An SI epidemic model for a vertically as well as horizontally transmitted disease is investigated when the fertility, natural mortality, and disease-induced mortality rates depend on age and the force of infection corresponds to a special form of intercohort transmission called proportionate mixing. We determine the steady states and obtain explicitly computable threshold conditions, and then perform stability analysis.
M. El-Doma. "Analysis of an age-dependent SI epidemic model with disease-induced mortality and proportionate mixing assumption: the case of vertically transmitted diseases." J. Appl. Math. 2004 (3) 235 - 254, 2 August 2004. https://doi.org/10.1155/S1110757X0430118X