  This paper introduces a novel hybrid mathematical modeling approach that
effectively couples Partial Differential Equations (PDEs) with Ordinary
Differential Equations (ODEs), exemplified through the simulation of
epidemiological processes. The hybrid model aims to integrate the spatially
detailed representation of disease dynamics provided by PDEs with the
computational efficiency of ODEs. In the presented epidemiological use-case,
this integration allows for the rapid assessment of public health interventions
and the potential impact of infectious diseases across large populations. We
discuss the theoretical formulation of the hybrid PDE-ODE model, including the
governing equations and boundary conditions. The model's capabilities are
demonstrated through detailed simulations of disease spread in synthetic
environments and real-world scenarios, specifically focusing on the regions of
Lombardy, Italy, and Berlin, Germany. Results indicate that the hybrid model
achieves a balance between computational speed and accuracy, making it a
valuable tool for policymakers in real-time decision-making and scenario
analysis in epidemiology and potentially in other fields requiring similar
modeling approaches.
