Extending SIRS epidemics to allow for gradual waning of immunity

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.