  We model the spread of a SIS infection on Small World and random networks
using weighted graphs. The entry $w_{ij}$ in the weight matrix W holds
information about the transmission probability along the edge joining node
$v_i$ and node $v_j$. We use the analogy between the spread of a disease on a
network and a random walk performed on this network to derive a master equation
describing the dynamics of the process. We find conditions under which an
epidemic does not break out and investigate numerically the effect of a
non-symmetric weight distribution of the initially infected individual on the
dynamics of the disease spread.
