Homogeneous Epidemic Systems in the Stable Population∗

In this paper, we develop a new approach to deal with asymptotic behavior of the age-structured homogeneous epidemic systems. For homogeneous systems, there is no attracting nontrivial equilibrium, instead we have to examine existence and stability of persistent solutions. Assuming that the host population dynamics is described by the stable population model, we rewrite the basic system into a system of ratio age distribution, which is the age profile divided by the stable age profile. If the host population has the stable age profile, the ratio age distribution system is reduced to a normalized system. Then we prove a linearized stability principle that the local stability [instability] of steady states of the normalized system implies the orbital stability [instability] of corresponding persistent solutions of the original homogeneous system.