The Stability Analysis Of SIR Epidemic Models With Birth Pulse

In traditional mathematical epidemic models, we always assume that the species reproduce throughout the year. Whereas it is often the case that births are seasonal or occur in regular pulse. Thus the continuous reproduction of popu-lation is then removed from the traditional models and replaced with a birth pulse. Vaccination is used to improve the individual’s immunity against infectious dis-ease. Considering the economic factors, we often use pulse vaccination strategy. In chapter two, an SIR model with the bilinear incidence rate and without verti-cal infection is developed. And we assume that pulse vaccination and birth pulse happen at different time. By using the stroboscopic map and Floquet theory, we prove the existence and local stability of the disease-free periodic solution. Then we analysis the best time of pulse vaccination. In chapter three, an SIR model with the bilinear incidence rate is developed, where we assume that birth pulse and pulse vaccination happen at the same time. We prove the existence and lo-cal stability of the disease-free periodic solution. At last the bifurcation of the nontrivial periodic solution is discussed. In chapter four, an SIR model with birth pulse and continuous vaccination is developed at first. Next we build an SIR mod-el with birth pulse and no vaccination. Then we make a comparison of birth pulse and continuous birth on the effect of infectious disease. At last, much numerical simulation is made to prove our theories.