Global behavior of an age-structured epidemic model

The global behavior of the general $s \to i \to s$ age-structured epidemic model in a population of constant size is obtained. It is shown that there is a sharp threshold which determines the existence and global stability of an endemic state; hence, periodic solutions are ruled out. The threshold is identified as the spectral radius of a positive linear operator. The analysis employs the theory of semigroups and positive operator methods, and is based on the formulation of the problem as an abstract differential equation in a Banach space.