A Coupled Spatial-Network Model: A Mathematical Framework for Applications in Epidemiology

Area of Science:

• Epidemiology
• Mathematical Modeling
• Network Science

Background:

• Network structures like transport routes demonstrably accelerate epidemic dissemination.
• Understanding disease spread across interconnected populations is crucial for public health.

Purpose of the Study:

• To introduce a novel compartmental modeling framework for simulating epidemic spread.
• To couple population centers, travel routes, and continuous geographical areas within a unified model.
• To analyze the dynamics of infectious disease transmission through complex networks.

Main Methods:

• Developed a hybrid modeling framework integrating ordinary differential equations (ODEs) for population centers, 1D equations for travel routes (edges), and 2D continuum equations for the general population.
• Implemented junction conditions to couple vertex ODEs with edge equations and boundary conditions to link domain equations with edges.
• Employed a numerical method combining spatial finite differences for edges and finite elements for the 2D domain.

Main Results:

• The model successfully simulates epidemic spread across interconnected geographical areas.
• Numerical solutions demonstrate exponential decay in infection rates over time post-initial spread.
• The cumulative infected population across all compartments (vertices, edges, domain) stabilizes to a time-invariant, spatially varying steady-state.

Conclusions:

• The proposed modeling framework provides a robust tool for studying network-mediated epidemic dynamics.
• Network structure plays a critical role in shaping the spatial and temporal patterns of disease spread.
• The model's steady-state solutions offer insights into the long-term distribution of infections in connected populations.