Beware R0 Variance

The big push now re Covid19 is to use “social distancing” to cut “R0”, the rate at which infection spreads. More precisely, R0 is the average number of other people that one infected person would infect, if they were not already infected. With no efforts to reduce it, estimates for natural R0 range from 2 to 15, with a best estimate perhaps around 4. The big goal is to get this number below 1, so that the pandemic is “suppressed” and goes away, and stays away, until a vaccine or other strong treatment, allowing most to escape infection. In contrast, if R0 stays above 1 we might “flatten the curve”, so that each infected person can get more medical resources when they are sick, but soon most everyone gets infected.

Apparently even with current “lockdown” efforts, all of 11 European nations studied now still have best estimate R0 over 2, with a median ~3.7. So they must do a lot more if they are to suppress. But how much more? My message in this post is that it is far from enough to push median R0 down below 1; one must also push down its variance.

Imagine a population composed of different relatively-isolated subpopulations, each with a different value of R0. Assume that few are infected, so that subpopulation pandemic growth rates are basically just R0. Assume also that these different R0 are distributed log-normally, i.e., the logarithm of R0 has a Gaussian distribution across subpopulations. This is (correctly) the usual distribution assumption for parameters bounded by zero below, as usually many small factors multiply together to set such parameters. The total effective R0 for the whole population is then found simply by integrating (via a lognormal) the effective growth over R0 subpopulations.

For example, assume that the R0 lognormal distribution has log mean (mu) -2 and sigma 1. Here the mode of the distribution, i.e., the most common R0 number, is 0.05, the median R0 is 0.14, only 5% of subpopulations have R0 above 0.70, and only 2% have R0 >1. Even so, if each of these subpopulations maintain their differing R0 over ten infection iterations, the mean growth factor R0 of the whole population is 20 per iteration!

As another example (for log mean -1, sigma 0.5), the R0 mode is 0.29, the median is 0.37, only 5% of subpopulations have an R0 over 0.85, only 2% have R0>1. Yet over ten infection iterations maintaining these same R0 factors per subpopulation, the mean growth factor R0 of the whole population is 1.28 per iteration. That is, the pandemic grows.

Of course these growth numbers eventually don’t apply to finite subpopulations, once most everyone in them gets infected. Because when most of a population is infected, then R0 no longer sets pandemic growth rates. And if these subpopulations were completely isolated from each other, then all of the subpopulations with R0<1 would succeed in suppressing. However, with even a modest amount of interaction among these populations, the highly infected ones will infect the rest.

The following graph tells a somewhat more general story. On the x-axis I vary the median value of R0 among the subpopulations, which sets the log-mean. For each such value, I searched for the log-sigma of the lognormal R0 distribution that makes the total average R0 for the whole population (over ten iterations) exactly equal to 1, so that the pandemic neither grows nor shrinks. Then on the graph I show the standard deviation, in R0 terms, that this requires, and the fraction of subpopulations that grow via R0>1.

As you can see, we consistently need an R0 standard deviation less than 0.21, and the lower the median R0, the lower a fraction of subpopulations with a positive R0 we can tolerate.

So, as long as there is substantial mixing in the world, or within a nation, it is far from enough to get the R0 for the median subpopulation below 1. You also need to greatly reduce the variation, especially the fraction of subpopulations in which the pandemic grows via R0>1. For example, when the median R0 is 0.5, you can tolerate less than 3% of subpopulations having an R0>1, just to hold the pandemic at a constant overall level. And to suppress in limited time, you need to go a lot further.

Different subpopulations with differing R0 seems plausible not just because our world has different nations, classes, cultures, professions, industries, etc., but because Covid19 policy has mostly been made at relatively local levels, varying greatly even within nations. In addition, most things that seem log-normally distributed actually have thicker than-lognormal tails, which makes this whole problem worse.

All of which is to say that suppressing a pandemic like this, with high R0 and many asymptomatic infected, after it has escaped its initial size and region, is very hard. Which is also to say, we probably won’t succeed. Which is to say: we need to set up a Plan B, such as variolation.

Spreadsheet for all this here.

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