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Two preprints recently make an important point: for any infection, including COVID-19, it is possible that herd immunity can be accomplished with more than 1/R0 of the population still susceptible. The first was by Gabriela Gomes et al. @LSTMnews…
Just after was one from a different perspective by Tom Britton @Stockholm_Uni and colleagues
They are complementary. Both consider the impact of individual heterogeneity. Gomes et al. consider a well-mixed model with individuals varying in their rates of exposure or probabiity of infection given exposure (susceptibility).
Britton et al. use social mixing matrices, which combine individual-level heterogeneity in (at least exposure) with age-structured mixing, so that one varies not only in HOW MANY exposures one has, but also TO WHOM.
This difference in approach leads to a common finding: the most exposed/susceptible people in the population are more likely to be infected, and their infection is a bigger "hit" to the virus's transmission because they were more efficient spreaders.
So virus transmission disproportionately removes those most useful to it from contributing to future transmission (if they become immune). The size of the effect varies depending on how much & what kind of heterogeneity occurs, but can plausibly be more than 10 percentage points
First a few technical points. The preprints make a valid and important point. An important thing to point out, as Britton emphasizes, is that the proportion that need to get effectively vaccinated at random in a population to achieve the "herd immunity threshold" remains 1-1/R0.
Naturally acquired immunity can get away with less because it is naturally targeted -- the high risk people matter most to transmission and are likely to get infected first. Vaccination (at random) doesn't do that.
The idea of vaccinating or treating "core groups" in infectious diseases (kids for flu or pneumococcal disease, highly active persons for STIs) is to try to gain that same advantage for our interventions -- more bang for the buck by intervening on the transmitters.
To do that we have to be able to identify them -- sometimes easier than others.
A related point: identifying those who are most exposed or susceptible to getting infected may be harder than it looks. People on an island may be low-exposure in the sense that the infection will likely come to them only late, but once it is there it can transmit efficiently.
To generalize -- those in some far off part of the transmission network may look like the "less susceptible" or "less exposed" but may be just as much in need of vaccine as others.
It was partly awareness of this effect that made me -- early on in this pandemic to say I though 20-60% might get infected before herd immunity (even earlier I said 40-70%) even though most estimates were R0>2 implying in the absence of heterogeneity that 50%+ would get infected.
Dr. Gomes in particular has been working on this issue for years now, and @ecoevo_kel (who coauthored the recent preprint) and I have even written about it with her.… .
Since the early days of the pandemic, many of us have gotten a little sloppy and started using the simple well-mixed model numbers without qualification -- half must get immune if R0=2, two-thirds if R0=3. These preprints are good reminders that this is not strictly true.
A more careful way to say it is that up to half/two thirds must become infected to get to herd immunity if R0 = 2 or 3, though likely less due to heterogeneity. However those proportions are correct if we are relying on vaccination for immunity , unless we can target vaccine.
Several caveats for those who think this fundamentally changes our policy thinking. 1 while any heterogeneity will have some effect we don’t know how much exists for COVID. A hallmark has been relatively even infection rates by age though each study shows some variation
2. The lower threshold depends on assuming the heterogeneity is permanent. Is a low risk person is always low risk. So if it depends on behavior for example it could change and thus increase the threshold back to theoretical max.
In an extreme case we could reach a level of immunity in the pop that suppressed transmission under the contact patterns in social distancing and find that with a new “peacetime” distribution of who contacts whom some of the formerly low exposure are now high exposure & at risk
3. There is respectable work that suggests R0 more in range of 4-5 or more… with faster doubling time than early reports and longer generation time than appreciated Bc most estimates included effects of control which shortens gen time
I don’t know if this is right or not but it has gotten little discussion on Twitter that I have seen and it pushes herd estimates up. Have heard Ruian Ke the senior author present and not found major problem with the totality of evidence for this view for certain pops.
4. For moderate R0 2-3 the 1-1/R0 threshold is sharply lower than the overshoot amount 1-f = exp(-R0 f).
5. As @bansallab points out the conclusions of the preprints reverse with certain network structures . I hope @bansallab takes it from here to explain further.
So bottom line these preprints make a v important point but I’m not celebrating. Hers immunity is still a long way off in most places.
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