Masking, models and reality (part 4)
What do SEIR models predict about interventions like mask mandates and how do those predictions compare to what actually happened?
This is the forth in a short series of posts:
Part 2: What do SEIR models predict will happen if R₀ changes?
Part 4: Modelling the effects of a mask mandate with 30% compliance using SEIR
Part 5: Comparing mask mandate model predictions with the real world
In the previous parts we have looked at how lowering R₀ or reducing the fraction of the population that is susceptible to infection affects the progress of an epidemic. In both cases we found that the width of the infectious wave increases as we lower R₀ and/or decrease the fraction of the population that is susceptible to infection. Now we will turn our attention to what happens if a proportion of the population is at reduced risk of infection…
Modelling the effects of a mask mandate with 30% compliance using SEIR
One of the interventions that politicians seem to advocate is the mask mandate, perhaps this is because politicians like to be seen to do something and making people wear masks is a very visible something. The question we would like to answer is whether this is actually having any effect!
So what effect do we expect? We can answer this question with an SEIR model by adding more compartments. We split the Susceptible, Exposed and Infectious compartments each into two, so that we have Susceptible&Complant, Susceptible&Non-compliant, Exposed&Compliant, etc… We then modify the R₀ for the Compliant sets by our hypothesised efficiency of masking.
I’m already risking scaring many readers off with all my graphs, so I will not write down the equations or show you the computer program that runs this modified SEIR, rather let’s get straight to the graphs. Using an SEIR model with the following parameter:
Basic reproduction number (R₀): 2.2
Length of incubation period: 5.2 days
Duration patient is infectious: 2.9 days
Initial infection level: 1 in 7,000,000
70% of the population is in the non-masking set of compartments and 30% of the population is in the masking set of compartments.
We will start off modelling what happens if masks do nothing… this is a way to check that my changes to the program have not broken anything:
Now if you remember in the previous parts using these parameters we had the Full Width at Half Maximum height (also known as FWHM) as 32.6 days and the maximum was on day 127… both of which are the same as with my modified model… PHEW! I didn’t break the code with my changes!
Now the question is what happens if masks have an effect. Let’s start by supposing that masks reduce the amount of virions you are exposed to by 30%… in this scenario a masked susceptible person will be exposed to 30% less virions and will be 30% less likely to receive the minimum infectious dose… also a masked infectious person will spread 30% less virions reducing the risk of infection for the unmasked susceptible and giving a double reduction for the masked susceptible…
OK that was quite a mouthful… here’s the graph where the masks reduce your risk of infection by 30%, i.e. the masked have a 70% risk of infection…
Once again, anything we do that reduces the impact of the epidemic widens the wave of infectious people, in this case 30% of people complying with a mask mandate where masks reduce your risk of infection by 30% increases the width of the wave from 32.6 days to 42.2 days.
We can repeat this for a mask that is 50% effective…
That’s great, the width has increased even more… now the FWHM is 50 days.
If you recall the 95 in an N95 mask stands for a 95% reduction in particles 0.3µm or larger… what would happen if that 95% corresponded to the efficiency of N95 masks at reducing transmission of a virus…
This seems awesome, even with just 30% of the population complying, assuming N95 masks reduce your risk of infection by 95% we have a really strong signal, the width of the curve is almost doubled from 32.6 to 61.9 days, if you remember from part 3 when 30% of the population was immune the FWHM was 62.0 days so if N95 masks reduced your risk of infection by 95% they would be almost as good as being immune and we should see a really strong signal by getting significantly wider waves!
For completeness, here’s a graph showing how the FWHM varies as we change the relative risk of the 30% of people complying with a mask mandate.
In the final part of this series we will look at some real world data to see if we can find the predicted increase in the Full Width at Half Maximum that would show masks work [SPOILER: we don’t]