Pulse Effect Epidemic Model

The impulsive di?erential equations have obtained much attention from many authors,and deeply developed during the past a few years. It is widely applied in various domains suchas biological technology, medicine dynamics, physics, economy, population dynamics andepidemiology and so on. It is well-known that many natural phenomena and human activitiesdo exhibit impulsive e?ects in the fields of epidemiology. In this paper, we investigate someepidemic models with impulsive e?ects. In the chapter 1, 2, We give the stability of periodicinfection-free solution, the uniform persistence of the infectious disease, and the existence ofepidemic periodic solution. In the chapter 3, we study the stability of equilibrium of discretetime SI and SIS with vertical transmission.In chapter 2, we study SIR epidemic model with birth pulse, impulsive vaccination,impulsive recruitment and standard incidence. Using the stroboscopic map and Floquettheory , we obtain the existence of periodic infection-free solution and local asymptoticstability. Above this threshold, we prove global asymptotic stability of periodic infection-freesolution by comparison theorem and impulsive di?erence inequation .At the same time,theuniform persistence of the infectious disease is proved. At last, numerical simulation showsthat our proof is correct.In chapter 3, we consider SIR epidemic model with impulsive constant vaccination.We obtain the existence of periodic infection-free solution and local and global asymptoticstability. Further, the existence of a nontrivial periodic solution (endemic) is considered byusing bifurcation theory.In chapter 4, by using the theories of discrete dynamics, we study and deal with thestability of free and endemic equilibrium in discrete time SI and SIS with vertical transmis-sion. At the same time, when the force of infection arises from the Poisson distribution inthe discrete time SIS epidemic model, the system can exhibit bistability(alternative stableequilibria)over a wide range of parameter values.