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

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.传染病流行通常呈现可预测的极限环（limit cycles）特征。理论研究指出，这类循环可因传播率（transmission rates）的高幅度季节性波动而被打破，进而催生确定性混沌（deterministic chaos）。然而，学界尚未观测到持续存在的确定性混沌现象，究其原因，部分在于传播率出现足够大幅波动的场景并不常见。即便出现此类波动，由此引发的深度流行低谷也会切断传播链，即便在大型城市中也会导致传染病流行彻底消亡。本研究展示了一条全新路径：无需借助传播率的高幅度波动，仅通过传播模式季节性特征的细微偏移，即可在局部区域催生持续存在的混沌性流行。我们的分析基于对美国与英国疫苗接种前时代（prevaccination era）80座主要城市的麻疹发病率（measles incidence）数据开展的对比研究。与英国观测到的规则极限环不同，美国城市的麻疹流行周期始终表现出自发偏移现象，进而形成混沌模式。我们证实，这些模式源于两国间夏季低传播期时长存在的微小系统性差异。本实证案例表明，疾病传播模式的微小扰动，即可从根本上改变传染病流行的规律性与时空连贯性（spatiotemporal coherence）。