Modeling of Asymptotically Periodic Outbreaks: a long-term SIRW2 description of COVID-19?

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.