A generalized mean-field game model for the dynamics of pedestrians with limited predictive abilities

Abstract

The model assumes that each individual in the crowd moves in a known domain, aiming at minimizing a given cost functional. Both the pedestrian dynamics and the cost functional itself depend on the position of the whole crowd. In addition, pedestrians are assumed to have predictive abilities, but limited in time, extending only up to θ time units into the future, where θ ∈ [0,∞) is a model parameter. 1) For θ = 0 (no predictive abilities), we recover the modeling assumptions of the Hughes's model, where people take decisions on the basis of the current position of the crowd only. 2) For θ → ∞, instead, we recover the standard mean field game (MFG) setting, where people are able to forecast the behavior of the others at any future time and take decisions on the basis of the current and future position of the whole crowd. 3) For very short values of θ (typically coinciding with a single time step in a discretein-time setting), we recover instead the MFG setting joined to the model predictive control technique. 4) For intermediate values of θ we obtain something different: as in the Hughes's model, the numerical procedure to solve the problem requires to run an off-line procedure at any fixed time t, which returns the current optimal velocity field at time t by solving an associated backward-in-time Hamilton-Jacobi-Bellman equation; but, differently from the Hughes's model, here the procedure involves a prediction of the crowd behavior in the sliding time window [t,t + θ), therefore the optimal velocity field is given by the solution of a forward-backward system which joins a Fokker-Planck equation with a Hamilton-Jacobi-Bellman equation as in the MFG approach. The fact that a different forward-backward system must be solved at any time t arises new interesting theoretical questions. Numerical tests will give some clues about the well-posedness of the problem.

A numerical method to calculate multiple epidemic waves in COVID-19 with a realistic total number of people involved

“…Then, the accurate prediction of the number of detected individuals is significant. Meanwhile, it is an interesting problem to investigate whether the optimal activities of each individual are determined to minimize the total cost (economic loss by both infection and self-restraint) using feedback control with the future cost, as demonstrated by the Hamilton-Jacobi-Bellman equation [12,13], or not. Therefore, the SIR type model with the time-dependent parameters might be closed in the framework of the data independent theory, if time-dependent parameters in the SIR type model are defined as a function of time on the basis of game theoretical behavior of individuals.…”

“…σ (k) β (k) ∈ R + is the rate of the reinfection, 1/γ (k) (Θ) = 9, 1/γ (k) (Θ) = 14, and Q (k) (t = 0, Θ) = R (k) (t = 0, Θ) = 0. We obtain N S (t, Θ), N I (t, Θ), N Q (t, Θ), and N R (t, Θ) using equation (13). Let σ (k) r be the rate of the reinfection of individuals who have already recovered from the infection and σ v be the rate of the infections of individuals who have become the recovered ones from the susceptible state ones due to vaccinations.…”

Preventing congestion in crowd dynamics caused by reversing flow

“…It is also useful to mention that models for pedestrians often stem from those developed in the context of vehicular traffic [18,21]. Moreover, there is a strict connection between pedestrian modeling and control theory, including mean-field games, see, e.g., [17,22,23] and reference therein.…”

Nonsmooth mean field games with state constraints

“…Up to the authors' knowledge, the first work to be fully dedicated to a mean field game model for crowd motion is [44], which proposes an MFG model for a two-population crowd with trajectories perturbed by additive Brownian motion and considers both their stationary distributions and their evolution on a prescribed time interval. Other works have later proposed MFG models for crowd motion taking into account different characteristics, such as [10], which considers the fast exit of a crowd and proposes a mean field game model which is studied numerically; [21], which is not originally motivated by the modeling of crowd motion but considers a MFG model with a density constraint, which is a natural assumption in some crowd motion models; [8], which presents numerical simulations for some variational mean field games related to crowd motion; or also [26], which provides a generalized MFG model for pedestrians with limited predictive abilities.…”