Generating function approach to the effective degree SIR model.

Area of Science:

• Epidemiology
• Network Science
• Mathematical Biology

Background:

• The effective degree SIR model analyzes disease spread on networks with lifetime immunity.
• Current models struggle with infinite degree distributions like scale-free networks.
• Existing methods often use ordinary differential equations for neighbor infection status.

Purpose of the Study:

• To develop a novel approach for the effective degree SIR model applicable to infinite degree networks.
• To establish the mathematical foundation for analyzing disease dynamics on complex contact networks.
• To provide a framework for studying disease-free steady states and thresholds in large-scale networks.

Main Methods:

• A generating function approach was used to transform the SIR model into a nonlinear partial differential equation.
• Existence and uniqueness of biologically relevant solutions were demonstrated.
• The model was shown to reduce to the Volz model under specific independence assumptions.

Main Results:

• A new partial differential equation formulation for the effective degree SIR model was derived.
• The mathematical existence and uniqueness of relevant solutions were proven.
• The connection to the Volz model was established, validating the new approach.

Conclusions:

• The proposed generating function method extends SIR modeling to infinite degree networks.
• This work provides a robust framework for analyzing epidemic thresholds and stability.
• The findings pave the way for more comprehensive studies of disease dynamics on complex systems.