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

Olsen, Lars Folke; Truty, G.L.; Schaffer, William M.

doi:10.1016/0040-5809(88)90019-6

Cited by 195 publications

(128 citation statements)

References 20 publications

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“…There are also papers which point to an inverse relationship between varicella and zoster incidence. Only two local studies of case reports, one from a general practice in England [12] and the other from a hospital in Sweden [13], describe a peak in the incidence of zoster in the summer months, which coincides with the yearly trough in varicella incidence, which has a seasonal peak in the winter months [6][7][8]. Neither analysis is particularly convincing.…”

Section: Varicella Inhibits Reactivationmentioning

confidence: 99%

The epidemiology of varicella–zoster virus infections: the influence of varicella on the prevalence of herpes zoster

1992

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SUMMARYThis paper uses mathematical models and data analysis to examine the epidemiological implications of possible immunologically mediated links between patterns of varicella and herpes-zoster incidence in human communities. A review of previously published reports does not clarify whether or not there is a relationship between the incidence of varicella and the incidence of zoster. However, new analysis of data collected by the Royal College of General Practitioners provides indirect evidence for the hypothesis that a high intensity of varicella transmission suppresses viral reactivation. The significance of this finding for proposed varicella vaccination campaigns is explored by a review of published data on the use of the vaccine. No significant difference is shown to exist between the risk of zoster caused by the vaccine and the wild virus. A mathematical model is then developed to take into consideration the influence of the prevalence of varicella on viral reactivation and the impact of vaccination with attenuated virus, which may be able to recrudesce. Under some conditions, mass application of such vaccines may have the impact of increasing zoster incidence. The results presented here indicate that, before starting any vaccination programme against varicella, its consequences need to be assessed in much more depth.

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Section: Varicella Inhibits Reactivationmentioning

confidence: 99%

The epidemiology of varicella–zoster virus infections: the influence of varicella on the prevalence of herpes zoster

1992

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“…With this relatively small number we expect stochastic e¡ects to be important, so we use a Monte-Carlo implementation of the model. This technique has often been used to model the spread of disease in small or severely £uctuating populations and it explicitly takes into account the fact that the jackal population is composed of N individual units (Olsen et al 1988;Anderson & May 1991;Grenfell 1992). Also, we make a further modi¢cation in line with ¢eld observations relating to the seasonality of the birth rate.…”

Section: (B) Implementation Of the Modelmentioning

confidence: 99%

Rabies in Zimbabwe: reservoir dogs and the implications for disease control

Rhodes

Atkinson

Anderson

et al. 1998

Phil. Trans. R. Soc. Lond. B

117

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Using detailed ¢eld study observations of the side-striped jackal (Canis adustus) and a simple stochastic model of the transmission dynamics of the virus and host demography, we discuss the epidemiology of rabies virus infection in the jackal population of Zimbabwe. Of the two jackal species in Zimbabwe, the other being the black-backed jackal (Canis mesomelas), the bulk of noti¢ed rabies cases are in side-striped jackals. Speci¢cally, we show that the side-striped jackal population itself does not seem able to support rabies infection endemically, i.e. without frequent reintroduction from outside sources of infection. We argue that this is probably because the overall average jackal population density is too low to maintain the chain of infection. This study suggests that the disease is regularly introduced to jackals by rabid dogs from populations associated with human settlements. Given the rapidly rising dog population in Zimbabwe, estimates are derived of the future incidence of jackal rabies based on di¡erent dog-vaccination scenarios.

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“…Indeed, it is known since long ago that nonlinear mechanical and electronic systems described by Duffing and Van der Pol equations have a very simple dynamic behavior in the constant parameter case, but become very complex (multiplicity of attractors, catastrophes, and chaos) when they are periodically perturbed (Guckenheimer and Holmes, 1986). Another important example in a different field is the classical SEIR epidemic model which has a globally stable equilibrium in the constant parameter case and a great number of modes of behavior in the periodically varying case (Schwartz and Smith, 1983;Aron and Schwartz, 1984;Kot et al, 1988;Olsen et al, 1988).…”

Section: Introductionmentioning

confidence: 99%

Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities

Rinaldi

Muratori

Kuznetsov

1993

Bulletin of Mathematical Biology

139

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ForewordThis paper presents the analysis of a periodically fbrced second order nonlinear dynamical system describing predator-prey communities. Six different seasonality mechanisms are identified and compared in terms of bifurcation diagrams. The analysis is carried out by means of an interactive package which detects Hopf, flip and fold bifurcations curves as well as codimension two bifurcation points. The results are in agreement with the general theory of periodically perturbed Hopf bifurcations. This work shows that complex environmental issues can be highlighted by suitably combining basic results of nonlinear system theory and powerful numerical techniques. Moreover, the two classical routes to chaos, namely, torus destruction and cascade of period doublings, are numerically detected. Since in the case of constant parameters the model cannot have multiple attractors, catastrophes, and chaos, the results support the conjecture that seasons can very easily give rise to complex population dynamics.

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

Cited by 195 publications

References 20 publications

The epidemiology of varicella–zoster virus infections: the influence of varicella on the prevalence of herpes zoster

The epidemiology of varicella–zoster virus infections: the influence of varicella on the prevalence of herpes zoster

Rabies in Zimbabwe: reservoir dogs and the implications for disease control

Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities

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