Invariant Epidemic Transient Decay From Radically Different Forms of Seasonal Forcing

Abstract

This thesis begins by analyzing whooping cough dynamics in London from 1664 to 1950 in chapter 2. We use a historical whooping cough mortality time-series from the London Bills of Mortality and the Registrar General’s Weekly Returns, in which a spectral analysis of the time-series reveals annual, biennial, triennial, and even quadrennial epidemic cycles. We originally sought to model and explain these transitions in the frequency structure of the whooping cough mortality data using the sinusoidally forced Susceptible-Infectious-Recovered (SIR) model [KM91]. The method of transition analysis previously used on historical disease-induced mortality time-series, including measles and smallpox [HE15, Kry11], relies on the existence of a period-doubling bifurcation in the basic reproduction number. Our analysis using this method on whooping cough, however, reveals the existence of only an annual attractor for relevant values of the basic reproduction number and amplitudes of forcing. Furthermore, the lack of bifurcations in relevant parameter spaces of our model for whooping cough led us to investigate the transient dynamics.

We explore the transient dynamics of the seasonally forced SIR model in chapter 3. Conveniently, we discover the transient periods of the associated annual attractor have potential to explain the transitions seen in the frequency structure of the whooping cough mortality data. We additionally consider a family of forcing functions when analyzing the transient dynamics. Prior to this work, it was unknown if the transient dynamics of the seasonally forced SIR model were invariant to the shape of seasonal forcing. Papst and Earn showed that key bifurcations of the standard SIR model are invariant to the shape of seasonal forcing if the amplitude of forcing is appropriately adjusted [PE19]. Our results from chapter 3 expand upon Papst and Earn's findings. We discover invariance in the decay of transient periods of the associated annual attractor from radically different shapes of seasonal forcing with appropriately adjusted amplitudes of forcing.