Recent years have witnessed an explosion of extensive geolocated datasets related to human movement, enabling scientists to quantitatively study individual and collective mobility patterns, and to generate models that can capture and reproduce the spatiotemporal structures and regularities in human trajectories. The study of human mobility is especially important for applications such as estimating migratory flows, traffic forecasting, urban planning, and epidemic modeling. In this survey, we review the approaches developed to reproduce various mobility patterns, with the main focus on recent developments. This review can be used both as an introduction to the fundamental modeling principles of human mobility, and as a collection of technical methods applicable to specific mobility-related problems. The review organizes the subject by differentiating between individual and population mobility and also between short-range and long-range mobility. Throughout the text the description of the theory is intertwined with real-world applications.

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Human Mobility: Models and Applications

These notes are devoted to a summary on the mean-field limit of large ensembles of interacting particles with applications in swarming models We first make a summary of the kinetic models derived as continuum versions of second order models for swarming We focus on the question of passing from the discrete to the continuum model in the Dobrushin framework We show how to use related techniques from fluid mechanics equations applied to first order models for swarming, also called the aggregation equation We give qualitative bounds on the approximation of initial data by particles to obtain the mean-field limit for radial singular (at the origin) potentials up to the Newtonian singularity We also show the propagation of chaos for more restricted set of singular potentials

The derivation of Swarming models: Mean-Field Limit and Wasserstein distances

Self‐Organization in Biological Systems.Princeton Studies in Complexity. ByScott Camazine,, Jean‐Louis Deneubourg,, Nigel R Franks,, James Sneyd,, Guy Theraulaz, and, Eric Bonabeau; original line drawings by, William Ristineand, Mary Ellen Didion; StarLogo programming by, William Thies. Princeton (New Jersey): Princeton University Press. $65.00. viii + 538 p + 8 pl; ill.; index. ISBN: 0–691–01211–3. 2001.

This paper presents a revisiting, with developments, of the so-called kinetic theory for active particles, with the main focus on the modeling of nonlinearly additive interactions. The approach is based on a suitable generalization of methods of kinetic theory, where interactions are depicted by stochastic games. The basic idea consists in looking for a general mathematical structure suitable to capture the main features of living, hence complex, systems. Hopefully, this structure is a candidate towards the challenging objective of designing a mathematical theory of living systems. These topics are treated in the first part of the paper, while the second one applies it to specific case studies, namely to the modeling of crowd dynamics and of the immune competition.

On the difficult interplay between life, "complexity", and mathematical sciences

These notes are devoted to a summary on the mean-field limit of large ensembles of interacting particles with applications in swarming models. We first make a summary of the kinetic models derived as continuum versions of second order models for swarming. We focus on the question of passing from the discrete to the continuum model in the Dobrushin framework. We show how to use related techniques from fluid mechanics equations applied to first order models for swarming, also called the aggregation equation. We give qualitative bounds on the approximation of initial data by particles to obtain the mean-field limit for radial singular (at the origin) potentials up to the Newtonian singularity. We also show the propagation of chaos for more restricted set of singular potentials.

/pdf/the-derivation-of-swarming-models-mean-field-limit-and-5e9m0c9oga.pdf

The derivation of swarming models: Mean-field limit and Wasserstein distances

We consider a self-propelled interacting particle system for the collective behavior of swarms of animals, and extend it with an attraction term called roosting force, as it has been suggested in Ref. 30. This new force models the tendency of birds to overfly a fixed preferred location, e.g. a nest or a food source. We include roosting to the existing individual-based model and consider the associated mean-field and hydrodynamic equations. The resulting equations are investigated analytically looking at different asymptotic limits of the corresponding stochastic model. In addition to existing patterns like single mills, the inclusion of roosting yields new scenarios of collective behavior, which we study numerically on the microscopic as well as on the hydrodynamic level.

/pdf/self-propelled-interacting-particle-systems-with-roosting-5a268zbrr6.pdf

Self-propelled interacting particle systems with roosting force

We introduce a novel first-order stochastic swarm intelligence (SI) model in the spirit of consensus formation models, namely a consensus-based optimization (CBO) algorithm, which may be used for the global optimization of a function in multiple dimensions. The CBO algorithm allows for passage to the mean-field limit, which results in a nonstandard, nonlocal, degenerate parabolic partial differential equation (PDE). Exploiting tools from PDE analysis we provide convergence results that help to understand the asymptotic behavior of the SI model. We further present numerical investigations underlining the feasibility of our approach.

A consensus-based model for global optimization and its mean-field limit

We consider interacting particle systems and their mean-field limits, which are frequently used to model collective aggregation and are known to demonstrate a rich variety of pattern formations. The interaction is based on a pairwise potential combining short-range repulsion and long-range attraction. We study particular solutions, which are referred to as flocks in the second-order models, for the specific choice of the Quasi-Morse interaction potential. Our main result is a rigorous analysis of continuous, compactly supported flock profiles for the biologically relevant parameter regime. Existence and uniqueness are proven for three space dimensions, while existence is shown for the two-dimensional case. Furthermore, we numerically investigate additional Morse-like interactions to complete the understanding of this class of potentials.

/pdf/explicit-flock-solutions-for-quasi-morse-potentials-4pgi3vi7sl.pdf

Explicit flock solutions for Quasi-Morse potentials

Our main objective is the modelling and simulation of complex production networks originally
introduced in [15, 16] with random breakdowns of individual processors. Similar to
[10], the breakdowns of processors are exponentially distributed. The resulting network
model consists of coupled system of partial and ordinary differential equations with Markovian switching
and its solution is a stochastic process. We show our model to fit into the framework of piecewise
deterministic processes, which allows for a deterministic interpretation of dynamics between a
multivariate two-state process. We develop an efficient algorithm with an emphasis on accurately
tracing stochastic events. Numerical results are presented for three exemplary networks, including
a comparison with the long-chain model proposed in [10].

/pdf/time-continuous-production-networks-with-random-breakdowns-3t75qglsih.pdf

Stephan Martin

Papers

Self-propelled interacting particle systems with roosting force

A consensus-based model for global optimization and its mean-field limit

A consensus-based model for global optimization and its mean-field limit

Explicit flock solutions for Quasi-Morse potentials

Time-continuous production networks with random breakdowns