  As the outbreak of COVID-19 enters its third year, we have now enough data to
analyse the behavior of the pandemic with mathematical models over a long
period of time. The pandemic alternates periods of high and low infections, in
a way that sheds a light on the nature of mathematical model that can be used
for reliable predictions. The main hypothesis of the model presented here is
that the oscillatory behavior is a structural feature of the outbreak, even
without postulating a time-dependence of the coefficients. As such, it should
be reflected by the presence of limit cycles as asymptotic solutions. This
stems from the introduction of (i) a non-linear waning immunity based on the
concept of immunity booster (already used for other pathologies); (ii) a fine
description of the compartments with a discrimination between individuals
infected/vaccinated for the first time, and individuals already
infected/vaccinated, undergoing to new infections/doses. We provide a
proof-of-concept that our novel model is capable of reproducing long-term
oscillatory behavior of many infectious diseases, and, in particular, the
periodic nature of the waves of infection. Periodic solutions are inherent to
the model, and achieved without changing parameter values in time. This may
represent an important step in the long-term modeling of COVID-19 and similar
diseases, as the natural, unforced behavior of the solution shows the
qualitative characteristics observed during the COVID-19 pandemic.
