Epidemic Models 1, Part 4

David A. Tanzer, July 18, 2020, in unit Epidemic Models 1.

The SIR model

Last time, we introduced the general idea of reactions. Now we turn to the SIR reaction network, which is a fundamental model:

SIR-network

Footnote: such diagrams are called ‘Petri nets’, after Carl Petri, who invented them in 1939, at age 13, to describe chemical reactions. They are now more widely applied.

Here we see that the three compartments are represented by the blue circles, and the two reaction processes are represented by the green squares which connect the circles.

Notice the infection reaction has a more complex structure, which involves four connecting arrows with the compartments, whereas the recovery reaction has only two connecting arrows. Here we look into the simpler reaction, leaving the more complex one until next time.

The recovery reaction

The recovery reaction has a simple form: it transfers people from one compartment to another, from Infected to Recovered. It proceeds by steps, with each step resulting in the transfer of one person. Put differently, in a step one infected person gets taken out of container Infected, goes through a metamorphosis to recover, and then gets sent to container Recovered. Each step is called a firing of the reaction.

Here is a summary of the reaction:

$$\mathrm{Infected} \xrightarrow{\mathit{recovery}} \mathrm{Recovered}$$

The formula tells us a couple of things:

• ‘recovery’ is a reaction that inputs from container Infected and outputs to Recovered.
• Each time it fires, the count of Infected goes down by one and Recovered goes up by one.

The reaction process can be likened to a pump; every time it fires, it removes a ball from one container and sends it to another container.

The recovery reaction in context

Let’s see how the recovery reaction fits into the SIR network as a whole. First review the diagram above, which has squares for the reactions infection and recover.

Now look again at the formula for recovery:

$$\mathrm{Infected} \xrightarrow{\mathit{recovery}} \mathrm{Recovered}$$

Now make the key observation that this formula is a kind of ‘miniature diagram’ that exactly matches with the right-hand side of the overall diagram above.

That is how the recovery reaction fits into the SIR model as a whole.

Outlook

We’ve just seen the germ of a basic concept in science, called reaction networks. The example was very simple, but it served its purpose, which was to set up an exact language for talking about reaction networks. We are now well prepared to look further into the subject.

Indeed this is just the beginning of the story, as reaction networks get for more complex and interesting; they turn up all over the place, in epidemiology, biology, chemistry, ecology, genetics and elsewhere. Consider that, from one point of view, life itself can be seen as a vast network, with countless reactions intertwined in myriad ways.

Next time we will cover the other reaction in the model, which is infection. Note from the SIR diagram that it has more connections than recovery does. By way of anticipation, try to guess at its formula.