  Accurate identification of effective epidemic threshold is essential for
understanding epidemic dynamics on complex networks. The existing studies on
the effective epidemic threshold of the susceptible-infected-removed (SIR)
model generally assume that all infected nodes immediately recover after the
infection process, which more or less does not conform to the realistic
situation of disease. In this paper, we systematically study the effect of
arbitrary recovery rate on the SIR spreading dynamics on complex networks. We
derive the theoretical effective epidemic threshold and final outbreak size
based on the edge-based compartmental theory. To validate the proposed
theoretical predictions, extensive numerical experiments are implemented by
using asynchronous and synchronous updating methods. When asynchronous updating
method is used in simulations, recovery rate does not affect the final state of
spreading dynamics. But with synchronous updating, we find that the effective
epidemic threshold decreases with recovery rate, and final outbreak size
increases with recovery rate. A good agreement between the theoretical
predictions and numerical results are observed on both synthetic and real-world
networks. Our results extend the existing theoretical studies, and help us to
understand the phase transition with arbitrary recovery rate.
