A Numerical Comparison of Petri Net and Ordinary Differential Equation SIR Component Models

TL;DR

This study compares Petri Net and ODE models of the SIR epidemic framework, demonstrating that with proper numerical techniques, Petri Nets can accurately replicate traditional differential equation results.

Contribution

It introduces a systematic numerical comparison between Petri Net and ODE SIR models, including a novel deterministic Petri Net implementation with dynamic transition weights.

Findings

Petri Net models achieve less than 1% error compared to ODEs.

Rescaling and rounding are crucial for numerical convergence.

Both stochastic and deterministic Petri Nets are valid for SIR modeling.

Abstract

Petri Nets are an increasingly used modeling framework for the spread of disease across populations or within an individual. For example, the Susceptible-Infectious-Recovered (SIR) compartment model is foundational for population epidemiological modeling and has been implemented in several prior Petri Net studies. While the SIR model is typically expressed as Ordinary Differential Equations (ODEs), with continuous time and variables, Petri Nets operate as discrete event simulations with deterministic or stochastic timings. We present the first systematic study of the numerical convergence of two distinct Petri Net implementations of the SIRS compartment model relative to the standard ODE. In particular, we introduce a novel deterministic implementation of the SIRS model using dynamic transition weights in the GPenSIM package and stochastic Petri Net models using SPIKE. We show how rescaling and rounding procedures are critical for the numerical convergence of Petri Net SIR models relative to the ODEs, and we achieve a relative root mean squared error of less than 1% compared to ODE simulations for biologically relevant parameter ranges. Our findings confirm that both stochastic and deterministic discrete time Petri Nets are valid for modeling SIR-type dynamics with appropriate numerical procedures, laying the foundations for larger-scale use of Petri Net models.