If we assume (key) that the probability of becoming infected is directly proportional to the number of people you interact with, and consider that the probability of spreading infection given that you are infected is obviously directly proportional to the number of people you interact with, the product gives a quadratic function:

[probability of becoming infected and spreading] = [probability of becoming infected] * [probability of spreading] ~ [people you meet]^2

Okay so something like:

probability of spreading = a * n probability of infected = b * n

where: a,b = some blackbox function for spreading/infection ratio? n = number of people you meet

Assuming both are independent events we get:

p = a * b * n^2

I can see how we get n^2 with that. The way I'm using a, and b seems clumsy though, is there a better way?

In math and (especially) physics it is common to express "proportionality" laws usually with a symbol that looks like LaTeX \propto. So for instance the activity A of a radioactive sample is written A \propto e^(-t/T) where t is time and T is the mean lifetime of a single particle. For convenience \propto is often transliterated as ~ when typing. So that becomes

p_catch ~ n

p_spread ~ n

p_relay = p_spread * p_catch ~ n^2.

Thanks, that's a much clearer way to think of it.

This isn't quite right. For one, the probability of being infected is not linear (it's capped at 1). If you do the math, the expected number of people you infect (given you are not infected) is roughly linear-ish.

> the probability of being infected is not linear

Correct, it's not. But it is roughly linear in the limit of small numbers of people with a small constant probability of becoming infected per interaction. (This assumption becomes problematic when you see "clustering" of highly social people with other highly social people.)

To be specific, if P is the probability of becoming infected when interacting with a single person, then the probability of becoming infected after interacting with N people is 1-(1-P)^N = NP - O(N^2 P^2). It's easy to see that the limiting infection probability is 1 in this simplified model, and that if N*P < 1 you're looking at close-to-linear growth.

Wouldn't it also vary by percent of population without immunity, perhaps demographics in the underlying population, and other environmental things (weather?). Seems like a (useful) aproximation.

If you have N nodes and completely connect the nodes, there are O(N^2) edges. This comes from the formula for the sum(1...N).

Edit: I’m not sure I interpreted the original statement correctly.

That's what I was about to answer. That's where squaring comes from, in this case. However, as others have noted, you could just sort by # of people contacted, since squaring just gives you larger numbers for no benefit in analysis here.