Post #1003: North Dakota, is this an outbreak threshold effect?

Posted on February 11, 2021

Source for all graphs:  Calculated from NY Times Github COVID data repository, data through 2/10/2021.

U.S. new cases continue to decline. We’re now 58% down from the peak.  But the rate of decline isn’t speeding up.  It’s still averaging just above 20% per week.



North Dakota continues to be an outlier.  New cases continue to fall, and as of 2/10/2021 the new-case rate in ND just slightly higher than in Hawaii.  This graph shows the states, since 1/1/2021, in logs (so that constant growth rate graphs as a straight line).

I really can’t say why North Dakota is doing so well.  I’d like to say that this is the end-game on herd immunity, but the data just don’t support that.  If that were the main driver of ND, then we’d be seeing some sort of changes in all the states with (estimated) population COVID-19 immunity levels near that of ND.

In fact, the more you look at that graph, the less you know about what’s driving this.  I’m back to Post #972, The Rain Falls on the Just and the Unjust.  How on earth can this totally diverse set of states all show virtually identical rates of decline in new cases, all starting at virtually the same time?

So I continue to scrabble for explanations.  We can chalk it all up to “seasonality”, right up to the point where ND breaks away from the pack.  At that point, it’s not clear what’s going on.

The only thing that plausibly distinguished North Dakota just prior to this recent rapid decline was a low absolute level of new cases per capita.  As in, low-double-digits.  A few tens of new cases, in the entire state.  And the fact that this is a small population in a low-population density state.  So what you have left of the pandemic in ND is a small number of cases, thinly spread.

Could that, by itself, be enough to explain what appears to be a collapse of the pandemic in ND?  Is there some low threshold of density of cases below which pandemics tend to collapse?

Turns out, the answer is maybe yes, just as a matter of arithmetic.  Or, at least, simulations of pandemics exhibit such a threshold.  Near as I can tell, the bottom line of this piece of research is that if cases get thinly enough spread, pandemics can die off just by mis-chance. From purely random “stochastic effects”.  And, assuming I’m reading this correctly, the higher the “k factor” of the disease (i.e., the more the infection spreads via clusters, not one-on-one), the higher that threshold.

Source:  Hartfield M, Alizon S. Introducing the outbreak threshold in epidemiology [published correction appears in PLoS Pathog. 2017 May 31;13(5):e1006415]. PLoS Pathog. 2013;9(6):e1003277. doi:10.1371/journal.ppat.1003277

N.B. “Log” above is natural (base e) log.

Reading the graph above, COVID-19 has a very low K factor, similar to the top line (SARS 2003).  And the R-nought is somewhere around 3.  That means that if you have fewer than about 5 people in a population, the odds favor the epidemic just sputtering out, instead of growing.

This is the first I’ve even heard of this.  But let me now remind you of a couple of things that seem to dovetail with this.

First, this seems consistent with the finding that only a small fraction of COVID-19-infected individuals goes on to spread the disease.  In prior posts, I’ve cited a figure of about one-in-seven persons infected with COVID-19 go on to infect someone else.  (That’s stated in Post #849 but without citation as to source.)  The other six are dead ends, in terms of maintaining the pandemic.

If those are the odds, then it’s pretty intuitive that with a small number of cases, you’ve got good chance that all of them are “duds”.  And so, every time you find yourself with some isolated population with just a few cases, there’s a good chance that the pandemic will simply die off, in that population, of its own accord, purely by chance.

A little numerical example is helpful.  If only 15% of cases end up spreading it, with anything below five active cases, the odds favor the pandemic simply dying out.  Only with five cases or more do you have a greater than 50% chance of passing the disease on to a next generation of cases.

Second, recall all the stories you’ve read that said, in effect, coronavirus may have been in fill-in-the-blank-location as early as fill-in-the-blank-date.  That, in hindsight, scientists can now find a stray case or two, very early on.  But that somehow, that stray case didn’t directly lead to an epidemic outbreak.  I am now guessing that those were examples of having cases below the outbreak threshold.  So, sure, there was a case that showed up early, but it’s not as if COVID was lurking in the population ever since.  More likely, that one case never succeeded in infecting anybody else, and the pandemic sputtered out.

So without doing even one more bit of research, I’ll just stop with that.  Possibly the outbreak is collapsing now in ND because they’ve fallen below the outbreak threshold.  Cases within isolated populations are simply becoming so thin on the ground that, one piece at a time, just by chance, you see zero infections in the next generation of cases.  And the pandemic comes to a halt.

And possibly, that’s just nonsense.  E.g., why hasn’t the same thing happened after months and months of low case counts in Hawaii?

Otherwise, I’ve got nothing in terms of explaining why, all of a sudden, in North Dakota (and only North Dakota) new cases have taken sharp turn downward.

So I give up.  Here’s a graph of what should be the top 10 states, in terms of the level of COVID immunity in the population.  (Many caveats apply.)  Until I see some peel off and follow North Dakota, then as far as I’m concerned, the rapid decline in North Dakota is a fluke.  I need to see something systematic across the states to be able to attribute this to some underlying cause.