Hi Gowers,

In this paper Sidoravicius, V. and  Kesten, H . A shape theorem for the spread of an infection. Annals of Mathematics, v. 167, p. 1-63, 2008.

the authors consider a different model, but they were interested in the problem shape like you. One time Vladas told me that about a model similar what you describe and also that there is a believe about universality for those models, perhaps they discuss this on this reference.

This is a link for the arxiv version of this
[paper.][1]
 


  [1]: http://arxiv.org/pdf/math/0312511v1