I am interested in critiquing my understanding of the simplest SIR epidemiological model and also praising a critique of an effort to extend the model and guide policy developed by some very smart economic theorists.
First the useful point is that this post by Noah Smith is brilliant. As is typical, Smith argues that the useful implications economic models depend on strong assumptions so economic theory isn’t very useful. He praises simple empirical work instead.
I will discuss Smith contra Acemoglu, Chernozhukov, Werning and Whinston and Smith pro Sergio Correia, Stephan Luck and Emil Verner after the jump, but really Smith is better at presenting Smith than I am.
It made me wonder. In the simplest model herd immunity stops an epidemic when 1-1/R0 of people have been infected. R0 as I recently learned and everyone now knows is the number of people who would catch a pathogen from one infected person if no one had any resistence. Over time people develop resistence so Rt < R0. If 1-1/R0 of people are resistent, then Rt =1. A bit later Rt<1 so each infected person will lead to a geometrical decreasing series of expected infections so total infections would be 1-1/R0 plus a (small) constant over the number infected at that critical time t. The SIR has susceptible, infected and resistent. The idea is that if one has not been exposed one is vulnerable. If one becomes infected, one carries and sheds the pathogen for a while and then one recovers. After one recovers one is immune and won't get it again. The key assumption in the model is that for every infected people R0 people are exposed (and infected if not immune) and that those people are chosen at random out of the entire population. It is necessary to assume that spread is equally likely from Mr A to Ms B if they share a house or live on opposite sides of the country. This is a silly assumption and the model is the old model used to teach kids and not, I'm sure, current research. It is also the model always used to guide public policy decisions (see me contra benchmark models http://rjwaldmann.blogspot.com/2016/10/benchmarks-model-and-hypotheses.html ) In population biology and evolutionary biology the silly assumption is called "pan mictic" in economics it is called "random matching". The assumption is made very often because doing without it can get one stuck in really hard math. I would like to put a few minutes of effort into trying to figure out if the random matching assumption affects the level of infection needed for herd immunity (of course everyone knows it matters a lot). Below I will always assume R0 =3. Model 2 the population is actually divided into N equal subpopulations and there is no spread from one to the other. The disease starts with one case in one sub population. It will spread until a few more than two thirds of that population has been infected. Spread will stop when 1/(3N) of the whole population is infected. the relaxation of random matching assumption reduceces the incidence needed for herd ommunity by the factor N. This works for any N. Model 3 very like model 2. Half of people have innate immunity to the virus. People transmit the virus to on average 6 other people (on average 3 have innate immunity). the virus will spread until 5 of 6 are immune. that means (5/6)-(1/2) = 1/3 must acquire immunity (by getting infected). So 1/3 not 2/3. OK can we be sure that the number who will get onfected is less than 2/3 ? Consider Model 4. people live one to a square of an invisible chess board (which is a really big square) they transmit the pathogen to those with whoù they share an edge. R0 = 3 (I get it from 1 neighbor and early in the epidemic give it to my other 3 not yet infected neighbors). How many people get infected ? All of them Katy. The currently infected are always in the border zone between the resistent and the vulnerable. So R0 = 3 implies herd immunity will stop the spread at some level which ranges from 1/(3N) for N as big as I like, to 100%. R0=3 and a priori reasoning without arbitrary assumptions which we know are false and make for convenience tells us nothing at all. Without some assumption about mixing, matching, and population structure, the core SIR assumptions have no implications. Maybe economists and epidemiologists have more in common than we thought.