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# Note on the application of Petri nets / compartmental models to epidemiology

edited July 2020

Here is a cross-post of some comments I made on the category theory community server:

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A note on the application of Petri nets to compartmental models in epidemiology, such as the commonly cited SIR model:

...these are simplifying models, which are predicated on the assumption that the epidemic process is Markovian. While this is a plausible assumption for the reaction Susceptible+Infected --> Infected+Infected, it is not plausible for the reaction Infected --> Recovered. The distribution of the infection period is hardly exponential. The simplest model for the infection period would be that it is a constant K. That would lead to a rate equation in which the derivative of the Recovered population equals the size of the Infected population from K days back.

Comment Source:> A note on the application of Petri nets to compartmental models in epidemiology, such as the commonly cited SIR model: > > > ...these are simplifying models, which are predicated on the assumption that the epidemic process is Markovian. While this is a plausible assumption for the reaction Susceptible+Infected --> Infected+Infected, it is not plausible for the reaction Infected --> Recovered. The distribution of the infection period is hardly exponential. The simplest model for the infection period would be that it is a constant K. That would lead to a rate equation in which the derivative of the Recovered population equals the size of the Infected population from K days back. 
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Quotation from there:

While network models capture contact more accurately, the assumption that the underlying stochastic transmission and recovery processes are memoryless (Keeling and Eames 2005; Volz 2008; House and Keeling 2011) remains restrictive. Of course, memoryless processes are mathematically more tractable and relatively simple to analyse when compared to models where the inter-event times are chosen from distributions other than the exponential. However, when compared to data, these assumptions are often violated. For example, diseases can exhibit unique and non-Markovian behaviour in terms of the strength and duration of infection. In this respect, the distribution of the infectious period is usually better approximated by some peaked distribution with a well defined mean, see e.g. Bailey (1954), Gough (1977), Wearing et al. (2005) and references therein.

Comment Source: * Sherborne, N., Miller, J.C., Blyuss, K.B. et al. [Mean-field models for non-Markovian epidemics on networks](https://link.springer.com/article/10.1007/s00285-017-1155-0), J. Math. Biol. 76, 755–778 (2018). Quotation from there: > While network models capture contact more accurately, the assumption that the underlying stochastic transmission and recovery processes are memoryless (Keeling and Eames 2005; Volz 2008; House and Keeling 2011) remains restrictive. Of course, memoryless processes are mathematically more tractable and relatively simple to analyse when compared to models where the inter-event times are chosen from distributions other than the exponential. However, when compared to data, these assumptions are often violated. For example, diseases can exhibit unique and non-Markovian behaviour in terms of the strength and duration of infection. In this respect, the distribution of the infectious period is usually better approximated by some peaked distribution with a well defined mean, see e.g. Bailey (1954), Gough (1977), Wearing et al. (2005) and references therein.
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edited July 2020

David Tanzer wrote:

The Markov assumption involved in SIR could be a workable approximation as the infection period is small in comparison to the duration of the epidemic. But it does suggest that the application SIR etc. deserves to reviewed in each empirical context as there is a built-in error term here.

Of course there is another simplifying assumption in these models, which is that the population is well-mixed. But that is commonly acknowledged. This one deserves to be included in the footnotes as well. The wikipedia article on compartmental models in epidemiology, for example, makes no mention of it. I will add something there...

Comment Source:David Tanzer wrote: > The Markov assumption involved in SIR could be a workable approximation as the infection period is small in comparison to the duration of the epidemic. But it does suggest that the application SIR etc. deserves to reviewed in each empirical context as there is a built-in error term here. > > Of course there is another simplifying assumption in these models, which is that the population is well-mixed. But that is commonly acknowledged. This one deserves to be included in the footnotes as well. The wikipedia article on [compartmental models in epidemiology](https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology), for example, makes no mention of it. I will add something there...
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Jules Hedges wrote:

Interesting..... I've been wondering whether Petri nets are actually used in real life epidemiology, or if that's a just a lie that keeps circulating. (From what little I've seen of epidemiology it seems to be mostly statistics)

Comment Source:Jules Hedges wrote: > Interesting..... I've been wondering whether Petri nets are actually used in real life epidemiology, or if that's a just a lie that keeps circulating. (From what little I've seen of epidemiology it seems to be mostly statistics)
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This is the main application area Stochastic Petri Nets (SPN)

https://youtu.be/1IPOIE0PvQY

"The mEPN scheme: an intuitive and flexible graphical system for rendering biological pathways" https://bmcsystbiol.biomedcentral.com/articles/10.1186/1752-0509-4-65

Overall architecture (2020): https://sci-hub.tw/10.1093/bib/bbaa046

Comment Source:This is the main application area Stochastic Petri Nets (SPN) https://youtu.be/1IPOIE0PvQY "The mEPN scheme: an intuitive and flexible graphical system for rendering biological pathways" https://bmcsystbiol.biomedcentral.com/articles/10.1186/1752-0509-4-65 Overall architecture (2020): https://sci-hub.tw/10.1093/bib/bbaa046