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DanielLC comments on Rationality Quotes October 2014 - Less Wrong
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So? A 60% reduction in the chances of getting the flu is still orders of magnitude better than a 100% reduction in the chances of getting ebola. Also, herd immunity isn't all-or-nothing. I'd expect giving everyone a 60% effective flu vaccine would still reduce the the probability of getting the flu by significantly more than 60%.
I hear that herd immunity only really works when the percentage of people vaccinated is in the high 90s, but IANAD.
According to the Wikipedia page on herd immunity, it seems to be that it generally has to be at about the 80s. But my point is that it's somewhat of a false dichotomy. Herd immunity is a sliding scale. Someone chose an arbitrary point to say that it happens or it doesn't happen. But there still is an effect at any size. IANAD, but I would expect a 60% reduction would still be enough for a significant amount of the disease to be prevented in the non-immune population. In fact, I wouldn't be surprised if it was higher. If you vaccinate 90% of the population, then herd immunity can't protect more than the remaining 10%.
You can treat herd immunity as a sliding scale, but you can treat it as a hard threshold as well.
In the hard threshold sense it means that if you infect a random individual in the immune herd, the disease does not spread. It might infect a few other people, but it will not spread throughout the entire (non-immunized) herd, it will die out locally without any need for a quarantine.
Mathematically, you need a model that describes how the disease spreads in a given population. Plug in the numbers and calculate the expected number of people infected by a sick person. If it's greater than 1, the disease will spread, if it's less then 1, the disease will die out locally and the herd is immune.
The spreading of deseases sounds like it would be modeled quite well using Percolation Theory, although on the applications page there is mention but no explanation of epidemic spread.
The interesting thing about percolation theory is that in that model both DanielLC and Lumifer would be right: there is a hard cutoff above which there is zero* chance of spreading, and below that cutoff the chance of spreading slowly increases. So if this model is accurate there is both a hard cutoff point where the general population no longer has to worry as well as global benefits from partial vaccination (the reason for this is that people can be ordered geographically, so many people will only get a chance to infect people that were already infected. Therefore treating each new person as an independent source, as in Lumifer's expected newly infected number of people model, will give wrong answers).
*Of course the chance is only zero within the model, the actual chance of an epidemic spread (or anything, for that matter) cannot be 0.
I think percolation theory concerns itself with a different question: is there a path from starting point to the "edge" of the graph, as the size of the graph is taken to infinity. It is easy to see that it is possible to hit infinity while infecting an arbitrarily small fraction of the population.
But there are crazy universality and duality results for random graphs, so there's probably some way to map an epidemic model to a percolation model without losing anything important?
The main question of percolation theory, whether there exists a path from a fixed origin to the "edge" of the graph, is equivalently a statement about the size of the largest connected cluster in a random graph. This can be intuitively seen as the statement: 'If there is no path to the edge, then the origin (and any place that you can reach from the origin, traveling along paths) must be surrounded by a non-crossable boundary'. So without such a path your origin lies in an isolated island. By the randomness of the graph this statement applies to any origin, and the speed with which the probability that a path to the edge exists decreases as the size of the graph increases is a measure (not in the technical sense) of the size of the connected component around your origin.
I am under the impression that the statements '(almost) everybody gets infected' and 'the largest connected cluster of diseased people is of the size of the total population' are good substitutes for eachother.
In something like the Erdös-Rényi random graph, I agree that there is an asymptotic equivalence between the existence of a giant component and paths from a randomly selected points being able to reach the "edge".
On something like an n x n grid with edges just to left/right neighbors, the "edge" is reachable from any starting point, but all the connected components occupy just a 1/n fraction of the vertices. As n gets large, this fraction goes to 0.
Since, at least as a reductio, the details of graph structure (and not just its edge fraction) matters and because percolation theory doesn't capture the idea of time dynamics that are important in understanding epidemics, it's probably better to start from a more appropriate model.
Maybe look at Limit theorems for a random graph epidemic model (Andersson, 1998)?
The statement about percolation is true quite generally, not just for Erdős-Rényi random graphs, but also for the square grid. Above the critical threshold, the giant component is a positive proportion of the graph, and below the critical threshold, all components are finite.
The example I'm thinking about is a non-random graph on the square grid where west/east neighbors are connected and north/south neighbors aren't. Its density is asymptotically right at the critical threshold and could be pushed over by adding additional west/east non-neighbor edges. The connected components are neither finite nor giant.
And indeed the table you mention does shows ranges rather than points. But even the bottom of those ranges are far above 60%.
How do you know there is no phase transition?
Retracted after reading Kyre's comment that what applies to measles doesn't necessarily apply to flu.