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

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, a generalized method for calculating R_0 directly from PN models has not been established. Thus, in this paper, we present a general method for calculating R_0 for Petri nets. Additionally, we show how a computational method implementing the next-generation algorithm in ODE models can also be applied to Petri net models. We also provide multiple examples of how to use this approach to calculate 0 for various SIR-type Petri net models.