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Crowds in Equations: An Introduction to the Microscopic Modeling of Crowds (Advanced Textbooks in Mathematics)

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The book contains self-contained descriptions of existing models, accompanied by critical analyses of their properties both from a theoretical and practical standpoint. It aims to develop 'modeling skills' within the readers, giving them the ability to develop their own models and improve existing ones. Written in connection with a full, open source Python Library, this project also enables readers to run the simulations discussed within the text.

The modeling of particle systems has raised a considerable activity in the last centuries. Physicists, mathematicians, and more recently computer scientists, have joined their efforts to formalize the laws which govern the motion of particles in interaction with one another. They built theoretical frameworks and numerical tools to evolve the “big picture”, i.e. induce general rules or macroscopic equations which would make it possible to describe the behavior of the considered system at a large scale, beyond the individual destinies of its components.

Since a few decades, scientists from various domains have extended the approach to systems of “active entities”, bird flocks, fish schools, crowds of mammals or insects, and, even further apart from passive particles, walking or driving human beings. Part of the frameworks developed for physical systems can be straightforwardly transposed to this new situation. In particular, the kinematic modeling will consist in representing a moving crowd by a time-varying vector which contains the positions of its individuals. Now assuming one is able to model the will of an individual and its interaction with others in an equation involving the positions and its derivatives, the model takes the form of a system of differential equations, for which a tremendous amount of theoretical and numerical tools have been developed. Yet, the modeling of living entities presents particular features which make it very peculiar among other activities in particle system modeling.
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