Childhood infections remain a public health concern as well as a modelling challenge, despite many decades of control measures and theoretical efforts. Large-scale vaccination programmes against some of these diseases started in the 1940s–1960s in developed countries and led to marked reductions of incidence levels, but in developing countries they are still a cause of significant levels of infant mortality [1
]. The resurgence of pertussis reported in developed countries after decades of high vaccine coverage [6
], as well as a recent upsurge of measles in eastern and southern Africa, prompted a renewed interest in assessing the efficacy of control policies for these childhood infections.
Such an assessment must rely on simulations based on mathematical models [11
], whose more recent versions highlight the importance of contact structure, stochasticity and seasonality as drivers of the observed temporal patterns of incidence [15
]. Concurrently, analytical tools to deal with these ingredients have been developed [23
Models of higher complexity involve many parameters, some of which are difficult to determine, leaving considerable room for data fitting [31
]. On the other hand, one of the features of childhood infections is the variability exhibited by different data records, even within those that correspond to a single disease in comparable social environments [4
]. Therefore, successful modelling of particular datasets reported in the literature [17
] has not closed the discussion about the key determinants of the observed dynamics. Measles is an exception: a parsimonious, stochastic, susceptible–infective–recovered (SIR)-based model has been shown to adequately describe disease incidence in large urban populations [15
At the opposite extreme, pertussis keeps defying mathematical modelling, as illustrated by recent efforts in different directions [10
]. In particular, several proposals explore the hypothesis about disease-induced and vaccine-induced immunity [10
], relying on assumptions about vaccine uptake and efficacy for the purpose of comparison with real data. However, we are still lacking a sound uncontroverted model for pre-vaccination pertussis.
Here, we have sought to contribute to the goal of using pre-vaccination data records to obtain information about the properties of disease-induced immunity. The paper focuses on exploring the influence of naive hosts' immune response in the long-term patterns of pertussis prevalence as given by the averaged power spectra of simulated time series corresponding to several decades. The power spectrum of the stochastic, seasonally forced, well-mixed SIR model is compared with those of four modifications of the model that reflect different assumptions about disease-induced immunity. These four different assumptions have been proposed in the literature in the framework of deterministic models that have all been shown to be compatible with available data. The five variants we consider deliberately avoid all the complications related to contact structure and spatial spread, as well as vaccine coverage and efficacy and waning of vaccine-induced immunity, in order to keep a relatively low number of free parameters in the model.
We show that the stochastic versions of the SIR model and its four variants have significantly different properties, which translate into quantitatively different prevalence and incidence power spectra. This opens the possibility of using the stochastic properties of long, well-resolved data records to constrain these and other variants of the model, with the power spectra of the pertussis incidence time series in large urban centres in the pre-vaccination era as the target long-term dynamics that the model should reproduce. We illustrate this strategy by applying it to two publicly available historical data records for pre-vaccination pertussis incidence.