  We analyze the dynamics of a population of independent random walkers on a
graph and develop a simple model of epidemic spreading. We assume that each
walker visits independently the nodes of a finite ergodic graph in a
discrete-time markovian walk governed by his specific transition matrix. With
this assumption, we first derive an upper bound for the reproduction numbers.
Then we assume that a walker is in one of the states: susceptible, infectious,
or recovered. An infectious walker remains infectious during a certain
characteristic time. If an infectious walker meets a susceptible one on the
same node there is a certain probability for the susceptible walker to get
infected. By implementing this hypothesis in computer simulations we study the
space-time evolution of the emerging infection patterns. Generally, random walk
approaches seem to have a large potential to study epidemic spreading and to
identify the pertinent parameters in epidemic dynamics.
