Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate

•Stochastic SIS model with nonlinear incidence rate is established by perturbing transmission coefficient.•The threshold value R˜0 is obtained to determine extinction and weak permanence of the disease in probability.•The sufficient condition on permanence in the mean of the disease with probability one is showed.•Numerical simulations are presented to illustrate some open problems.

In this paper, a class of stochastic SIS epidemic models with nonlinear incidence rate is investigated. It is shown that the extinction and persistence of the disease in probability are determined by a threshold value R˜0. That is, if R˜0<1 and an additional condition holds then disease dies out, and if R˜0>1 then disease is weak permanent with probability one. To obtain the permanence in the mean of the disease, a new quantity R̂0 is introduced, and it is proved that if R̂0>1 the disease is permanent in the mean with probability one. Furthermore, the numerical simulations are presented to illustrate some open problems given in Remarks 1–3 and 5 of this paper.