I wrote a simulation where some people are bigger spreaders than others (I used a geometric distribution), and I indeed found cases where 20 percent infected was enough to send R dropping to zero with no other interventions.
I had never thought that a number as low as 25 percent (say) would be enough, but your logic convinced me it was plausible, and the simulation confirmed it.
Zvi, what specific distribution were you thinking? I can find one where the top 20 percent makes up 80 percent of interactions, but if you have one in mind, I’ll try that one in the sim and see what happens.
Oops! There was a problem with my simulation, where the random numbers were repeating. I fixed it, and the results changed. It now took about 45% infected before R dropped below 1. That’s for a geometric (exponential) distribution of spreaders. For a uniform distribution, it should take 75% with R0=4.
It’s hard to figure out what geometric distribution gives the equivalent initial R0=4 by trial and error, but maybe I’ll calculate it by expectation just to get the 45% firmed up better.
When I tried a more spread-out distribution, I didn’t get that much below 45% for anything plausible. I actually squared the relative weightings (so if A had 4x as many chances to spread as B, he now has 16x), and I don’t think it dropped below 40%. Too lazy do walk over to double-check my notes as I write this.
I wrote a simulation where some people are bigger spreaders than others (I used a geometric distribution), and I indeed found cases where 20 percent infected was enough to send R dropping to zero with no other interventions.
I had never thought that a number as low as 25 percent (say) would be enough, but your logic convinced me it was plausible, and the simulation confirmed it.
Zvi, what specific distribution were you thinking? I can find one where the top 20 percent makes up 80 percent of interactions, but if you have one in mind, I’ll try that one in the sim and see what happens.
Oops! There was a problem with my simulation, where the random numbers were repeating. I fixed it, and the results changed. It now took about 45% infected before R dropped below 1. That’s for a geometric (exponential) distribution of spreaders. For a uniform distribution, it should take 75% with R0=4.
It’s hard to figure out what geometric distribution gives the equivalent initial R0=4 by trial and error, but maybe I’ll calculate it by expectation just to get the 45% firmed up better.
When I tried a more spread-out distribution, I didn’t get that much below 45% for anything plausible. I actually squared the relative weightings (so if A had 4x as many chances to spread as B, he now has 16x), and I don’t think it dropped below 40%. Too lazy do walk over to double-check my notes as I write this.