Dynamics of a delayed SEIQ epidemic model

In this work we consider an epidemic model that contains four species susceptible, exposed, infected and quarantined. With this model, first we find a feasible region which is invariant and where the solutions of our model are positive. Then the persistence of the model and sufficient conditions associated with extinction of infection population are discussed. To show that the system is locally asymptotically stable, a Lyapunov functional is constructed. After that, taking the delay as the key parameter, the conditions for local stability and Hopf bifurcation are derived. Further, we estimate the properties for the direction of the Hopf bifurcation and stability of the periodic solutions. Finally, some numerical simulations are presented to support our analytical results.