Simple agent-based models and their continuum limit

Theodore Kolokolnikov
Dalhousie University

We discuss several very different ABM models and their continuum limits.
First, consider the following agent-based model of coronavirus spread: people move randomly and infection occurs with some nonzero probability when an infected individual comes within a certain ``infection radius'' of a susceptible individual. The question is how the infection radius affects the reproduction number. At low infection rates, this model leads to the classical S-I-R ODE model as its continuum limit. However higher infection rates lead to a saturation effect, which we compute explicitly using basic probability theory. Its continuum limit It leads to an S-I-R type model with a specific saturation term. We also show that this modified model gives a much better fit to the real-world data than the classical SIR model.
Next, we will look at a very simple stochastic model of bacterial aggregation which leads to a novel fourth-order nonlinear PDE in its continuum limit. This PDE admits soliton-type solutions corresponding to bacterial aggregation patterns, which we explicitly construct.
If time allows, we will consider a spatial model of wealth exchange which leads to novel integro-differential equations.
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