The article investigates a stochastic SIRS epidemic model by perturbing the natural death rate. A stochastic threshold $R_s$ that determines the dynamics of the model is given. More concretely, if $R_s<1$, then the disease will die out; if $R_s>1$, then the model exhibits an ergodic stationary distribution. Furthermore, a Gaussian approximation of the stationary distribution is given using the stochastic sensitivity technique. For the visual description of the distribution, a confidence ellipsoid is presented by the visualization geometric method of confidence domains. The confidence ellipsoid is helpful for us to estimate the equilibrium region of the stochastic model. Moreover, several numerical analyses are presented to verify the results.