Hopf Bifurcation in a Delayed Epidemic Model with Vaccination

In this study, we construct a mathematical epidemic model with the vaccinated compartment as an infinite dimensional system with time delay. We assume that the rate at which susceptible individuals are vaccinated depends on the history of the infectious population. We obtain a sufficient condition for the Hopf bifurcation, which induces a nontrivial periodic solution. Such a periodic solution could explain the recurrent epidemic waves that have been observed in many infectious diseases.