Discontinuity waves as tipping points: applications to biological & sociological systems.

*(English)*Zbl 1304.35413Summary: The ‘tipping point’ phenomenon is discussed as a mathematical object, and related to the behaviour of nonlinear discontinuity waves in the dynamics of topical sociological and biological problems. The theory of such waves is applied to two illustrative systems in particular: a crowd-continuum model of pedestrian (or traffic) flow; and an hyperbolic reaction-diffusion model for the spread of the hantavirus infection (a disease carried by rodents). In the former, we analyse propagating acceleration waves, demonstrating how blow-up of the wave amplitude might indicate formation of a ‘human-shock’, that is, a ‘tipping point’ transition between safe pedestrian flow, and a state of overcrowding. While in the latter, we examine how travelling waves (of both acceleration and shock type) can be used to describe the advance of a hantavirus infection-front. Results from our investigation of crowd models also apply to equivalent descriptions of traffic flow, a context in which acceleration wave blow-up can be interpreted as emergence of the ‘phantom congestion’ phenomenon, and ‘stop-start’ traffic motion obeys a form of wave propagation.

##### MSC:

35L60 | First-order nonlinear hyperbolic equations |

35K57 | Reaction-diffusion equations |

35L67 | Shocks and singularities for hyperbolic equations |

92D30 | Epidemiology |

90B20 | Traffic problems in operations research |

##### Keywords:

traffic modelling; crowd dynamics; hantavirus; SIS epidemic model; hyperbolic reaction-diffusion equations
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\textit{J. Bissell} and \textit{B. Straughan}, Discrete Contin. Dyn. Syst., Ser. B 19, No. 7, 1911--1934 (2014; Zbl 1304.35413)

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