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...n zones with restricted access for pedestrians (e.g. closed rooms, prohibited access areas, inner concrete structures)4, whose union we call Ωs. Let us also assume that the remaining region, say Ωp, which is defined by Ωp := Ω − Ωs, is connected. Consequently, Ωp is accessible to pedestrians. The exits of Ω – target that each pedestrian wants to reach – are assumed to belong to the boundary of Ωp. The way we imagine the heterogeneity of Ω is sketched in Figure 1. In this framework, we choose for the following working plan: Firstly, we extend the multiscale approach developed by Piccoli et al. [10] (see also the context described in [21] and [22]) to the case of counterflow5 of pedestrians; then we allow the pedestrian dynamics to take place in the heterogeneous domain Ω, and finally, we include an implicit velocity law for the pedestrians motion. The main reason why we choose the counterflow scenario [also called bidirectional 3Note that, for instance, Campanella et al. [8] give a different meaning to heterogeneity: they mainly refer to lack of homogeneity in the speed distributions of pedestrians. In [7] the geometric heterogeneities - obstacles - are introduced in the microscopic mod...

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...h people can even die due to overcompression (e.g. on Jamarat Bridge in Saudi Arabia, see SYSTEM OF CONTINUITY EQUATIONS WITH NON-LOCAL FLOW 3 [15]). A similar model was introduced in Piccoli & Tosin =-=[11, 21]-=-, where the authors instead of an isotropic convolution kernel, consider a nonlocal functional taking into account the direction in which the pedestrians are looking. In [6], the authors study the sca...

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...e eikonal equation with 1/f(ρ) in the right hand side, while in the latter case ν(x) = 0 and I(ρ) is the gradient of the solution to an elliptic partial differential equation. The model introduced in =-=[13, 29]-=- relies on this measure valued conservation law: ∂tµ+ div (µ v) = 0 where v = ν(x) + ∫ R2 f(|x− y|)ϕ ( (y − x) · ν(x) ) x− y ‖x− y‖ dµt (y) . Here the unknown is a map µ : [0, T ]→M(RN ;R) assigning a...

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...e in [8]). In general, CM or KM are more efficient than IBM for large crowds because their computational time does not increase with the number of agents. However, they suffer from different drawbacks, such as a reduced validity range due to the necessary recourse to closure relations, as detailed below. Nonetheless, CM are invaluable tools for large-scale analysis and prediction of crowd behavior. Therefore, it is important to firmly base the 2 derivation of CM on their small-scale IBM counterpart. The literature on the derivation of CM from microscopic models (IBM or CA) is scarce (see e.g. [1, 16, 18, 22]). The present paper addresses this question and intends to propose a hierarchy of KM and CM based on the IBM developed in [54]. The psychological literature shows that pedestrians can estimate the positions and velocities of moving obstacles such as other pedestrians with fairly good accuracy [23]. Therefore, the subjects are able to process this information in order to determine the dangerousness level of an encounter [67]. Taking these considerations into account, the heuristic-based model of [54] proposes that pedestrians follow a heuristic rule composed of two phases: a perception phase a...

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...lticellular micro-organisms, inspired a very rich mathematical literature [35, 36, 52], see also the very recent work [4] and references therein. Human motion, including pedestrian and crowd modeling =-=[21, 22]-=-, for instance in evacuation process simulations, has been a matter of intensive research, connecting also with new developments such as mean field games, see [40] and the overview in its Section 2. C...

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...lar discrete models is that they are very flexible in engineering ad-hoc assumptions. In addition, they allow agent (person or car) tracking rather than only counting densities. A new hybrid approach =-=[16, 40]-=- describes a combined microscopic and macroscopic modeling in a multiscale framework using a measure-theoretic approach. The crowd velocity is composed as a sum of a macro-scale desired velocity and a...

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...stems [20, 21, 22, 29, 32, 33, 54], towards modeling phenomena in biology, such as cell aggregation and motion [4, 34, 35, 45], animal motion [3, 8, 11, 18, 15, 16, 22, 38, 41, 42, 49, 53, 59], human =-=[19, 24, 50]-=- and synthetic agent behavior and interactions, such as cooperative robots [12, 37, 43, 52]. As it is very hard to be exhaustive in accounting all the developments of this very fast growing field, we ...