Recurrent epidemic waves in a delayed epidemic model with quarantine

In this paper, we are concerned with an epidemic model with quarantine and distributed time delay. We define the basic reproduction number $$ and show that if $$, then the disease-free equilibrium is globally asymptotically stable, whereas if $$, then it is unstable and there exists a unique endemic equilibrium. We obtain sufficient conditions for a Hopf bifurcation that induces a nontrivial periodic solution which represents recurrent epidemic waves. By numerical simulations, we illustrate stability and instability parameter regions. Our results suggest that the quarantine and time delay play important roles in the occurrence of recurrent epidemic waves.