Petri net

Reaction network \((S, T, s, t)\)

• Eg: SIRS infectious disease model
  • \(S + I \xrightarrow{\iota} 2I\)
  • \(I \xrightarrow{\rho} R \xrightarrow{\lambda} S\)
  • species:
    • \(S\): susceptible
    • \(I\): infected
    • \(R\): resistant
  • reactions:
    • \(\iota\): infection
    • \(\rho\): recovery
    • \(\lambda\): loss of resistance
• definition:
  • a finite set of species
  • reactions go between "complexes", which is finite linear combinations of these species with natural number coefficients
• rate constant and rate equation:
  • rate constant: a positive number called can be attached to each reaction
  • rate equation: now a reaction network determines a system of differential equations saying how the concentrations of the species change over time
  • Eg
    • \(\frac{dS}{dt} = r_\lambda R-r_\iota SI\)
    • \(\frac{dI}{dt}=r_\iota SI-r_\rho I\)
    • \(\frac{dR}{dt} = r_\rho I-r_\lambda R\)
  • interesting properties:
    • existence and uniqueness of steady state solutions
• Mathematics definition:
  • consists of:
    • a finite set \(S\)
      • : elements are called species
    • a finite set \(T\)
      • : elements are called transitions
    • functions \(s, t: T \rightarrow N^S\)
      • \(N^S\)'s elements are called complexes, which is finite linear combinations of these species with natural number coefficients.
      • any transition \(\tau \in T\) has a source \(s(\tau)\) and a target \(t(\tau)\)
      • if \(s(\tau) = \kappa\) and \(t(\tau) = \kappa'\), we write \(\tau: \kappa \rightarrow \kappa'\)
  • the set of complexes relevant to a given reaction network is
    • : \(K = im(s) \cup im(t) \subseteq N^S\)
  • graph:
    • a reaction network gives a graph
      • : vertices set of \(K\)
      • : an edge for each transition \(\tau: \kappa \rightarrow \kappa'\)
    • it can have multiple edges or self-loops, thus sometimes called:
      • a directed multigraph
      • or a quiver

Petri Net

• bipartite directed graph
  • : 2 kinds of vertices, species and reactions
  • : edges
    • into a reaction, specifying its input
    • out, specifying its output

• Terminology
  • in Petri net literature, species are called "places", and reactions are called "transitions"
  • so Petri net is sometimes called place-transition net or P/T net
• stochastic Petri net: when each reaction has a rate constant attached
• open Petri net
  • inputs and outputs: species can flow in or out
  • open rate equation: the usual one with extra terms describing inflows and outflows