Good job looking for cruxes! I agree with you that quantifying a differential in exposures would help nail down how much we should favor vaccination (or not), but the idea behind the probabilities I laid out was getting at the risk of inducing asymptomatic-spread. At the most unfavorable to vaccination (like how I also assumed vaccination leads to only asymptomatic disease), asymptomatics generate N infections from N exposures with p=1 and symptomatics generate exposures with p=0 (because they quarantine), so we can just look at the risk of inducing asymptomatic-spread without additional layers of calculation.
Though that does indeed depend on the key probabilities going into calculating the risk. If p>>5%, then additional calculation would be warranted, and calibrating the probabilities better would be more important. For example, it’s clear just looking at the conditional probabilities that the turning point is when the relative risk of infection depending on vaccination equals the reciprocal of the relative risk of being asymptomatic conditional on infection depending on vaccination—that is, if vaccinateds are twice as likely to be asymptomatic conditional on infection than unvaccinateds (wow, the RR is a little under 2, but let’s call it 2, Fig 3), we prefer vaccination as long as vaccination cuts the risk of infection by at least half (vaccine effectiveness >= 50%). Any less than half, and then we can’t just prefer vaccination out of hand and have to go through and calculate. And then figuring out the actual differential in exposures (and viral loads!) would be relevant too.
I agree with you that the probabilities I’m focusing on are in a much narrower time frame and that widening it out, p will lift off from the rate estimated in the clinical trials (about a 2 month window). As the vaccine effectiveness rate approaches 0%, then indeed we can’t prefer vaccination out of hand. How would that happen? As you suggest, with a long enough time window, the attack rates could equalize at 100%. I don’t actually see that happening (I expect the vaccines don’t only provide probabilistic protection of around 85% but, at least for some, effective immunity). But vaccine effectiveness could reach the reciprocal of the relative risk of being asymptomatic conditional on infection depending on vaccination well before getting to 0%. If you think vaccine effectiveness for the long term will fall below 50%, then we have some more calculating to do. Seeing as effectiveness has stayed about as high as models would tell you, falling below 50% only seems like a real possibility with Omicron, and my guess is we’ll either get a new shot to avoid lower effectiveness [1] [2] or learn that 3 doses work against it.
My prior on vaccine effectiveness staying over 50% even in the long term is strong enough, and the extra research and calculation that would otherwise be required to address this further is daunting enough, that I’ll leave it at that. I don’t want to say the burden of proof is on either of us here, since ultimately it depends on which prior is “deemed” the prior.
I want to reiterate that your general point that a vaccine might not have the public good value we assume it has is legit. We are used to diseases that generate symptomatic infections with high p, so any reduction in symptomatic infection is noticeable and contributes to stopping the spread. If a vaccine pushes infections to “hide” in asymptomatic ones instead (because the disease generates symptomatic infections with low-moderate p), and asymptomatic infections are still highly transmissible, the public good value is not quite so certain, generally speaking.
Good job looking for cruxes! I agree with you that quantifying a differential in exposures would help nail down how much we should favor vaccination (or not), but the idea behind the probabilities I laid out was getting at the risk of inducing asymptomatic-spread. At the most unfavorable to vaccination (like how I also assumed vaccination leads to only asymptomatic disease), asymptomatics generate N infections from N exposures with p=1 and symptomatics generate exposures with p=0 (because they quarantine), so we can just look at the risk of inducing asymptomatic-spread without additional layers of calculation.
Though that does indeed depend on the key probabilities going into calculating the risk. If p>>5%, then additional calculation would be warranted, and calibrating the probabilities better would be more important. For example, it’s clear just looking at the conditional probabilities that the turning point is when the relative risk of infection depending on vaccination equals the reciprocal of the relative risk of being asymptomatic conditional on infection depending on vaccination—that is, if vaccinateds are twice as likely to be asymptomatic conditional on infection than unvaccinateds (wow, the RR is a little under 2, but let’s call it 2, Fig 3), we prefer vaccination as long as vaccination cuts the risk of infection by at least half (vaccine effectiveness >= 50%). Any less than half, and then we can’t just prefer vaccination out of hand and have to go through and calculate. And then figuring out the actual differential in exposures (and viral loads!) would be relevant too.
I agree with you that the probabilities I’m focusing on are in a much narrower time frame and that widening it out, p will lift off from the rate estimated in the clinical trials (about a 2 month window). As the vaccine effectiveness rate approaches 0%, then indeed we can’t prefer vaccination out of hand. How would that happen? As you suggest, with a long enough time window, the attack rates could equalize at 100%. I don’t actually see that happening (I expect the vaccines don’t only provide probabilistic protection of around 85% but, at least for some, effective immunity). But vaccine effectiveness could reach the reciprocal of the relative risk of being asymptomatic conditional on infection depending on vaccination well before getting to 0%. If you think vaccine effectiveness for the long term will fall below 50%, then we have some more calculating to do. Seeing as effectiveness has stayed about as high as models would tell you, falling below 50% only seems like a real possibility with Omicron, and my guess is we’ll either get a new shot to avoid lower effectiveness [1] [2] or learn that 3 doses work against it.
My prior on vaccine effectiveness staying over 50% even in the long term is strong enough, and the extra research and calculation that would otherwise be required to address this further is daunting enough, that I’ll leave it at that. I don’t want to say the burden of proof is on either of us here, since ultimately it depends on which prior is “deemed” the prior.
I want to reiterate that your general point that a vaccine might not have the public good value we assume it has is legit. We are used to diseases that generate symptomatic infections with high p, so any reduction in symptomatic infection is noticeable and contributes to stopping the spread. If a vaccine pushes infections to “hide” in asymptomatic ones instead (because the disease generates symptomatic infections with low-moderate p), and asymptomatic infections are still highly transmissible, the public good value is not quite so certain, generally speaking.