Abstract: Firstly,  the basic reproduction number R₀ of the SIVS (Susceptible-Infectious-Immune-Susceptible) epidemic model is solved by the method of next generation matrix, through the threshold, the disease-free equilibrium point always exists, and the endemic equilibrium point only exists when R₀>1, furthermore, the conditions of extinction and persistence of the disease are determined. Secondly, the stability and the bifurcation situations of the model at the equilibrium point are proved by the properties of Jacobian matrix, Jury criterions and the construction of Lyapunov function. The results show that when R₀<1, the disease-free equilibrium point is globally asymptotically stable, and the transcritical bifurcation occurs when R₀=1. When R₀>1, the endemic equilibrium point is locally asymptotically stable, if the limitation on contact rate β in reality is ignored, the model will produce the period-doubling bifurcation and even chaotic phenomena at the endemic equilibrium point. Finally, numerical simulation and sensitivity index method are used to verify the  theoretical analysis results, it is concluded that improving the vaccination rate and recovery rate can effectively reduce the incidence of the disease.

Key words: discrete model, SIVS model, vaccination, stability, bifurcation