Hybrid PDE-ODE Models for Efficient Simulation of Infection Spread in Epidemiology

TL;DR

The paper develops a hybrid PDE-ODE framework that couples a spatial SEIR diffusion-reaction PDE with a region of homogeneous mixing modeled by ODEs to enable fast yet spatially informed simulations of infection spread. The method preserves local spatial detail where needed while reducing computational load in less-critical regions, using a domain-decomposition coupling across a boundary with flux- and density-continuity conditions. Key contributions include a concrete Hybrid PDE-ODE formulation, a Levenberg-Marquardt–based parameter identification workflow, and extensive synthetic and real-world evaluations in Lombardy and Berlin that reveal how boundary placement and Allee effects shape transmission dynamics. The work demonstrates substantial speed gains with acceptable accuracy, and offers practical guidance for boundary design, potential extensions to transportation dynamics, and public-health decision support in real-time scenario analysis.

Abstract

This paper introduces a novel hybrid model combining Partial Differential Equations (PDEs) and Ordinary Differential Equations (ODEs) to simulate infectious disease dynamics across geographic regions. By leveraging the spatial detail of PDEs and the computational efficiency of ODEs, the model enables rapid evaluation of public health interventions. Applied to synthetic environments and real-world scenarios in Lombardy, Italy, and Berlin, Germany, the model highlights how interactions between PDE and ODE regions affect infection dynamics, especially in high-density areas. Key findings reveal that the placement of model boundaries in densely populated regions can lead to inaccuracies in infection spread, suggesting that boundaries should be positioned in areas of lower population density to better reflect transmission dynamics. Additionally, regions with low population density hinder infection flow, indicating a need for incorporating, e.g., jumps in the model to enhance its predictive capabilities. 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.

Figures (18)

• Figure 1: Infectious density of PDE model (\ref{['PDE-model']}) for times $t \in \{0,29,30,59\}$ in rectangular domain. The Gaussian-shaped spread of infection is evident due to the Gaussian-shaped population density.
• Figure 2: Number of infectious individuals of full-PDE and hybrid models in rectangular domain for times $t \in [0,59]$ (left) and maximum average deviation from full-PDE model (right). The infectious count decreases with a higher percentage of the ODE domain. This is attributed to the inclusion of the Allee term, which spatially modifies the infection rate in the PDE domain while remaining constant in the ODE domain.
• Figure 3: Accuracy (mean absolute error) of the full-PDE model (\ref{['PDE-model']}) and hybrid model (\ref{['Eq:hybrid_model']}) in rectangular domain. We can observe an approximately linear growth.
• Figure 4: Runtimes of the full-PDE (\ref{['PDE-model']}), hybrid (\ref{['Eq:hybrid_model']}) and full-ODE models in rectangular domain. The runtime initially increases (/finally decreases) due to the additional (/absence of) implementation costs associated with the boundary conditions of the hybrid model.
• Figure 5: Extreme cases of the hybrid model (\ref{['Eq:hybrid_model']}) in rectangular domain: the number of infectious individuals is initially equal to the total population number (left) and zero (right). The outcomes for the ODE region appear visually unaffected by the location of the population in the PDE domain, whereas the outcomes for the PDE region vary significantly.