  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.
