- Darren J. Wilkinson,
*Stochastic Modelling for Systems Biology*, Taylor & Francis, New York, 2006. Good introduction to stochastic Petri nets, with applications to gene expression and the [[Lotka-Volterra equations]] for predator-prey interactions.

Chapter on case studies has the following applications, with simulation lab exercises:

- Dimerization kinetics
- Michaelis-Menten enzyme kinetics
- An auto-regulatory genetic network
- The lac operon

In this thread we'll look into the one on dimerization kinetics.

]]>All told, I propose an Azimuth quest with the following focus:

- Pursuit of applications of Petri nets to stochastic as well as deterministic epidemiology

This is wide open.

Here is one idea I had, which I posted to the Azimuth blog:

Modeling each country separately leaves holes in the overall model for a pandemic. E.g. if the curve goes down, travel restrictions are lifted, and then it goes back up due to whatâ€™s happening in other countries. Compartmental models use ODEs and assume a well-mixed population. What about a multi-level approach, where each country or well-mixed region has a compartmental model with its own parameters. Then there could be transitions between the compartments in different countries, reflecting flows due to travel. This looks like a potential application of composition of open networks. Perhaps a good composition rule could produce an aggregated, abstracted compartmental model for the whole globe. Or help us in other ways to understand the dynamics of the whole.

Posted to:

- How scientists can help fight Covid-19, John Baez, Azimuth blog, March 31.

Spelled out, the suggestion is to apply the open Petri net framework that John, Jade and Blake have been developing to the composition of global pandemic networks from smaller regional networks.

Open Petri nets, John C. Baez, Jade Master, Aug 2018.

A compositional framework for reaction networks, John C. Baez, Blake S. Pollard, April 2017.

- Darren J. Wilkinson,
*Stochastic Modelling for Systems Biology*, Taylor & Francis, New York, 2006. Good introduction to stochastic Petri nets, with applications to gene expression and the [[Lotka-Volterra equations]] for predator-prey interactions.

This has a short chapter called case studies, with the following applications and simulation challenges:

- Dimerisation kinetics
- Michaelis-Menten enzyme kinetics
- An auto-regulatory genetic network
- The lac operon

I was hoping for a more basic text to start with, and one that covered a wider range of applications, but for the time being we can use it as a working text. We can discuss some of the simulation challenges on a new thread, and who ever is up for it can try their hand at coding them up and sharing their results.

These are small, homework level simulations, which can be done on any laptop in anyone's language of choice.

@DanielGeisler if you are interested in digging into this subject, I recommend getting this textbook. Although oriented to biology, it gives a broader theoretical foundation.

]]>Hi, we have seen sufficient interest in Petri nets to start a reading group. In particular @DanielGeisler is interested in working to get this organized - thanks Daniel!

]]>- S(t) = susceptible individuals at time t
- I(t) = infected individuals
- R(t) = recovered individuals

For a general Petri net, these are called *species*.

There is a process called *infection*, which takes as "inputs" one infected individual and one susceptible individual, and "outputs" two infected individuals. This process can be written:

\[\mathit{infection}: S + I \rightarrow I + I\]

Another process called *recovery* inputs an infected individual and outputs a recovered individual:

\[\mathit{recovery}: I \rightarrow R\]

The processes are also referred to as transitions, or reactions.

Each process has an associated *transition rate*, which is a kind of "speed coefficient" for the reaction.

- John C. Baez and Jacob Biamonte, Quantum techniques for stochastic mechanics, arXiv:1209.3632 [quant-ph]. Text includes treatment from the ground up of Petri nets, both stochastic and deterministic. Example includes SI, SIR and SIRS models.

]]>The classical deterministic susceptible-infectious-recovered (SIR) model has played an important role in the analysis of epidemic systems with large populations. However, when population numbers become small e.g. in a hospital ward, a stochastic analysis will be vital. The stochastic Petri-Nets using Gillespie algorithm for modeling the transmission of airborne infections in enclosed spaces is present to be incorporated into an SIR epidemic model with a short incubation period to simulate the transmission dynamics of airborne infectious diseases in indoor environments. The stochastic model not only allows the long-term impact of infection control measures and enables the evaluation of environmental factors, but also depicts the probability of an outbreak of an airborne infection. A quantitative performance study was carried out to demonstrate how to limit the probability rate of outbreak of infection.

- John Baez, Network theory (Part 1), Azimuth Blog, March 4, 2011. Introduction of the idea of green mathematics.
- John Baez, Green mathematics, Azimuth Forum, August 2015. Describes a category-theoretic setting for green math.
- John C. Baez and Brendan Fong, A Compositional Framework for Passive Linear Networks, arXiv:1504.05625v6 [math.CT], Nov 2018.
- John C. Baez and Jacob Biamonte, Quantum techniques for stochastic mechanics, arXiv:1209.3632 [quant-ph]. Text includes treatment from the ground up of Petri nets, both stochastic and deterministic. Example includes SI, SIR and SIRS models.