Effective degree network disease models.

Area of Science:

• Epidemiology
• Network Science
• Mathematical Biology

Background:

• Traditional disease models often assume homogeneous mixing, which may not reflect real-world network structures.
• Understanding disease dynamics on networks is crucial for public health interventions.

Purpose of the Study:

• To introduce and validate an effective degree approach for modeling infectious disease spread on networks.
• To apply this approach to both Susceptible-Infectious-Susceptible (SIS) and Susceptible-Infectious-Recovered (SIR) disease models.
• To derive and analyze disease threshold conditions for these network models.

Main Methods:

• Formulating effective degree models as systems of ordinary differential equations.
• Tracking susceptible and infectious neighbors for each individual.
• Conducting numerical simulations on random graphs.
• Comparing model results with stochastic processes and percolation theory.

Main Results:

• Effective degree models accurately capture initial growth rates, endemic equilibria (SIS), and epidemic peaks (SIR) compared to stochastic simulations.
• Derived formulas for disease threshold conditions for both SIS and SIR effective degree models.
• The SIS model threshold is higher than predicted by percolation theory, suggesting easier disease invasion.
• The SIR model threshold matches percolation theory predictions.

Conclusions:

• Effective degree models provide a robust framework for analyzing disease spread on networks.
• The SIS and SIR effective degree models exhibit distinct disease threshold conditions, unlike homogeneous mixing models.
• These findings have implications for predicting and controlling infectious diseases in structured populations.