Mathematical analysis of an age-structured SIR epidemic model with vertical transmission

In this paper, we consider a mathematical model for the spread of a directly transmitted infectious disease in an age-structured population. We assume that infected population is recovered with permanent immunity or quarantined by an age-specific schedule, and the infective agent can be transmitted not only horizontally but also vertically from adult individuals to their newborns. For simplicity we assume that the demographic process of the host population is not affected by the spread of the disease, hence the host population is a demographic stable population. First we establish the mathematical well-posedness of the time evolution problem by using the semigroup approach. Next we prove that the basic reproduction ratio is given as the spectral radius of a positive operator, and an endemic steady state exists if and only if the basic reproduction ratio $R_0$ is greater than unity, while the disease-free steady state is globally asymptotically stable if $R_0 < 1$. We also show that the endemic steady states are forwardly bifurcated from the disease-free steady state when $R_0$ crosses the unity. Finally we examine the conditions for the local stability of the endemic steady states.