Dynamics of the deterministic and stochastic SIQS epidemic model with non-linear incidence

The deterministic and stochastic SIQS models with non-linear incidence are introduced. For deterministic model, the basic reproductive rate R-0 is derived. Moreover, if R-0 <= 1, the disease-free equilibrium is globally asymptotically stable and if R-0 > 1, there exists a unique endemic equilibrium which is globally asymptotically stable. For stochastic model, sufficient condition for extinction of the disease that is regardless of the value of R-0 is presented. In addition, if the intensities of the white noises are sufficiently small and R-0 > 1, then there exists a unique stationary distribution to stochastic model. Numerical simulations are also carried out to confirm the analytical results. (C) 2014 Elsevier Inc. All rights reserved.