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I'd say there are two different approaches to understanding and modelling the way population heterogeneity affects the attack rate and/or herd immunity threshold for an epidemic: top-down or bottom-up. •1/19
By “top-down” I mean that we divide the population into subpopulation groups with different dynamics (contact rates, contagiousness…). For example, it could be urban/rural, young/old, geographical, or some more complex socio-economic classes. •2/19
In an S(E)IR model, these subpopulation groups might be modeled as compartments. (Each one is still treated as homogeneous internally.) The full equations seem somewhat intractable analytically: https://twitter.com/gro_tsen/status/1260226391838261249 — but we can still say some things. •3/19
Simplest approximation: if we assume inter-group contacts to be much less important than intra-group contacts (i.e., social mixing matrix is mostly diagonal), then each group mostly has its own SIR dynamic, with its own R₀ (reproduction number). •4/19
But as I like to recall, in a sum of exponentials you only see the exponential with the fastest growth. So the epidemic's dynamic is controlled by that of the fastest-growing group (e.g., urban, maybe NYC in the US). https://twitter.com/gro_tsen/status/1249738537206984704 •5/19
So basically the R₀ you see for the overall population is the R₀ of the subpopulation with the highest value. But if this subpopulation is a small proportion of the whole, you overestimate the attack rate and/or herd immunity threshold: … •6/19
… because when that (initially fastest-growing) group hits sufficient immunity, it stops being the fastest-growing group and suddenly we're down to a smaller R₀, and the same happens again, in cascade. Simplest case described here: https://twitter.com/gro_tsen/status/1249738504222969864 •7/19
So this is the “top-down” reason why population heterogeneity decreases the attack rate or herd immunity threshold (or more accurately, causes the simplistic estimate by initially observed R₀ to be an overestimate when applied to the entire population). •8/19
From what I see, this “top-down” reason is what's analysed in the preprint mentioned here (and in many others): https://twitter.com/gro_tsen/status/1258891431097372675 •9/19
This is also what I modeled in a very simplistic, but still informative, way here: https://twitter.com/gro_tsen/status/1259932295609409536 •10/19
The “bottom-up” approach, on the other hand, is concerned with the way variations at the individual level (in number of contacts, contagiousness or, more importantly than contagiousness, susceptibility), affects the attack rate. •11/19
Here the appropriate mathematical model seems to be one based on graphs (or “networks” as some people like to call them). The general idea is simple: individuals with more contacts are infected earlier, so they are made immune earlier… https://twitter.com/gro_tsen/status/1241746000261283846 •12/19
… and by being effectively removed earlier, they contribute to starving the epidemic quicker. From what I see, this “bottom-up” approach is the one being tracked in the preprint mentioned here: https://twitter.com/gro_tsen/status/1258891432879894529 •13/19
I wrote two looooooong threads on modelling this approach using random graphs: the first is here: https://twitter.com/gro_tsen/status/1258835372315901952 — •14/19
— and the second (written earlier, but probably best read afterward) is here: https://twitter.com/gro_tsen/status/1258482878779965440 •15/19
Note that these are about attack rates, though, and not herd immunity thresholds (the attack rate is how many people eventually get infected if the epidemic runs its course; the herd immunity threshold is lower, because it's merely the turning point: … •16/19
… people keep getting infected past this turning point where the infected proportion peaks, so there's an “overshoot”, which I explained around here: https://twitter.com/gro_tsen/status/1236326809412796416 — except I didn't know the term “herd immunity” when I wrote this — anyway). •17/19
It's not clear to me how one would approach computing the herd immunity threshold in the bottom-up approach, because the graph-based model doesn't really know about time or dynamics, so it's not easy to use it for anything other than the full attack rate. •18/19
But another thing which still puzzles me, is how these “top-down” and “bottom-up” approaches I just describe can or will meet in the middle: whether they're two descriptions of the same thing or two different phenomena (subgroup variations and individual variations). •19/19
