Nonlinear time series analysis of empirical population dynamics

Drepper, Friedhelm R.; Engbert, Ralf; Stollenwerk, Nico

doi:10.1016/0304-3800(94)90016-7

Cited by 21 publications

(17 citation statements)

References 16 publications

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“…), though we would argue that measles provides a less cluttered path between data and mechanistic models than most other systems. Childhood infections have also been a key test bed for understanding dynamical systems and associated statistical methodologies at the population level (Schaffer and Kot 1985, Casdagli 1991, Rand and Wilson 1991, Stone 1992, Drepper et al 1994, Grenfell et al 1994, Stone et al 1996, Finkenstädt and Grenfell 2000. In this paper, we presented a modeling framework that weds the theoretical and statistical approaches through a mechanistic, stochastic model for measles epidemics (see also Finkenstädt and Grenfell 2000).…”

Section: Discussionmentioning

confidence: 99%

Dynamics of Measles Epidemics: Estimating Scaling of Transmission Rates Using a Time Series SIR Model

Bjørnstad¹,

Finkenstädt²,

Grenfell³

2002

Ecological Monographs

279

439

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Before the development of mass-vaccination campaigns, measles exhibited persistent fluctuations (endemic dynamics) in large British cities, and recurrent outbreaks (episodic dynamics) in smaller communities. The critical community size separating the two regimes was ϳ300 000-500 000. We develop a model, the TSIR (Time-series Susceptible-Infected-Recovered) model, that can capture both endemic cycles and episodic outbreaks in measles. The model includes the stochasticity inherent in the disease transmission (giving rise to a negative binomial conditional distribution) and random immigration. It is thus a doubly stochastic model for disease dynamics. It further includes seasonality in the transmission rates. All parameters of the model are estimated on the basis of time series data on reported cases and reconstructed susceptible numbers from a set of cities in England and Wales in the prevaccination era . The 60 cities analyzed span a size range from London (3.3 ϫ 10 6 inhabitants) to Teignmouth (10 500 inhabitants). The dynamics of all cities fit the model well. Transmission rates scale with community size, as expected from dynamics adhering closely to frequency dependent transmission (''true mass action''). These rates are further found to reveal strong seasonal variation, corresponding to high transmission during school terms and lower transmission during the school holidays. The basic reproductive ratio, R 0 , is found to be invariant across the observed range of host community size, and the mean proportion of susceptible individuals also appears to be constant. Through the epidemic cycle, the susceptible population is kept within a 3% interval. The disease is, thus, efficient in ''regulating'' the susceptible population-even in small cities that undergo recurrent epidemics with frequent extinction of the disease agent. Recolonization is highly sensitive to the random immigration process. The initial phase of the epidemic is also stochastic (due to demographic stochasticity and random immigration). However, the epidemic is nearly ''deterministic'' through most of the growth and decline phase.

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Section: Discussionmentioning

confidence: 99%

Dynamics of Measles Epidemics: Estimating Scaling of Transmission Rates Using a Time Series SIR Model

Bjørnstad¹,

Finkenstädt²,

Grenfell³

2002

Ecological Monographs

279

439

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“…These formulations, which we have previously used to explore the dynamics of measles epidemics in England and Wales in the prevaccination period (12,29,35), incorporate age structure, simple metapopulation structure, and environmental or demographic stochasticity. The models are extended versions of the standard susceptible/exposed/ infective/recovered (SEIR) model, which has been exhaustively analyzed in the mathematical epidemiology literature (10,11,13,(36)(37)(38)(39)(40)(41)(42). We use a more realistic age-structured (RAS) measles model (12,16,35), which takes the basic epidemiological structure of the SEIR model and adds age structure and a more detailed seasonal pattern to it.…”

Section: Data Sets and Modelsmentioning

confidence: 99%

Impact of vaccination on the spatial correlation and persistence of measles dynamics.

Grenfell

1996

Proc. Natl. Acad. Sci. U.S.A.

151

142

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The onset of measles vaccination in England and Wales in 1968 coincided with a marked drop in the temporal correlation of epidemic patterns between major cities. We analyze a variety of hypotheses for the mechanisms driving this change. Straightforward stochastic models suggest that the interaction between a lowered susceptible population (and hence increased demographic noise) and nonlinear dynamics is sufficient to cause the observed drop in correlation. The decorrelation of epidemics could potentially lessen the chance of global extinction and so inhibit attempts at measles eradication.The combination of practical importance and theoretical interest has made measles in urban populations one of the most studied epidemiological systems, but spatial and stochastic effects remain less well explored (1). Practically, the continuing failure of mass vaccination campaigns to eradicate measles in developed countries and the importance of measles mortality among children in developing countries has led to continuing interest. Theoretically, recent work showing that nonlinear dynamics can reduce the extinction rate of spatially subdivided populations (2, 3) may be testable with the body of theory and data gathered to evaluate the persistence of measles and other childhood infections under mass vaccination (4-7).The basic dynamics of measles epidemics have been extensively studied (8-13). Large measles epidemics exhaust a pool of susceptible children, who gain lifelong immunity after having the disease; the interepidemic period is determined by the severity of the epidemic and the length of time taken to build up the pool of susceptibles again from new births. The seasonal pattern of aggregation of children in schools is also important in determining the size and pattern of epidemics (14-16). Furthermore, seasonality strongly influences the recurrent epidemic behavior of measles, where there is strong evidence for nonlinear (and possibly chaotic) effects (17-23).In small isolated communities, large epidemics are often followed by extinction of disease as the chain of transmission breaks down (8,24,25). The critical community size (8,24,26), the threshold population size above which measles can persist through interepidemic troughs, may depend on the spatial structure and connectedness of the regional population. The persistence behavior of measles is of particular interest in connection within the eradication of a disease by mass vaccination. Most theoretical estimates focus on the invasibility threshold, the level of control at which a disease will not only go extinct but also be unable to reinvade the population (13). One can also locally eradicate infection simply by driving it below its persistence threshold, the point at which the disease is likely to go extinct during the troughs between epidemics (27). The persistence threshold is greater than the invasibility threshold and hence easier to reach, but eradication through local extinction is fragile; reintroduction of the disease sparks a new epidemic, which...

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“…Measles epidemics in human populations have been a subject of investigations for a long time ( , [London & Yorke, 1973a], [Dietz, 1976]), since rather good empirical time series are available, and various aspects of recent paradigmatic theories like deterministic chaos in prevaccination dynamics ( [Schwartz & Smith, 1983], [Schenzle, 1984], [Aron & Schwartz, 1984], [Schaffer, 1985], [Schaffer & Kott, 1985], [Olsen & Schaffer, 1990], [May & Sugihara, 1990], [Rand & Wilson, 1991], [Grenfell, 1992], [Bolker & Grenfell, 1993], [Drepper et al, 1994]) and criticality in island populations have been investigated ( [Rhodes & Anderson, 1996], [Rhodes et al, 1997]). …”

Section: Measles Around Criticalitymentioning

confidence: 99%

Recent Advances in Applied Probability

2005

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Nonlinear time series analysis of empirical population dynamics

Cited by 21 publications

References 16 publications

Dynamics of Measles Epidemics: Estimating Scaling of Transmission Rates Using a Time Series SIR Model

Dynamics of Measles Epidemics: Estimating Scaling of Transmission Rates Using a Time Series SIR Model

Impact of vaccination on the spatial correlation and persistence of measles dynamics.

Recent Advances in Applied Probability

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