Wednesday, July 22, 2020

A Density Query

One factor relevant to different rates of Covid deaths in different countries is population density. People often talk about herd immunity as if it required the same percentage of immune people everywhere, but that obviously is not true. In a society where the average individual only encounters someone else every three months, herd immunity starts at zero percent immune. The more people interact, the higher the fraction that must be immune in order that each infected person will, on average, pass the disease on to fewer than one more.

Frequency of interaction depends in part on population density, but not the density calculated by dividing population by area. Adding a million square miles with nobody living there to a country sharply reduces the average population density defined in that way but has no effect on the average density experienced by an individual. What the usual definition is measuring is  density per acre averaged over acres. What we want is the density averaged over people, giving heavier weight to acres that have more people on them. That difference is the reason that people in a country like the U.S., large parts of which are nearly empty, tend to greatly overestimate how crowded it is — almost all the places they observe are places with people in them.

Does anyone know of a source for the form of population density I want? It might turn out that some countries had a much higher density relative to others than the conventional figure implies.


Unknown said...

This seems somewhat similar to a thought i had about campaign donations, where it could be interesting knowing the "average" donation size from the perspective of the dollar. that is, what is the expected size of donation a dollar came in, if i pick a random donated dollar.
In that this too makes large concentrations more significant

Ricardo Cruz said...

I wonder whether the most important factor is not simply the way deaths are counted. For example, even within the UK, there is a large discrepancy in the way deaths by covid19 are counted. There is currently a controversy over the fact that England is counting anyone who has been diagnosed and later died as a covid death, no matter how long the test was performed and even if the person had been considered cured. Other UK nations are using smaller timeframes of 28 days. Other countries only count covid deaths if the person has been found to be seropositive near death or after a post-mortum analysis, while other countries only count death as being caused by covid if the doctor thinks it was a contributor, as was typically done in respiratory and other diseases.

SB said...

This whole thing is reminiscent of the "the other line moves faster" phenomenon. If there's any difference in line speed, you would expect the average person to spend more time in slow-moving lines than fast-moving lines, and therefore (averaged over people and time) the other line really does move faster.

Likewise, in a democratic government with (effectively) binary elections, the average person's preferred candidate wins more often than loses, because by definition more people are on the winning side.

Anyway, one could measure what you're talking about by taking population densities of a bunch of different small regions, then averaging them weighted by population. That is,

\Sum_{region X}(pop(X)^2 / area(X))
\Sum_{region X}(pop(X))

This has the same units (population/area) as the classical definition, and should be meaningfully comparable between countries. But it depends on the choice of regions: if there's only one region, it's just the classical definition, and the more smaller regions you have, the more it differs from that definition. So if you wanted to make a meaningful comparison across countries or states, you'd need to pick a uniform way to pick regions within them. Should regions within a country/state be roughly the same area, or roughly the same population, or does it not matter much? Should regions chosen for one country/state be roughly the same area, or roughly the same population, or roughly the same number, as those chosen for another country/state, or does it not matter much? Time for some experiments....

Anonymous said...

The term is "population-weighted density"

William H. Stoddard said...

I don't know a source, but it sounds as if you're looking for something like a harmonic mean . . .

Steve Witham said...

Density per person makes more sense than just density. But, disease-spreading still might not be proportional to that. There might be power laws involved. Like, (making this up) the sum of (population ^ 1.7 x density ^ 2.6) ^ .35, that people have found from surveying lots of statistics of the same kind of problem.

Like Zipf's Law situations with that feeling of a statistical uniformity that one can only lamely guess at the mechanism behind.

But it might be in the code on the github page of some big epidemology lab.

There are also things

In chaos or complexity theory there's something called "mixing" that's related.