Impact of population mixing: The tractability problem

During graduate school, once upon a time, I asked my professor an obvious question: “The assumptions of this model are clearly wrong, so why are we learning it?”

He answered, without a hint of irony, “Yes, but it’s tractable”.

The model relied on the assumption of ordinal utility and the details do not really matter. But my point was, simply, teaching us models based on incorrect assumptions is about as useless as teaching us about a geocentric universe. That professor taught me a valuable lesson: All modellers are wrong, some are honest.

Generally, that’s why I don’t write about theoretical models. They tend towards simplicity to the point of absurdity. The old quip that “some are useful” neglects the antipodal “most are not useful”, and, as we have seen with the coronavirus and climate change models, can blur or even obscure people’s world views to the point where they become dangerous, society-devouring fanatics.

But a couple people pointed out a paper from David Fisman titled Impact of population mixing between vaccinated and unvaccinated subpopulations on infectious disease dynamics, so I decided to give it a read.

The paper points out that “[s]imple mathematical models can often provide important insights into the behaviour of complex communicable diseases systems”. The model is definitely simple, but the insights garnered from it are about as useful as a modeller playing tic-tac-toe with himself.

Here is the setup:

1. Agents in the model can be vaccinated or unvaccinated.

2. They can exist within three states: “susceptible to infection”, “infected and infectious”, or “recovered from infection with immunity”.

3. They treat vaccinated individuals as “all-or-nothing”, which essentially means vaccinated individuals either get a perfectly sterilizing vaccine or a saline solution.

4. Contact between groups can range from completely random to highly assortative (the unspoken assumption of a vaccine passport system).

5. The R0 does not change and vaccine immunity does not wane.

6. The modellers did not imagine that a scenario could exist where vaccine effectiveness could be negative and chose parameters accordingly.

The results are obvious because their entire model is just overly complicated algebra: the vaccinated get infected at higher rates when unvaccinated are presented and unvaccinated do worse with unvaccinated. This model is so insightful that I had to double check the credentials of the modellers because I could have sworn they were 3rd year undergraduate students.

If you set up the model in the same way, but assume vaccine effectiveness is strictly negative, then you get the opposite result. Unfortunately for the modellers, negative effectiveness is all we have been seeing in most data sources lately. They never bothered to parameterize based on the real world data we have been seeing for months and instead chose fantastical vaccine effectiveness numbers that never existed for omicron (see below regarding the first omicron outbreak).

Media lies: -212% VE against omicron in South Africa?

Nor did the authors even bother to stress test their model despite ample real world data available to do so. A simple difference-in-differences or well designed regression discontinuity would go a long way towards validating their model. Do they really expect us Neanderthals with “antivaccine sentiment” to do it for them?

But forget all of that. They are essentially making an argument for a passport system and ignored waning immunity, they ignored negative effectiveness (in both the first few weeks and now with omicron in the general vaccinated population respectively), they ignored sub-characteristics of populations, behavioral differences, population densities, non-sterilizing vaccination, and eventually widespread societal impacts. They ignored a thousand other variables all so they could have a simple, tractable model that tells vaccine enthusiasts what they want to hear.

The thing about tractability when dealing with human beings is we are not so easily controlled.