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

## TL;DR

This paper compares how well Petri net and ODE models simulate disease spread, showing that Petri nets can accurately model SIR dynamics with proper numerical methods.

## Contribution

The paper introduces a novel deterministic Petri net implementation of the SIRS model and demonstrates numerical convergence with ODEs.

## Key findings

- A deterministic Petri net implementation of the SIRS model achieves less than 1% error compared to ODE simulations.
- Rescaling and rounding procedures are critical for numerical convergence in Petri net SIR models.
- Both stochastic and deterministic Petri nets can accurately model SIR dynamics with appropriate numerical methods.

## 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 variable 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.