Mathematical modeling of spatio-temporal population dynamics and application to epidemic spreading

TL;DR

This paper develops mathematical models to efficiently simulate spatio-temporal population dynamics and epidemic spreading, introducing metapopulation approaches that reduce computational costs while maintaining key dynamical features.

Contribution

It presents two novel metapopulation modeling approaches derived from agent-based models, including a stochastic and a piecewise deterministic version, with a mathematical derivation and application to COVID-19.

Findings

Metapopulation models significantly reduce computational costs.

Stochastic model derived via Galerkin projection from ABMs.

Framework successfully applied to COVID-19 spread modeling.

Abstract

Agent based models (ABMs) are a useful tool for modeling spatio-temporal population dynamics, where many details can be included in the model description. Their computational cost though is very high and for stochastic ABMs a lot of individual simulations are required to sample quantities of interest. Especially, large numbers of agents render the sampling infeasible. Model reduction to a metapopulation model leads to a significant gain in computational efficiency, while preserving important dynamical properties. Based on a precise mathematical description of spatio-temporal ABMs, we present two different metapopulation approaches (stochastic and piecewise deterministic) and discuss the approximation steps between the different models within this framework. Especially, we show how the stochastic metapopulation model results from a Galerkin projection of the underlying ABM onto a finite-dimensional ansatz space. Finally, we utilize our modeling framework to provide a conceptual model for the spreading of COVID-19 that can be scaled to real-world scenarios.