Petri nets in epidemiology

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.

Keywords: Basic reproduction number; Ordinary differential equations; Petri nets.

Fig. 1

The formation of water

Fig. 2

The SIR model

Fig. 3

The SIS model

Fig. 4

The SIRS model

Fig. 5

A simplified logistic Malthusian model

Fig. 6

The SEIR model

Fig. 7

The SEAIR model

Fig. 8

The SCIR model

Fig. 9

The SIWR model

Fig. 10

A SIR model for Malaria

Fig. 11

First model of vaccination

Fig. 12

Second model of vaccination

Fig. 13

The SIR model with quarantine

Fig. 14

Petri net associated with an ODE with one compartment

Fig. 15

Petri net associated with an ODE with two compartments

Fig. 16

The algorithm applied to the ODEs of the SIR model

Fig. 17

Susceptible, infection-process and infection modules for the SEAIR model

Fig. 18

Susceptible, infection-process and infection modules for the model of Malaria

Fig. 19

A path between the compartments $x=x_{i_1}$ and $x^{\prime }=x_{i_k}$ passing through the transition $\beta$

Fig. 20

The SEIR model of Saika-Bora