Stability and Hopf bifurcation for an epidemic disease model with delay

A predator–prey system with disease in the prey is considered. The stability of the positive equilibrium and the existence of Hopf bifurcation with delay τ in the term of degree 2 is investigated, where τ is regarded as a parameter. It is found that there are stability switches, and Hopf bifurcations occur when the delay τ passes through a sequence of critical values. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions is derived.