So what does small world theory have to do with pandemics?

The answer is according to Duncan Watts the author of Six Degrees quite a lot. One way to model how a pandemic progresses is using percolation theory, originally developed by Flory and Stockmayer in the 1940’s. Sites (people) are connected to one another by bonds (a means of infection transmission) forming a network. Each site has a probability of being ‘occupied’, that represents it’s susceptability to infection. While individual bonds have a probability of allowing flow, that represents the infectiousness of the infection. Starting at some point (patient zero) an outbreak will spread from site to site through any open bonds with the resulting set of infected sites called a cluster.

In between the extremes of complete infection and complete resistance is a region in which many clusters of differing sizes can potentially exist and an epidemic will occur if a cluster exists that runs through the entire population of sites. The presence of this ‘percolation cluster’ makes an epidemic (or pandemic) possible, the size of the epidemic is in turn determined by the so called ‘correlation length’ of the percolating cluster.

So what does all this have to do with small world theory? In the late 90’s Duncan Watts and Mark Newman of the Santa Fe Institute started working on the how the shape of a network could affect the point at which an oubreak turned into an epidemic. They demonstrated that the small world effect (the presence of a small number of random shortcuts in an otherwise lattice type network) could dramatically increase correlation length and turn a minor outbreak into a pandemic. The difference that these random shortcuts make is that the presence of even only a few can radically decrease the number of jumps that an infection has to make to move across the network even though most sites are clustered locally.

Unfortunately for us rapid international travel makes such short cuts very easy. As a comparison, in the 14th century the bubonic plague, constrained to spread from town to town, took three years to burn it’s way across Europe, in a few short weeks the current A(N1H1) flu virus has reached 41 nations. Importantly in a small world pandemics also tend to go through a period of slow growth which transitions into explosive growth to be finally followed by a burnout phase. For the current A(N1H1) flu we can see this pattern in the initial outbreak in Mexico followed by the flu virus using international travellers to ‘short cut’ to other countries where slower burning local outbreaks then occur.

The importance of this new way of looking at pandemics is not simply in it’s ability to predict, but in the insights that it gives us into how diseases spread. Thinking about the spread of an infection in terms of networks of infection, clusters, percolation thresholds and correlation lengths offers us new perspectives on current flu pandemic control measures. As Duncan Watts points out, in a small world while a pandemic is in it’s slow growth phase the opportunity is there to combat it through focusing upon the reduction and elimination of random short cuts.

The presence of outbreaks in major transportation hubs are therefore of much more concern than those in more isolated areas as they offer more chance for the flu virus to find a short cut. Where local outbreaks have occurred the identification of the local cluster and prevention of shortcuts out becomes critical, for example by placing general restrictions on international or interstate travel within a particular outbreak area (similar to traditional travel restrictions placed upon areas affected by foot and mouth disease).

The conclusion to all this is that the methods use to combat a pandemic need to address both localised transmission as well as possible short cuts.

References

Newman, M.E.J., & Watts, D.J. Scaling and percolation in the small-world network model., Physical Review E, 60, 7332-7342 (1999).

Watts. D.J., Six degrees: The science of a connected age., Vintage Pub. (2004).