Nonlinear Oscillations in Epidemic Models

Constant-parameter epidemic models for a closed population (described by integral and integrodifferential equations) are analyzed to determine whether time delays can give rise to periodic oscillations. Delays can destabilize the steady state in one finite delay case and in one infinite delay case considered. Hopf bifurcation techniques are used to show the existence of locally asymptotically stable periodic solutions for certain parameter values. Inclusion of time delays in another related model does not alter the local asymptotic stability of the endemic steady state.