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

Trevor Reckell, Beckett Sterner, Petar Jevtić

TL;DR

This work provides a universal, computational framework (NGMPN) to compute the basic reproduction number $R_0$ directly from Petri Net representations, compatible with both deterministic Variable Arc Weight PN (VAPN) and stochastic SPN formulations. By partitioning infected and non-infected places and constructing infection and transition matrices $F$ and $V$, the method computes $R_0$ as the dominant eigenvalue of $FV^{-1}$ and validates it across diverse models, including SIRS, SEIR, SEEIR, SVEIR, COVID-19, nonlinear dynamics, and multi-patch systems. The authors demonstrate analytical concordance with traditional ODE-derived $R_0$ expressions and provide numerical verification against simulation-based estimates, achieving high accuracy (low relative RMSE). The NGMPN framework thus bridges ODE-based epidemiology with modular PN representations, enabling rigorous, scalable analysis of complex, policy-sensitive infectious-disease dynamics.

Abstract

The basic reproduction number ($R_0$) is an epidemiological metric that represents the average number of new infections caused by a single infectious individual in a completely susceptible population. The methodology for calculating this metric is well-defined for numerous model types, including, most prominently, Ordinary Differential Equations (ODEs). The basic reproduction number is used in disease modeling to predict the potential of an outbreak and the transmissibility of a disease, as well as by governments to inform public health interventions and resource allocation for controlling the spread of diseases. A Petri Net (PN) is a directed bipartite graph where places, transitions, arcs, and the firing of the arcs determine the dynamic behavior of the system. Petri Net models have been an increasingly used tool within the epidemiology community. However, no generalized method for calculating $R_0$ directly from PN models has been established. Thus, in this paper, we establish a generalized computational framework for calculating $R_0$ directly from Petri Net models. We adapt the next-generation matrix method to be compatible with multiple Petri Net formalisms, including both deterministic Variable Arc Weight Petri Nets (VAPNs) and stochastic continuous-time Petri Nets (SPNs). We demonstrate the method's versatility on a range of complex epidemiological models, including those with multiple strains, asymptomatic states, and nonlinear dynamics. Crucially, we numerically validate our framework by demonstrating that the analytically derived $R_0$ values are in strong agreement with those estimated from simulation data, thereby confirming the method's accuracy and practical utility.

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

TL;DR

This work provides a universal, computational framework (NGMPN) to compute the basic reproduction number directly from Petri Net representations, compatible with both deterministic Variable Arc Weight PN (VAPN) and stochastic SPN formulations. By partitioning infected and non-infected places and constructing infection and transition matrices and , the method computes as the dominant eigenvalue of and validates it across diverse models, including SIRS, SEIR, SEEIR, SVEIR, COVID-19, nonlinear dynamics, and multi-patch systems. The authors demonstrate analytical concordance with traditional ODE-derived expressions and provide numerical verification against simulation-based estimates, achieving high accuracy (low relative RMSE). The NGMPN framework thus bridges ODE-based epidemiology with modular PN representations, enabling rigorous, scalable analysis of complex, policy-sensitive infectious-disease dynamics.

Abstract

The basic reproduction number () is an epidemiological metric that represents the average number of new infections caused by a single infectious individual in a completely susceptible population. The methodology for calculating this metric is well-defined for numerous model types, including, most prominently, Ordinary Differential Equations (ODEs). The basic reproduction number is used in disease modeling to predict the potential of an outbreak and the transmissibility of a disease, as well as by governments to inform public health interventions and resource allocation for controlling the spread of diseases. A Petri Net (PN) is a directed bipartite graph where places, transitions, arcs, and the firing of the arcs determine the dynamic behavior of the system. Petri Net models have been an increasingly used tool within the epidemiology community. However, no generalized method for calculating directly from PN models has been established. Thus, in this paper, we establish a generalized computational framework for calculating directly from Petri Net models. We adapt the next-generation matrix method to be compatible with multiple Petri Net formalisms, including both deterministic Variable Arc Weight Petri Nets (VAPNs) and stochastic continuous-time Petri Nets (SPNs). We demonstrate the method's versatility on a range of complex epidemiological models, including those with multiple strains, asymptomatic states, and nonlinear dynamics. Crucially, we numerically validate our framework by demonstrating that the analytically derived values are in strong agreement with those estimated from simulation data, thereby confirming the method's accuracy and practical utility.

Paper Structure

This paper contains 13 sections, 54 equations, 8 figures.

Table of Contents

  1. Introduction
  2. A Brief Overview of Petri Nets
  3. Next-Generation Matrix for Petri Nets
  4. Next-Generation Matrix for Ordinary Differential Equations
  5. Constructing the Next-Generation Matrix for Petri Nets
  6. Case Studies
  7. SIRS
  8. SEIR
  9. SEEIR
  10. SVEIR
  11. Basic COVID-19 Model
  12. Nonlinear System
  13. Patch System Overview

Figures (8)

  • Figure S1: This VAPN is used as a guide to help visualize some of the concepts described in the Next-Generation Matrix for Petri Nets formulation. The legend on the right describes the elements within the PN. The places of this VAPN can be labeled $(P_1,P_2,P_3,P_4,P_5)$ or equivalently $(N_1,I_1,I_2,I_3,N_2)$, where the second label gives more description on the infectious state of the place and is used in the diagram. Red outline correspond to use in the corresponding $\mathcal{F}$ matrix and green outline correspond to use in the corresponding $\mathcal{V}$ matrix.
  • Figure S2: Two Petri Net implementations of the SIRS ODE model given in Equations \ref{['eq:S_SIRS']}--\ref{['eq:R_SIRS']}. Option (a) uses a discrete-time, variable arc weight PN. Option (b) uses a continuous-time, stochastic Markov Chain PN. The arcs and arc weights used in forming matrix $F_i(x)$ are outlined and highlighted in red, and the arcs and arc weights used in forming matrix $V_i(x)$ are outlined and highlighted in green.
  • Figure S3: Two Petri Net implementations of the SEIR ODE model given in Equations \ref{['eq:S_SEIR']}--\ref{['eq:R_SEIR']}. Option (a) uses a discrete-time, variable arc weight PN. Option (b) uses a continuous-time, stochastic Markov Chain PN. The arcs and arc weights used in forming matrix $F_i(x)$ are outlined and highlighted in red, and the arcs and arc weights used in forming matrix $V_i(x)$ are outlined and highlighted in green.
  • Figure S4: This SEEIR Petri Net model is mapped directly from Equations \ref{['eq:S_EEIR']}--\ref{['eq:R_EEIR']}. The arc weights used in forming matrix $F_i(x)$ are outlined in red, and the arc weights used in forming matrix $V_i(x)$ are outlined in green.
  • Figure S5: A variable arc weight SVEIR Petri Net mapped from Equations \ref{['eq:S_SVEIR']}--\ref{['eq:R_SVEIR']}. The arc weights used in forming matrix $F_i(x)$ are outlined in red, and the arc weights used in forming matrix $V_i(x)$ are outlined in green.
  • ...and 3 more figures