A minimal model of boosting and waning iin a recurrent seasonal epidemic

We propose a model of the immunity to a cyclical epidemic disease taking account not only of seasonal boosts during the infectious season, but also of residual immunity remaining from one season to the next. The focus is on the exponential waning process over successive cycles, imposed on the temporal distribution of infections or exposures over a season. This distribution, interacting with the waning function, is all that is necessary to reproduce, in mathematically closed form, the mechanical cycle of boosting and waning immunity characteristic of recurrent seasonal infectious disease. Distinct from epidemiological models predicting numbers of individuals moving between infectivity compartments, our result enables us to directly estimate parameters of waning and the infectivity distribution. We can naturally iterate the cyclical process to simulate immunity trajectories over many years and thus to quantify the strong relationship between residual immunity and the time elapsed between annual infectivity peaks.