If you restrict it to the vaccinated population and the unvaccinated population, then you can consider the vaccinated population fully populated, and the unvaccinated population entirely unvaccinated.
Vaccinated group:
Rt = 6 * ( 1 - (1 * .8)) = 1.2
Unvaccinated group:
Rt = 6 * ( 1 - (0 * .8)) = 6
The question of "how much less" is hard to parse, but the point is that even among the vaccinated group (in this scenario), the virus is still spreading and growing exponentially.
I suppose if Rt was clearly below 1 for the vaccinated group, that might mean something, but if the groups intermingle, that advantage would be lost pretty quickly.
No, that’s not what it means. It means that comparing the absolute numbers of people infected in both groups - vaccinated people and unvaccinated people - or even the percentage of the total number of infections is not valid. You need to take the ratio of people infected relative to the group size.
The numbers are so different that it doesn't matter very much how you compare. It would be different if only 5% of unvaccinated people contract the disease, or if the vaccine reduced the likelihood of infection by only 5%.
> the absolute number of infected individuals in the Vaccinated (boosted) group is likely to be at least as high as in the Unvaccinated
But if the vaccinated represent a much larger share of the population, the same absolute number represents a much smaller proportion who become infected.
I think you are still missing my point. You are using a circular definition of R0. Where did you get that number 6? Is that for a fully unvaccinated population? Are partially vaccinated population? If so, what percentage? Your calculation makes absolutely no sense, it's circular. You can't calculate R0 using R0.
I'm sure you're getting weary at this point, but I really need your help understanding what you're saying here, because to me it seems supremely untrue.
And it makes intuitive sense that once you infect <= 1.00 people, growth is no longer exponential. So I don't think that point is wrong at all.
> The idea that vaccination and epidemics need the same percent of immunity is completely 100% wrong. An active epidemic is going to infect way more people before dying out than the simple, naive herd immunity calculation will give you. In fact, while an r0 of 2 indicates that you get herd immunity with just 50% of the population, a typical active epidemic with the same r0 would infect 80+% before dying out.
This doesn't make any sense to me. Vaccination is just an artificial way to build immunity. So I don't understand your active epidemic point at all. Or put another way, as the proportion of % recovered goes up, we start approaching the eventual limit where `1/1-R_0`->1. That is to say, in your example we start with R_0 = 2, then as people recover that value slowly starts decreasing until it hits 1 at the herd immunity threshold. It's not clear to me what the discrepancy you're detecting is here.
I'm curious, is there any information on the confidence interval for this 95% number?
Intuitively, I would expect the confidence to be pretty low - after all, if tomorrow 2 more people in the vaccine group happen to get sick, the relative risk reduction will go from 20x to 15x.
I'm honestly not sure, maybe we are really just talking past each other. If there are no unvaccinated people left the relative risk reduction would be zero, that just follows from the definition. The absolute number of hospitalisations would still be meaningful. In the perfect world that everyone was vaccinated you would still see an age dependent decrease of effectiveness over time and relative to alpha. The real data from Israel seems to show that two vaccine doses in a very compliant population are not enough to effectively prevent the number of cases, hospitalisations and deaths to rise without additional other measures (boosters, masks, etc.).
Imagine a study where half gets vax and half doesn't. If population is pristine at outset at end of study VE is 1-ratio between people who get COVID in the two arms.
Now, if there is any prior exposure to the disease among participants (this isn't always known, esp. for something like COVID), and if there is any natural immunity that accrues, this will pull that ratio towards 1 (VE=0)
Yes, I brought up that equation merely to illustrate how vaccination alone can impact Rt downward - for a while it was unclear to me exactly how "partial vaccination" (before herd immunity) was a benefit; that helped me see that it really just mean it slows down the doubling rate until you get to Rt=1. But overall, Rt is impacted by all forms of mitigation including natural immunity.
The way I actually use that equation in my personal dashboards is to estimate what Rt "would" be, at today's mitigation levels, if no one had gotten vaccinated. So for instance, Portland's Rt is currently 0.92 (according to one model). By plugging in Portland's vaccination rate and efficacy estimates, you can estimate that Portland's Rt would be around 2 today if not for the vaccinations, if all other mitigation were the same.
On the one hand, 2 is a lot better than those estimates of 5 to 9.5, which means that we're impacted a lot by current mitigation practices (masks, distancing) and natural immunity. On the other hand, 2 is huge! Given estimates on infectious periods, that means that currently our cases halve every 90 days or so, but 2 would mean cases would be doubling every week or so. Gargantuan difference. So just an illustration that vaccination has a big impact and matters a lot.
Thanks for bothering :) And sure, I agree it's implied that people are getting equal exposure to the virus but vaccinated are getting infected less often, but that's hard to quantify. And you are right, my initial take was completely off mark because of this.
Still, I am guessing there's at list a single digit multiplier (2-3x) for risky behaviour of the unvaccinated group: they are more likely to also shun masks, mingle with more people, etc. But it's a guess only.
So again, my problem is with ignoring all of that and spouting some numbers out — they are always going to be misleading. CDC did not do an analysis for a reason.
Switching to proportion is actually the wrong thing to do. If you look at the data from a place with high vaccination rates (such as King County), even low probability of a break through case among the vaccinated will translate to high proportion.
Edge case that demonstrates the fallacy of looking at the proportion is: let's say you have 100 people, 90 of them vaccinated, 10 not vaccinated. Let's then say that vaccinated have 10% of getting infected, and the unvaccinated have 50% of getting infected. You'll then get 9 vaccinated people that got sick, + 5 unvaccinated people that got sick, for an almost 66% proportion! But that doesn't change the fact that as a vaccinated person, your chances of getting infected are 5x smaller compared to the unvaccinated.
I thought this was simple math. The reproduction value R of the original type was estimated to be around 3 (one sick person infects on average three other persons). So to get this below 1, we need a vaccination rate of around 2/3 (1-1/R). The delta variant has a higher R of 6-8, so we may need as much as 90% of the population to be vaccinated. It's completely plausible and has nothing to do with lying.
> If getting vaccinated reduced your odds of spreading the virus by 90%, and 92% of the population were double-vaccinated, then the majority of the spread would be...
> Among, and by the double-vaccinated. (8.28% vs 8%)
I don't know how you're getting those numbers.
If baseline spread is 100% unvaccinated spreading to 100% unvaccinated, then 92% vaccinated spreading 10% to 92% vaccinated amounts to 8.464% of baseline, 92% vaccinated spreading 10% to 8% unvaccinated is 0.736% of baseline, 8% unvaccinated spreading to 92% vaccinated is 7.36% of baseline and 8% unvaccinated spreading to 8% unvaccinated is 0.64% of baseline. The total sums to 17.2% of baseline, of which vaccinated to vaccinated spread amounts to 49.2%.
(It's not terribly important since the numbers are made-up anyway, but I'd like to know whether I made a mistake somewhere.)
Vaccinated group:
Rt = 6 * ( 1 - (1 * .8)) = 1.2
Unvaccinated group:
Rt = 6 * ( 1 - (0 * .8)) = 6
The question of "how much less" is hard to parse, but the point is that even among the vaccinated group (in this scenario), the virus is still spreading and growing exponentially.
I suppose if Rt was clearly below 1 for the vaccinated group, that might mean something, but if the groups intermingle, that advantage would be lost pretty quickly.
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