I recommend everyone interested in percolation theory to also check https://meltingasphalt.com/interactive/going-critical/ and play with critical thresholds interactively!

This was fascinating! It started with some simple simulations, but led into some very enlightening generalizations about why culture spreads more rapidly in cities than in the countryside, and why new language spreads more rapidly among adolescents.

Great link, thanks for sharing!

Thank you!

And, to give an example we've probably all become familiar with, this this is equivalent to how a disease becomes an epidemic as the R value crosses 1.

That seems more like a truism: If on average one case leads to more than one other (R > 1), the number of cases will grow. The R number therefore appears to be nothing more than a measure of growth or decline. What am I missing?

As far as I understand it R is not a rate because it does not include time. Diseases that take on average a few days for you to pass on verses a few months will have different growth rates. For example, an STI might have an R factor of 3, but the rate is completely dependent on how often people have unprotected sex.

Good point, R is often referred to as a rate both in media and academic publications, but "number" or "value" appears to be more correct. Fixed.

R is not a measure of anything (i.e. real world data), it's more of a prediction based on a mathematical modeling procedure. It's not constant for a pathogen like Sars-CoV2, as it depends also on other conditions (environmental factors like temp/humidity, and social factors (norms like handshaking, etc.).

The models used to generate R-numbers have some assumptions which may or may not be valid: (1) rectangular and stationary age distribution, and (2) homogeneous mixing of the population. Thus, the result is a fairly rough estimate.

A major use is in getting a decent estimate of what percentage of a population needs to be vaccinated in order to halt the spread of a viral infection. However, this supposes 'sterilizing vaccination', i.e. vaccinated individuals are not asymptomatic carriers and spreaders of the infection. While this was the case for the smallpox vaccine, it doesn't seem to be the case with all known Covid19 vaccines, where there are many breakthrough cases (even though symptoms are reduced and hospitalization is minimal).

This is essentially what is happening in the percolation case as well.

What I mean is that R varies by disease (or for a given disease, according to interventions, etc).

The size of the total epidemic varies sharply as R is changed in this way.

When you say R, are you referring to R_0? Because R_0 is not a mathematical predictor of percolation.

I think there's lots of ways to formulate this to see how it's percolation. I'm not a percolation expert though so please call me out if you are, and I'm wrong:

To sketch here:

Consider R the reproductive number of a disease (or R_e, the effective reproductive number, to be specific). This depends on both the inherent contagiousness of the disease, but also on the behavior of the population. If people choose to have fewer contacts (or bars are closed) then R decreases.

Let's say covid is spreading. We ask people to limit their contacts, and see if this stops spread.

We can think of this as trying to remove edges from the contact graph that the disease spreads on.

The contact graph becoming connected or unconnected as we remove edges is clearly percolation.

(Subject to some modeling assumptions about edges being removed at random, but these assumptions are common to both Erdos renyi graph models and SIR style compartmental models).

Make sense?

Sudden thresholds were just recently relevant with disease transmission and recoverability too.