Epidemic processes on digraphs of random mappings

A random mapping ( T n ; q ) of a finite set V , V = {1,2,…, n }, into itself assigns independently to each i ∊ V its unique image j ∊ V with probability q if i = j and with probability P = (1- q )/( n −1) if i ≠ j . Three versions of epidemic processes on a random digraph G T representing ( T n ; q ) are studied. The exact probability distributions of the total number of infected elements as well as the threshold functions for these epidemic processes are determined.