We consider a nonautonomous eco-epidemiological model with general
functions for predation on infected and uninfected preys as well as
general functions associated to the vital dynamics of the susceptible
prey and predator populations. We obtain persistence and extinction
results for the infected prey based on assumptions on auxiliary systems
constructed from the disease-free system. We moreover consider an
iterative process that can improve the extinction results. We apply our
results to general eco-epidemiological models that include several
examples existent in the literature.