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

The basic reproduction number ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>R</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:mrow> </mml:math> ) 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>R</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:mrow> </mml:math> directly from PN models has been established. Thus, in this paper, we establish a generalized computational framework for calculating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>R</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>R</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:mrow> </mml:math> values are in strong agreement with those estimated from simulation data, thereby confirming the method's accuracy and practical utility.