  We study the dynamics of information (or virus) dissemination by $m$ mobile
agents performing independent random walks on an $n$-node grid. We formulate
our results in terms of two scenarios: broadcasting and gossiping. In the
broadcasting scenario, the mobile agents are initially placed uniformly at
random among the grid nodes. At time 0, one agent is informed of a rumor and
starts a random walk. When an informed agent meets an uninformed agent, the
latter becomes informed and starts a new random walk. We study the broadcasting
time of the system, that is, the time it takes for all agents to know the
rumor. In the gossiping scenario, each agent is given a distinct rumor at time
0 and all agents start random walks. When two agents meet, they share all
rumors they are aware of. We study the gossiping time of the system, that is,
the time it takes for all agents to know all rumors. We prove that both the
broadcasting and the gossiping times are $\tilde\Theta(n/\sqrt{m})$ w.h.p.,
thus achieving a tight characterization up to logarithmic factors. Previous
results for the grid provided bounds which were weaker and only concerned
average times. In the context of virus infection, a corollary of our results is
that static and dynamically moving agents are infected at about the same speed.
