In this paper, the dynamical behavior and probability density function for a stochastic epidemic model with incomplete and temporal immunization and nonlinear incidence are investigated. Firstly, for the corresponding deterministic model, we show that the global stability of disease-free equilibrium and the uniform persistence of positive solutions are fully determined by the basic regeneration number $R_{0}$ . Subsequently, for the stochastic model, a threshold value $R_{0}^{s}$ for stochastic extinction is first proposed. That is, the disease dies out with probability one if $R_{0}^{s}<1$ . Then, a new threshold value $\tilde{R}_{0}^{S}$ for the stochastic persistence and existence of stationary distribution is defined. Namely, the disease is persistent in the mean and any positive solution is ergodic and has a unique stationary distribution if $\tilde{R}_{0}^{S}>1$ . Particularly, a novel analysis technique is introduced in the proof of persistence. Furthermore, the approximate expression of the log-normal probability density function around the quasi-stationary state of the stochastic model is calculated by introducing a new threshold condition $\hat{R}_{0}^{s}>1$ . A new calculation method of density function is proposed. From the expressions of $R_{0}^{s}$ and $\tilde{R}_{0}^{S}$ as well as the main results obtained in this paper, we see that not only the white noises have very strong effects for the dynamical behavior of the model, but all other parameters also have very strong effects for the dynamics of the model. Particularly, direct at immunization we see that when the vaccination has the incomplete and temporal immunization, then it is protective effect and the role of controlling disease epidemics will be relatively weak. Therefore, increasing the all-right protective effect of vaccination is very important to control the epidemic of infectious diseases. Finally, the numerical examples and simulations are presented to validate the main results.
