Discrete-time models of infectious diseases: a project in memory of Aziz-Abdul Yakubu
This talk is dedicated to the late Aziz-Abdul Yakubu who passed away suddenly and unexpectedly last August, 2022. To honor Aziz’s memory, I will briefly discuss a project on which he and I were working at the time of his death, naming a chapter on infectious disease models to appear in a book on discrete-time models in population dynamics co-authored with Saber Elaydi. In this chapter we provide a general framework for the formulation of such models and give some general theorems concerning disease-free equilibria and the fundamental bifurcation of endemic equilibria that occurs upon the destabilization of a disease-free equilibrium. This basic bifurcation theorem will be related to the reproduction number R₀, for which we provide a general definition. We also give an extension of this fundamental bifurcation phenomenon for periodic disease-free and endemic cycles. I will illustrate the application of the theorems using a few specific disease models.