We investigate the complex dynamics of an epidemic model with nonlinear incidence rate of saturated mass action which depends on the ratio of the number of infectious individuals to that of susceptible individuals. We first deal with the boundedness, dissipation, persistence, and the stability of the disease-free and endemic points of the deterministic model. And then we prove the existence and uniqueness of the global positive solutions, stochastic boundedness, and permanence for the stochastic epidemic model. Furthermore, we perform some numerical examples to validate the analytical findings. Needless to say, both deterministic and stochastic epidemic models have their important roles.