Stochastic model of movement of a group of individuals in a space with boundaries taking into account their social behavior

I. V. Derevich ; A. A. Panova

Matematičeskoe modelirovanie, Tome 35 (2023) no. 6, pp. 51-62.

Résumé

A stochastic model of random movement of a small group of individuals in a confined space with internal obstacles is proposed. The social behavior of individuals in the group is taken into account, which reduces the likelihood of their close physical contact and collisions with internal obstacles. The equations of motion of individuals are written in the form of a system of ordinary stochastic differential equations (SODE). The direction and speed of the desired movement of an individual are described by a random process structured in time. The social behavior and interaction of individuals with obstacles is modeled by effective potential. The SODE system is integrated based on modified Runge–Kutta algorithms. Examples are given of the movement of a small group in a closed gallery with columns in poor visibility conditions, during evacuation from the gallery in case of panic. The viral infection scenario is illustrated by a reduction in the relative distance between an infected individual and susceptible group members.

Détail

Keywords: stochastic ordinary differential equations, modified Runge–Kutta algorithm, social dynamics model, color random process, viral infection.

Bibliographie

[1] M. E. Stepantsov, “Matematicheskaia model napravlennogo dvizheniia gruppy liudei”, Mat. Model., 16:3 (2004), 43–49 | MR | Zbl

[2] A. A. Samartsev, V. A. Kushnikov, V. A. Ivachenko, i dr., “Multiagentnaia model protsessa evakuatsii ludei is pomeshcheniia pri vozniknovenii chrezvychainych situatsii”, Upravlenie technicheskimi sistemami i technologicheskimi protsessami, 2018, no. 72b, 217–244

[3] A. M. Aptukov, D. A. Bratsun, A. V. Lyushnin, “Modelirovanie povedeniia panikuiushchei tolpy v mnogourovnevom razvetvlennom pomeshchenii”, Computernye issledovaniia i modelirovanie, 5:3 (2013), 491–508

[4] E. A. Kotkova, “Sistemno-dinamicheskaia model rasprostraneniia paniki pri evakuatsii iz obshchestvennykh zdanii”, Nauch. analit. zhurn. «Vestnik Saint-Petersburg un-ta GPS MCHS Rossii», 2022, no. 1, 182–194

[5] X. Yang, Z. Wu, Y. Li, “Difference between real-life escape panic and mimic exercises in simulated situation with implications to the statistical physics models of emergency evacuation: The 2008 Wenchuan earthquake”, Physica A, 390 (2011), 2375–2380 | DOI

[6] A. Mohammadi, M. T.U. Chowdhury, S. Yang et al, “Developing levels of pedestrian physical distancing during a pandemic”, Safety Science, 134 (2021), 105066 | DOI

[7] L. Gosce, A. Johansson, “Analytical modelling of the spread of a disease: From corridors to transport networks”, Transportation Research Procedia, 2 (2014), 513–517 | DOI

[8] E. Papadimitriou, G. Yannis, J. Golias, “A critical assessment of pedestrian behaviour models”, Transportation Research Part F, 12 (2009), 242–255 | DOI

[9] S. Zhou, D. Chen, W. Cai, L. Luo et al, “Crowd Modeling and Simulation Technologies”, ACM Transactions on Modeling and Computer Simulation, 20:4 (2010), 1–35 | DOI | MR

[10] F. Martinez-Gil, M. Lozano, I. Garcia-Fernandez et al., “Modeling, Evaluation, and Scale on Artificial Pedestrians: A Literature Review”, ACM Computing Surveys, 50:5 (2017), 1–35 | DOI

[11] D. C. Duives, W. Daamen, S. P. Hoogendoorn, “State-of-the-art crowd motion simulation models”, Transportation Research Part, 37 (2013), 193–209 | DOI

[12] W. Yu, A. Johansson, “Modeling crowd turbulence by many-particle simulations”, Physical Review E, 76 (2007), 046105 | DOI

[13] D. Helbing, P. Molnar, “Social force model for pedestrian dynamics”, Physical Review E, 51:5 (1995), 4282–4286 | DOI

[14] D. Helbing, I. Farkas, T. Vicsek, “Simulating dynamical features of escape panic”, Nature, 407:28 (2000), 487–490 | DOI

[15] M. Ye. Stepantsov, “Kletochnye avtomaty v modelirovanii sotsialno-economicheskikh system”, Proektirovanie budushchego. Problemy tsifrovoi realnosti, trudy 2-i Mezhdunarodnoi konf. (7-8 fevralia 2019, Moskva), IPM im. M.V. Keldysha, M., 2019, 138–143 | DOI

[16] S. Wei-Guo, Y. Yan-Fei, W. Bing-Hong et al, “Evacuation behaviors at exit in CA model with force essentials: A comparison with social force model”, Physica A, 371 (2006), 658–666 | DOI

[17] A. Schadschneider, A. Seyfried, “Validation of CA models of pedestrian dynamics with fundamental diagrams”, Cybernetics Systems: An International J., 40 (2009), 367–389 | DOI | Zbl

[18] A. Kirchner, K. Nishinari, A. Schadschneider, “Friction effects and clogging in a cellular automaton model for pedestrian dynamics”, Physical Review E, 67 (2003), 056122 | DOI

[19] L. F. Henderson, “The statistics of crowd fluids”, Nature, 229:5 (1971), 381–383 | DOI

[20] R. L. Hughes, “The flow of human crowds”, Annu. Rev. Fluid Mech., 35 (2003), 169–182 | DOI | MR | Zbl

[21] S. P. Hoogendoorn, F. Wageningen-Kessels, W. Daamen et al, “Continuum modelling of pedestrian flows: From microscopic principles to self-organized macroscopic phenomena”, Physica A, 416:15, 684–694 | MR

[22] S. P. Hoogendoorn, P. H. L. Bovy, “Pedestrian route-choice and activity scheduling theory and models”, Transportation Research Part B, 38 (2004), 169–190 | DOI

[23] I. V. Derevich, A. K. Klochkov, “Fluctuatsii skorosti chastitsy v viazkom gase so sluchainoi skorostiu v vide summy dvukh korrelirovannykh tsvetnykh shumov”, Matematika i matematicheskoe modelirovanie, 2020, no. 1, 33–49 | MR

[24] K. Debrabant, A. Rø"ßler, “Classification of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations”, Mathematics Computers in Simul., 77:4 (2008), 408–420 | DOI | MR | Zbl

[25] I. V. Derevich, A. K. Klochkov, “Modeling the motion of particles in the potential force field with allowance for the random velocity fluctuations of a medium”, J. of Eng. Physics and Thermophysics, 95 (2022), 1089–1100 | DOI