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.