Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination

A new SEIRS epidemic disease model with two profitless delays and nonlinear incidence are proposed, and the dynamic behaviors of the model under pulse vaccination are analyzed. Using the discrete dynamical system determined by the stroboscopic map, we show that there exists a ‘infection-free’ periodic solutions, further, we show that the ‘infection-free’ periodic solution is globally attractive when the period of impulsive effect is less than some critical value. Using a new modeling method, we obtain sufficient condition for the permanence of the epidemic model with pulse vaccination. We show that time delays, pulse vaccination and nonlinear incidence can bring different effects on the dynamic behaviors of the model by numerical analysis. Our results also show the delays are “profitless”. In this paper, the main feature is to introduce two discrete time delays and impulse into SEIRS epidemic model and to give pulse vaccination strategies.