Exit problem of stochastic SIR model with limited medical resource

Nonlinearity and randomness are both the essential attributes for the real world, and the case is the same for the models of infectious diseases, for which the deterministic models can not give a complete picture of the evolution. However, although there has been a lot of work on stochastic epidemic models, most of them focus mainly on qualitative properties, which makes us somewhat ignore the original meaning of the parameter value. In this paper we extend the classic susceptible-infectious-removed (SIR) epidemic model by adding a white noise excitation and then we utilize the large deviation theory to quantitatively study the long-term coexistence exit problem with epidemic. Finally, in order to extend the meaning of parameters in the corresponding deterministic system, we tentatively introduce two new thresholds which then prove rational.