Mathematical modeling and efficient simulation of epidemic metapopulation dynamics

Abstract

Infectious disease modeling faces an inherent tension between physical realism and computational feasibility. Granular metapopulation models offer interpretability and the capacity to resolve complex behavioral dynamics but can become computationally demanding when coupled with explicit transport dynamics on dense networks. Conversely, purely statistical or black-box machine learning approaches enable rapid inference, although often at the cost of substantial training time, but frequently lack causal explanation. In this thesis, we present a set of metapopulation and machine learning methods that address several open issues in infectious disease modeling - related to necessary spatial resolution, the inclusion of important aspects such as mobility and behavior, and computational efficiency. <br/>

First, we establish a mechanistic baseline by proposing a generalized piecewise-continuous graph-based metapopulation ordinary differential equation (ODE) framework (<strong>Manuscripts 1-2</strong>). We introduce explicit transit dynamics that model infection risk during travel, resolving the error due to instantaneous travel of classical models, and incorporate risk-mediated behavioral feedback to analyze the interplay between local mitigation and national suppression strategies. <br/>

To overcome the computational bottlenecks of explicitly tracking traveling subpopulations, which typically leads to a quadratic growth of the state space in densely connected networks, we develop a stage-aligned numerical method (<strong>Manuscript 3</strong>). By solving an aggregated global ODE system that scales only linearly with the number of regions, and algebraically computing detailed traveler states on-the-fly during Runge-Kutta integration, the dimension of the expensive ODE integration is effectively decoupled from the network connectivity. We mathematically prove that this method yields numerical solutions identical to those of the fully resolved system, enabling runtime accelerations of up to 76-fold in the evaluated scenarios without any loss of accuracy. <br/>

Furthermore, building on the piecewise-continuous graph-based framework, we generate extensive training data for Graph Neural Network (GNN) surrogates (<strong>Manuscript 4</strong>). By learning the causal structure of the mechanistic backbone, these surrogates emulate complex intervention scenarios with sub-second execution times, establishing a tool for interactive intervention analysis. Addressing the parallel challenge of data privacy in real-world surveillance, we investigate decentralized learning strategies (<strong>Manuscript 5</strong>). We demonstrate that federated learning enables feasible forecasting utility even under reasonable differential privacy guarantees. <br/>

All presented methods are implemented in the open-source platform <em>MEmilio</em> (<strong>Manuscript 6</strong>), which serves as a highly scalable infrastructure for future pandemic preparedness and unifies the contributions of this thesis into a single operational system, combining computational scalability, rapid surrogate modeling, and privacy-preserving surveillance.