Persistent Chaos of Measles Epidemics in the Prevaccination United States Caused by a Small Change in Seasonal Transmission Patterns

Abstract

Epidemics of infectious diseases often occur in predictable limit cycles. Theory suggests these cycles can be disrupted by high amplitude seasonal fluctuations in transmission rates, resulting in deterministic chaos. However, persistent deterministic chaos has never been observed, in part because sufficiently large oscillations in transmission rates are uncommon. Where they do occur, the resulting deep epidemic troughs break the chain of transmission, leading to epidemic extinction, even in large cities. Here we demonstrate a new path to locally persistent chaotic epidemics via subtle shifts in seasonal patterns of transmission, rather than through high-amplitude fluctuations in transmission rates. We base our analysis on a comparison of measles incidence in 80 major cities in the prevaccination era United States and United Kingdom. Unlike the regular limit cycles seen in the UK, measles cycles in US cities consistently exhibit spontaneous shifts in epidemic periodicity resulting in chaotic patterns. We show that these patterns were driven by small systematic differences between countries in the duration of the summer period of low transmission. This example demonstrates empirically that small perturbations in disease transmission patterns can fundamentally alter the regularity and spatiotemporal coherence of epidemics.

Statistical physics of vaccination

“…However, the model fails to capture a number of further characteristics, ranging from the sustained biennial pre-vaccination oscillations of measles in US and UK cities (see again Fig. 5), to the more complicate dynamics observed in other contexts, both prior and after mass vaccination [91][92][93]. Motivated by the fact that both the weather and certain social processes, namely school-recruitment following the yearly school calendar, have one year periodicity, an important strain of the literature has focused on 1 − year periodic β(t) in endemic SIR models.…”

“…Motivated by the fact that both the weather and certain social processes, namely school-recruitment following the yearly school calendar, have one year periodicity, an important strain of the literature has focused on 1 − year periodic β(t) in endemic SIR models. If the corresponding period T β is around 1 year, then nonlinear resonance may appear, inducing phenomena such as biennal and multi-year periodicity, and chaos [91,[93][94][95][96][97][98][99] that partially explain a number of observed epidemiological time-series.…”

Drivers of measles mortality: the historic fatality burden of famine in Bangladesh

“…Several smaller outbreaks occurred between 1970 and 1991. Unlike historical measles data from England and Wales and elsewhere [24], [30], the outbreak patterns in Matlab were erratic, and annual cycles were only evident for a short period between 1976 and 1988 (Supplementary Figure S1). This may be due to differences between mortality and incidence data.…”

“…The estimated transmission rates peaked in March and was low between May and October (Supplementary Figure S4). A reduction in the transmission during the summer months has been well-documented for measles epidemics in major European and American cities prior to the start of vaccination, and has been attributed to the role of school terms [24], [30]–[32]. In Matlab, the decline in transmission during summer was not significantly lower than the mean transmission across the year.…”

A Comparison of Measles Vaccination Regimes in a Stochastic, Spatial SIR Model

“…Dalziel et al fit data on measles in England, Wales, and the US during the pre-vaccine era to a TSIR model. They were able to model "persistent chaotic epidemics" in the US from "small shifts in seasonal transmission patterns" [Dalziel et al, 2016].…”