Sunday, February 14, 2021

Have We Reached Herd Immunity?

I have been looking at the number of new cases per day, both for Santa Clara County and for the U.S., and the pattern is striking. Over the past month, number of new cases per day in the U.S. has gone down by a factor of about 2.4, for the county by a factor of about 3.5. Death rates are also down, although by much less, but one would expect death rates to lag case rates by a few weeks. I don't think the explanation can be the weather, since it's still winter.  I don't think it can be vaccination because there has not yet been enough of it to substantially affect the case rate.

The other obvious explanation is that the previous high was created by a lower level of precaution due to XMas and New Years. But that explains only a reduction in the rate of increase, not a reduction in the level, since all those people infected during the holidays were then around to infect other people. The fact that the level is falling means that each infected person is passing the infection on to fewer than one person, which is the definition of herd immunity.

If behavior is held constant, herd immunity ought to first appear as a constant rate of infection, each person passing the disease on to one other, then gradually become a shrinking rate. It looks as though we reached herd immunity under non-holiday behavior something close to two months ago, infection rates kept going up due to the holiday bump, and by the time that ended we were far enough into herd immunity (with non-holiday behavior) so that rates were falling. 

I was still surprised that they were falling so fast until I looked at how many cases had occurred recently. I haven't made an exact calculation, but it looks as though, for both the county and the country as a whole, the surge in cases over the two months of the peak  roughly doubled the cumulative total. It wouldn't be that surprising if that had pushed us well past the start of herd immunity.

There are two qualification to be made to the optimistic conclusion that the pandemic is almost over. One is that the requirement for herd immunity depends, among other things, on how people behave. If everyone concludes the pandemic is over and drops all precautions against passing on Covid, cases might start increasing again. The other is that there are at least two new variants of Covid now spreading through the U.S., and we cannot be sure that people immune to the old variant will be equally immune to the new. If, to take the most pessimistic possibility, the protection provided by having had Covid turns out to have no effect one of the variants, we are back to ground zero and in trouble.

My guess, however, is that neither will happen, and that in another few months the case rate will be back to last spring's lows. And falling.


Karl said...

I've been analyzing a much smaller data set. Specifically, the count of employees of the LA Department of Water and Power who have contracted Covid. My curiosity was piqued when my boss described the increase in cases in December as "exponential".

Was it? Did it really obey Y'=kY?
AS it turns out, no. The rate of new cases had approximately tripled, but it stayed at the new level.
Counting back two weeks, I found that the increase in cases occurred two weeks after the day before Thanksgiving, and was remaining elevated well into December. Hypothesis: The number of new cases will drop on or about January 15, two weeks after the New Year's Eve and Day festivities.
And although the signal was a bit noisy, the rate of new cases has dropped. It's averaging between 6 and 7 per day, down from 15, and the difference reached statistical significance weeks ago.

Since the total number of cases is about 10% of the work force, I doubt it's herd immunity. Instead, it's probably people going back to being more careful.

David Friedman said...

"Since the total number of cases is about 10% of the work force"

That's the number of symptomatic cases that were caught. The CDC has been estimating total cases as a substantial multiplier — early on 11, more recently 7 — of the number of known cases. 70% could be herd immunity.

People being more careful would reduce the rate of increase, but in order for the number of new cases each day to be going down, each infected person has to be passing the disease to fewer than one person, which is the definition of herd immunity.

Anonymous said...

It does seem like the US has reached herd immunity (only under present social distancing restrictions).

Why are COVID levels in every country dropping though? All reached herd immunity at the same time? Every state and every country at once? I don't understand it - Bobboccio

Brad said...


I think this definition of herd immunity, while interesting, is not the most commonly accepted:

One is that the requirement for herd immunity depends, among other things, on how people behave. If everyone concludes the pandemic is over and drops all precautions against passing on Covid, cases might start increasing again.

If you look at the Wiki article on herd immunity, it does not contemplate behavior.

Per your own point about behavior, R0 is below 1 not because people are immune to the infection, but because people are still engaging in behavior that keeps them from exposing themselves (if they have COVID) to others and keeping themselves from being exposed to others who may have COVID.

I do think that we're getting closer and closer to herd immunity and the vaccines will accelerate that. But I honestly believe that if we simply went back to "normal" tomorrow, i.e. congregating in small and/or large groups, traveling, ending WFH for the industries which are doing it, etc, that R0 would then exceed 1 again and cases would rise. It sounds like, from the quoted text, that you agree that it MIGHT rise, assuming that perhaps not enough people are immune for a return to "normal" behavior.

I would also highlight that most of the people I've heard suggesting that we may have reached herd immunity are doing so from the perspective of arguing against the various restrictions on behavior that are in place. If you are conceding that your definition of herd immunity is behavior-dependent, it belies that changing behavior would increase R0, perhaps again above 1, which would suggest that we're not herd-immune in a normal behavior scenario. So why would one argue that we should reduce restrictions based on that definition herd immunity of the result would mean that we're not herd-immune under the new behavior?

I personally believe that it was the holiday behavior which led to the rise in cases. While everyone has lockdown fatigue, it's a lot easier to forego a trip to a restaurant when cases are high than it is to tell your parents that you refuse to see them for Thanksgiving/Christmas with the unspoken message being "my own family is potentially 'unclean' and a threat.' I think people got bullied by their families and the social convention of the holidays and simply chose to take the risk rather than upset family dynamics. Once the holidays were over, I think people hunkered down a little because they were no longer faced with family events that they were "expected" to attend.

tl;dr version: If herd immunity is behavior dependent, then it really doesn't matter if we've reached it under "lockdown" behavior, because nobody wants to remain locked down. The only herd immunity that matters is the one under "normal" behavior, because that's what we all want to get back to.

Anonymous said...

You looked only a recent history. The fact that the epidemic has had 3 waves rules out a simple SIR model. Behavior has changed a lot. In that sense we have achieved herd immunity twice before. That is a silly definition of herd immunity. It only makes sense against a fixed level of behavior. If behavior were a simple function of government impositions, you could ask if we had achieved immunity with respect to a stringent level of lockdown and whether each new wave was the result of relaxing restrictinos. But I think a detailed look at the rules and the progression rules that out. Behavior has adjusted over a wide range and thus there is no useful baseline to define herd immunity. And it has adjusted mainly due to perceived risk, which itself is driven by discrete mandates, but mainly through fine gradations of news. The national nature of news has caused simultaneous downturn in every state, despite diverse outbreaks, answering Bobboccio's question. Surely some states have not achieved immunity, even if others have.

This is a negative feedback system producing oscillations and thus it is very hard to tell when it uses up the population and loses the ability to cause another outbreak.

(I think that this is pretty much the same as Brad's comment.)

naivetheorist said...

see today's WSJ f

or a concurring opinion.

Michael Wolf said...

Have you looked into Dr. Hope-Simpson's research on seasonality at all? I too thought it just meant "winter," but his lifelong study on epidemic influenza taught me it's more complicated than that. I haven't seen any mainstream media source discuss it, but it's been going around the unapproved, anti lockdown crowd for a while. It does seem to have a lot of consilience with covid.

Take a look at Hold2LLC's graphs on twitter as well as some general web searches for Hope-Simpson regarding covid.

Here's a jumping off point for Hold2LLC:

He's been graphing it for almost 9 months now: