Seasonally forced disease dynamics explored as switching between attractors
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)
- et al.
Nearly one-dimensional dynamics in an epidemic
J. Theor. Biol.
(1985) - et al.
Dynamical complexity in age-structured models of the transmission of measles virus
Math. BioSci.
(1996) - et al.
Seasonality and period-doubling bifurcations in an epidemic model
J. Theor. Biol.
(1984) Chaotic response of nonlinear oscillators
Phys. Rep.
(1982)- et al.
Transitions in the parameter space of a periodically forced dissipative system
Chaos Solitons Fractals
(1996) - et al.
Period and non-periodic responses of a periodically forced Hodgkin–Huxley oscillator
J. Theor. Biol.
(1984) Multiple attractors and resonance in periodically forced population models
Physica D
(2000)- et al.
(Meta)population dynamics of infectious diseases
TREE
(1997) Modelling the persistence of measles
Trends MicroBiol.
(1997)- et al.
The impact of immunisation on pertussis transmission in England and Wales
Lancet
(2000)
An age-structured model for pertussis transmission
Math. Biosci.
Oscillations and chaos in epidemics: a nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark
Theor. Popul. Biol.
Distinguishing error from chaos in ecological time series
Philos. Trans. Roy. Soc. Lond. B
Chaotic stochasticity — a ubiquitous source of unpredictability in epidemics
Proc. Roy. Soc. Lond. B
Population biology of infectious diseases, part II
Nature
An age-structured model of pre- and post-vaccination measles transmission
IMA J. Math. Appl. Med. Biol.
Epidemics: models and data
J. Roy. Stat. Soc. A
Seasonality and extinction in chaotic metapopulations
Proc. Roy. Soc. Lond. B
Disease extinction and community size: modeling the persistence of measles
Science
Opposite patterns of synchrony: in sympatric disease metapopulations
Science
A simple model for complex dynamical transitions in epidemics
Science
Chaos and complexity in measles models: a comparative numerical study
IMA J. Math. Appl. Med. Biol.
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