Backward Bifurcation for Pulse Vaccination

Abstract

In this talk we investigate an SIVS epidemic model with imperfect vaccine,thus vaccinated individuals can also contract the infection. We considerpulse vaccination, that means we vaccinate a large fraction of populationat xed time intervals.It is known that in some vaccination models, backward bifurcation occursand multiple subthreshold endemic equilibria exist, thus the behaviourof solution depends on the initial value as well [1]. We know that pulsevaccination can be more eective than constant vaccination, thus it is aninteresting question to study whether backward bifurcation can arise in apulse vaccination model.We proved that backward bifurcation can occur in pulse vaccination.First we found the disease-free periodic solution, which is locally asymptoticallystable in the whole phase space if the control reproduction numberless than one; if the control reproduction number greater than one, thenthe infection is strongly uniformly persistent in the population.We performed the complete bifurcation analysis of a xed point equation,where the most important tool was the Lyapunov-Schmidt methodand we obtained a sucient and necessary condition for the existence ofsubcritical bifurcation.