Posts Tagged 'h1n1'

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.

Why H1N1 spread so fast

I heard a talk last week at the Intelligence and Security Informatics Conference (ISI2009) about the models used for disease spread, and I realized why the WHO (and everyone else) were surprised by the speed with which the H1N1 flu spread. These models have different assumptions about the probability of spread from one person to another, how much time each individual is infectious, ill, recovering, and immune or not. But they tend to have one underlying assumption about spread, and that is that it’s a planar phenomenon. Spread is usually modelled as a differential equation, a kind of model that if 500 people are in a school and the probability of infection is 10% in a day, then 50 people will become infected.

The problem with these models is that they don’t take into account the “six degrees of separation” phenomenon. Although most people mix with only a small number of people who are geographically close, enough others mix with people who are geographically far away. As a result, after 3 transmissions, the infection hasn’t reached half the world’s population — but it has reached half way around the world!! Failing to take into account the connectivity between people makes the models far, far too conservative about spread.

Including the graph structure that connects people shows that quarantine mechanisms cannot possibly work. These long-distance connections apply at all scales, not just between countries. So if there’s an outbreak in a single city block, there will be some people who have travelled a few miles away before the infection is detected; in an outbreak in a city, there will be some people who have been to another city; and so on.

Of course, the work on “six degrees of separation” was based on communication, which does not always imply transmission. So the constants might be a but larger; but it seems clear that the pass-the-parcel (pass-the-virus) graph can’t have much larger diameter.