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

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) over tilde (0). That is, if (R) over tilde (0) < 1 and an additional condition holds then disease dies out, and if <(R)over tilde>(0) > 1 then disease is weak permanent with probability one. To obtain the permanence in the mean of the disease, a new quantity (R) over cap (0) > 1 is introduced, and it is proved that if (R) over cap (0) > 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. (C) 2016 Elsevier B.V. All rights reserved.