Epidemic Models 1, Part 5

David A. Tanzer, July 26, 2020

The SIR model (cont’d)

Last time, we introduced the SIR model, and looked at the formula for one of its reactions, recovery. Here we complete the effort, by looking into the more complex reaction, infection.

SIR-network

The infection reaction

Recall that the recovery reaction has the formula:

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

The infection reaction has more arrows connecting it to the compartments than the recovery reaction does, and this will be shown by the formula for the reaction. Specifically, infection has two input arrows, one from compartment Susceptible and one from Infected, and it has two output arrows to compartment Infected. All of these statements are reflected in the formula:

$$\mathrm{Susceptible} + \mathrm{Infected} \xrightarrow{\mathit{infection}} \mathrm{Infected} + \mathrm{Infected}$$

In words, the formula tells us that infection is a reaction where one susceptible person and one infected person interact, resulting in a situation with two infected people.

Now let’s apply the pump metaphor to this reaction. Recall that we said that the recovery reaction was like a pump, which when fired removes a ball from container Infected and transfers it to Recovered.

So, what happens when the pump for the infection reaction fires? Well, it removes one ball from Susceptible and one from Infected, and then puts two back into Infected. The net result of this is that the population count in Susceptible goes down by one and the count in Infected goes up by one.

However, it is not a simple transfer, as the interaction between two input compartments is a key feature of this reaction. As we will see later, this has an important bearing on the dynamics of the infection reaction, in particular, on the rate at which infections occur under different circumstances.

Summarizing the reaction formulas

Taking a cue from chemistry, we shorten the formulas to use single letters:

$$S + I \xrightarrow{\mathit{infection}} I + I$$

$$I \xrightarrow{\mathit{recovery}} R$$

Compare these two formulas to the SIR reaction network diagram above. Each formula exactly describes a different part of the diagram. The two formulas together convey exactly the same information as the diagram. Whereas the diagram is a ‘right-brain’ representation of the SIR model, the formulas are a ‘left-brain’ representation. But the content is the same.

There is one more abbreviation that can be applied to the formula for infection: $I + I$ can be replaced by $2I$. Putting it all together, we arrive at the final form of the reaction formulas for the SIR model:

$$S + I \xrightarrow{\mathit{infection}} 2I$$

$$I \xrightarrow{\mathit{recovery}} R$$

Outlook

We have gotten somewhere. We now have covered the basic language used to talk about compartmental models and reaction networks. This language includes both a diagram notation (‘Petri nets’) and an equivalent representation with formulas. And we applied it to understand the ‘wiring’ for a basic model in epidemiology, SIR.

Next time, we will revisit our menu of compartmental models, and concisely explain how these “engines” are built. These are all variations on the theme that we have just covered for SIR – now that we have the conceptual toolbox, let’s apply it. Included in this tour will be an explanation of the SEIR mechanism, which is a basis for modeling epidemics like COVID-19.