Abstract
We present some ideas on how to extend a kinetic-type model for crowd dynamics to account for an infectious disease spreading. We focus on a medium size crowd occupying a confined environment where the disease is easily spread. The kinetic theory approach we choose uses tools of game theory to model the interactions of a person with the surrounding people and the environment, and it features a parameter to represent the level of stress. It is known that people choose different walking strategies when subjected to fear or stressful situations. To demonstrate that our model for crowd dynamics could be used to reproduce realistic scenarios, we simulate passengers in one terminal of Hobby Airport in Houston. In order to model disease spreading in a walking crowd, we introduce a variable that denotes the level of exposure to people spreading the disease. In addition, we introduce a parameter that describes the contagion interaction strength and a kernel function that is a decreasing function of the distance between a person and a spreading individual. We test our contagion model on a problem involving a small crowd walking through a corridor.
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References
J. P. Agnelli, F. Colasuonno, and D. Knopoff. A kinetic theory approach to the dynamics of crowd evacuation from bounded domains. Mathematical Models and Methods in Applied Sciences, 25(01):109–129, 2015.
V. V. Aristov. Biological systems as nonequilibrium structures described by kinetic methods. Results in Physics, 13:102232, 2019.
M. Asano, T Iryo, and M. Kuwahara. Microscopic pedestrian simulation model combined with a tactical model for route choice behaviour. Transportation Research Part C: Emerging Technologies, 18(6):842–855, 2010.
B. Aylaj, N. Bellomo, L. Gibelli, and A. Reali. A unified multiscale vision of behavioral crowds. Mathematical Models and Methods in Applied Sciences, 30(1):1–22, 2020.
S. Bandini, S. Manzoni, and G. Vizzari. Agent based modeling and simulation: An informatics perspective. Journal of Artificial Societies and Social Simulation, 12(4):4, 2009.
N. Bellomo and A. Bellouquid. On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms. Networks and Heterogeneous Media, 6(3):383–399, 2011.
N. Bellomo, A. Bellouquid, L. Gibelli, and N Outada. A quest towards a mathematical theory of living systems. Modeling and Simulation in Science, Engineering and Technology. (Birkhäuser, 2017).
N. Bellomo, A. Bellouquid, and D. Knopoff. From the microscale to collective crowd dynamics. SIAM Multiscale Modeling & Simulation, 11(3):943–963, 2013.
N. Bellomo, R. Bingham, M. K. Chaplain, G. Dosi, G. Forni, D. A. Knopoff, J. Lowengrub, R. Twarock, and M. E. Virgillito. A multiscale model of virus pandemic: Heterogeneous interactive entities in a globally connected world. Mathematical Models and Methods in Applied Sciences, 30(8):1591–1651, 2020.
N. Bellomo and L. Gibelli. Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds. Mathematical Models and Methods in Applied Sciences, 25(13):2417–2437, 2015.
N. Bellomo and L. Gibelli. Behavioral crowds: Modeling and Monte Carlo simulations toward validation. Computers & Fluids, 141:13–21, 2016.
N. Bellomo, L. Gibelli, and N. Outada. On the interplay between behavioral dynamics and social interactions in human crowds. Kinetic and Related Models, 12(2):397–409, 2019.
N. Bellomo, K. J. Painter, Y. Tao, and M. Winkler. Occurrence vs. absence of taxis-driven instabilities in a May–Nowak model for virus infection. SIAM Journal on Applied Mathematics, 79(5):1990–2010, 2019.
M. Chraibi, U. Kemloh, A. Schadschneider, and A. Seyfried. Force-based models of pedestrian dynamics. Networks and Heterogeneous Media, 6(3):425–442, 2011.
M. Chraibi, A. Tordeux, A. Schadschneider, and A. Seyfried. Modelling of Pedestrian and Evacuation Dynamics, pages 649–669. (Springer-New York, 2019).
E. Cristiani, B. Piccoli, and A. Tosin. Multiscale Modeling of Pedestrian Dynamics. (Springer-Cham, 2014).
J. Dai, X. Li, and L. Liu. Simulation of pedestrian counter flow through bottlenecks by using an agent-based model. Physica A: Statistical Mechanics and its Applications, 392(9):2202–2211, 2013.
R. Glowinski. Finite element methods for incompressible viscous flow, in Handbook of numerical analysis, P. G. Ciarlet, J. L. Lions (Eds), volume 9. (North-Holland, 2003).
D. Helbing and P. Molnár. Social force model for pedestrian dynamics. Physical review. E, 51:4282–4286, 1998.
R. L. Hughes. The flow of human crowds. Annual Review of Fluid Mechanics, 35(1):169–182, 2003.
D. Kim, K. O’Connell, W. Ott, and A. Quaini. A kinetic theory approach for 2D crowd dynamics with emotional contagion. Submitted, 2020. https://arxiv.org/abs/2012.08108.
D. Kim and A. Quaini. A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles. Kinetic & Related Models, 12(6):1273–1296, 2019.
D. Kim and A. Quaini. Coupling kinetic theory approaches for pedestrian dynamics and disease contagion in a confined environment. Mathematical Models and Methods in Applied Sciences, 30(10):1893–1915, 2020.
R. J. LeVeque. Numerical Methods for Conservation Laws. Lectures in Mathematics ETH ZĂĽrich, Department of Mathematics Research Institute of Mathematics. (Springer, 1992).
S. Liu, S. Lo, J. Ma, and W. Wang. An agent-based microscopic pedestrian flow simulation model for pedestrian traffic problems. IEEE Transactions on Intelligent Transportation Systems, 15(3):992–1001, 2014.
A. Schadschneider and A. Seyfried. Empirical results for pedestrian dynamics and their implications for modeling. Networks and Heterogeneous Media, 6(3):545–560, 2011.
A. Shende, M. P. Singh, and P. Kachroo. Optimization-based feedback control for pedestrian evacuation from an exit corridor. IEEE Transactions on Intelligent Transportation Systems, 12(4):1167–1176, 2011.
A. U. K Wagoum, A. Tordeux, and W. Liao. Understanding human queuing behaviour at exits: An empirical study. Royal Society Open Science, 4(1):160896, 2017.
L. Wang, M. B. Short, and A. L. Bertozzi. Efficient numerical methods for multiscale crowd dynamics with emotional contagion. Mathematical Models and Methods in Applied Sciences, 27(1):205–230, 2017.
B. Zhou, X. Wang, and X. Tang. Understanding collective crowd behaviors: Learning a mixture model of dynamic pedestrian-agents. In 2012 IEEE Conference on Computer Vision and Pattern Recognition, pages 2871–2878, 2012.
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This work has been partially supported by NSF through grant DMS-1620384.
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Kim, D., Quaini, A. (2021). A Kinetic Theory Approach to Model Crowd Dynamics with Disease Contagion. In: Bellomo, N., Gibelli, L. (eds) Crowd Dynamics, Volume 3. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-91646-6_7
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