  We obtain conditions for eradication and permanence of infection for a
nonautonomous SIQR model with time-dependent parameters, that are not assumed
to be periodic. The incidence is given by functions of all compartments and the
threshold conditions are given by some numbers that play the role of the basic
reproduction number. We obtain simple threshold conditions in the autonomous,
asymptotically autonomous and periodic settings and show that our thresholds
coincide with the ones already established. Additionally, we obtain threshold
conditions for the general nonautonomous model with mass-action, standard and
quarantine-adjusted incidence.
