Finding patient zero in susceptible-infectious-susceptible epidemic processes

Finding the source of an epidemic is important, because correct source identification can help to stop a budding epidemic or prevent new ones. We investigate the backward equations of the N-intertwined mean-field approximation susceptible-infectious-susceptible (SIS) process. The backward equations allow us to trace the epidemic back to its source on networks of sizes up to at least N=1500. Additionally, we show that the source of the "more realistic" Markovian SIS model cannot feasibly be found, even in a "best-case scenario," where the infinitesimal generator Q, which completely describes the epidemic process and the underlying contact network, is known. The Markovian initial condition s(0), which reveals the epidemic source, can be found analytically when the viral state vector s(t) is known at some time t as s(0)=s(t)e^{-Qt}. However, s(0) can hardly be computed, except for small times t. The numerical errors are largely due to the matrix exponential e^{-Qt}, which is severely ill-behaved.