Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates

In this paper, two delayed SEIR epidemic models with continuous and impulsive vaccination and saturating incidence are investigated. The dynamical behaviors of the disease are analyzed. For continuous vaccination, we obtain a basic reproductive number R1R1 and prove that if R1≤1R1≤1 then the disease-free equilibrium is globally attractive and if R1>1R1>1 then the disease is permanent by using the Lyapunov functional method. For impulsive vaccination, we obtain two thresholds R∗R∗ and R∗R∗ and prove that if R∗<1R∗<1 then the disease-free periodic solution is globally attractive and if R∗>1R∗>1 then the disease is permanent by using the comparison theorem of impulsive differential equation and the Lyapunov functional method. Lastly, we compared the effects of two vaccination strategies.