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.

Diffusion occupation time before exiting

Using the approach of D. Landriault et al. and B. Li and X. Zhou, for a one-dimensional time-homogeneous diffusion process X and constants c 〈 a 〈 b 〈 d, we find expressions of double Laplace transforms of the form Ex[e--θTd--λ∫o Td1a 〈Xs〈b ds; Td 〈 Tc], where Tx denotes the first passage time of level x. As applications, we find explicit Laplace transforms of the corresponding occupation time and occupation density for the Brownian motion with two-valued drift and that of occupation time for the skew Ornstein- Uhlenbeck process, respectively. Some known results are also recovered.

Exit identities for diusion processes observed at Poisson arrival times

For diffusion processes,we extend various two-sided exit identities to the situation when the process is only observed at arrival times of an independent Poisson process.The results are expressed in terms of solutions to the differential equations associated with the diffusions generators.