  Decline of immunity is a phenomenon characterized by immunocompromised host
and plays a crucial role in the epidemiology of emerging infectious diseases
(EIDs) such as COVID-19. In this paper, we propose an age-structured model with
vaccination and reinfection of immune individuals. We prove that the
disease-free equilibrium of the model undergoes backward and forward
transcritical bifurcations at the critical value of the basic reproduction
number for different values of parameters. We illustrate the results by
numerical computations, and also find that the endemic equilibrium exhibits a
saddle-node bifurcation on the extended branch of the forward transcritical
bifurcation. These results allow us to understand the interplay between the
decline of immunity and EIDs, and are able to provide strategies for mitigating
the impact of EIDs on global health.
