Invariant predictions of epidemic patterns from radically different forms of seasonal forcing

Abstract

Seasonal variation in environmental variables, and in rates of contact among individuals, are fundamental drivers of infectious disease dynamics. Unlike most periodically-forced physical systems, for which the precise pattern of forcing is typically known, underlying patterns of seasonal variation in transmission rates can be estimated approximately at best, and only the period of forcing is accurately known. Yet solutions of epidemic models depend strongly on the forcing function, so dynamical predictions-such as changes in epidemic patterns that can be induced by demographic transitions or mass vaccination-are always subject to the objection that the underlying patterns of seasonality are poorly specified. Here, we demonstrate that the key bifurcations of the standard epidemic model are invariant to the shape of seasonal forcing if the amplitude of forcing is appropriately adjusted. Consequently, analyses applicable to real disease dynamics can be conducted with a smooth, idealized sinusoidal forcing function, and qualitative changes in epidemic patterns can be predicted without precise knowledge of the underlying forcing pattern. We find similar invariance in a seasonally forced predator-prey model, and conjecture that this phenomenon-and the associated robustness of predictions-might be a feature of many other periodically forced dynamical systems.

How can contemporary climate research help understand epidemic dynamics? Ensemble approach and snapshot attractors

“…It has also been demonstrated that for certain values of system parameters the nonautonomous models of recurrent epidemics (measles, mumps, rubella, H5N1 avian influenza) show chaotic behavior 6,[19][20][21] . Traditional SIR-like epidemic models are dissipative nonlinear low-dimensional systems with constant, periodic, quasiperiodic 22 , or term-time 3,23 external forcing whose dynamics a) Electronic mail: tkovacs@general.elte.hu is governed by (chaotic) attractors in phase space. Furthermore, even if the long-term dynamics is regular and the final state of the system is a fixed-point attractor the route to this condition might be rather complex.…”

“…Furthermore, even if the long-term dynamics is regular and the final state of the system is a fixed-point attractor the route to this condition might be rather complex. Many studies examined the role of finite-time irregularity in ecological models [24][25][26][27] as well as in epidemic dynamics 9,23,28,29 concluding the relevance of transient behavior.…”

Patterns of smallpox mortality in London, England, over three centuries

“…However, how 493 interventions influence the frequency structure and seasonality of epidemic time series 494 over decades and centuries is much more subtle [56, 74,75]. While preliminary work has 495 been promising [76], careful estimation [77,78] and analysis [75,79]…”

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“…Moreover, the time series is so long that the underlying pattern of seasonal forcing probably changed substantially. Changes in seasonal forcing can be accommodated in predictions [16,106], and estimation of underlying seasonal variation in transmission is becoming easier [107]; however, obtaining a convincing mechanistic explanation for all the structure we have identified in London's smallpox dynamics represents a major challenge.…”

“…Ultimately, we anticipate that meeting this challenge will involve further developments of the existing theory of transitions in epidemic patterns [13][14][15][16]106], coupled with state-of-theart methods of inference to obtain parameter estimates for dynamical models [108][109][110][111][112].…”