# How much universality is there for contact processes?

A couple of weeks ago I had to pick my daughter up from her nursery because of suspected chicken pox. It turned out to be a false alarm, but while I was waiting at the doctor's surgery to establish that, it occurred to me to wonder how it was known that the gestation period before spots appear is five days, and that somebody with the disease is infectious for up to nine days after getting it. Presumably the answer is boring: that there are enough situations where you can pinpoint when infection must have taken place to make it possible to determine the gestation and infection periods fairly accurately.

But suppose that were not the case, and instead you wanted to deduce the infection period from the global behaviour of the spread of the disease. A rough version of my question is whether this is possible.

Here's a more precise version (but this is just one model and I don't mind answers that apply to different but similar models). Let's define a contact process as follows. At any one time, each point in $\mathbb{Z}^2$ is either diseased or healthy. If you get the disease at time $t$, then there is some probability distribution $\mu$ on $[t,\infty]$ for your four immediate neighbours: their probability of catching the disease from you during the time interval $[a,b]$ is $\mu([a,b])$. I'm thinking of $\mu(\{\infty\})$ as the probability that your neighbour doesn't catch the disease from you. And let's say that if two of your neighbours are infected, then your chances of catching the disease from one of them is independent of your chances of catching it from the other. Finally, let's assume that $\mu$ is the same for everybody (apart from the translation by $t$ to take account of when a point becomes infected).

A simple example of a distribution $\mu$ would be half the uniform distribution on $[t,t+1]$ plus half a point mass at $\infty$. That would represent the situation where if you get the disease at time $t$ then your neighbour's chance of getting the disease from you is 1/2 and if your neighbour does get the disease from you then the time of infection is uniformly distributed over $[t,t+1]$.

With this set-up, my question is this: how much can you tell about the probability distribution $\mu$ from the global spread of the disease? That's still not a completely precise question, and I think I may lack the expertise to make it completely precise, but it's the usual picture that applies to models of this kind, where you look at everything from a great distance so that you can't see the small-scale structure (so, for example, I can't just empirically test lots of pairs of neighbours to build up a picture of the probability distribution).

A different way of asking the question, which explains the title, is this. It is a well-known and fascinating phenomenon that many probabilistic models like this have global behaviour that is very insensitive to the details of their local behaviour. So I would expect, for example, that all compactly supported probability distributions for which $\mu(\infty)$ is the same would have similar global behaviour: perhaps the only parameter that mattered would be the expected time to become infected, or something like that, which would govern how quickly the disease spread. (It's not obvious to me that that is the right parameter, by the way.) But perhaps I'm wrong about this. It might, for instance, be that if the time of infection is sharply concentrated in two places, then the disease spreads in two "waves", one faster and one slower. And if that's the case, then perhaps you can work out virtually the entire distribution from the global behaviour.

I'm asking this question out of idle curiosity and nothing more. I'd just be interested to know what is known (or at least believed to be true).

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Well, for any $\mu$ there is a convex limit shape the normalized diseased cluster tend to. It seems like you're asking whether the function taking $\mu$ to the limit shape is injective, or what can be said about it. I don't think much is known about this - there are more basic questions still unanswered (like whether this shape is a disc, if $\mu$ is exponential). –  Ori Gurel-Gurevich Oct 14 '11 at 17:01