| Traditional mathematical models have invariably assumed that the species reproducethroughout the year, whereas it is often the case that births are seasonal or occur in regularpulses. Thus the continuous reproduction of population is then removed from the traditionalmodels and replaced with a birth pulse. In this paper, we proposed three epidemic modelswith birth pulses.In chapter 2, an SIR model with the bilinear incidence rate and constant recruitmentis developed. Furthermore, we assume that the infectious have a lower fertility than thesusceptible and recovered due to the e?ect of the disease. So a distinguishing feature ofthe model considered here is that the susceptible, infectious and recovered have di?erentbirth rate, which makes the model is more realistic. Using the discrete dynamical systemdetermined by stroboscopic map, we obtain the exact cycle of system in the absence of theinfection. Also we proved the local and global stability of the period infection-free solution byapplying the Floquet theory and impulsive di?erential inequality. Last section, we comparethe e?ect of the constant and pulse birth to eliminate the disease. When the pulse intervalis less than the critical value Tc, the pulse birth is easier than the constant one to eliminatethe disease.In chapter 3, we mainly consider an SIRS infectious model with the nonlinear incidencerate and a general periodic vaccination to the susceptible. Similarly we obtain the existenceand local stability of periodic infection-free solution. The basic reproduction number Rf0 isdefined. Using the compared theory we prove the periodic infection-free solution is globallystable when Rf0 < 1. Furthermore, we find that the disease is uniformly weak-persistentwhen Rf0 > 1.In chapter 4, an SISV model with standard incidence and vertical transmission is con-sidered. By the compared theory we get the global stability of the periodic infection-freesolution and prove the disease is uniformly persistent when the threshold Rv0 > 1. |
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