Article

A weakly coupled model of differential equations for thief tracking

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

In this work we introduce a novel model for the tracking of a thief moving through a road network. The modeling equations are given by a strongly coupled system of scalar conservation laws for the road traffic and ordinary differential equations for the thief evolution. A crucial point is the characterization at intersections, where the thief has to take a routing decision depending on the available local information. We develop a numerical approach to solve the thief tracking problem by combining a time-dependent shortest path algorithm with the numerical solution of the traffic flow equations. Various computational experiments are presented to describe different behavior patterns.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... Recent publications couple microscopic and macroscopic models to analyze the path of a single vehicle that travels along a road, see [1,6,13]. Unlike strongly-coupled models [6], a weakly-coupled approach [1] assumes that the traffic influences the vehicle but not vice versa when the trajectory of the vehicle is given by an ordinary differential equation. In the case of congestion, the right-hand side of this ordinary differential equation becomes discontinuous and we cannot expect a classical solution any more. ...
... Therefore, we take a closer look at the travel time on a road e ∈ E and the waiting time at a junction v with e ∈ δ + (v). The definition of the travel time is based on the formulation in [13]. Figure 2 shows an illustration of the notation that will be utilized in the following. ...
Article
Full-text available
This article deals with the modeling for an individual car path through a road network, where the dynamics is driven by a coupled system of ordinary and partial differential equations. The network is characterized by bounded buffers at junctions that allow for the interpretation of roundabouts or on‐ramps while the traffic dynamics is based on first‐order macroscopic equations of Lighthill‐Whitham‐Richards (LWR) type. Trajectories for single drivers are then influenced by the surrounding traffic and can be tracked by appropriate numerical algorithms. The computational experiments show how the modeling framework can be used as navigation device.
... Recent publications couple microscopic and macroscopic models to analyze the path of a single vehicle that travels along a road, see [1,6,13]. Unlike strongly-coupled models [6], a weakly-coupled approach [1] assumes that the traffic influences the vehicle but not vice versa when the trajectory of the vehicle is given by an ordinary differential equation. In the case of congestion, the right-hand side of this ordinary differential equation becomes discontinuous and we cannot expect a classical solution any more. ...
... Therefore, we take a closer look at the travel time on a road e ∈ E and the waiting time at a junction v with e ∈ δ + (v). The definition of the travel time is based on the formulation in [13]. Figure 2 shows an illustration of the notation that will be utilized in the following. ...
Preprint
This article deals with the modeling for an individual car path through a road network, where the dynamics is driven by a coupled system of ordinary and partial differential equations. The network is characterized by bounded buffers at junctions that allow for the interpretation of roundabouts or on-ramps while the traffic dynamics is based on first-order macroscopic equations of Lighthill-Whitham-Richards (LWR) type. Trajectories for single drivers are then influenced by the surrounding traffic and can be tracked by appropriate numerical algorithms. The computational experiments show how the modeling framework can be used as navigation device.
Chapter
The first step in the investigation of transport models for aggregation and movement, is represented by the study of one-equation models. To emphasise the complexity of these models, we start with a variety of hyperbolic models for car traffic and pedestrian traffic (since the models for collective movement of pedestrians are a natural extension of the car traffic models, and moreover traffic-like aspects can be found in many biological systems). Next, we discuss models for animal movement that incorporate constant or linear velocity functions. We review also models with reaction terms describing the inflow/outflow of cars and populations. In the context of animal movement, we present in more detail an analytical investigation of the speed of travelling waves. We conclude with a very brief discussion of numerical approaches for advection equations.
Article
Full-text available
Hotspots of crime localized in space and time are well documented. Previous mathematical models of urban crime have exhibited these hotspots but considered a static or otherwise suboptimal police response to them. We introduce a program of police response to hotspots of crime in which the police adapt dynamically to changing crime patterns. In particular, they choose their deployment to solve an optimal control problem at every time. This gives rise to a free boundary problem for the police deployment's spatial support. We present an efficient algorithm for solving this problem numerically and show that police presence can prompt surprising interactions among adjacent hotspots.
Article
Full-text available
We introduce a numerical method for tracking a bus trajectory on a road network. The mathematical model taken into consideration is a strongly coupled PDE-ODE system: the PDE is a scalar hyperbolic conservation law describing the traffic flow while the ODE, that describes the bus trajectory, needs to be intended in a Carathéodory sense. The moving constraint is given by an inequality on the flux which accounts for the bottleneck created by the bus on the road. The finite volume algorithm uses a locally non-uniform moving mesh which tracks the bus position. Some numerical tests are shown to describe the behavior of the solution.
Article
Full-text available
Given a discrete sample of event locations, we wish to produce a probability density that models the relative probability of events occurring in a spatial domain. Standard density estimation techniques do not incorporate priors informed by spatial data. Such methods can result in assigning significant positive probability to locations where events cannot realistically occur. In particular, when modelling residential burglaries, standard density estimation can predict residential burglaries occurring where there are no residences. Incorporating the spatial data can inform the valid region for the density. When modelling very few events, additional priors can help to correctly fill in the gaps. Learning and enforcing correlation between spatial data and event data can yield better estimates from fewer events. We propose a non-local version of maximum penalized likelihood estimation based on the H(1) Sobolev seminorm regularizer that computes non-local weights from spatial data to obtain more spatially accurate density estimates. We evaluate this method in application to a residential burglary dataset from San Fernando Valley with the non-local weights informed by housing data or a satellite image.
Article
Full-text available
This article is devoted to traffic flow networks including traffic lights at intersections. Mathematically, we consider a nonlinear dynamical traffic model where traffic lights are modeled as piecewise constant functions for red and green signals. The involved control problem is to find stop and go configurations depending on the current trafic volume. We propose a numerical solution strategy and present computational results.
Article
Full-text available
Objectives The main objective of this study was to see if the characteristics of offenders’ crimes exhibit spatial patterning in crime neutral areas by examining the relationship between simulated travel routes of offenders along the physical road network and the actual locations of their crimes in the same geographic space. Method This study introduced a Criminal Movement model (CriMM) that simulates travel patterns of known offenders. Using offenders’ home locations, locations of major attractors (e.g., shopping centers), and variations of Dijkstra’s shortest path algorithm we modeled the routes that offenders are likely to take when traveling from their home to an attractor. We then compare the locations of offenders’ crimes to these paths and analyze their proximity characteristics. This process was carried out using data on 7,807 property offenders from five municipalities in the Greater Vancouver Regional District (GVRD) in British Columbia, Canada. Results The results show that a great proportion of crimes tend to be located geographically proximal to the simulated travel paths with a distance decay pattern characterizing the distribution of distance measures. Conclusion: These results lend support to Crime Pattern Theory and the idea that there is an underlying pattern to crimes in crime neutral areas.
Article
Full-text available
Motivated by empirical observations of spatio-temporal clusters of crime across a wide variety of urban settings, we present a model to study the emergence, dynamics, and steady-state properties of crime hotspots. We focus on a two-dimensional lattice model for residential burglary, where each site is characterized by a dynamic attractiveness variable, and where each criminal is represented as a random walker. The dynamics of criminals and of the attractiveness field are coupled to each other via specific biasing and feedback mechanisms. Depending on parameter choices, we observe and describe several regimes of aggregation, including hotspots of high criminal activity. On the basis of the discrete system, we also derive a continuum model; the two are in good quantitative agreement for large system sizes. By means of a linear stability analysis we are able to determine the parameter values that will lead to the creation of stable hotspots. We discuss our model and results in the context of established criminological and sociological findings of criminal behavior.
Article
Full-text available
This paper treats five discrete shortest-path problems: (1) determining the shortest path between two specified nodes of a network; (2) determining the shortest paths between all pairs of nodes of a network; (3) determining the second, third, etc., shortest path; (4) determining the fastest path through a network with travel times depending on the departure time; and (5) finding the shortest path between specified endpoints that passes through specified intermediate nodes. Existing good algorithms are identified while some others are modified to yield efficient procedures. Also, certain misrepresentations and errors in the literature are demonstrated.
Article
Full-text available
We introduce a model for gas flows in pipeline networks based on isothermal Euler equations. We model the intersection of multiple pipes by posing an additional assumption on the pressure at the interface. We give a method to obtain solutions to the gas network problem, and present numerical results for sample networks.
Article
Full-text available
In this paper we introduce a computation algorithm to trace car paths on road networks, whose load evolution is modeled by conservation laws. This algorithm is composed by two parts: computation of solutions to conservation equations on each road and localization of car position resulting by interactions with waves produced on roads. Some applications and examples to describe the behavior of a driver traveling in a road network are showed. Moreover a convergence result for wave front tracking approximate solutions, with BV initial data on a single road is established.
Article
Full-text available
We present a weakly nonlinear analysis of our recently developed model for the formation of crime patterns. Using a perturbative approach, we nd amplitude equations that govern the development of crime \hotspot" patterns in our system in both the 1D and 2D cases. In addition to the supercritical spots already shown to exist in our previous work, we prove here the existence of subcritical hotspots that arise via subcritical pitchfork bifurcations or transcritical bifurcations, depending on geometry. We present numerical results that both validate our analytical ndings and conrm the existence of these subcritical hotspots as stable states. Finally, we examine the dierences between these two types of hotspots with regard to attempted hotspot suppression,
Article
Full-text available
We present a new class of macroscopic models for pedestrian flows. Each individual is assumed to move towards a fixed target, deviating from the best path according to the instantaneous crowd distribution. The resulting equation is a conservation law with a nonlocal flux. Each equation in this class generates a Lipschitz semigroup of solutions and is stable with respect to the functions and parameters defining it. Moreover, key qualitative properties such as the boundedness of the crowd density are proved. Specific models are presented and their qualitative properties are shown through numerical integrations.
Article
Full-text available
We present a strict Lyapunov function for hyperbolic systems of conservation laws that can be diagonalized with Riemann invariants. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. It is shown that the derived boundary control allows to guarantee the local convergence of the state towards a desired set point. Furthermore, the control can be implemented as a feedback of the state only measured at the boundaries. The control design method is illustrated with an hydraulic application, namely the level and flow regulation in an horizontal open channel
Article
Full-text available
We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408--463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 1892--1917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.
Article
We discuss continuous traffic flow network models including traffic lights. A mathematical model for traffic light settings within a macroscopic continuous traffic flow network is presented, and theoretical properties are investigated. The switching of the traffic light states is modeled as a discrete decision and is subject to optimization. A numerical approach for the optimization of switching points as a function of time based upon the macroscopic traffic flow model is proposed. The numerical discussion relies on an equivalent reformulation of the original problem as well as a mixed-integer discretization of the flow dynamics. The large-scale optimization problem is solved using derived heuristics within the optimization process. Numerical experiments are presented for a single intersection as well as for a road network.
Article
Motivated by a problem of traffic flow, we consider an ordinary differential equation whose vector field depends upon the solution of a conservation law. The Cauchy problem related to the O.D.E. does not fit in the results currently available in the literature. Here we address its well posedness.
Article
In this article, an evacuation model describing the egress in case of danger is considered. The underlying evacuation model is based on continuous network flows, while the spread of some gaseous hazardous material relies on an advection-diffusion equation. The contribution of this work is twofold. First, we introduce a continuous model coupled to the propagation of hazardous material where special cost functions allow for incorporating the predicted spread into an optimal planning of the egress. Optimality can thereby be understood with respect to two different measures: fastest egress and safest evacuation. Since this modeling approach leads to a pde/ode-restricted optimization problem, the continuous model is transferred into a discrete network flow model under some linearity assumptions. Second, it is demonstrated that this reformulation results in an efficient algorithm always leading to the global optimum. A computational case study shows benefits and drawbacks of the models for different evacuation scenarios.
Article
In this paper, a nonlinear mathematical model has been proposed and analyzed to study the effect of police force in controlling crime in a society with variable population size. In the modeling process, it is considered that immigration rate of susceptibles as well as criminals, is altered by police force. It is considered that the police force not only captures the criminals but also lessen their inflow in the region under consideration. A particular case of the proposed model has also been analyzed by assuming that there is no immigration of criminals into the society. The model analysis reveals that by ceasing immigration of criminals and maintaining a sufficient number of baseline police force, the crime in the society can be controlled completely. An explicit expression for critical number of baseline police force has been derived analytically. Numerical simulation and sensitivity analysis is also carried out to investigate the influence of certain key parameters on the dynamics of crime in the society.
Book
Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.
Article
We present a concise study of the time-dependent shortest path problem, its theoretical properties, and its solution algorithms. Time-dependent networks, in which the travel time along each arc is a known function of the departure time along the arc, arise in many practical applications, particularly those related to vehicular transportation. Since the general problem is at least NP-hard, we focus entirely on the case of FIFO networks, in which commodities travel through arcs in a First-In-First-Out manner. This special case is very common in practice and enables the development of rich theoretical properties and efficient solution algorithms. Our aim is to present a unified framework which encompasses a wide range of problem variants in both discrete and continuous time, which ties together past work and recent developments. 1
Article
The authors bring the "person" back into criminology by focusing on understanding individual differences in criminal conduct and recognizing the importance of personal, interpersonal, and community factors. What results is a truly interdisciplinary general personality and social psychology of criminal behavior that is open to a wide variety of factors that relate to individual differences - a perspective with both theoretical and practical significance in North America and Great Britain.The book is now organized into four parts: (1) The Theoretical Context and Knowledge Base to the Psychology of Criminal Conduct, (2) The Major Risk/Need Factors of Criminal Conduct, (3) Applications, and (4) Summary and Conclusions. Chapters include helpful Resource Notes that explain important concepts. A selection of technical notes, separated from the general text, allows the advanced student to explore complex research without distracting readers from the main points. Resource notes throughout explain important concepts. Technical notes at the back of the book allow the advanced student to explore complex research without distracting readers from the main points. An acronym index is also provided.
Article
This paper is devoted to the proof of the well posedness of a class of ordinary differential equations (ODEs). The vector field depends on the solution to a scalar conservation law. Forward uniqueness of Filippov solutions is obtained, as well as their Hölder continuous dependence on the initial data of the ODE. Furthermore, we prove the continuous dependence in C ⁰ of the solution to the ODE from the initial data of the conservation law in L ¹ . This problem is motivated by a model of traffic flow.
Article
Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx) The analysis of criminal behavior with mathematical tools is a fairly new idea, but one which can be used to obtain insight on the dynamics of crime. In a recent work 33 Short et al. developed an agent-based stochastic model for the dynamics of residential burglaries. This model produces the right qualitative behavior, that is, the existence of spatio-temporal collections of criminal activities or "hotspots," which have been observed in residential burglary data. In this paper we prove local existence and uniqueness of solutions to the continuum version of this model, a coupled system of partial differential equations, as well a continuation argument. Furthermore, we compare this PDE model with a generalized version of the Keller-Segel model for chemotaxis as a first step to understanding possible conditions for global existence vs. blow-up of the solutions in finite time.
Article
This paper contains an extension of an optimal control model considered by Sethi (Sethi, S. P. 1979. Optimal Pilfenng policies for dynamic continuous thieves. Management Sci. 25 (6, June) 535–542.) to a differential game situation. It is assumed that Sethi's concave thief, i.e., a risk-averter, plays against the police, whose objective function incorporates convex costs of law enforcement, a one-shot utility at the time of arrest and a continuous utility or cost rate after the thief is arrested. The probability that the thief is caught by time t is influenced not only by the pilfering rate (as is the case in Sethi's model) but also by the instrumental variable of the police, namely its rate of law enforcement. Our aim is to analyze noncooperative Nash solutions of the differential game sketched above. Due to the special structure of the Hamiltonians, a system of two nonlinear differential equations for the control variables of both players can be derived. The system is solved explicitly under special assumptions. It is shown that the optimal rate of law enforcement of police increases monotonically whereas for the Nash-optimal pilfering rate three cases may be characterized in which increase, decrease or constancy occur. Thus the Nash-optimal solutions show in each case monotonic behavior, i.e., changes in trend never occur. Moreover, the dependency of the terminal values of control can be described. Finally, an interpretation of the model as market entrance game is sketched.
Article
A simple theory of traffic flow is developed by replacing individual vehicles with a continuous “fluid” density and applying an empirical relation between speed and density. Characteristic features of the resulting theory are a simple “graph-shearing” process for following the development of traffic waves in time and the frequent appearance of shock waves. The effect of a traffic signal on traffic streams is studied and found to exhibit a threshold effect wherein the disturbances are minor for light traffic but suddenly build to large values when a critical density is exceeded.
Article
This paper uses the method of kinematic waves, developed in part I, but may be read independently. A functional relationship between flow and concentration for traffic on crowded arterial roads has been postulated for some time, and has experimental backing (§2). From this a theory of the propagation of changes in traffic distribution along these roads may be deduced (§§2, 3). The theory is applied (§4) to the problem of estimating how a ‘hump’, or region of increased concentration, will move along a crowded main road. It is suggested that it will move slightly slower than the mean vehicle speed, and that vehicles passing through it will have to reduce speed rather suddenly (at a ‘shock wave’) on entering it, but can increase speed again only very gradually as they leave it. The hump gradually spreads out along the road, and the time scale of this process is estimated. The behaviour of such a hump on entering a bottleneck, which is too narrow to admit the increased flow, is studied (§5), and methods are obtained for estimating the extent and duration of the resulting hold-up. The theory is applicable principally to traffic behaviour over a long stretch of road, but the paper concludes (§6) with a discussion of its relevance to problems of flow near junctions, including a discussion of the starting flow at a controlled junction. In the introductory sections 1 and 2, we have included some elementary material on the quantitative study of traffic flow for the benefit of scientific readers unfamiliar with the subject.
Article
We introduce a model that describes heavy traffic on a network of unidirectional roads. The model consists of a system of initial-boundary value problems for nonlinear conservation laws. We uniquely formulate and solve the Riemann problem for such a system and, based on this, then show existence of a solution to the Cauchy problem.
Article
This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from the conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions; hence we choose to have some fixed rules for the distribution of traffic plus optimization criteria for the flux. We prove existence of solutions to the Cauchy problem and we show that the Lipschitz continuous dependence by initial data does not hold in general, but it does hold under special assumptions. Our method is based on a wave front tracking approach [A. Bressan, Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 2000] and works also for boundary data and time-dependent coefficients of traffic distribution at junctions, including traffic lights.
Book
This book is devoted to primarily continuous models for a special class of supply chains often called production or supply networks. The aim is to present a mathematical description of different phenomena appearing in planning and managing supply chains. We address both the mathematical modeling as well as techniques for simulation and optimization purposes. The problem of a continuous description of supply chains and production networks dates back to the early 60's and started with the work of [8, 30]. Significantly, the models were proposed in particular for large volume production on complex networks where a discrete description might fail. Since then, many methods and ideas have been developed concerning the modeling of different features of supply chains, including the efficient simulation and the optimization of product flows among suppliers and customers. In recent years continuous and homogeneous product flow models have been introduced, for example, in [2, 13, 23, 26, 27, 28, 29, 37, 41, 42]. These models have been built in close connection to other transport problems like vehicular traffic flow and queuing theory. Hence, this suggests that the obtained models should be given by partial differential equations for the product flow, similar to those of gas dynamics. Depending on the problem at hand, these equations are possibly accompanied by ordinary differential equations describing the load of inventories. Also some optimization techniques have been proposed in order to answer questions arising in supply chain planning [31, 50, 57]. Starting from a network formulation, we derive equations for a continuous description of homogeneous product flows. The derivation is based on first principles, but the final equations are closely related to discrete event simulations of supply chains. Additionally, we present extensions to include more realistic phenomena. Such extensions consist of systems of partial differential equations or coupled partial and ordinary differential equations. The book surveys the underlying fundamentals and provides evolved mathematical techniques for simulation and efficient optimization of the presented models.
Article
We study a model of vehicular traffic flow, represented by a coupled system formed by a scalar conservation law, describing the evolution of cars density, and an ODE, whose solution is the position of a moving bottleneck, i.e., a slower vehicle moving inside the cars’ flow. A fractional step approach is used to approximate the coupled model, and convergence is proved by compactness arguments. Finally, the limit of such an approximating sequence is proved to solve the original PDE-ODE model.
Traffic light control: A case study, Discrete and Continuous Dynamical Systems-Series
  • S Göttlich
  • U Ziegler
S. Göttlich and U. Ziegler, Traffic light control: A case study, Discrete and Continuous Dynamical Systems-Series S, 7 (2014), 483–501.