A shape theorem for the spread of an infection
Abstract
We consider the following interacting particle system: There is a ``gas'' of particles, each of which performs a continuous time simple random walk on the ddimensional lattice. These particles are called Aparticles and move independently of each other. We assume that we start the system with a Poisson number of particles at each lattice site x, with the number of particles at different x's i.i.d. In addition, there are a finite number of Bparticles which perform the same continuous time simple random walks as the Aparticles. A and Bparticles are interpreted as individuals who are healthy or infected, respectively. The Bparticles move independently of each other. The only interaction is that when a Bparticle and an Aparticle coincide, the latter instantaneously turns into a Bparticle. Let B(t) be the set of sites visited by a Bparticle during [0,t]. We show that B(t) grows linearly in time and has an asymptotic shape; more precisely, there exists a nonrandom convex, compact set B_0 such that almost surely, for all 0 < a <1, (1a)tB_0 is contained in B(t) and B(t) is contained in (1+a)tB_0 eventually.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2003
 arXiv:
 arXiv:math/0312511
 Bibcode:
 2003math.....12511K
 Keywords:

 Probability;
 60K35 (Primary);
 60J15 (Secondary)
 EPrint:
 59 pages in AMSTex format plus 1 figure in eps format