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Some aspects on kinetic modeling of evacuation dynamics
JID:PLREV AID:784 /DIS
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Comment
Some aspects on kinetic modeling of evacuation dynamics Comment on “Human behaviours in evacuation crowd dynamics: From modelling to “big data” toward crisis management” by Nicola Bellomo et al. Juan Calvo, Juanjo Nieto ∗ University of Granada, Departamento de Matemática Aplicada, 18071 Granada, Spain Received 18 August 2016; accepted 19 August 2016
Communicated by J. Fontanari
The management of human crowds in extreme situations is a complex subject which requires to take into account a variety of factors. To name a few, the understanding of human behaviour, the psychological and behavioural features of individuals, the quality of the venue and the stress level of the pedestrian need to be addressed in order to select the most appropriate action during an evacuation process on a complex venue. In this sense, the mathematical modeling of such complex phenomena can be regarded as a very useful tool to understand and predict these situations. As presented in [4], mathematical models can provide guidance to the personnel in charge of managing evacuation processes, by means of helping to design a set of protocols, among which the most appropriate during a given critical situation is then chosen. An extensive review of the recent literature about evacuation dynamics can be found in [4]. The different scales at which modeling approaches have been proposed and the different features of the social behaviour of humans are highlighted in this contribution. Different points of view on the subject, ranging from psychological works to numerical simulations and also including pedestrian movement references, fluid and traffic connections, analogies with the modeling of biological mutations, mechanical kinetic theory approaches and active particles modeling, are presented in this review (see for instance the works [1–3,5–9,11–14], which are discussed in [4]). A mesoscopic model treated in [7] is featured, where most of the aforementioned aspects are included and where simulations on several venues are carried out. Actually, we want to elaborate on this comment on several aspects of the model proposed in [4,7]. First of all, we focus on the parameter β modeling the stress level or anxiety of the walkers under evacuation. It contains all the psychological aspects that can influence walker’s dynamics and plays a key role on the individual dynamic described by operators A. Author claim that “it should be treated as an additional internal variable”, on which we agree. In fact, we think that this anxiety parameter can be replicated by the activity variable (customarily denoted by u) available on the kinetic theory of active particles, a theory which is incidentally mentioned in [4, Sec. 3]. This DOI of original article: http://dx.doi.org/10.1016/j.plrev.2016.05.014. * Corresponding author.
E-mail addresses: [email protected] (J. Calvo), [email protected] (J. Nieto). http://dx.doi.org/10.1016/j.plrev.2016.08.008 1571-0645/© 2016 Elsevier B.V. All rights reserved.
JID:PLREV AID:784 /DIS
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J. Calvo, J. Nieto / Physics of Life Reviews ••• (••••) •••–•••
amounts to identify u = β, and hence the heterogeneity and dynamics of the (in this approach) active particles should be also modeled within this theory [1,8]. As regards the parameter α (which includes the whole information of the quality of the venue), we think that it does not really play the role it deserves on the modeling presented in [4]. We suggest that the quality of the venue (or the perception of such a quality by any of the walkers) should be included on the internal parameter β (or u). Actually, α does only appear in combination with β as γ = αβ on the descriptions of operators A, precisely to increase the probability to accelerate, which could be related to the perception by the walkers that the conditions of the venue are good. We also have to mention that the criteria used by the walkers under evacuation to choose their preferred direction of motion θ could create a rambling behaviour. This can be noticed on some numerical simulations. Actually, we think that it could explain the surprising behaviour obtained on [4, Fig. 3.(d)], after the door is closed, where some of the walkers come back to the venue under evacuation. Note that θ is given by ν (p) = (cos(θ ), sin(θ )) on [4, Eq. (7)] as a convex “vectorial” combination of three directions ν (t) , ν (s) , and ν (v) (associated respectively to the target, the stream and the crowd-avoidance direction). Intuitively speaking, when the angles formed by these basic directions ν (t) , ν (s) and ν (v) are small, the former convex combination works as a weighted mean and could be regarded a perturbation of the main direction. On the contrary, if these angles are greater than π/2 (say), this combination is very unstable and the walker under scrutiny could pick almost any direction (quite true when two of these reference directions form an angle close to π ), producing an artificial behaviour. Finally, we would like to draw attention on the fact that kinetic models can also replicate clustering phenomena by means of coagulation mechanisms. Clustering tendencies in evacuation dynamics could arise for instance in the presence of a leaders-and-followers dynamics; this hints at what is termed in [4] as “functional subsystems”. We think that a possible way of accounting for these features is to consider a two-species (isolated individuals and clusters) kinetic model with suitable coagulation kernels; arguably an extension of the model introduced in [10] could provide an adequate framework to tackle such matters. Acknowledgements JC and JN are partially supported by MINECO (Spain), Project MTM2014-53406-R, and Junta de Andalucía, Project P12-FQM-954. JC is also partially supported by “Plan Propio de Investigación, programa 9” (funded by Granada University and FEDER funds) and by Project MTM2015-71509-C2-1-R (funded by the Spanish MINECO and European FEDER funds). References [1] Bellomo N, Bellouquid A, Nieto J, Soler J. Multicellular biological growing systems: hyperbolic limits towards macroscopic description. Math Models Methods Appl Sci 2007;17:1675–93. [2] Bellomo N, Bellouquid A, Nieto J, Soler J. On the asymptotic theory from microscopic to macroscopic growing tissue models: an overview with perspectives. Math Models Methods Appl Sci 2012;22:1130001. [3] Bellomo N, Bellouquid A, Nieto J, Soler J. On the multiscale modeling of vehicular traffic: from kinetic to hydrodynamics. Discrete Contin Dyn Syst, Ser B 2014;19:1869–88. [4] Bellomo N, Clarke D, Gibelli L, Townsend P, Vreugdenhil BJ. Human behaviours in evacuation crowd dynamics: from modelling to “big data” toward crisis management. Phys Life Rev 2016. http://dx.doi.org/10.1016/j.plrev.2016.05.014 [in this issue]. [5] Bellomo N, Colasuonno F, Knopoff D, Soler J. From a systems theory of sociology to modeling the onset and evolution of criminality. Netw Heterog Media 2015;10:421–41. [6] Bellomo N, Elaiw A, Althiabi AM, Alghamdi A. On the interplay between mathematics and biology hallmarks toward a new systems biology. Phys Life Rev 2015;12:44–64. [7] Bellomo N, Gibelli L. Behavioral crowds: modeling and Monte Carlo simulations toward validation. Comput Fluids 2016. http://dx.doi.org/10.1016/j.compfluid.2016.04.022. In press. [8] Bellomo N, Knopoff D, Soler J. On the difficult interplay between life, “complexity”, and mathematical sciences. Math Models Methods Appl Sci 2013;23:1861–913. [9] Bellomo N, Soler J. On the mathematical theory of the dynamics of swarms viewed as complex systems. Math Models Methods Appl Sci 2012;22:1140006. [10] Calvo J, Jabin PE. Large time asymptotics for a modified coagulation model. J Differ Equ 2011;250:2807–37. [11] Hughes RL. The flow of human crowds. Annu Rev Fluid Mech 2003;35:169–82. [12] Le Bon G. The crowd. A study of the popular mind. Dover Pub.; 2002. [13] Nieto J. The kinetic theory of active particles as a biological systems approach. Phys Life Rev 2015;12:81–2. [14] Wijermans N. Understanding crowd behaviour. PhD thesis. University of Groningen; 2011.
[m3SC+; v1.235; Prn:24/08/2016; 9:50] P.1 (1-2)
Available online at www.sciencedirect.com
ScienceDirect Physics of Life Reviews ••• (••••) •••–••• www.elsevier.com/locate/plrev
Comment
Some aspects on kinetic modeling of evacuation dynamics Comment on “Human behaviours in evacuation crowd dynamics: From modelling to “big data” toward crisis management” by Nicola Bellomo et al. Juan Calvo, Juanjo Nieto ∗ University of Granada, Departamento de Matemática Aplicada, 18071 Granada, Spain Received 18 August 2016; accepted 19 August 2016
Communicated by J. Fontanari
The management of human crowds in extreme situations is a complex subject which requires to take into account a variety of factors. To name a few, the understanding of human behaviour, the psychological and behavioural features of individuals, the quality of the venue and the stress level of the pedestrian need to be addressed in order to select the most appropriate action during an evacuation process on a complex venue. In this sense, the mathematical modeling of such complex phenomena can be regarded as a very useful tool to understand and predict these situations. As presented in [4], mathematical models can provide guidance to the personnel in charge of managing evacuation processes, by means of helping to design a set of protocols, among which the most appropriate during a given critical situation is then chosen. An extensive review of the recent literature about evacuation dynamics can be found in [4]. The different scales at which modeling approaches have been proposed and the different features of the social behaviour of humans are highlighted in this contribution. Different points of view on the subject, ranging from psychological works to numerical simulations and also including pedestrian movement references, fluid and traffic connections, analogies with the modeling of biological mutations, mechanical kinetic theory approaches and active particles modeling, are presented in this review (see for instance the works [1–3,5–9,11–14], which are discussed in [4]). A mesoscopic model treated in [7] is featured, where most of the aforementioned aspects are included and where simulations on several venues are carried out. Actually, we want to elaborate on this comment on several aspects of the model proposed in [4,7]. First of all, we focus on the parameter β modeling the stress level or anxiety of the walkers under evacuation. It contains all the psychological aspects that can influence walker’s dynamics and plays a key role on the individual dynamic described by operators A. Author claim that “it should be treated as an additional internal variable”, on which we agree. In fact, we think that this anxiety parameter can be replicated by the activity variable (customarily denoted by u) available on the kinetic theory of active particles, a theory which is incidentally mentioned in [4, Sec. 3]. This DOI of original article: http://dx.doi.org/10.1016/j.plrev.2016.05.014. * Corresponding author.
E-mail addresses: [email protected] (J. Calvo), [email protected] (J. Nieto). http://dx.doi.org/10.1016/j.plrev.2016.08.008 1571-0645/© 2016 Elsevier B.V. All rights reserved.
JID:PLREV AID:784 /DIS
2
[m3SC+; v1.235; Prn:24/08/2016; 9:50] P.2 (1-2)
J. Calvo, J. Nieto / Physics of Life Reviews ••• (••••) •••–•••
amounts to identify u = β, and hence the heterogeneity and dynamics of the (in this approach) active particles should be also modeled within this theory [1,8]. As regards the parameter α (which includes the whole information of the quality of the venue), we think that it does not really play the role it deserves on the modeling presented in [4]. We suggest that the quality of the venue (or the perception of such a quality by any of the walkers) should be included on the internal parameter β (or u). Actually, α does only appear in combination with β as γ = αβ on the descriptions of operators A, precisely to increase the probability to accelerate, which could be related to the perception by the walkers that the conditions of the venue are good. We also have to mention that the criteria used by the walkers under evacuation to choose their preferred direction of motion θ could create a rambling behaviour. This can be noticed on some numerical simulations. Actually, we think that it could explain the surprising behaviour obtained on [4, Fig. 3.(d)], after the door is closed, where some of the walkers come back to the venue under evacuation. Note that θ is given by ν (p) = (cos(θ ), sin(θ )) on [4, Eq. (7)] as a convex “vectorial” combination of three directions ν (t) , ν (s) , and ν (v) (associated respectively to the target, the stream and the crowd-avoidance direction). Intuitively speaking, when the angles formed by these basic directions ν (t) , ν (s) and ν (v) are small, the former convex combination works as a weighted mean and could be regarded a perturbation of the main direction. On the contrary, if these angles are greater than π/2 (say), this combination is very unstable and the walker under scrutiny could pick almost any direction (quite true when two of these reference directions form an angle close to π ), producing an artificial behaviour. Finally, we would like to draw attention on the fact that kinetic models can also replicate clustering phenomena by means of coagulation mechanisms. Clustering tendencies in evacuation dynamics could arise for instance in the presence of a leaders-and-followers dynamics; this hints at what is termed in [4] as “functional subsystems”. We think that a possible way of accounting for these features is to consider a two-species (isolated individuals and clusters) kinetic model with suitable coagulation kernels; arguably an extension of the model introduced in [10] could provide an adequate framework to tackle such matters. Acknowledgements JC and JN are partially supported by MINECO (Spain), Project MTM2014-53406-R, and Junta de Andalucía, Project P12-FQM-954. JC is also partially supported by “Plan Propio de Investigación, programa 9” (funded by Granada University and FEDER funds) and by Project MTM2015-71509-C2-1-R (funded by the Spanish MINECO and European FEDER funds). References [1] Bellomo N, Bellouquid A, Nieto J, Soler J. Multicellular biological growing systems: hyperbolic limits towards macroscopic description. Math Models Methods Appl Sci 2007;17:1675–93. [2] Bellomo N, Bellouquid A, Nieto J, Soler J. On the asymptotic theory from microscopic to macroscopic growing tissue models: an overview with perspectives. Math Models Methods Appl Sci 2012;22:1130001. [3] Bellomo N, Bellouquid A, Nieto J, Soler J. On the multiscale modeling of vehicular traffic: from kinetic to hydrodynamics. Discrete Contin Dyn Syst, Ser B 2014;19:1869–88. [4] Bellomo N, Clarke D, Gibelli L, Townsend P, Vreugdenhil BJ. Human behaviours in evacuation crowd dynamics: from modelling to “big data” toward crisis management. Phys Life Rev 2016. http://dx.doi.org/10.1016/j.plrev.2016.05.014 [in this issue]. [5] Bellomo N, Colasuonno F, Knopoff D, Soler J. From a systems theory of sociology to modeling the onset and evolution of criminality. Netw Heterog Media 2015;10:421–41. [6] Bellomo N, Elaiw A, Althiabi AM, Alghamdi A. On the interplay between mathematics and biology hallmarks toward a new systems biology. Phys Life Rev 2015;12:44–64. [7] Bellomo N, Gibelli L. Behavioral crowds: modeling and Monte Carlo simulations toward validation. Comput Fluids 2016. http://dx.doi.org/10.1016/j.compfluid.2016.04.022. In press. [8] Bellomo N, Knopoff D, Soler J. On the difficult interplay between life, “complexity”, and mathematical sciences. Math Models Methods Appl Sci 2013;23:1861–913. [9] Bellomo N, Soler J. On the mathematical theory of the dynamics of swarms viewed as complex systems. Math Models Methods Appl Sci 2012;22:1140006. [10] Calvo J, Jabin PE. Large time asymptotics for a modified coagulation model. J Differ Equ 2011;250:2807–37. [11] Hughes RL. The flow of human crowds. Annu Rev Fluid Mech 2003;35:169–82. [12] Le Bon G. The crowd. A study of the popular mind. Dover Pub.; 2002. [13] Nieto J. The kinetic theory of active particles as a biological systems approach. Phys Life Rev 2015;12:81–2. [14] Wijermans N. Understanding crowd behaviour. PhD thesis. University of Groningen; 2011.