# Associating a probability to disease propagation in regions of a map

I am trying to model disease propagation. I am considering a map separated into regions. One region is infected. There is data over time, with the number of people travelling into and out of the infected region and which region they travel to (a table of transport of individuals over the whole map). This is an O-D matrix (wiki description).

There is also some data about the prevalence of this disease in the region.

The questions I would like to ask are:

1. What are some classical options for modelling disease propagation to different regions with this spatial data which will produce easy to interpret probabilities? (eg. SIS/SIR models which require simulations that I can just average over, or a time series model such as ARM with some measured parameters?)
2. What specifics of the disease must I know before hand? (eg. parameters of infection in different circumstances?)
3. What is the common name for these types of applications in research? (eg. temporal-spatial disease modelling? etc.)

## 2 Answers

Alright, first, I actually think your question is more suited for the new "Computational Science" site that should be coming into Public Beta through Area 51 soon. There are a number of issues behind the modeling of infectious diseases that are not really within the scope of a statistical analysis site.

Answering your questions in order:

1. "Easy to interpret probabilities" is somewhat vague - probabilities of what? The disease moving to that region? Likelihood of being infected, as expressed through something like the final prevalence of disease?

There are some common ways to model this problem however. The first is an extension of the classic SIR type model, which are commonly known as "meta-population" models. Essentially, rather than a single set of SIR equations, you have a series of them, one for each region, with parameters in the model governing the interaction between the populations (in your case, map regions). The can be deterministic, at which point very little math is needed, or stochastic, which produces nice distributions of results for further statistical analysis.

Another, less "classic" way as mentioned by @Spacedman, is to use an agent-based model to track individuals. This kind of model is somewhat more difficult to implement, but has the advantage of producing individual level data that can be analyzed using more conventional statistical techniques.

You could also simulate this by representing the map as a set of nodes in a network, and modeling the spread of disease over that network using something like a percolation model.

As you can see, approaches to your problem abound.

2. In terms of specifics, it depends very much on your disease system, what you're trying to model, and what type of assumptions you're willing to settle for. Along with how much either programming or mathematical complexity you're willing to tolerate. It depends so much that this question is essentially unanswerable, on the scale of "What variables should I put in a regression model".

The answer is likely: A fair number. By way of example, even for a model that doesn't have geographic spread, but could, one model I'm working on has roughly 23 parameters. The agent based version has more. My best advice is to consult an expert.

3. Spatio-temporal models, meta-population models, spatially discrete models, disease percolation models...there are tons of names. I'd say meta-population models are probably the one which will yield a number of example models the most swiftly, but there are lots of different names. One way to search is also to search for models of the disease you're interested in, to see if there's an approach you could replicate.

• @chl I saw the private beta, just haven't had a chance to log on. Long weekend >.> Dec 5, 2011 at 21:28

Sounds like you want to do "Agent-based modelling" - if you are computing with the behaviour of individuals. [Personally I prefer the term 'simulation' to 'modelling' when it's not what a statistician would call a 'model'...]

I suspect the simplest example is just to have a finite time-step, and rates of transfer between zones, and then pick which people move, they have infection probabilities and everyone has a susceptibility probability. Work out who catches the disease in the next time step.

What you then get is a non-deterministic (because you have some random number generator) simulation that you can run hundreds of times to get simulation uncertainties on the spread of your disease. I'm not sure how easy it is to compare these kinds of models with reality though...