  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.
