Concerning Contagion

A measles outbreak has been in the news lately, and since the disease was previously considered eradicated in the US, I wanted to try a simple model of disease spread with and without vaccination. To represent a population, I used Python's NetworkX package, which gives us nodes (people) connected by edges (contact that can spread disease). In addition to vaccination status, I wanted to see how different levels of contact affected spread, so I generated the layouts using the Watts-Strogatz Graph, which takes a given number of nodes n, a community size k, and a rewiring parameter pwire. The nodes are arranged in a ring, with each connected to its k nearest neighbors. Then we randomly switch some of those connections according to pwire to randomize the interactions a bit.

Turning now to the nodes of these networks, the model for people I chose tracked a few qualities: Current infection, vaccination/immunity status, susceptibility to infection, and infectious period. Based on some stats I found from the CDC, I decided after being infected a person would be sick for 22 days, but only contagious for days 10-18. While contagious, each day a person's neighbors can become infected according to their susceptibility (0.5% for immune, 50% otherwise). I started the model with 10% of people infected, and ran until all cases were gone or a maximum of 300 steps (days). I ran several different values for k and pwire, as well as different vaccination levels.

After running the simulations, I tried to come up with some summary statistics. The first thing that came to mind was the fraction of people that get sick:

The x-axis shows the k parameter for each run, which represents the number of people each person has contact with. For the lower values of k, this shows clear separation in the different levels of vaccination – Low vaccination leads to high infection rates. With enough connection though, everyone encounters an unvaccinated person and gets sick.

That's just the number of people who ever get sick, but we can also look at how long the disease spreads before being eradicated:

Again we see a big uptick for high community size, to the point where disease is still spreading after 300 steps. For the smallest size, we again see separation in the vaccination rates, but there's also an effect from the pwire parameter. This is easier to understand if we look at a map of one of these communities:

Blue dots are people who have never been ill, red are currently sick, and green are vaccinated, or recovered. The twisting shape shows how the pwire parameter has connected a few people on opposite sides of the initial ring. The black dot in the plot above shows that for low pwire, the disease has to make its way around the whole ring, increasing the time to eradication. We can also look at a case with larger communities and more interconnection:

In this case, almost everyone immediately gets sick, which you might imagine would reduce the time before the disease is eradicated, but the increased connection also results in many more breakthrough infections.

I'm really impressed with the complex behavior this simple model was able to show! That said, I fully recognize it's just a toy, and should not be used for setting policy. I'm not headed to that farm upstate just yet...