Dynamics of an SLIR model with nonmonotone incidence rate and stochastic perturbation.

In this paper we study an SLIR epidemic model with nonmonotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectives is getting larger. By carrying out a global analysis of the model and studying the stability of the disease-free equilibrium and the endemic equilibrium, we show that either the number of infective individuals tends to zero or the disease persists as time evolves. For the stochastic model, we prove the existence, uniqueness and positivity of the solution of the model. Then, we investigate the stability of the model and we prove that the infective tends asymptotically to zero exponentially almost surely as R0 < 1. We also proved that the SLIR model has the ergodic property as the fluctuation is small, where the positive solution converges weakly to the unique stationary distribution.