  We study the stochastic susceptible-infected-susceptible model of epidemic
processes on finite directed and weighted networks with arbitrary structure. We
present a new lower bound on the exponential rate at which the probabilities of
nodes being infected decay over time. This bound is directly related to the
leading eigenvalue of a matrix that depends on the non-backtracking and
incidence matrices of the network. The dimension of this matrix is N+M, where N
and M are the number of nodes and edges, respectively. We show that this new
lower bound improves on an existing bound corresponding to the so-called
quenched mean-field theory. Although the bound obtained from a recently
developed second-order moment-closure technique requires the computation of the
leading eigenvalue of an N^2 x N^2 matrix, we illustrate in our numerical
simulations that the new bound is tighter, while being computationally less
expensive for sparse networks. We also present the expression for the
corresponding epidemic threshold in terms of the adjacency matrix of the line
graph and the non-backtracking matrix of the given network.
