Dynamics of a kind of stochastic SIRS models with two different nonlinear incidences

A new kind of stochastic SIRS models with two different nonlinear incidences are extended. The obtained results can be expressed in two dimensions. In mathematics, the threshold values R1s and R2s which ensure permanent or extinct disease are presented, respectively. More concretely, when Ris>1(i=1,2), the two diseases are persistence in mean. When Ris<1 or Ris>1(i=1,2), the two diseases will either be extinct or be permanent, respectively. What's more interesting is the numerical results which show that the two diseases go to extinction at a large time, when R1s=1 and R2s=1. Furthermore, the sufficient conditions for the diseases that are extinct and permanent are given. In addition, the stochastic boundedness of the model is investigated. Epidemiologically, we can conclude that large noise fluctuations can effectively prevent the outbreak of disease, which can supply us beneficial methods to control the spreading of two diseases. Finally, a few numerical simulations are presented to demonstrate and verify the findings obtained.