One of the simplest and most widely used mathematical models to describe epidemics is the so-called SIR model. The model considers the population compartmentalized into three groups: healthy people (S), infected people (I), and recovered and immune people (R). It also predicts that there are two possible transitions between these groups: one occurs when a healthy person becomes infected because he knew another who was infected – both become infected – and the other occurs when an infected person becomes infected.

The Petri net has compartments (groups of people) and transitions (group changes).

This model can be analyzed mathematically using a tool called a Petri net, a type of net that has compartments (groups of people) and transitions (changes from one group to another), with weighted arrows to describe the relationships between compartments and transitions. (see figure). Starting from the Petri net with the parameters that indicate the speed with which the transitions are made (estimated experimentally), the model describes the evolution of the compartments and, therefore, of the epidemic.

Regarding Petri nets, there was an old problem in theoretical computation open since the 1980s that, unexpectedly, was solved thanks to the Covid modeling work carried out by the researcher from the Universitat Autònoma de Barcelona (UAB), Joachin Kock. The research was recently published in the prestigious journal ACM newspaper.

### individuals instead of groups

When Kock started modeling COVID-19 using Petri nets early in the pandemic, he wanted to experiment with considering people not as statistical groups but as individuals, an idea inspired by theoretical computation. In this domain, Petri nets consider a certain number of tokens in each compartment, which move according to transitions, so that a transition can occur if there are enough tokens in the input compartments: tokens are consumed in the input compartments and new ones are generated produced in output bins.

Relationships between compartments and transitions in the SIR model / Center for Mathematical Research (CRM) at UAB

In this discipline, the most important use of Petri nets is as a model for concurrent computation, but this study presented an old unresolved problem. This resides in the fact that there are two different ways of reasoning mathematically about these networks as a model of competition, two ‘semantics’ that could not be reconciled: an algebraic semantics and a geometric one, with their advantages and disadvantages.

To find a solution, the UAB professor reviewed the theory of Petri nets and found that it was necessary to modify the definition itself.

Simulating COVID-19 with the Petri nets that are used on computers to consider individuals “was not a good idea from an epidemiological point of view”, explains Kock, “because the formalism of these nets did not allow tracking people individually. ” This “obstruction” turned out to be the same one that prevented the reconciliation of algebraic and geometric semantics, which brought scientists back to the old problem of Petri nets.

To find a solution, the UAB professor reviewed the entire theory of Petri nets and discovered that it was necessary to slightly modify the definition itself so that these nets admit parallel arrows instead of weights, that is, to pass from a number that represents the weight of an arrow for a set of arrows with that number of elements.

Abstract mathematics made possible the transfer of knowledge and experience from one science to another.

Joachim Kock

“In homotopy theory, one of my areas of research, this kind of consideration is common,” says Kock. “In this case, when introducing sets of arrows, ways of rearranging them appear and symmetries that do not exist if we take into account only the weight as a number, as occurs in the conventional definition”.

### Experimenting with ideas without knowing where they lead

What was missing in conventional Petri nets was exactly the access to information on the symmetries of a net that Kock’s definition provides. semantic, algebraic and geometric”, in his own words.

The new definition proposed by the researcher from the Department of Mathematics at UAB has already been used by other researchers (Evan Patterson and his team in Berkeley, USA) to develop a computer program based on Petri nets, with the aim of modeling epidemics such as Covid .

“So the circle closes. Abstract mathematics has made it possible to transfer knowledge and experience from one science to another, in this case unexpectedly. I looked for one thing and ended up finding something very different. Sometimes it’s productive to experiment with ideas you don’t know exactly where they’re going to take you,” concludes Kock.

Reference:

J.Kock. Whole Grain Petri Nets and Processes. ACM newspaper (2023).