We study the following model for the spread of a rumor or infection: There is a ``gas'' of so-called $A$-particles, each of which performs a continuous time simple random walk on $\Bbb Z^d$, with jumprate $D_A$. We assume that ``just before the start'' the number of $A$-particles at $x$, $N_A(x,0-)$, has a mean $\mu_A$ Poisson distribution and that the $N_A(x,0-), \, x \in \Bbb Z^d$, are independent.
In addition, there are $B$-particles which perform continuous time simple random walks with jumprate $D_B$. We start with a finite number of $B$-particles in the system at time 0. The positions of these initial $B$-particles are arbitrary, but they are non-random. The $B$-particles move independently of each other. The only interaction is that when a $B$-particle and an $A$-particle coincide, the latter instantaneously turns into a $B$-particle. \cite {KSb} gave some basic estimates for
the growth of the set $\wt B(t):= \{x \in \Bbb Z^d:$ a $B$-particle visits $x$ during $[0,t]$\}. In this article we show that if $D_A=D_B$, then $B(t) = \wt B(t) + [-\frac 12, \frac 12]^d$ grows linearly in time with an asymptotic shape, i.e., there
exists a non-random set $B_0$ such that $(1/t)B(t) \to B_0$, in a sensewhich will be made precise. Joint work with H. Kesten.
