Thread by @joel_c_miller

@mlipsitch has a thread about a pair of recent preprints (note - not peer reviewed) which show that due to heterogeneities, immunity acquired through infection can lead to "herd immunity" even if the usual threshold of 1 - 1/R has not been infected:

https://twitter.com/mlipsitch/status/1258827506930667523?s=20

I have not read these preprints closely yet, but I want to offer some of my own comments on these results. To be clear, I think the main points of the modeling are correct.

First, let me highlight the "friendship paradox". Before I explain it, here are two references:

opinionator.blogs.nytimes.com/2012/09/17/fri… (by @stevenstrogatz)

jstor.org/stable/2781907

To see it, look at this population. You can check that half of them have 1 partner, while the other half have 3. Now choose one of these individuals, and then choose a random partner. The partner is 3 times as likely to have 'degree' 3 as 1. Exercise: prove this.

For the disease, a degree 1 person that it reaches is a dead end, while degree 3 people can lead to 2 further transmissions.

Since the average person can only contribute 1 new infection, you'd think that if the transmission probability p<1, then disease will die out.

The problem is that to the disease, the population initially looks like 3/4 of the population has degree 3, and so it turns out that it can still spread widely.

So R0 =/= p ave(K) [what you might naturally expect]

Rather R0 = p ave(K^2-K)/ave(K)

In the population I showed, this is p(5 - 2)/2 = 1.5 p. So as long as p>1/1.5, spread is possible.

Gven that we've found R0, the "herd immunity threshold" should be 1-1/R0. And this is correct if the vaccines are randomly distributed (but not if we can target degree 3 people)

Even if we can't find out who those people are, the disease will preferentially find them.

(Note that this often leads to a smaller epidemic than a homogeneous model predicts b/c all those low degree people have a good chance of escaping)

If we can mitigate the spread somehow, the disease will not get to full size, but it will disproportionately remove high risk individuals.

So we would expect that after a first wave with strong interventions, the rebound will be less than a homogeneous model predicts.

And additionally, we can prevent the rebound altogether at lower cost than our naive homogeneous models predict.

Basically, after the first wave, the people that the disease encounters in a second wave are likely to be the same it saw the first time around.

We want herd immunity among them, not among the general population.

For this reason I'm less concerned about the fact that New York serosurveys have selected a biased sample - the people showing up to grocery stores. So long as they sample the people the disease will see, it's giving very valuable data.

Here are some caveats I have:

1) I believe interventions will have a larger impact on higher degree individuals. So the first wave will not be so efficient on getting high degree individuals as a first model would predict.

2)I believe interventions are adopted heterogeneously, so some of the high degree individuals will not have high degree in the first wave.

3) I am not convinced that the survey data most modelers use to parametrize our models capture the hot spots of transmission that we have seen. I don't think prisons were surveyed, or meat packing plants. I don't know about aged care facilities.

I think mixing (particularly for a respiratory disease) in these and other hotspots is actually rather homogeneous. So I don't know how this plays out.

Points 1 & 2 can be added to a model fairly simply. 3 is a little harder.

The overall message to me doesn't change: Herd immunity is the consolation prize for having a largely uncontrolled epidemic spread through the population. Failure to contain results in many, many deaths.

However, I anticipate that some will argue that this supports the idea: "let's isolate the high risk individuals and let the epidemic spread in the rest of the population, because it's easier to get there than many people thought".

I am not convinced.

1) Most places have tried to isolate the high risk individuals while keeping a small epidemic in the rest of the population. The isolation has usually failed. I don't think allowing a larger epidemic in the rest of the population makes this easier.

2) If we remove all high risk individuals and their direct household contacts, I don't think we get to the herd immunity threshold.

3) I have seen evidence that susceptibility to infection increases with age. So I question whether we reach herd immunity if we successfuly isolate high risk individuals.

4) The high risk individuals also mix preferentially. The aged-care facilities are the clearest example. This means that herd immunity in the general population doesn't provide herd immunity within their communities.

So fundamentally, anyone who argues we should "just isolate the high risk groups" needs to explain how they plan to isolate them.

"I saw a model that rescaled the transmission rate in older individuals" is not an answer to my question.

So what are the implications of these studies? I agree with the primary results.

An epidemic with fairly large interventions in place gets us to herd immunity at a smaller threshold than we'd predict from a homogenous model.

This moves the threshold. But I don't think it gives a new paradigm (and I know many people have been accounting for the fact that the homogeneous prediction for the threshold is an over-estimate).

Grrr - I forgot to give credit - the first preprint is from Tom Britton, Frank Ball & Pieter Trapman, all of whom I know and respect highly. The second is by Gabriela Gomes and 8 other authors. I don't know them, but I'm familiar with Dr Gomes's work, and it's good.