Seasonality and mixed vaccination strategy in an epidemic model with vertical transmission

First vaccination (vaccinate at birth) and pulse vaccination are two methods used to control the spread of diseases as well as the elimination of them. Owing to the seasonal fluctuations in transmission of many diseases, we propose an impulsive SIRS epidemic model with periodic saturation incidence and vertical transmission. The effects of periodic varying contact rate and mixed vaccination strategy on eradication of infectious diseases are studied. A threshold for a disease to be extinct or endemic is established. Our results imply that the diseases will die out eventually if the basic reproduction number is less than unity, whereas the diseases will persist if the basic reproduction number is larger than unity. Finally, numerical simulations support our analytical conclusions.