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 R0R0 is derived. Moreover, if R0⩽1R0⩽1, the disease-free equilibrium is globally asymptotically stable and if R0>1R0>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 R0R0 is presented. In addition, if the intensities of the white noises are sufficiently small and R0>1R0>1, then there exists a unique stationary distribution to stochastic model. Numerical simulations are also carried out to confirm the analytical results.