Chance and chaos in population biology—Models of recurrent epidemics and food chain dynamics

Engbert, Ralf; Drepper, Friedhelm R.

doi:10.1016/0960-0779(94)90028-0

Cited by 67 publications

(66 citation statements)

References 19 publications

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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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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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“…Even small amounts as used in the simulations depress the onset of period 3 attractors which only appear at significantly higher magnitudes of forcing. This effect, whereby immigration can destroy longer period attractors, has been reported before (Engbert & Drepper 1994;Ferguson et al 1996;Alonso et al 2007). One can speculate that the lack of immigration in the deterministic analysis of Nguyen & Rohani (2008) might explain their findings; alternatively there may be other stable solutions of the deterministic equations, but that the noise is never strong enough to move the system towards them.…”

Section: Discussionmentioning

confidence: 84%

“…We also include a small immigration term h, allowing for infectious imports (Hagenaars et al 2004;Alonso et al 2007). The formulation we use corresponds to a commuter model, where a susceptible has the possibility of catching the disease while away from the main population, and then returning (Engbert & Drepper 1994). Vaccination is simple to incorporate, as it can be shown in the deterministic version of the model that vaccinating a proportion p of the population at birth scales b by (1 2 p), and this remains true for the stochastic version (Earn et al 2000).…”

Section: The Staged Seir Modelmentioning

confidence: 99%

Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough

Black

McKane

2010

J. R. Soc. Interface.

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Although many stochastic models can accurately capture the qualitative epidemic patterns of many childhood diseases, there is still considerable discussion concerning the basic mechanisms generating these patterns; much of this stems from the use of deterministic models to try to understand stochastic simulations. We argue that a systematic method of analysing models of the spread of childhood diseases is required in order to consistently separate out the effects of demographic stochasticity, external forcing and modelling choices. Such a technique is provided by formulating the models as master equations and using the van Kampen system-size expansion to provide analytical expressions for quantities of interest. We apply this method to the susceptible-exposed -infected -recovered (SEIR) model with distributed exposed and infectious periods and calculate the form that stochastic oscillations take on in terms of the model parameters. With the use of a suitable approximation, we apply the formalism to analyse a model of whooping cough which includes seasonal forcing. This allows us to more accurately interpret the results of simulations and to make a more quantitative assessment of the predictions of the model. We show that the observed dynamics are a result of a macroscopic limit cycle induced by the external forcing and resonant stochastic oscillations about this cycle.

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“…(4.72) were "rst considered by Dietz [100], who assumed that the contact rate b varies periodically with the period equal to one year and found analytically periodic oscillations of the model variables. Later these equations were studied in detail by Olsen and Scha!er [101] and Engbert [102]. It was shown that periodic variation of the contact rate can result not only in periodic oscillations of childhood infections but in chaotic behavior as well.…”

Section: Oscillations In a Standard Model For Childhood Epidemics Indmentioning

confidence: 99%

Changes in the dynamical behavior of nonlinear systems induced by noise

Landa

McClintock

2000

Physics Reports

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Chance and chaos in population biology—Models of recurrent epidemics and food chain dynamics

Cited by 67 publications

References 19 publications

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

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

Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough

Changes in the dynamical behavior of nonlinear systems induced by noise

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