Post #1134, COVID-19 trend to 5/3/2021, and a look at hospitalization and mortality rates

There’s no change in trend.  We’re 19 days past the peak of the U.S. fourth wave.  There is a slow downward trend in new COVID-19 cases per day.  We’re now 30% below the peak rate.  New case counts are declining steeply in the Northeast and are stable-to-declining in the other regions.

Continue reading Post #1134, COVID-19 trend to 5/3/2021, and a look at hospitalization and mortality rates

Post #1131: Herd immunity: Why aren’t we there yet?

Warning:  This is a long and somewhat technical post.  There aren’t really any results to speak of.  If you don’t have a strong interest in the topic of herd immunity, there’s nothing much here for you.

With that out of the way, the short answer is that we really should be getting close to herd immunity now.  But there’s no sign of it, and it’s sure starting to look overdue at this point.  I provide some state-level estimates showing that, below.

Other than saying “we’ll get there when we get there”, can I point to anything that might plausibly explain why were NOT seeing herd immunity yet?

I don’t think it’s the data. The vaccine counts are tough to argue with.  And while we can dither over exactly how many people have had COVID-19 (versus the number formally diagnosed), I don’t think that’s the hangup, either.

At this point, my guess (and it is just a guess) is that the problem is the far-too-simple model that epidemiologists use to estimate what is required for herd immunity.  I can’t really say what’s wrong with it.  But I can say that if it’s correct, and the first estimates of the infectiousness of COVID-19 (the “R-nought”) were ballpark, then it’s getting to the point where there’s really no way to explain why we’re not seeing herd immunity yet, within that standard, simple model.

My best guess?  Non-homogeneity of the population.  The standard model for herd immunity relies on an assumption of homogeneity.  In effect, it assumes that immune individuals, still-at-risk individuals, and their interactions, are all randomized.  That’s the case where, on average, many immune individuals stand between the still-vulnerable individuals and the infected individuals, stopping spread of disease.  But if that’s not true — if the natural breaks within our society result in clustering the infected and the non-immune together — it seems to me that a pandemic can keep going well past the point where the averages suggest we should have reached herd immunity.

Continue reading Post #1131: Herd immunity: Why aren’t we there yet?

Post #1127: COVID-19 trends to 4/28/2021, and a NY Times article on the end of the pandemic that gets the math wrong.

Trend?  No change.  The U.S. average daily new COVID-19 cases per 100,000 continues to fall slowly.  It’s falling in most regions.  There’s no new crisis area to take the place of Michigan. Continue reading Post #1127: COVID-19 trends to 4/28/2021, and a NY Times article on the end of the pandemic that gets the math wrong.

Post #1124: COVID-19 trend to 4/26/2021


If I had to sum it the current trend, it would be slow progress but no consistency.  For the U.S. as a whole, new COVID-19 cases per day continue to fall slowly.  But if you plot the individual states, there’s nothing approaching a clear, common trend across the states.

You can see that below.

(I’ve rebased these plots to start on 3/1/2021, just before the start o the U.S. fourth wave of COVID-19.  I’ve also added about 11,000 cases to New Jersey today, as they dropped roughly that many duplicate cases in their last day of data.)


Source for this and all other graphs of new cases:  Calculated from The New York Times. (2021). Coronavirus (Covid-19) Data in the United States. Retrieved 4/27/2021.  The NY Times U.S. tracking page can be found at

There’s a slight downward trend.  There is no new crisis area to take the place of Michigan.  If you plot the individual states, there’s no obvious downward trend that is visible to the eye.

Otherwise, it’s just business as usual. 

There are no outbreaks in the five states with the highest percentage of the U.K. variant.  But there’s also no particular downward trend either.

Cases continue to trend upward in Oregon and Washington.  Colorado is still an outlier among Mountain states.  The Northeast area is generally trending downward.  Basically, same as it has been for the past couple of weeks.

The elderly continue to get vaccinated, albeit at a slower pace than in the past.  Between these two snapshots, the vaccinated fraction of the elderly increased by just over 0.2 percentage points per day.  That increase is less than it was, say, four weeks ago.

It’s just a very odd time.  The vaccines are clearly working, but we can’t quite get to the point of shutting this down via herd immunity.  And it still looks as if we’ve passed a peak in our vaccination rate, although we won’t know for sure until use of the J and J vaccine picks up again.

Source: CDC.

What does the low rate of “breakthrough” infections reveal?

In a sense, the U.S. population now splits into two pieces:  Vaccinated and not.  New COVID-19 cases are almost unheard-of within the vaccinated population.  You can see rates reported at this link.  If I’ve done the math right, the reported rate of COVID-19 infections among fully-vaccinated individuals in Michigan works out to 0.2 cases / 100,000 / day.  Less than under one-hundredth the rate for the Michigan population as a whole, for that period.

That low rate shows that this is due both to medicine and to behavior.  Based on medicine alone (the vaccine randomized clinical trials), you’d expect to see one-tenth the number of infections.  I.e., the vaccines are about 90% effective in preventing infection.  But, in fact, you are seeing vastly fewer infections than that.  Plausibly, that’s because the people who are getting vaccinated are mainly those who have been taking this pretty seriously all along, and are not taking infection risks.

That low rate of “breakthrough” infections is good news.  But it means that essentially all the cases you see reported are from the shrinking slice of the population that has not been vaccinated.  And, by inference, increasingly contains people who aren’t being careful about COVID-19 hygiene.

So, for example, CDC reports that 81.7% of the elderly are vaccinated (shown above).  And yet, separately, they report roughly 7 cases / 100,000 / day for the elderly.  (Shown below as roughly 50 cases/ week.)

Source:  CDC

But this means that, roughly speaking, that’s 7 cases per day among the 18.3 percent of the population that hasn’t been vaccinated. Or a rate of nearly 40 infections / 100,000 /day in the remaining un-vaccinated elderly population.  If they were a state, the un-vaccinated elderly would rank second only to Michigan in terms of their current incidence of COVID-19.

And so, I think this needs to be said plainly.  Given the miniscule rate of “breakthrough” COVID-19 infections, the entire pandemic is being perpetuated by the increasingly small fraction of the population that won’t get vaccinated and won’t take COVID-19 hygiene seriously.

I’m now fully vaccinated.  And yet, I’m still wearing a NIOSH-certified N95 mask when I go to the store.  Not because I’m paranoid, but because I’m smart enough to calculate the odds and do the cost/benefit calculation.  I’m going to have to keep wearing a mask until the incidence in the community is far lower than it is now.

The more people refuse the vaccine, the longer it’s going to take to end this in the U.S.  And so, in a sense, the well-being of the many is held hostage by the stupidity of the few.  That’s not going to change until we finally have enough people immune, either via vaccination or via infection.  All we can do is wait for that to play out.

Post #1122: COVID-19 trend to 4/24/2021

The incidence of new COVID-19 cases per day is down about 17% from the peak of the U.S. fourth wave.  Or, at least, from what I hope was the peak, about ten days ago.

Currently, new case rates are falling about 12 percent per week.

Source for this and all other graphs of new cases:  Calculated from The New York Times. (2021). Coronavirus (Covid-19) Data in the United States. Retrieved 4/25/2021.  The NY Times U.S. tracking page can be found at

Michigan now clearly appears to be getting over its recent outbreak.  The count of new cases per day remains at a high level, but it’s trending sharply downward.  Most Midwest states also have a modest downward trend in new cases per day.

No other crisis areas appear on the horizon, but a handful of states still have a persistent upward trend in new COVID-19 cases per day.  Among these, Oregon and Washington stand out for having more than a month of steadily rising new case counts.

It’s difficult to guess what the issue is there.  First, they have a lower-than-average proportion of the U.K. variant.  As of the two weeks ending 3/27/2021, 26% of the new cases in that region were the U.K. variant, versus nearly 47% for the U.S. as a whole (per this CDC web page).

Source:  US CDC.  Arrows and red text added.

Second, they have a higher-than-average proportion of the population vaccinated.  Certainly, the vaccination rate there is as high or higher than in Texas, Florida, or Tennessee, states that all have much higher proportion of the U.K. variant, but no pronounced upward trend in new cases per day.  (Below, darker = higher vaccination rate).

Source:  US CDC.

One thing these states have in common is that they resumed all high school sports in late February.  (Per this reference, Oregon, and this reference, Washington).  The timing of that — about two weeks before new cases began to rise — fits the data.  And yet, there seems to be no news reporting that points to high schools as a main area of COVID-19 spread, as there was for Michigan.

As a bit of due diligence, let me find the data for one of those states and plot the trend in new cases, by age group.   And that’s clearly not the issue.

But one final thing these states have in common, and have in common with Michigan, is that up to now, they’ve had a relatively small share of the population infected with COVID-19.  The chart below is old (as of mid-January), but I’m sure it’s still shows the approximate relative ranking.  These two states have a much lower proportion of the total population with immunity to COVID-19, and a much higher proportion who are still susceptible, compared to the U.S.  Thus, as with Michigan, the likely story here is that they are now paying for their past success in suppressing spread of the virus.   They are farther from herd immunity than is the average U.S. state.

The only truly dark cloud for the U.S. fourth wave of COVID-19 is a falling rate of COVID-19 vaccinations.  You can’t trust the news reporting, because they are always in the business of finding and reporting the outliers.  But the CDC plot of doses administered, for the U.S. as a whole, by date reported to the CDC, does in fact show a recent peak.

The extent to which this is attributable to the “pause” in administrations of the J and J vaccine is not clear.  It looks one could plausibly attribute most of it, but not all of it, to that J and J vaccine pause, based on data from CDC:

Source:  New York Times.

That’s just something to keep an eye on.  Will that rate bounce back up, now that the J and J vaccine is available again?  Or have we actually passed the peak rate of vaccinations.  Given that there’s no hint of hitting herd immunity yet, in any U.S. state, a continued slowdown in vaccinations suggests that the end of the pandemic in the U.S. may be a long, drawn-out process.


Post #1119: COVID-19 trends, update to 4/22/2021.

Looks like the U.S. fourth wave is fizzling out.

You may have noticed that the news media are no longer screaming about Michigan.  That’s because things are getting somewhat better there, and that doesn’t make the news.

Source for this and all other graphs of new cases:  Calculated from The New York Times. (2021). Coronavirus (Covid-19) Data in the United States. Retrieved 4/23/2021.  The NY Times U.S. tracking page can be found at

More than that, the entire Midwest now seems to have topped at just about exactly the same time as Michigan. That’s the sort of synchronous behavior that makes me ask whether there’s still a large element of seasonality to this current wave of COVID-19.  Think about it.  Those states have widely-varying histories (fraction of population already infected), varying rates of vaccination, and varying policies toward (e.g.) re-opening of schools.  And yet, they all appear to peak within a few days of one another?  It’s tough for me to believe that there isn’t some underlying factor causing that behavior.

Be that as it may, more than four weeks into this fourth wave, the U.S. daily new case rate stands just 14% above the low point of the prior wave (marked in red below).  That works out to a growth rate of just 3 percent per week.  That’s not even remotely comparable to the prior three U.S. waves of COVID-19.

I keep using the term “fizzle out” to describe this U.S. fourth wave.  There are no new hotspots to take the place of Michigan.  On the other hand, the virus isn’t going away.  Not anywhere.  Or, at least, not in any state, not even those states that had a large fraction of the population immune via infection.

Here’s a graph of what was, at some time in the past, my top ten candidates for herd immunity, based on having a large share of the population immune via prior infection with COVID-19.   There’s absolutely nothing to suggest that any state is even close to herd immunity.

Tribe immunity

If I contrast what’s just been happening at William and Mary (Post #1117), versus what has happened in Michigan, I’m starting to see some depth to the herd immunity issue.

The W&M student body is largely isolated from the rest  of the world.  As of ten days ago, three-quarters of them had been vaccinated.  And they now appear to have very nearly eradicated COVID-19 within the student body, having had one positive case in the last 4500 or so tests, in the current round of “census” testing of every student on campus.

In Michigan, by contrast, the crisis arose because the virus found itself a fresh, largely uninfected population:  High school students.  The decision to re-open high schools created a pathway for spread of COVID-19 in a population that had largely been protected from it, or, at least, mostly uninfected by it, up to now.

W&M fits the classic model of herd immunity.  That’s a single, isolated, well-mixed population.  With a high vaccination rate, and almost uniformly good COVID-19 hygiene, they have all-but-eliminated the spread of COVID-19 on campus.

But in Michigan, we have numerous sub-groups of the population, and what we have seen of late is that the pandemic has remained alive by rotating to a new targeted group.  Fresh victims, as it were.  There, a large population of unvaccinated individuals  was newly exposed to a situation where transmission was likely.  And so the rapid spread of the virus continued.

Here’s the point:  Michigan wasn’t a flare-up of the virus in the entire population.  It’s not as if they saw a spike across-the-board.  It was mostly due to the spread of the virus to a new target group, high school students, and to a lesser degree, grade-school students and young adults.  My recollection is that for older adults, there wasn’t much of an increase at all in new cases per day.

Michigan had that sub-population of “fresh victims” standing by.  And Michigan saw that flare-up at a time when, according to the standard model of epidemics, infection rates should be tapering off.  Michigan seems to defy the rules, but that’s because the rules don’t really fit the situation in Michigan.

By contrast, within the W&M student body, there is no group of fresh victims for the virus to turn to.  That’s a more-or-less a homogeneous mass of people, living in its own little world.  The refer themselves as the Tribe, and that’s oddly appropriate in this context.  They are a most-vaccinated tribe, and the results are following the standard theory of epidemics.  Tribe immunity, if you will.

Prostatectomies per 100,000 men;  COVID-19 cases per 100,000 non-immune individuals.

In the field of public health, some disease rates are not calculated on a per-person basis.  For example, you won’t see figures for prostatectomies per 100,000 persons.  You will see figures for prostactomies per 100,000 men.  That’s for the obvious reason that women are irrelevant to that calculation, and it makes more sense to calculate incidence of surgery per 100,000 potentially eligible for that surgery.  (You might not think that matters, but if you stratify by age, you’ll find that men account for just 40% of the population age 75+).

A recent Washington Post opinion piece offered a truly profound insight into the current situation with COVID-19.  (Opinion: This is the most dangerous moment to be unvaccinated, by Robert M. Wachter, April 19, 2021).  Or, at least, I found it to be truly profound.  Let me summarize the gist of it, with a little twist.

If you start counting up all the people who are now immune to COVID-19, you realize that the rate of new infections among those who are still capable of being infected is really pretty horrendous.  

The current U.S. new-infection rate doesn’t look too bad, but that’s because we calculate it as new infections divided by the total population. When we do that, we get an average of about 20 new cases / 100,000 population / day.

But when you think about it, that’s like prostatectomies per capita, not per male.  It’s not really an accurate picture of what’s going on.  A large fraction of the population is now immune to  COVID-19, starting with more than half of the adult population having been vaccinated. 

Let me bring my “herd immunity” chart up to date, and then discuss that point.

Source:  Calculated, with a separate assumption as to the ratio of total infections to reported infections, based on the CDC COVID Data Tracker as of 4/22/2021.

By my best-guess estimate, more than two-thirds of the U.S. population should be fully immune to COVID-19 at this point.  And so, when you see the U.S. new-case rate of 20/100K/day,  based on the total population, you should mentally triple that, and say, that’s 60 new cases /100,000 population /day within the population that’s still at risk for infection.

Once you get that in perspective, you see the current situation in a new light.  The U.K. variant is raging within the population still at risk.  The only reason we don’t see that is that two-thirds of the population is no longer at risk.  And when we average those two populations together — 60/100K/day and 0/100K/day — we end up with the seemingly-OK published value of 20/100K/day.

That gets back to the fundamental question for the end of the pandemic:  If two-thirds of us are immune, and we continue to engage in COVID-19 hygiene, why is there still no sign of herd immunity?  That immunity by itself should be able to handle a virus with a basic replication factor (R-nought) of 3.  Toss in the COVID-19 hygiene, and that plausibly should handle the U.K. variant, with an estimated R-nought of maybe 3.5.  Why isn’t that happening yet?

I think the explanation is that the new, more infectious U.K. variant is now finding fresh victims.  It’s jumping to people who would not have been infected, in their normal course of business with the older, less-infectious variant.  As a result, cases are skyrocketing among the portion of the population that remains at risk for infection.

So the U.S. as a whole, with the new U.K. variant being spread, perhaps does not fit the standard model of a pandemic.  It’s not like William and Mary, with a homogenous and well-mixed population.  It’s more like Michigan, with pockets of fresh victims ready to be infected, if only some pathway opens up for the virus to reach them.  And, plausibly, the greater infectiousness of the U.K. variant is the pathway by which the virus continues to spread within the remaining small, non-immune population.

In conclusion:  I haven’t quite figured out what this means for the end of the pandemic.  But, at least, I think this explains why we’re not seeing a swift and clean end of the pandemic, as seems to be occurring at William and Mary.  All across the country, the introduction of the more-infectious U.K. variant means that we’re finding the equivalent of Michigan’s high school students.  We’re finding fresh victims who would not otherwise have become infected.  And we’re now going to have to wait for that process to work itself out before we finally get the total level of immunity in the population both high enough, and homogeneous enough, to suppress further transmission of the virus.

Post #1117: William and Mary, another good day, the math of herd immunity.

Today’s test results were 1-for-1772.  Three days of census testing are highlighted in yellow below.  In short, they’ve tested three-quarters of campus enrollment and found one COVID-19 case.

Again, obviously good news.

This is a follow-up to my last post, and in this I try to explain why you might not want to, or be able to, relax the William and Mary COVID-19 hygiene rules just yet.  Despite this good news. Continue reading Post #1117: William and Mary, another good day, the math of herd immunity.