Study on the threshold of a stochastic SIR epidemic model and its extensions

• This paper provides a simple but effective method for estimating the threshold.
• The threshold is identified for any size of the noises, without constraint: the noise is small.
• When the disease is persistent, the system is proved to be convergent in time mean.

This paper provides a simple but effective method for estimating the threshold of a class of the stochastic epidemic models by use of the nonnegative semimartingale convergence theorem. Firstly, the threshold R0SIR is obtained for the stochastic SIR model with a saturated incidence rate, whose value is below 1 or above 1 will completely determine the disease to go extinct or prevail for any size of the white noise. Besides, when R0SIR>1, the system is proved to be convergent in time mean. Then, the threshold of the stochastic SIVS models with or without saturated incidence rate are also established by the same method. Comparing with the previously-known literatures, the related results are improved, and the method is simpler than before.