Tuesday, August 31, 2021

Vaccination Arithmetic

A good deal of the current discussion of vaccination takes it for granted that it is in almost everyone's interest to get vaccinated, hence that failure to get vaccinated is evidence of false beliefs or irrational behavior. To see why this is not true for everyone, it is worth looking at some numbers.

According to the CDC, the estimated infection fatality rate is 0.05 percent for 18-to-49-year-olds. I start my calculations with someone who is certain to get infected and has a life expectancy of thirty years. Thirty years is 262,800 hours, so the reduction in life expectancy is .0005x262,800=131 hours.

If you believe your chance of getting infected is only .1, not unreasonable if you regard the current wave as the last of the epidemic, that reduces it to 13 hours. If you are 25, which according to one source gives an IFR of .01, that takes it down to less than three hours. Saving that may not be worth the time and trouble of two injections, a likely few days of not very serious side effects and some small risk of more substantial side effects. The same is more true for younger ages or people in particularly good health.

It may be objected that a .1 probability of getting infected is unreasonably low, but that depends what you believe about vaccine effectiveness. If you are optimistic about vaccines, you should expect the current wave to be the last serious one. If you are pessimistic about vaccines, the risk of infection is much higher but the benefit of vaccination lower.

It may also be objected that I am looking only at death. I don't have, I don't think anyone has, good data on long term effects of getting Covid and recovering. The short term effects range from zero for asymptomatic infection to several weeks in the hospital for almost lethal infection. Including that would reduce the number of people for whom vaccination is not a clear benefit but not, I think, to zero. 

And, on the other side, I am ignoring people who have already had Covid, hence have protection comparable to, perhaps better than, that provided by vaccination. Vaccination apparently increases the protection, but not by much. The CDC estimates  that about a third of the population have had symptomatic Covid, so that is a large group for which the benefit should be reduced by at least an order of magnitude.

I am 76, so a similar calculation implies that I should be vaccinated, and I am. But I do not agree with the claim that everyone should obviously be vaccinated as well.

All of this is in terms of the self-interest of the individual. Vaccination also reduces the spread of the disease, benefiting others, although by how much is not clear. That is an additional argument for getting vaccinated but one whose size is harder to estimate.


Dave Orr said...

I think any reasonable discussion needs to talk about variance.

Most insurance is -EV (or else insurance companies wouldn't make money), and yet people (rationally) buy it, and in fact I think it's basically mandatory for home owners with good reason. People can't afford the variance associated with their house burning down.

Similarly, you may be overpaying in hours lost to vaccine recovery versus the EV of hours lost to COVID, but the point is not to gain or lose some hours on the margin, the point is to avoid the very worst outcomes where you lose all your hours, or make them much worse.

Anonymous said...

> That is an additional argument for getting vaccinated but one whose size is harder to estimate.

An economic analysis of just-a-little-drunk driving in the absense of punitive laws probably looks similar.

Anonymous said...

It's a prisoner's dilemma. The value of the vaccine is decreased each time someone else gets the vaccine. If everyone else gets it, it's not worth the very small cost of getting it, because you almost certainly won't get Covid anyway. If no one gets it, then the majority of people get Covid, unless they completely seclude themselves. The obvious, sensible thing to do is for everyone to agree that everyone should get the vaccine as soon as possible.

SB said...

Another interaction between self-interest and public interest: if I can avoid getting seriously ill, I not only improve my own life directly but also reduce the burden I place on hospital facilities, thus increasing other people's likelihood of surviving.

Sal Canzoneri said...

On the topic of natural immunity I’d point out that it isn’t at all a given that natural immunity is as good. I read a recent paper where the authors argued that mRNA vaccines offer better protection against variants because they target recognition of binding receptor domain (ie the spike protein) which is common to all coronaviruses while natural immunity may involve recognition of other parts of the virus.

The authors argue that it’s more likely that mutations are less likely in the RBD than other parts of the virus so mRNA vaccines at least should prove more effective againat future mutations than natural immunity

SB said...

It also occurs to me that a fixed life expectancy of 30 years is a strange assumption, especially since you're treating the subject's age as a variable elsewhere in the analysis, and since you at 79 have presumably less than a 30 year life expectancy, while the hypothetical 25-year-old, absent other information, has probably at least a 60 year life expectancy. What happens if you look up a chart of (say, US) life expectancy at various ages and recalculate your examples that way?

David Friedman said...

@Sal: The only actual study of the effectiveness of natural immunity vs vaccine I have seen is a recent Israeli study that found, for the vaccine they are using, that natural immunity was substantially more effective.

@SB: I was trying there to do a simple calculation that could be tweaked in either direction to allow for a variety of ways it was oversimpified.

Average Radical said...

Greenwald made a similar but orthogonal point that it seems hypocritical that we're only applying this to SARS-CoV-2. Why not to driving also? Why not make a national speed limit of 25mph?

Life has many tradeoffs and we should be having a nuanced, grown-up discussion about those tradeoffs for different groups and contexts. Instead, we have a hypochondriac Marxist totalitarianism creeping in with a one-size-fits-all policy through force.

Philo said...

Some people should, clearly, get vaccinated; for some people it is clearly not worth the bother. And there must be a large number who fall into the gray area between, where it may or may not be in their interest to get vaccinated but the difference in expected value is so small and so hard to estimate that they might do well to flip a coin to decide. It is certainly inappropriate to browbeat such people, even those for whom vaccination is, in fact, slightly in their interest.

RKN said...

I don't have, I don't think anyone has, good data on long term effects of getting Covid and recovering.

You need to read around more.

And, on the other side, I am ignoring people who have already had Covid, hence have protection comparable to, perhaps better than, that provided by vaccination. Vaccination apparently increases the protection, but not by much.

I wouldn't characterize 1.5 orders of magnitude as "not much."

David Friedman said...


If I correctly read the preprint, people who had been both infected and vaccinated had about half the chance of reinfection of those who had only been infected. Those who had only been infected had about one seventh the risk of those who had been vaccinated. They don't give their results in terms of overall effectiveness. If we assume vaccination effectiveness of 93%, hence risk of 7%, that makes the effectiveness of natural immunity (from previous infection) 99% and natural immunity plus vaccination 99.5%. I would describe that improvement as not much.

Source: https://www.medrxiv.org/content/10.1101/2021.08.24.21262415v1.full.pdf

How do you get your 1.5 orders of magnitude out of the article you linked to?

RKN said...


1.5 refers to the immunological improvement (B-cell) after vaccination in convalescent individuals, which is expected to increase durable immunity (see Discussion section). How does that square with your claim - Vaccination [in that cohort] increases protection not by much?

I'll concede the paper doesn't translate the molecular gain to a traditional measure of vaccine effectiveness, but evidence of a non-insignificant increase in immune system response is what previously-infected people should be focused on when deciding whether or not to get vaccinated.

And the paper I cited is not a preprint. It was published in Nature 6/2021.

You may also find informative: https://www.cdc.gov/mmwr/volumes/70/wr/mm7032e1.htm

Pastro said...

Interesting thoughts and discussion.
Since you mentioned the positive externality effect of vaccine in protecting the community beside the individual, have you given some thoughts on the possible (and controversial) negative effect of vaccination during a pandemic, as a potential evolutionary driver for "vaccine avoiding" variants (immune escape topic)?
I'm not an expert in the topic, but it seems that the predictions of people that were discredited for proposing the immune escape theory, became truth (or some aspect of the pandemic of variants predictions); and in my view models that can predict an outcome are worth to be considered seriously, no matter how crazy they may seem.
I'd love to hear your opinion on this particular aspect and possible "negative externality".

Cathy Raymond said...

I don't think David said that the paper RKN cites was a preprint; I read his remarks to mean that he read that paper when it was in preprint. If I'm wrong, David, please correct me.