Seasonally forced disease dynamics explored as switching between attractors

https://doi.org/10.1016/S0167-2789(00)00187-1Get rights and content

Abstract

Biological phenomena offer a rich diversity of problems that can be understood using mathematical techniques. Three key features common to many biological systems are temporal forcing, stochasticity and nonlinearity. Here, using simple disease models compared to data, we examine how these three factors interact to produce a range of complicated dynamics. The study of disease dynamics has been amongst the most theoretically developed areas of mathematical biology; simple models have been highly successful in explaining the dynamics of a wide variety of diseases. Models of childhood diseases incorporate seasonal variation in contact rates due to the increased mixing during school terms compared to school holidays. This ‘binary’ nature of the seasonal forcing results in dynamics that can be explained as switching between two nonlinear spiral sinks. Finally, we consider the stability of the attractors to understand the interaction between the deterministic dynamics and demographic and environmental stochasticity. Throughout attention is focused on the behaviour of measles, whooping cough and rubella.

Introduction

Epidemiology has been one of the most successful quantitative branches of ecology. In particular, the study of childhood infections using the SIR (susceptible–infectious–recovered) family of models has yielded many mathematically interesting results [1], [2], [3], [4] and shown good agreement with observations of disease case reports [5], [6], [7], [8], [9], [10], [11], [12]. The rich variety of dynamics observed arise through three-way interactions between nonlinearities (due to the mixing of susceptible and infectious individuals), stochasticity (both demographic and environmental) and temporal forcing (caused by changes in the average contact rate).

The deterministic SIR model [5], [7], [13] and more complex variations on this theme [6], [11], [14] all demonstrate qualitatively similar dynamical properties. In general, this family of models possess a globally stable fixed point attractor in a homogeneous environment, and shows large amplitude oscillations when seasonally forced, often producing period-doubling cascades as the level of seasonality is increased [15], [16], [17]. Numerous authors have observed the signature of seasonal forcing in the number of reported cases [18], [19] while others have found the magnitude and form of seasonal forcing to be important for obtaining both quantitative and qualitative agreement with observations [6], [9], [13], [20].

Periodically forced nonlinear oscillators have been studied extensively in the mathematical literature [21], [22], [23], [24], [25], [26], with many examples motivated by biological processes [27], [28], [29], [30], [31], [32]. However, much of the theory relates to limiting cases where either the forcing is small, or the period is close to the resonant frequency of the oscillator. For realistic disease models we are not afforded such convenience, the period is fixed at 1 year and the level of forcing can be considerable. We must therefore apply our nonlinear techniques and understanding to the results of numerical simulations. The methodology used in this paper has similarities to the work on phase-resetting maps for kicked oscillators [33]; in our models the kicks occur periodically due to the change between school terms and holidays.

An added problem in understanding the observed case reports of any disease is the effects of noise, which in the disease dynamics can come from two main sources, external fluctuations or internal stochasticity due to the chance occurrence of each event. Stochasticity is epidemiologically important as it allows us to assess the persistence of a disease, with chance events causing the population level of the pathogen to hit zero. Such localised extinctions are a key feature of small populations [9], [11], [34], [35] and understanding their frequency is vital for successful vaccination [36]. Dynamically, stochasticity can force the orbit away from the deterministic attractor leading to transient behaviour playing a far larger role [4]. Section 4 considers the effects of stochasticity on the system, and in particular concentrates on the behaviour of whooping cough, measles and rubella in an attempt to explain the observed dynamics.

Section snippets

The basic model and biological paradigm

For simplicity of the mathematics and figures, we shall restrict our attention to the SIR model with constant population size — this reduces the system to two dimensions, as S+I+R is constant. In fact, even the more complex SEIR model [7] which incorporates an incubation period and RAS model [14] which includes age-structure both demonstrate rapid convergence close to a two-dimensional attractor; hence we believe our conclusions are likely to be generic. Traditionally, SIR models have assumed a

Switching between attractors

The concept of switching between term and holiday fixed points shall now be used to consider the dynamics for two very different ranges of seasonality, β1. When β1 is small it is possible to perform some rigorous analysis; however, for more realistic amounts of seasonality, we must rely on numerical simulation.

Stochasticity and large deviations

In order to explain why noise has a greater effect on the dynamics of whooping cough than measles we shall at first consider the stability of the attractor; the usual method of doing this is to calculate the Lyapunov exponents. Fig. 4 shows the local Lyapunov exponents around the orbit (see Appendix B for the method of calculation). It is clear that both attractors have some regions where the Lyapunov exponent is positive and hence perturbations grow. However, measles actually has more of these

Conclusion

The SIR model (and other related models of infectious diseases) are amongst the simplest and yet most accurate of all biological models [5], [8], [9]. The rapid mixing within a large human population means that the mass-action assumption for disease transmission is a good approximation, and the presence of this single nonlinearity means that these models are well suited for detailed mathematical study [4], [13], [16]. The dynamical study of disease behaviour has benefited both subjects, pushing

Acknowledgements

This research was supported by the Royal Society (MJK and PR) and the Wellcome Trust (BTG). We wish to thank Robert MacKay for his help, as well as Colin Sparrow for his useful comments on this manuscript.

References (47)

  • H.W. Hethcote

    An age-structured model for pertussis transmission

    Math. Biosci.

    (1997)
  • L.F. Olsen et al.

    Oscillations and chaos in epidemics: a nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark

    Theor. Popul. Biol.

    (1986)
  • G. Sugihara et al.

    Distinguishing error from chaos in ecological time series

    Philos. Trans. Roy. Soc. Lond. B

    (1990)
  • D.A. Rand et al.

    Chaotic stochasticity — a ubiquitous source of unpredictability in epidemics

    Proc. Roy. Soc. Lond. B

    (1991)
  • R.M. May et al.

    Population biology of infectious diseases, part II

    Nature

    (1979)
  • D. Schenzle

    An age-structured model of pre- and post-vaccination measles transmission

    IMA J. Math. Appl. Med. Biol.

    (1984)
  • R.M. Anderson, R.M. May, Infectious Diseases of Humans, Oxford University Press, Oxford,...
  • D. Mollison et al.

    Epidemics: models and data

    J. Roy. Stat. Soc. A

    (1993)
  • B.T. Grenfell et al.

    Seasonality and extinction in chaotic metapopulations

    Proc. Roy. Soc. Lond. B

    (1995)
  • M.J. Keeling et al.

    Disease extinction and community size: modeling the persistence of measles

    Science

    (1997)
  • P. Rohani et al.

    Opposite patterns of synchrony: in sympatric disease metapopulations

    Science

    (1999)
  • D.J.D. Earn et al.

    A simple model for complex dynamical transitions in epidemics

    Science

    (2000)
  • B.M. Bolker

    Chaos and complexity in measles models: a comparative numerical study

    IMA J. Math. Appl. Med. Biol.

    (1993)
  • Cited by (227)

    View all citing articles on Scopus
    View full text