  We investigate a diffusive, stage-structured epidemic model with the
maturation delay and freely-moving delay. Choosing delays and diffusive rates
as bifurcation parameters, the only possible way to destabilize the endemic
equilibrium is through Hopf bifurcation. The normal forms of Hopf bifurcations
on the center manifold are calculated, and explicit formulae determining the
criticality of bifurcations are derived. There are two different kinds of
stable oscillations near the first bifurcation: on one hand, we theoretically
prove that when the diffusion rate of infected immature individuals is
sufficiently small or sufficiently large, the first branch of Hopf bifurcating
solutions is always spatially homogeneous; on the other, fixing this diffusion
rate at an appropriate size, stable oscillations with different spatial
profiles are observed, and the conditions to guarantee the existence of such
solutions are given. These phenomena are investigated by calculating the
corresponding eigenfunction of the Laplacian at the first Hopf bifurcation
point.
