  We analyse an extended SIRS epidemic model in which immunity at the
individual level wanes gradually at exponential rate, but where the waning rate
may differ between individuals, for instance as an effect of differences in
immune systems. The model also includes vaccination schemes aimed to reach and
maintain herd immunity. We consider both the informed situation where the
individual waning parameters are known, thus allowing selection of vaccinees
being based on both time since last vaccination as well as on the individual
waning rate, and the more likely uninformed situation where individual waning
parameters are unobserved, thus only allowing vaccination schemes to depend on
time since last vaccination. The optimal vaccination policies for both the
informed and uniformed heterogeneous situation are derived and compared with
the homogeneous waning model (meaning all individuals have the same immunity
waning rate), as well as to the classic SIRS model where immunity at the
individual level drops from complete immunity to complete susceptibility in one
leap. It is shown that the classic SIRS model requires least vaccines, followed
by the SIRS with homogeneous gradual waning, followed by the informed situation
for the model with heterogeneous gradual waning. The situation requiring most
vaccines for herd immunity is the most likely scenario, that immunity wanes
gradually with unobserved individual heterogeneity. For parameter values chosen
to mimic COVID-19 and assuming perfect initial immunity and cumulative immunity
of 12 months, the classic homogeneous SIRS epidemic suggests that vaccinating
individuals every 15 months is sufficient to reach and maintain herd immunity,
whereas the uninformed case for exponential waning with rate heterogeneity
corresponding to a coefficient of variation being 0.5, requires that
individuals instead need to be vaccinated every 4.4 months.
