Novel travel time aware metapopulation models and multi-layer waning immunity for late-phase epidemic and endemic scenarios
Authors: Zunker Henrik , Schmieding René , Kerkmann David , Schengen Alain , Diexer Sophie , Mikolajczyk Rafael , Meyer-Hermann Michael , Kühn Martin J. 
Published: March 2024

In the realm of infectious disease control, accurate modeling of the transmission dynamics is pivotal. As human mobility and commuting patterns are key components of communicable disease spread, we introduce a novel travel time aware metapopulation model. Our model aims to enhance estimations of disease transmission. By providing more reliable assessments on the efficacy of interventions, curtailing personal rights or human mobility behavior through interventions can be minimized. The proposed model is an advancement over traditional compartmental models, integrating explicit transmission on travel and commute, a factor available in agent-based models but often neglected with metapopulation models.

Our approach employs a multi-edge graph ODE-based (Graph-ODE) model, which represents the intricate interplay between mobility and disease spread. This granular modeling is particularly important when assessing the dynamics in densely connected urban areas or when heterogeneous structures across entire countries have to be assessed. The given approach can be coupled with any kind of ODE-based model.

In addition, we propose a novel multi-layer waning immunity model that integrates waning of different paces for protection against mild and severe courses of the disease. As this is of particular interest for late-phase epidemic or endemic scenarios, we consider the late-phase of SARS-CoV-2 in Germany.

The results of this work show that accounting for resolved mobility significantly influences the pattern of outbreaks. The improved model provides a refined tool for predicting outbreak trajectories and evaluating intervention strategies in relation to mobility by allowing us to assess the transmission that result on traveling. The insights derived from this model can serve as a basis for decisions on the implementation or suspension of interventions, such as mandatory masks on public transportation. Eventually, our model contributes to maintaining mobility as a social good while reducing exuberant disease dynamics potentially driven by travel activities.

Author summary

As human contacts and contact networks are key to the development and prediction of infectious disease spread, travel and commuting activities are important components to be considered in mathematical-epidemiological modeling. Two, often contrasting modeling approaches, based on subpopulations and based on individuals can provide insights of different granularity but also come at different levels of complexity.

With this article, we extend a recently introduced Graph-ODE-based model by the explicit introduction of mobility-based infection models in which we allow focused nonpharmaceutical interventions, like face mask mandates in public transport, and in which we can explicitly keep track of secondary cases induced by travel activities, a component mostly not available with equation-based models. In addition, we introduce a novel multi-layer waning immunity model particularly suitable for late-phase epidemic or endemic scenarios.

On a daily level and geographically small scale, the newly proposed model often develops similarly, although our results show that complex mobility networks can lead to substantially different disease dynamics in the entirety of a federal state or country. The proposed model thus enables a better understanding of infectious disease dynamics through mobility. It allows for targeted numerical investigations and thus leads to more appropriate real-world interventions.