Influenza and small-world graphs

A paper in JAMA about the effectiveness of vaccination against the flu received some play in the media last week (for example, the New York Times). The paper’s result is that vaccination of children reduced the overall rate of infection in the population, a demonstration of the herd effect — vaccinated individuals break chains of transmission that would otherwise reach further and to more people.

The problem is that the research was carried out in 49 small Hutterite communities (groups that live as independently as possible from the rest of the world). The social contact graph in such a community is (a) small, and (b) almost fully connected. The social contact graph for those of us who don’t live in such communities is very different. Every day I am in close proximity to many people who live about 10 miles from me, a smaller number who live 100 miles from me, a few who live 1000s of miles from me, and one or two who live 10,000 miles from me. In other words, the graph that I, and almost all of you, live in has lots of locality (most of the people we see live close to us) but also long edges, and the long edges are frequent enough that the diameter of the graph as a whole is small (so-called 6 degrees of separation).

In such a small-world graph, results about protection by vaccination in small, richly connected graphs say nothing. In a small-world graph, if I get a dread disease, the several people who live 1000s of miles from me might have been infected and carried it home by tomorrow, and there’s a chance that someone will have carried it 10,000 miles by tomorrow too.

Unless epidemiologists understand that small-world graphs are unintuitive but realistic, there are going to be continuing misunderstandings about how diseases behave on a macro level, and continuing surprises that they spread across the globe in a flash. Quarantine, for example, has to be total or nothing if it’s to serve any purpose.

Computer networks too are small-world networks (although with a characteristically different internal structure) so these same issues of infection apply to worms and the same calculus of innoculation applies.


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