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September  2016, 11(3): 447-469. doi: 10.3934/nhm.2016004

A weakly coupled model of differential equations for thief tracking

Simone Göttlich 1, and  Camill Harter 1,

1. 

Department of Mathematics, University of Mannheim, D-68131 Mannheim, Germany

Received  April 2015 Revised  November 2015 Published  August 2016

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.
Keywords: numerical simulations., Traffic flow networks, thief trajectories.
Mathematics Subject Classification: Primary: 90B20; Secondary: 65M0.
Citation: Simone Göttlich, Camill Harter. A weakly coupled model of differential equations for thief tracking. Networks and Heterogeneous Media, 2016, 11 (3) : 447-469. doi: 10.3934/nhm.2016004
References:
[1]

D. A. Andrews and J. Bonta, The Psychology Of Criminal Conduct, Anderson Publishing, Ltd., 2010.

[2]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41.

[3]

G. Bretti and B. Piccoli, A tracking algorithm for car paths on road networks, SIAM Journal on Applied Dynamical Systems, 7 (2008), 510-531. doi: 10.1137/070697768.

[4]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM Journal on Mathematical Analysis, 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.

[5]

R. M. Colombo and A. Marson, Conservation laws and ODEs: A traffic problem, Springer, (2003), 455-461.

[6]

R. M. Colombo and A. Marson, A hölder continuous ODE related to traffic flow, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 133 (2003), 759-772. doi: 10.1017/S0308210500002663.

[7]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp. doi: 10.1142/S0218202511500230.

[8]

J. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control, 52 (2007), 2-11. doi: 10.1109/TAC.2006.887903.

[9]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization Of Supply Chains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898717600.

[10]

B. C. Dean, Shortest Paths in FIFO Time-Dependent Networks: Theory and Algorithms, Technical report, MIT Department of Computer Science, 2004.

[11]

M. L. Delle Monache and P. Goatin, A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow, Discrete and Continuous Dynamical Systems-Series S, 7 (2014), 435-447. doi: 10.3934/dcdss.2014.7.435.

[12]

S. E. Dreyfus, An appraisal of some shortest-path algorithms, Operations Research, 17 (1969), 395-412. doi: 10.1287/opre.17.3.395.

[13]

G. Feichtinger, A differential games solution to a model of competition between a thief and the police, Management Science, 29 (1983), 686-699. doi: 10.1287/mnsc.29.6.686.

[14]

A. F. Filippov and F. M. Arscott, Differential Equations With Discontinuous Righthand Sides: Control Systems, Springer, 1988. doi: 10.1007/978-94-015-7793-9.

[15]

M. Garavello and B. Piccoli, Traffic Flow On Networks, American Institute of Mathematical Sciences Springfield, MO, USA, 2006.

[16]

S. Göttlich, S. Kühn, P. Ohst, S. Ruzika and M. Thiemann, Evacuation dynamics influenced by spreading hazardous material, Netw. Heterog. Media, 6 (2011), 443-464. doi: 10.3934/nhm.2011.6.443.

[17]

S. Göttlich and U. Ziegler, Traffic light control: A case study, Discrete and Continuous Dynamical Systems-Series S, 7 (2014), 483-501. doi: 10.3934/dcdss.2014.7.483.

[18]

S. Göttlich, M. Herty and U. Ziegler, Modeling and optimizing traffic light settings in road networks, Computers & Operations Research, 55 (2015), 36-51. doi: 10.1016/j.cor.2014.10.001.

[19]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM Journal on Mathematical Analysis, 26 (1995), 999-1017. doi: 10.1137/S0036141093243289.

[20]

G. Jiang, D. Levy, C. Lin, S. Osher and E. Tadmor, High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 35 (1998), 2147-2168. doi: 10.1137/S0036142997317560.

[21]

C. Lattanzio, A. Maurizi and B. Piccoli, Moving bottlenecks in car traffic flow: A PDE-ODE coupled model, SIAM Journal on Mathematical Analysis, 43 (2011), 50-67. doi: 10.1137/090767224.

[22]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.

[23]

A. K. Misra, Modeling the effect of police deterrence on the prevalence of crime in the society, Applied Mathematics and Computation, 237 (2014), 531-545. doi: 10.1016/j.amc.2014.03.136.

[24]

A. A. Reid, R. Frank, N. Iwanski, V.Dabbaghian and P. Brantingham, Uncovering the spatial patterning of crimes: A criminal movement model (CriMM), Journal of Research in Crime and Delinquency, 51 (2014), 230-255. doi: 10.1177/0022427813483753.

[25]

P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.

[26]

N. Rodriguez and A. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior, Math. Models Methods Appl. Sci., 20 (2010), 1425-1457. doi: 10.1142/S0218202510004696.

[27]

M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behaviour, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029.

[28]

M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM Journal on Applied Dynamical Systems, 9 (2010), 462-483. doi: 10.1137/090759069.

[29]

J. T. Woodworth, G. O. Mohler, A. L. Bertozzi and P. J. Brantingham, Non-local crime density estimation incorporating housing information, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130403, 15pp. doi: 10.1098/rsta.2013.0403.

[30]

J. R. Zipkin, M. B. Short and A. L. Bertozzi, Cops on the dots in a mathematical model of urban crime and police response, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 1479-1506. doi: 10.3934/dcdsb.2014.19.1479.

show all references

References:
[1]

D. A. Andrews and J. Bonta, The Psychology Of Criminal Conduct, Anderson Publishing, Ltd., 2010.

[2]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41.

[3]

G. Bretti and B. Piccoli, A tracking algorithm for car paths on road networks, SIAM Journal on Applied Dynamical Systems, 7 (2008), 510-531. doi: 10.1137/070697768.

[4]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM Journal on Mathematical Analysis, 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.

[5]

R. M. Colombo and A. Marson, Conservation laws and ODEs: A traffic problem, Springer, (2003), 455-461.

[6]

R. M. Colombo and A. Marson, A hölder continuous ODE related to traffic flow, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 133 (2003), 759-772. doi: 10.1017/S0308210500002663.

[7]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp. doi: 10.1142/S0218202511500230.

[8]

J. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control, 52 (2007), 2-11. doi: 10.1109/TAC.2006.887903.

[9]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization Of Supply Chains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898717600.

[10]

B. C. Dean, Shortest Paths in FIFO Time-Dependent Networks: Theory and Algorithms, Technical report, MIT Department of Computer Science, 2004.

[11]

M. L. Delle Monache and P. Goatin, A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow, Discrete and Continuous Dynamical Systems-Series S, 7 (2014), 435-447. doi: 10.3934/dcdss.2014.7.435.

[12]

S. E. Dreyfus, An appraisal of some shortest-path algorithms, Operations Research, 17 (1969), 395-412. doi: 10.1287/opre.17.3.395.

[13]

G. Feichtinger, A differential games solution to a model of competition between a thief and the police, Management Science, 29 (1983), 686-699. doi: 10.1287/mnsc.29.6.686.

[14]

A. F. Filippov and F. M. Arscott, Differential Equations With Discontinuous Righthand Sides: Control Systems, Springer, 1988. doi: 10.1007/978-94-015-7793-9.

[15]

M. Garavello and B. Piccoli, Traffic Flow On Networks, American Institute of Mathematical Sciences Springfield, MO, USA, 2006.

[16]

S. Göttlich, S. Kühn, P. Ohst, S. Ruzika and M. Thiemann, Evacuation dynamics influenced by spreading hazardous material, Netw. Heterog. Media, 6 (2011), 443-464. doi: 10.3934/nhm.2011.6.443.

[17]

S. Göttlich and U. Ziegler, Traffic light control: A case study, Discrete and Continuous Dynamical Systems-Series S, 7 (2014), 483-501. doi: 10.3934/dcdss.2014.7.483.

[18]

S. Göttlich, M. Herty and U. Ziegler, Modeling and optimizing traffic light settings in road networks, Computers & Operations Research, 55 (2015), 36-51. doi: 10.1016/j.cor.2014.10.001.

[19]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM Journal on Mathematical Analysis, 26 (1995), 999-1017. doi: 10.1137/S0036141093243289.

[20]

G. Jiang, D. Levy, C. Lin, S. Osher and E. Tadmor, High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 35 (1998), 2147-2168. doi: 10.1137/S0036142997317560.

[21]

C. Lattanzio, A. Maurizi and B. Piccoli, Moving bottlenecks in car traffic flow: A PDE-ODE coupled model, SIAM Journal on Mathematical Analysis, 43 (2011), 50-67. doi: 10.1137/090767224.

[22]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.

[23]

A. K. Misra, Modeling the effect of police deterrence on the prevalence of crime in the society, Applied Mathematics and Computation, 237 (2014), 531-545. doi: 10.1016/j.amc.2014.03.136.

[24]

A. A. Reid, R. Frank, N. Iwanski, V.Dabbaghian and P. Brantingham, Uncovering the spatial patterning of crimes: A criminal movement model (CriMM), Journal of Research in Crime and Delinquency, 51 (2014), 230-255. doi: 10.1177/0022427813483753.

[25]

P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.

[26]

N. Rodriguez and A. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior, Math. Models Methods Appl. Sci., 20 (2010), 1425-1457. doi: 10.1142/S0218202510004696.

[27]

M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behaviour, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029.

[28]

M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM Journal on Applied Dynamical Systems, 9 (2010), 462-483. doi: 10.1137/090759069.

[29]

J. T. Woodworth, G. O. Mohler, A. L. Bertozzi and P. J. Brantingham, Non-local crime density estimation incorporating housing information, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130403, 15pp. doi: 10.1098/rsta.2013.0403.

[30]

J. R. Zipkin, M. B. Short and A. L. Bertozzi, Cops on the dots in a mathematical model of urban crime and police response, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 1479-1506. doi: 10.3934/dcdsb.2014.19.1479.

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