Petri nets in epidemiology

TL;DR

This paper introduces a geometric Petri net approach to derive the basic reproduction number in epidemiological models, linking system structures to the next-generation matrix method.

Contribution

It establishes a correspondence between ODE systems and Petri nets, providing a geometric interpretation of the next-generation matrix method in epidemiology.

Findings

Petri nets can represent key epidemiological model structures.

The basic reproduction number depends on three Petri net substructures.

Geometric description of the next-generation matrix method.

Abstract

This work provides a geometric version of the next-generation matrix method for obtaining the basic reproduction number of an epidemiological model. We exhibit a certain correspondence between any system of ODEs and Petri nets. We observe that any epidemiological model has the basic structures found in the SIR model of Kermack-McKendrick. This means that the basic reproduction number depends only on three substructures inside the Petri net, which are also given by three Petri nets inside, representing the susceptible population, the infection process, and the infected population. The five assumptions of the next-generation matrix method given by van den Driessche-Watmough can be described geometrically using Petri nets. Thus, the next-generation matrix results in a matrix of flows between the infection compartments with a dominant eigenvalue given by the basic reproduction number.