  We derive an analytical expression for the critical infection rate r_c of the
susceptible-infectious-susceptible (SIS) disease spreading model on random
networks. To obtain r_c, we first calculate the probability of reinfection, pi,
defined as the probability of a node to reinfect the node that had earlier
infected it. We then derive r_c from pi using percolation theory. We show that
pi is governed by two effects: (i) The requirement from an infecting node to
recover prior to its reinfection, which depends on the disease spreading
parameters; and (ii) The competition between nodes that simultaneously try to
reinfect the same ancestor, which depends on the network topology.

Disordered Systems and Neural Networks;Data Analysis, Statistics and Probability