In this paper, an epidemic dynamical model with vaccination is proposed. Vaccination of both newborn and susceptible is included in the present model. The impact of the vaccination strategy with the vaccine efficacy is explored. In particular, the model exhibits backward bifurcations under the vaccination level, and bistability occurrence can be observed. Mathematically, a bifurcation analysis is performed, and the conditions ensuring that the system exhibits backward bifurcation are provided. The global dynamics of the equilibrium in the model are also investigated. Numerical simulations are also conducted to confirm and extend the analytic results.
(Liming Cai, Zhaoqing Li, Xinyu Song, Global analysis of an epidemic model with vaccination, Journal of Applied Mathematics and Computing, 2018(57), 605-628, 2018. )