Petri nets in epidemiology

Area of Science:

• Epidemiology
• Mathematical Biology
• Theoretical Computer Science

Background:

• The basic reproduction number (R0) is a critical metric in epidemiology for understanding disease spread.
• Traditional methods for calculating R0 can be complex and computationally intensive.
• Petri nets offer a graphical modeling framework with potential for simplifying complex systems.

Purpose of the Study:

• To develop a geometric interpretation of the next-generation matrix method for calculating the basic reproduction number.
• To establish a correspondence between epidemiological models (ODEs) and Petri nets.
• To simplify the calculation and understanding of R0 through a visual, structural approach.

Main Methods:

• Utilizing Petri nets to represent epidemiological models, specifically focusing on the SIR model structure.
• Mapping the five assumptions of the next-generation matrix method to geometric properties within the Petri net framework.
• Analyzing the flow between infection compartments in the Petri net to derive the next-generation matrix.

Main Results:

• A direct correspondence was established between systems of ordinary differential equations (ODEs) and Petri nets.
• The basic reproduction number was shown to depend on three fundamental substructures within the Petri net representation.
• The next-generation matrix was geometrically interpreted as a matrix of flows between infection compartments.

Conclusions:

• Petri nets provide a powerful geometric framework for understanding and calculating the basic reproduction number.
• This geometric approach simplifies the analysis of epidemiological models by visualizing core components.
• The dominant eigenvalue of the flow matrix in the Petri net corresponds to the basic reproduction number.