were females. However, in fiscal year 2010, the number of female referrals more than doubled compared to previous years. The average age of children arriving in the United States and being referred to foster care was 16.02 for UAC and 16.38 for URM. The majority of children coming into care migrated from Honduras, Guatemala, and El Salvador. Children from these countries also reported a high incidence of violence in their home country as a reason for migration. Other reported reasons for migrating remained constant over the study years and included escaping violence in their homes, escaping abusive situations, reuniting with family, and seeking better educational and employment opportunities. However, children also have begun presenting with more complex needs and higher incidences of trauma, mental health issues, and substance use histories. Therefore, it was not surprising to find that about 85 percent of children in the study sample reported having some type of traumatic experience prior to entering ORR custody.Although the majority of trauma experiences occurred when children were in their home country, the number of children who experienced trauma, such as kidnapping or sexual or physical assault, during their journey to the United States increased throughout the study period. Consequently, it was not surprising to find that youth are coming into care with higher incidences of mental health and substance abuse problems. Although the specific mental health disorders remained consistent across the study years, the number of children in the sample with a diagnosed mental health disorder steadily increased, from 13 percent in fiscal year 2008 to 38 percent in fiscal year 2011. Interestingly, half the children identified with a mental health disorder at the time of referral came from Honduras; a majority of those children were male. The study also found a steady increase in reported substance use from 17 percent of the study sample in fiscal year 2008 to 33 percent in fiscal year 2011. Children reported using substances to alleviate mental health symptoms such as depression; however, none of the children received a formal diagnosis of substance abuse or dependence. Further, the study found that the average length of stay for youth in ORR-funded facilities decreased from almost eight months in fiscal year 2008 to less than six and a half months in the beginning months of fiscal year 2011.Children reported alarming stories about witnessing violence or being victims of crime while in their home country or during their journey to the United States. The common diagnoses of post-traumatic stress disorder, depression, and adjustment disorder demonstrate that the children had difficulty processing their trauma. Further, while the incidences of children with criminal histories and gang associations coming into care varied over the study period, it appears that children were more likely to be victims of crime and gang violence than perpetrators.RecommendationsEnsure foster care referral documentation includes all pertinent case information.Reviews of documents provided to USCCB/MRS during the foster care referral process revealed that the quality and quantity of information collected varied greatly across ORR-funded facilities and staff. Having a sound understanding of the child's background and trauma history, along with any current behavioral, mental health, or legal immigration case considerations, is vital to ensuring a successful foster care placement.Plan for placements for children close to age 18.The majority of children not placed into foster care were those approaching their eighteenth birthday. When children turn 18 without immigration relief or foster care placement, they are at risk of becoming homeless, exploited, and/or deported. Child welfare standards encourage the use of concurrent permanency planning that involves identifying and working toward a child's primary permanency goal (that is, legal relief) while simultaneously identifying and working on a secondary goal (that is, alternatives to detention if legal relief is not obtained).Increase availability of legal services across UAC foster care network.Consistently over the four years of the study, a number of UAC were not placed in foster care because they were so close to 18 at the time of referral. The availability of legal services for all UAC would ensure they have early representation to pursue immigration relief well before they reach the age of 18, which is especially important for those with complicated cases.Develop continuum of care, including therapeutic and group home options to increase placement match.Developing a continuum of care -- particularly creating more therapeutic and group home options -- is critical in meeting the varying needs of this population and ensuring placement availability for children with high needs. A number of children were not placed due to their behavioral and/or mental health issues. Therefore, it is crucial that ORR and the states support and provide resources for higher levels of care within the URM network.Conduct further research on UAC population.Although trauma, mental health, and substance abuse issues are commonly addressed with children in foster care in the United States, few scholarly articles address points raised in this paper. Similarities exist among children in domestic and federally funded foster care; however, risks associated with the migration journey to the United States add multiple layers for consideration when working with UAC. In particular, further study is recommended on the outcomes of these children once they have transitioned out of ORR custody

Uncertainty damping in kinetic traffic models by driver-assist controls

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

Multiphase modeling and qualitative analysis of the growth of tumor cords

In this paper a macroscopic model of tumor cord growth is developed, relying on the mathematical theory of deformable porous media. Tumor is modeled as a saturated mixture of proliferating cells, extracellular fluid and extracellular matrix, that occupies a spatial region close to a blood vessel whence cells get the nutrient needed for their vital functions. Growth of tumor cells takes place within a healthy host tissue, which is in turn modeled as a saturated mixture of non-proliferating cells. Interactions between these two regions are accounted for as an essential mechanism for the growth of the tumor mass. By weakening the role of the extracellular matrix, which is regarded as a rigid non-remodeling scaffold, a system of two partial differential equations is derived, describing the evolution of the cell volume ratio coupled to the dynamics of the nutrient, whose higher and lower concentration levels determine proliferation or death of tumor cells, respectively. Numerical simulations of a reference two-dimensional problem are shown and commented, and a qualitative mathematical analysis of some of its key issues is proposed.Comment: 34 pages, 18 figure

Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints

This paper is concerned with mathematical modeling of intelligent systems, such as human crowds and animal groups. In particular, the focus is on the emergence of different self-organized patterns from non-locality and anisotropy of the interactions among individuals. A mathematical technique by time-evolving measures is introduced to deal with both macroscopic and microscopic scales within a unified modeling framework. Then self-organization issues are investigated and numerically reproduced at the proper scale, according to the kind of agents under consideration.Comment: 24 pages, 13 figure

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

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

From individual behaviour to an evaluation of the collective evolution of crowds along footbridges

This paper proposes a crowd dynamic macroscopic model grounded on microscopic phenomenological observations which are upscaled by means of a formal mathematical procedure. The actual applicability of the model to real world problems is tested by considering the pedestrian traffic along footbridges, of interest for Structural and Transportation Engineering. The genuinely macroscopic quantitative description of the crowd flow directly matches the engineering need of bulk results. However, three issues beyond the sole modelling are of primary importance: the pedestrian inflow conditions, the numerical approximation of the equations for non trivial footbridge geometries, and the calibration of the free parameters of the model on the basis of in situ measurements currently available. These issues are discussed and a solution strategy is proposed.Comment: 23 pages, 10 figures in J. Engrg. Math., 201