  The dynamics of strongly immunizing childhood infections is still not well
understood. Although reports of successful modeling of several incidence data
records can be found in the literature, the key determinants of the observed
temporal patterns have not been clearly identified. In particular, different
models of immunity waning and degree of protection applied to disease and
vaccine induced immunity have been debated in the literature on pertussis. Here
we study the effect of disease acquired immunity on the long term patterns of
pertussis prevalence. We compare five minimal models, all of which are
stochastic, seasonally forced, well-mixed models of infection based on
susceptible-infective-recovered dynamics in a closed population. These models
reflect different assumptions about the immune response of naive hosts, namely
total permanent immunity, immunity waning, immunity waning together with
immunity boosting, reinfection of recovered, and repeat infection after partial
immunity waning. The power spectra of the output prevalence time series
characterize the long term dynamics of the models. For epidemiological
parameters consistent with published data for pertussis, the power spectra show
quantitative and even qualitative differences that can be used to test their
assumptions by comparison with ensembles of several decades long
pre-vaccination data records. We illustrate this strategy on two publicly
available historical data sets.
