Concerning Covid

Vaccination

If the question is whether I should get vaccinated in order to protect myself, the main issue is effectiveness against hospitalization or death. If the issue is whether I should get vaccinated to protect other people, the main issue is by how much getting vaccinated reduces the chance I will get infected and infect others.

Judged by current evidence, vaccines provide strong protection against death or hospitalization,¹ much weaker protection against infection. A study in Qatar found the Pfizer vaccine's protection against infection down to about 20% by the fifth month. That is the lowest figure I have seen; other estimates of the effectiveness of vaccination against infection over time vary, but 50% is fairly typical. People are likely to take fewer precautions after they have been vaccinated — certainly I did — and their infection is more likely to be asymptomatic; if you don’t know you are infected and contagious you have no reason to take any special precautions against infecting others. Allowing for both effects, it is not clear whether my getting vaccinated reduces your risk.

Evidence that while vaccination may reduce contagion it does not reduce it very much is the lack of any clear connection in international data between vaccination rate and infection rate. Israel, for instance, had steeply increasing infection rates from late June to early September of 2021, the UK from late May to late July, periods in which both had relatively high vaccination rates — higher than Czechia, which had declining infection rates until late June, roughly constant for the next two months. I have not yet seen any careful analysis of the international data trying to separate the effects of climate, vaccination rate, and natural immunity from previous infection, but the effect of vaccination alone is not large enough to be obvious from a casual look at the data.

If hospitalization for Covid gets high enough to crowd hospitals, as it did in Italy and a few parts of the U.S., my hospitalization imposes a significant cost on other people. That is an argument for requiring the vulnerable elderly to be vaccinated but a very weak argument for universal vaccination, especially  weak for requiring children to get vaccinated.² My conclusion is that getting vaccinated is prudent, especially for the old, but that the argument for pressuring other people to get vaccinated is much weaker than was being claimed through most of the pandemic.

Getting Herd Immunity Wrong

The initial calculations of the requirement for herd immunity were based on the implicit assumption that everyone was equally at risk, an assumption that is, for both behavioral and biological reasons, unlikely to be true. What happens if we drop that assumption?

Assume, for simplicity, that half the population consists of people vulnerable to the disease and half, for behavioral or biological reasons, invulnerable. Observing the early spread of the disease we find that, on average, each infected person passes the disease on to two others and conclude that we will only reach herd immunity when half the population have had the disease. That conclusion is wrong because the relevant figure is not what fraction of the population has become immune but what fraction of the vulnerable population has. In my simple model, half the vulnerable population is only a quarter of the total population, so we reach herd immunity much earlier than the simple calculation implies. 

The real world distribution of vulnerability will be much more complicated than that but the qualitative conclusion still holds. The most vulnerable will be most likely to get the disease, so the average vulnerability of those who have not yet gotten it will decline over time. Hence herd immunity will come sooner than the simple calculation implies.

So far I have only considered differences in how easily individuals can get the disease. There will be similar differences in how easily they can transmit it. The two will tend to correlate — someone who spends a lot of time in loud conversation with lots of others will be more likely than average to get the disease and, if he gets it, more likely to pass it on. So, over time, the probability of transmission will fall as those most likely to transmit are selectively removed from the pool of potential transmitters.

All of this tells us that herd immunity will come earlier than the simple calculation implies but not how much earlier. That depends on the actual distribution of vulnerability, of probability of transmission, and the correlation between the two.

A further problem with the simple calculation is that it ignores behavioral changes due to the pandemic itself. Part of the point of lockdowns was the attempt to temporarily push the transmission rate below one, driving the virus far enough down so that it could be controlled by a test and trace approach. In most, perhaps all, countries that imposed a lockdown that did not happen, but even without a lockdown the existence of the pandemic changes behavior in ways that should reduce the transmission rate below its initial value.

Our family was an extreme example. I was over seventy and retired, my wife over sixty, and we live with our two adult children. When I became convinced that the pandemic was real I cancelled the last two talks of a European speaking trip and flew home. From then until a vaccine became available I was, if I remember correctly, more than ten feet from our property only twice, on neither occasion out of my car and close to other people.

Biasing the Conversation

A commenter on a blog where I had described the implication of dropping the assumption that everyone was equally vulnerable pointed me at a comment in the editors' blog of Science magazine on a paper making a fancier version of my argument:

we were concerned that forces that want to downplay the severity of the pandemic as well as the need for social distancing would seize on the results to suggest that the situation was less urgent. We decided that the benefit of providing the model to the scientific community was worthwhile.

That implies that the editors believed part of their job was filtering the scientific literature that reaches the public in order to bias the public perception in the direction they approved of, although in this case they decided not to do so. It follows that one cannot take the published scientific literature on a controversial issue as giving an unbiased picture of the actual science, which is disturbing but not surprising.

What Happened

So much for the theory of herd immunity, what about the real world evidence? If we should have reached herd immunity when enough people were either vaccinated or immune due to being already infected, why is Covid still with us?

The answer is that we did reach herd immunity, many times over in many countries, as shown by the steep declines that follow peaks in the graph of infection rates for the UK and similar graphs for other countries. But we didn’t keep it

In an environment where many people are resistant to the current strain of the virus, new strains less easily recognized by defenses trained on the current strain have a competitive advantage. Once a strain to which we were less resistant became dominant we no longer had herd immunity, so contagion went back up.

Other things changed as well. It turned out that protection, whether by vaccination or infection, declined over time. That implies that if the virus had not mutated we would have ended up in an equilibrium with a roughly constant infection rate, just high enough to hold average resistance at the level where infections neither increased nor decreased. If the rate got higher than that increasing resistance would push it back down, if lower decreasing resistance would push it back up.

The environment changed as well, most obviously with the seasons, which could affect the infection rate either directly or through their effect on human behavior. Looking at the graph, the UK infection rate seems to have had a tendency to peak in both winter and summer, fall in spring and fall. Perhaps someone by now has done a more careful analysis, using data from different countries with different climates, to explain the patterns. The signal obviously has a lot of noise, since multiple things were changing, which would make doing that hard but perhaps not impossible.

Herd immunity was an elegant idea but one that depended on static models. Insofar as we defeated Covid it was not by eliminating it but by sharply reducing its costs through the combined effect on hospital and mortality rates of vaccination and improved treatment, in particular Paxlovid.

Also by giving up on lockdowns, a cure that turned out to be worse than the disease — as shown by the low excess mortality rate in Sweden, the one country that did not take any drastic measures to prevent contagion.

1. US data from earlier this year show a mortality rate for the unvaccinated about ten times as high as for those vaccinated with the newer bivalent booster, four times as high as for those vaccinated with the older booster.

2. According to CDC figures, ages 0-17 have so far accounted for about three percent of all Covid associated hospitalizations.