title: Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough.
abstract: Although many stochastic models can accurately capture the qualitative epidemic patterns of many childhood diseases, there is still considerable discussion concerning the basic mechanisms generating these patterns; much of this stems from the use of deterministic models to try to understand stochastic simulations. We argue that a systematic method of analysing models of the spread of childhood diseases is required in order to consistently separate out the effects of demographic stochasticity, external forcing and modelling choices. Such a technique is provided by formulating the models as master equations and using the van Kampen system-size expansion to provide analytical expressions for quantities of interest. We apply this method to the susceptible-exposed-infected-recovered (SEIR) model with distributed exposed and infectious periods and calculate the form that stochastic oscillations take on in terms of the model parameters. With the use of a suitable approximation, we apply the formalism to analyse a model of whooping cough which includes seasonal forcing. This allows us to more accurately interpret the results of simulations and to make a more quantitative assessment of the predictions of the model. We show that the observed dynamics are a result of a macroscopic limit cycle induced by the external forcing and resonant stochastic oscillations about this cycle.
authors.affiliation: Theoretical Physics Group, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK.
authors.forename: Andrew J
authors.initials: AJ
authors.lastname: Black
authors.forename: Alan J
authors.initials: AJ
authors.lastname: McKane
mesh_terms: D002648:Child; D003141:Communicable Diseases / Q000453:epidemiology*; D004196:Disease Outbreaks; D004198:Disease Susceptibility / Q000453:epidemiology; D006801:Humans; D008954:Models, Biological; D014917:Whooping Cough / Q000453:epidemiology*