# zbMATH — the first resource for mathematics

A shape theorem for the spread of an infection. (English) Zbl 1202.92077
This paper sharpens our former results in Ann. Probab. 33, No. 6, 2402–2462 (2005; Zbl 1111.60074), for the following model for the spread of a rumor or infection: There is a “gas” of so-called A-particles, each which performs a continuous time simple random walk on $$\mathbb{Z}^d$$, with jump rate $$D_A$$. The number of A-particles at $$x$$ just before the start, $$N_A(x,0-)$$, are mutually independent over $$x$$ and have a mean $$\mu_A$$ Poisson distribution. In addition there are B-particles which perform continuous time simple random walks with jump rate $$D_B$$. A finite number of B-particles are started in the system at time 0. The positions of these initial B-particles are arbitrary and non-random. The B-particles move independent of each other. The only interaction occurs when a B-particle and an A-particle coincide; the latter instantaneously turns into a B-particle.
The paper studies the growth of the set $$\bar{B}(t) = \{ x \in \mathbb{Z}^d : \text{a B-particle visits }x$$ during $$[0,t]$$

##### MSC:
 92D30 Epidemiology 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics 60G50 Sums of independent random variables; random walks
Full Text: