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Predicting the Effectiveness of Population Replacement Strategy Using Mathematical Modeling

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Published on: July 4, 2007

The Basic Reproduction Number for Petri Net Models: A Next-Generation Matrix Approach.

Trevor Reckell¹, Beckett Sterner², Petar Jevtić¹

¹School of Mathematical and Statistical Sciences, Arizona State University, 901 S. Palm Walk, Tempe, AZ 85287-1804, USA.

Applied Sciences (Basel, Switzerland)

|June 29, 2026

Summary

This study introduces a new computational framework for calculating the basic reproduction number (R0) directly from Petri Net models, enhancing epidemiological analysis. The method accurately predicts disease spread and informs public health interventions.

Keywords:

SIR model disease spread epidemiology modeling next-generation method ordinary differential equations petri nets reproduction number

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Area of Science:

Epidemiology
Computational Biology
Graph Theory

Background:

The basic reproduction number (R0) is a key metric in epidemiology for predicting disease outbreaks and guiding public health interventions.
Current methods for calculating R0 are well-established for Ordinary Differential Equations (ODEs) but lack a generalized approach for Petri Net (PN) models.
Petri Nets are increasingly utilized in epidemiology for modeling dynamic systems, highlighting the need for direct R0 calculation methods.

Purpose of the Study:

To establish a generalized computational framework for calculating the basic reproduction number (R0) directly from Petri Net (PN) models.
To adapt the next-generation matrix method for compatibility with various PN formalisms, including Variable Arc Weight Petri Nets (VAPNs) and stochastic continuous-time Petri Nets (SPNs).

Main Methods:

Developed a generalized computational framework for R0 calculation from Petri Net models.
Adapted the next-generation matrix method to work with both deterministic (VAPNs) and stochastic (SPNs) Petri Net formalisms.
Applied the framework to diverse epidemiological models, including those with multiple strains, asymptomatic states, and nonlinear dynamics.

Main Results:

Successfully established a generalized method for calculating R0 directly from Petri Net models.
Demonstrated the framework's versatility across complex epidemiological models.
Numerically validated the framework, showing strong agreement between analytically derived R0 values and simulation data.

Conclusions:

The developed framework provides an accurate and practical method for calculating R0 from Petri Net models.
This advancement facilitates more robust epidemiological analysis and supports informed public health decision-making.
The method's compatibility with multiple Petri Net formalisms enhances its utility in diverse disease modeling scenarios.