Masking, models and reality (part 3)
What do SEIR models predict about interventions like mask mandates and how do those predictions compare to what actually happened?
This is the third in a short series of posts:
Part 2: What do SEIR models predict will happen if R₀ changes?
Part 3: What do SEIR models predict will happen if the fraction of the population that is susceptible to infection is reduced?
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
Part 2 looked at how SEIR models change if R₀ changes and we noticed that the Full Width at Half Maximum (also known as FWHM) of the wave of Infectious people gets smaller as R₀ gets bigger.
What do SEIR models predict will happen if the fraction of the population that is susceptible to infection is reduced?
As before, we start by 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
Also this time we should note that the initial fraction of the population that is susceptible is 100%. With these parameters we get the following results:
The Full Width at Half Maximum of the Infectious curve is 32.6 days.
If we change the model so that only 90% of the population was initially susceptible then we get the following results:
Notice that the Blue (S for Susceptible) line starts at 0.9 this time and the Green (R for Recovered) line starts at 0.1.
This time the FWHM has increased to 38.0 days, also the susceptible line ends up slightly higher than the previous time, indicating not as many people were infected.
This is usually explained by the recovered acting as a buffer reducing the impact of the Infectious. Think of the infectious person wandering around meeting people… if they only meet susceptible people they will infect twice as many people than they would if half the people they meet are a immune, so the presence of immune people should, according to the models, increase the width of the peak.
What about if only 70% of people were initially susceptible…
This time we see that the infectious curve has been significantly flattened and the infectious are spread out over a longer time. The Full Width at Half Maximum is 62.0 days in this case, which is the same width as you would get if R₀ was 1.54 and 100% of the population was susceptible…
It is not a coincidence that 70% of 2.2 is 1.54. It turns out that the effective R₀ is the ideal R₀ multiplied by the fraction of the population that is susceptible, so reducing the fraction of the population that is initially susceptible will decrease the effective R₀ and therefore will decrease the width of the wave.
In the next part we will look at what would happen if 30% instead of being immune from infection, are just at a reduced risk of infection like we hypothesise might be the case if a mask mandate was in place and those 30% were actually complying with the mandate.