Hopf bifurcation of a delayed SIQR epidemic model with constant input and nonlinear incidence rate

An SIQR epidemic model with nonlinear incidence rate and two delays is studied under the assumption that a susceptible of the host population has a constant input. Local stability and existence of Hopf bifurcation are analyzed by regarding combination of the time delay due to the latent period of disease and the time delay due to the period that the infective and quarantined individuals need to be cured as the bifurcation parameter. Furthermore, the properties of the Hopf bifurcation are determined by using the normal form method and center manifold theory. Some numerical simulations are also carried out in order to verify our theoretical findings.