Fisman et al.'s main conclusion does not follow from their model
References
[1] David N. Fisman, Afia Amoako, Ashleigh R. Tuite. Impact of population mixing between vaccinated and unvaccinated subpopulations on infectious disease dynamics: implications for SARS-CoV-2 transmission. CMAJ 2022;194:E573-E580.
[2] Denis G. Rancourt, Joseph Hickey. OCLA Statement on CMAJ Fisman et al. Article Claiming Disproportionate Infection Risk from Unvaccinated Population, and on Negligent Media Reporting. Ontario Civil Liberties Association, 27 April 2022: https://ocla.ca
Fisman et al. [1]’s main conclusion – that risk of infection among vaccinated people can be disproportionately attributed to unvaccinated people – does not follow from the model presented [2].
Their ad hoc parameter Ψ – defined as “the fraction of all infections among vaccinated people that derived from contact with unvaccinated people, divided by the fraction of all contacts [involving vaccinated people] that occurred with unvaccinated people” – is incorrectly asserted to represent “a normalized index of the degree to which risk in one group may be disproportionately driven by contact with another.”
The assertion is incorrect because the model as presented is blind as to whether the “contacts” in the normalizing denominator of Ψ are infectious or benign, irrespective of vaccination status.
In the model, most “contacts” are benign (not involving an infectious person and a susceptible person), whether vaccinated or unvaccinated. This means that the normalizing denominator of Ψ cannot be assumed to represent “contacts driving infection”, as advanced by Fisman et al.
It is easy to see that the ad hoc parameter Ψ is nonsensical, from figures in their paper:
a. Fig. 2A shows Ψ dropping dramatically with increasing reproduction number. This would mean that unvaccinated people threaten vaccinated people proportionately less when the presumed pathogen is more infectious. The state should not worry about unvaccinated people if the pandemic is sufficiently virulent?
b. Fig. 2B shows Ψ approaching large values as the mixing coefficient η approaches 1. This would mean that unvaccinated people are proportionately more of a threat to vaccinated people as the two groups are more and more isolated from each other, up to complete isolation. This is an absurd result.
The obvious parameter that Fisman et al. could have reported is the numerator of Ψ, which is “the fraction of all infections among vaccinated people that derived from contact with unvaccinated people”.
We plot this “numerator of Ψ”, for parameters used by Fisman et al., versus the mixing coefficient η, and for different population fractions of unvaccinated people, here: https://ocla.ca/wp-content/uploads/2022/04/plot-numer-Psi-v-eta-R0-6-1.jpg
We see that there is no indication of disproportionate infections caused by unvaccinated people, and that the “the fraction of all infections among vaccinated people that derived from contact with unvaccinated people” is bound by the relative populations of vaccinated and unvaccinated susceptible individuals for random mixing, and goes more and more quickly to a value of zero as isolation between the two groups increases, as it must.
These are trivial results. The only way to get the simple model to say what Fisman et al. have said is to concoct and misinterpret an ad hoc parameter (Ψ).