I’m trying to think through how “increase in R” interacts with population heterogeneity (a frequent theme of this newsletter, but not mentioned in this particular one!)
This is me thinking out loud. Following is some paragraphs of math which readers can skip, my conclusions are in the last two paragraphs.
Imagine a histogram of everyone in a city. The horizontal axis is “how many microcovids of exposure does this person get per week?” but normalized to (i.e. divided by) the prevailing COVID rate in the city. The vertical axis is “fraction of the population” (again, this is a histogram). Heterogeneity means that the histogram is spread out—some people are almost completely isolated (a peak around 0) while others are out partying, way far to the right.
Actually, let’s omit all the immune people from our histogram. So OK, now there are a lot fewer people way out on the right, because most those people have caught COVID already. There are still some people way out on the right, either because they’ve been lucky, or because they recently jumped rightward—having run out of ability or willingness to isolate as much as before.
Call that histogram H.
Now multiply H by “a line through the origin with slope 1” (y=x). That gives us a function F(x) = x · H(x). While H had a big peak of hardcore isolators on the left side (near 0 microcovids / week), F mostly doesn’t have that peak, it got squashed down by the other factor. Conversely, out on the right, the curve for F is pulled way up, relative to H.
The area under F is proportional to R, the reproduction number.
Now what?
Herd immunity would look like reducing the area under H, which in turn reduces the area under F. Remember, immune people don’t get included in H. Vaccines would push down all parts of H, whereas community spread would disproportionately push down the right part of H, which has a bigger impact on F per person.
Lockdowns would look like squeezing H leftwards, which then reduces the area under F.
Now, let’s say it’s November 2020 in some USA city, and R just jumped up from 0.9 to 1.1. What’s the story? There are different possibilities. One story could be: everyone uniformly increased their microcovids by 20%. We grabbed the H curve and stretched it to the right, like a rubber band. Another story could be: 1% of the population just said “f*** it” and increased their microcovids by 2,000%. A new peak appears in H, way off to the right, pulling weight from elsewhere. The consequences of these two stories are different. In the first story, we need maybe 5-15% of the population to get infected (or 20% to get vaccinated) to get back to R=0.9. In the second story, just that 1% will quickly get infected and immune, and then we’re rapidly back at R=0.9.
Now we learn that, under current conditions, the new strain is, let’s say, 1.65× as infectious as the old strain. So we know that the area under F increased by 1.65×. The simplest story would be: every person has 1.65× as much microcovids as before (again, I mean “microcovids normalized to the prevailing infection rate”). That’s not quite true, but I suspect it’s close enough. So the new strain has stretched H horizontally, pulling it rightward by a factor of 1.65 (and squished it vertically to keep the same area under it). As usual, we have community spread pulling weight disproportionately out of the right side of H, and vaccinations pulling weight more uniformly out of H. How does it play out? I dunno, it’s kinda hard to say.
So where are we at? There’s a subset of people in the USA that are still almost completely isolated, including (in my narrow experience) a subset of New York Times liberals and Bernie progressives, a subset of people with serious comorbidies who can work from home or are retired, etc. They’re still way out on the left, not contributing any appreciable weight to F (i.e., they’re not contributing to the community spread = R). If their microcovids go up by 1.65×, well, they will still not really be contributing any weight to F. Then there is a big tail of other non-immune people with progressively more exposure, who constitute the bulk of community spread. If their microcovids go up by 1.65×, well, a lot of those people will catch COVID-19 who otherwise wouldn’t, before the area under F shrinks below 1 and cases start going down.
So if you compare March-April 2021 (in cities where the new strain has caught fire) to November 2020, say, we have 1.65× more exposure holding behavior fixed, but on the other hand, we’re talking about the behavior about the 50-70th percentile least isolated people (say), instead of the 70-90th percentile, because the latter already caught it in November-January. I don’t have a good sense for how those balance out, but it doesn’t seem so crazy to me to propose that a late-spring peak would be comparable to the peak we’re in now, as opposed to much larger. Maybe each peak infects 20% of the population, or something? I don’t really know, I just made that number up. But if we wind up with 50% infected at the end, well, I think I have a decent shot at not being one of those 50%, and it would make sense to keep trying...
I’m trying to think through how “increase in R” interacts with population heterogeneity (a frequent theme of this newsletter, but not mentioned in this particular one!)
This is me thinking out loud. Following is some paragraphs of math which readers can skip, my conclusions are in the last two paragraphs.
Imagine a histogram of everyone in a city. The horizontal axis is “how many microcovids of exposure does this person get per week?” but normalized to (i.e. divided by) the prevailing COVID rate in the city. The vertical axis is “fraction of the population” (again, this is a histogram). Heterogeneity means that the histogram is spread out—some people are almost completely isolated (a peak around 0) while others are out partying, way far to the right.
Actually, let’s omit all the immune people from our histogram. So OK, now there are a lot fewer people way out on the right, because most those people have caught COVID already. There are still some people way out on the right, either because they’ve been lucky, or because they recently jumped rightward—having run out of ability or willingness to isolate as much as before.
Call that histogram H.
Now multiply H by “a line through the origin with slope 1” (y=x). That gives us a function F(x) = x · H(x). While H had a big peak of hardcore isolators on the left side (near 0 microcovids / week), F mostly doesn’t have that peak, it got squashed down by the other factor. Conversely, out on the right, the curve for F is pulled way up, relative to H.
The area under F is proportional to R, the reproduction number.
Now what?
Herd immunity would look like reducing the area under H, which in turn reduces the area under F. Remember, immune people don’t get included in H. Vaccines would push down all parts of H, whereas community spread would disproportionately push down the right part of H, which has a bigger impact on F per person.
Lockdowns would look like squeezing H leftwards, which then reduces the area under F.
Now, let’s say it’s November 2020 in some USA city, and R just jumped up from 0.9 to 1.1. What’s the story? There are different possibilities. One story could be: everyone uniformly increased their microcovids by 20%. We grabbed the H curve and stretched it to the right, like a rubber band. Another story could be: 1% of the population just said “f*** it” and increased their microcovids by 2,000%. A new peak appears in H, way off to the right, pulling weight from elsewhere. The consequences of these two stories are different. In the first story, we need maybe 5-15% of the population to get infected (or 20% to get vaccinated) to get back to R=0.9. In the second story, just that 1% will quickly get infected and immune, and then we’re rapidly back at R=0.9.
Now we learn that, under current conditions, the new strain is, let’s say, 1.65× as infectious as the old strain. So we know that the area under F increased by 1.65×. The simplest story would be: every person has 1.65× as much microcovids as before (again, I mean “microcovids normalized to the prevailing infection rate”). That’s not quite true, but I suspect it’s close enough. So the new strain has stretched H horizontally, pulling it rightward by a factor of 1.65 (and squished it vertically to keep the same area under it). As usual, we have community spread pulling weight disproportionately out of the right side of H, and vaccinations pulling weight more uniformly out of H. How does it play out? I dunno, it’s kinda hard to say.
So where are we at? There’s a subset of people in the USA that are still almost completely isolated, including (in my narrow experience) a subset of New York Times liberals and Bernie progressives, a subset of people with serious comorbidies who can work from home or are retired, etc. They’re still way out on the left, not contributing any appreciable weight to F (i.e., they’re not contributing to the community spread = R). If their microcovids go up by 1.65×, well, they will still not really be contributing any weight to F. Then there is a big tail of other non-immune people with progressively more exposure, who constitute the bulk of community spread. If their microcovids go up by 1.65×, well, a lot of those people will catch COVID-19 who otherwise wouldn’t, before the area under F shrinks below 1 and cases start going down.
So if you compare March-April 2021 (in cities where the new strain has caught fire) to November 2020, say, we have 1.65× more exposure holding behavior fixed, but on the other hand, we’re talking about the behavior about the 50-70th percentile least isolated people (say), instead of the 70-90th percentile, because the latter already caught it in November-January. I don’t have a good sense for how those balance out, but it doesn’t seem so crazy to me to propose that a late-spring peak would be comparable to the peak we’re in now, as opposed to much larger. Maybe each peak infects 20% of the population, or something? I don’t really know, I just made that number up. But if we wind up with 50% infected at the end, well, I think I have a decent shot at not being one of those 50%, and it would make sense to keep trying...