David A. Tanzer, July 27, 2020, in unit Epidemic Models 1.

A diversity of compartmental models

Last time, we finished our pilot study of the SIR model. By taking variations on this theme, we are now in a position to grasp the structure of a diverse range of compartmental models. So we begin a small tour. Having absorbed the general ideas through SIR, these other models will be low-hanging fruits to comprehend.

Today we’ll get warmed up with with some straightforward extensions of SIR, called SIRD, SIRS and SIRSD.

Here I’ll be describing the models with the formulas alone. I will leave it as an exercise to the reader to sketch out the diagrams. As shown in the previous article, the formulas and diagrams present the same content, and one can always derive the diagrams from the formulas, and vice-versa.

Note this is not a truly magnanimous move on my part, as the practical reason why I left out the diagrams is that I don’t have the time at the moment to be fiddling around with Google Drawings. Because I’m feeling somewhat guilty about this, I’ll give some hints on how to sketch the diagrams.

The SIRS model

SIRS is the SIR model plus a third reaction, which is a process sending people back from Recovered to Susceptible. This is the unfortunate process for some diseases in which the immunity conferred by recovery eventually gets lost. Here are the reactions for it:

$$\mathrm{Susceptible} + \mathrm{Infected} \xrightarrow{\mathit{infection}} 2 \ \mathrm{Infected}$$

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

$$\mathrm{Recovered} \xrightarrow{\mathit{loss\ of\ immunity}} \mathrm{Susceptible}$$

Drawing hint: just take the diagram for SIR from the last article, and a reaction-box at the bottom, going back from compartment Recovered to Susceptible.

The SIRD model

This is a variant of the SIR model which adds a compartment for people who died: SIRD = Susceptible, Infected, Recovered, Dead.

$$\mathrm{Susceptible} + \mathrm{Infected} \xrightarrow{\mathit{infection}} 2 \ \mathrm{Infected}$$

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

$$\mathrm{Infected} \xrightarrow{\mathit{death}} \mathrm{Dead}$$

Drawing hint: Modify the SIR diagram by adding compartment D for Dead. Add an additional reaction-box called ‘death’ going from Infected to Dead.

For many applications, SIRD adds an extra level of complexity to the model, without adding any substantial information to the picture. Whether a person has recovered or died, as far as the epidemic wave itself is concerned, they are removed from the picture and have no further effect – because the dynamics of the infection are determined just by the susceptible and infected sub-populations. For this reason, SIR is alternatively described as having compartments Susceptible, Infected, Removed.

This highlights a more general point about modeling. Models help us to understand reality, but they are always simplifications. Part of the art of modeling is knowing when there is something to be gained by the complexity of extra detail.

The SIRDS model

This one combines the ideas of the preceding two models, SIRD and SIRS. You can think of it as an extension of SIRD, which also includes the loss of immunity reaction which is shown in SIRS. In other words, an infected person may recover or die (the SIRD idea), and a recovered person make lose immunity and become susceptible again (the SIRS idea).

Exercise

Based on the preceding description of the SIRDS model:

• Work out the reaction formulas for SIRDS
• Sketch the equivalent wiring diagram for SIRDS

Next time we’ll look at the SEIR model.

Copyright © 2020, David A. Tanzer. All Rights Reserved.