  We consider an epidemiological SIR model with an infection rate depending on
the recovered population. We establish sufficient conditions for existence,
uniqueness, and stability (local and global) of endemic equilibria and consider
also the stability of the disease-free equilibrium. We show that, in contrast
with classical SIR models, a system with a recovery-dependent infection rate
can have multiple endemic stable equilibria (multistability) and multiple
stable and unstable saddle points of equilibria. We establish conditions for
the occurrence of these phenomena and illustrate the results with some
examples.
