Global dynamics behaviors for new delay SEIR epidemic disease model with vertical transmission and pulse vaccination

A robust SEIR epidemic disease model with a profitless delay and vertical transmission is formulated, and the dynamics behaviors of the model under pulse vaccination are analyzed. By use of the discrete dynamical system determined by the stroboscopic map, an ‘infection-free’ periodic solution is obtained, further, it is shown that the ‘infection-free’ periodic solution is globally attractive when some parameters of the model are under appropriate conditions. Using the theory on delay functional and impulsive differential equation, the sufficient condition with time delay for the permanence of the system is obtained, and it is proved that time delays, pulse vaccination and vertical transmission can bring obvious effects on the dynamics behaviors of the model. The results indicate that the delay is ‘profitless’. Project supported by the National Natural Science Foundation of China (No. 10471117)