In this paper, a generalized stochastic SIR epidemic model with vaccination rules is presented and the threshold behavior of the proposed epidemic model is investigated. Firstly, the stability of the equilibrium of the deterministic system is considered and the corresponding conditions are obtained. Secondly, the threshold of a stochastic SIR system for the extinction and the permanence in mean of epidemic disease are investigated. The results show that a larger stochastic disturbance can cause infections diseases to go to extinction. However, for a relatively small stochastic disturbance, the evolutionary dynamics of the epidemic diseases are overwhelmingly depend on the incidence function. This implies that the stochastic disturbance and the incidence function play an important role in diseases control. To test the theoretical results, a series of numerical simulations of these cases with respect to different noise disturbance coefficients are conducted.

<abstract><p>In this paper, a generalized delay-induced $ SIRS $ epidemic model with nonlinear incidence rate, latency and relapse is proposed. Our epidemic model is a generalized one, and the published epidemic models are the special cases of ours under some conditions. By using LaSalle's invariance principle and Lyapunovi's direct method, the dynamical behaviors are investigated and the results show that the disease free-equilibrium $ Q_0 $ is globally asymptotically stable if the basic reproduction number $ R_0 &lt; 1 $ for any time delay. However, if the basic reproduction number $ R_0 &gt; 1 $, there exists a unique endemic equilibrium $ Q_* $ which is locally asymptotically stable under some conditions. Moreover, the effects of latency and relapse on the transmission dynamics of the diseases are analyzed by some numerical experiments which conducted based on $ ODE45 $ in Matlab.</p></abstract>