Chances are, if you find me talking about small worlds, I’m usually doing so in the context of a social network. One of the interesting things about the small worlds model is its wide applicability to many organic networked systems; a social network is just one of these. For today’s post, we’re going to analyze transversals in small worlds, in the context of epidemics.
Why epidemics, you ask? If you haven’t noticed, a shortage of flu vaccine is dominating headlines, causing a good deal of anxiety among vulnerable and non-vulnerable populations, and roiling the presidential race. At the outset, I must admit I’ve never really been one to buy into flu vaccine hysteria; in fact, frenzies in recent years have been somewhat amusing to me.
The situation this year is markedly different from those of years past, where shortage was relative and supply was tight, but sufficient. At last count, the total number of vaccinations available to the public was an optimistic 40M, 42M short of the 82M goal, and 30M short of the 70M shots required to vaccinate all at-risk populations.
A small world, as defined by Watts and Strogatz, incorporates two key properties:
The clustering coefficient, defined as an average of connections a node makes to other nodes in the world.
The characteristic path length, defined as number of path transversals one must make to encounter other nodes in the world.
Put a little more plainly, think of your close friends. If you’ve got a lot of close friends, with whom you spend most of your time, your group would have a high clustering coefficient. (Note though that the model is concerned with averages, which allows us to create population-level estimates.) Now, thinking in another direction, imagine your friend’s friend’s friend. The path to this friend is three transversals, i.e. you’d need two contacts to get in touch with this particular person. (Again, please note that path length is also an average in the model.)
Visualize if you will, the network of your friends. Imagine their network of friends, and so on and so forth, until you can’t see out any further. What you see probably looks something like a network, connected by lines, some intersecting, some not. In a small world, we’re concerned with both regular (your friends) connections, and also the connections you might make randomly, say, by striking up a conversation at a coffee shop (these are considered random in a small world network).
You might wonder how flu vaccines fit into all of this. As I’ve said, in a small world, we’re concerned with the people we know (regular), and the people we come into contact randomly (random). We realize we’re more likely to associate for longer periods of time with our regular connections, but these random connections also are valuable. In fact, when evaluating our small world in the context of a transmissible disease, all of our connections have roughly the role: transmission vectors.
How then does a supply shock to flu vaccine supply affect our small worlds model? In this particular network, vaccines affect path length. For example, think about all the people you had 10-minute conversations with today. Lets assume, for my illustration’s sake, that this conversation represents the opportunity for virus transmission. Now, try and imagine all of the people you talked to for ten minutes, then going on and talking to someone else for ten minutes, and so on and so forth. A scary thought at the surface, right?
We’ve established that small worlds are defined by the clustering coefficient and path length of nodes, so lets investigate how vaccines affect the model. I argue, perhaps obviously, that vaccination increases the path length between nodes in our cluster, often so effectively that random connections represent the only transmission vector. This is an ideal circumstance: when those who you most frequently come in contact with are vaccinated, and those who they come in contact with are vaccinated, and so on and so forth, we might just chalk it up to dumb luck that we chose to have a random ten minute conversation with a transmission vector. In fact, its not dumb luck, its just a small world.
The illustration does not end there, though. Imagine a world of clustered nodes (you and your friends or coworkers), where considerably less people are vaccinated. The path length in this world decreases so that, according to the model, there are higher chances that a regular or random connection could introduce the virus to your cluster. And, as your clustering coefficient rises (characteristic to small worlds), your incidence for transmission increases.
Imagine that you and your coworkers work in a small, ten-person office. While it remains true that you and your coworkers spend most of the day interacting with each other and each other only, you do interact in other clusters, and with random persons. Now, if there are less vaccinations, and more transmission vectors, the chance of you bringing back the virus to your work cluster are much higher.
If you’ve made it this far, you’re probably asking yourself why I just spent all that time illustrating something that is pretty much conventional wisdom. The fact is, our worlds are small, and this reveals why the flu vaccine is so important.
Imagine the case of a highly contagious disease, such as smallpox, emerging in a population. What would epidemiologists do? First, they would quarantine the individual, virtually increasing that individual’s path length to other nodes to infinity (no outside contact). For all intents and purposes, this is what a vaccination scheme does: it artificially raises our path length for transmission between nodes in a network. By making it harder to find vulnerable nodes and clusters of nodes on a network, the vaccine not only keeps us healthy, but it keeps others healthy.
Lets now assume the current situation, where only 60% of the at-risk population will be vaccinated. This, again, is an artificial shock to our small worlds model, but it can tell us quite a bit. Watts and Strogatz modeled log path length (p) with respect to path length (0) for probabilities ~0 to 1 that the world was random. The way this data plays out is that as the world gets more regular (higher chance of transmission), path length is affected strongly. The model points out that the shock is not proportional. A 40% increase does not mean that 40% more people could get the flu; in a small world, a 40% increase would create an exponential increase in flu transmissions.
Therein lies the real cause for alarm in our current shortage. We tend to think of the flu on a personal level – I should get vaccinated so I don’t get sick – and while this works out on a micro level, we often fail to realize that we are getting vaccinated so that we don’t get others sick. How will this eventually play out? The small worlds models predict a grave situation; ultimately, the nature of the particular strain of flu that presents itself will determine how well it can exploit the gaps in our small world model caused by the vaccine supply shock. Nevertheless, it is of great concern, and I’m scared of the flu for the first time ever.
Tags: networks
