  SIRS epidemic models assume that individual immunity (from infection and
vaccination) wanes in one big leap, from complete immunity to complete
susceptibility. For many diseases immunity on the contrary wanes gradually,
something that's become even more evident during COVID-19 pandemic where also
recently infected have a reinfection risk, and where booster vaccines are given
to increase immunity. This paper considers an epidemic model allowing for such
gradual waning of immunity (either linear or exponential waning) thereby
extending SIRS epidemics, and also incorporates vaccination. The two versions
for gradual waning of immunity are compared with the classic SIRS epidemic,
where the three models are calibrated by having the same \emph{average
cumulative immunity}. All models are shown to have identical basic reproduction
number $R_0$. However, if no prevention is put in place, the exponential waning
model has highest prevalence and the classic SIRS model has lowest. Similarly,
the amount of vaccine supply needed to reach and maintain herd immunity is
highest for the model with exponential decay of immunity and lowest for the
classic SIRS model. consequently, if truth lies close to exponential (or
linear) decay of immunity, expressions based on the SIRS epidemic will
underestimate the endemic level and the critical vaccine supply will not be
sufficient to reach and maintain herd immunity. For parameter choices fitting
to COVID-19, the critical amount of vaccine supply is about 50% higher if
immunity wanes linearly, and more than 150% higher when immunity wanes
exponentially, as compared to the classic SIRS epidemic model.
