Spreading of infectious diseases on complex networks with non-symmetric   transmission probabilities
  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. 
Britta Daudert, Bai-Lian Li