Nearly one dimensional dynamics in an epidemic
Journal of Theoretical BiologyPublished 1 January 1985
William M. Schaffer, Mark Kot
Citations226
SJR quartileQ2
SJR score0.53
SNIP0.71
Generate an AI Snapshot to get a quick, structured summary of this paper.
Study Snapshot
ObjectiveStudy objective
MethodsResearch methodology
PopulationPopulation studied
Sample sizeSample sizes
OutcomesStudy outcomes here
ResultsStudy results comes here
LimitationsResearch study limitations comes here
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
TL;DR
The incidence of measles in New York City and Baltimore was studied using recently developed techniques in nonlinear dynamics to suggest almost two dimensional, chaotic flows whose essential attributes are captured by one dimensional, unimodal maps.
Abstract
The incidence of measles in New York City and Baltimore was studied using recently developed techniques in nonlinear dynamics. The data, monthly case reports for the years 1928-1963, suggest almost two dimensional, chaotic flows whose essential attributes are captured by one dimensional, unimodal maps. The effects of noise, inevitable in ecological and epidemiological systems are discussed.
Keywords
MathematicsMedicineEconomics, Econometrics and Finance
Applied mathematical sciencesNonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
16,015 Citations1983John Guckenheimer, Philip Holmes
NatureSimple mathematical models with very complicated dynamics
6,948 Citations1976Robert M. May
This is an interpretive review of first-order difference equations, which can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations.
Journal of the American Statistical AssociationSpectral Analysis and its Applications.
5,019 Citations1969Walter Freiberger, Gwilym M. Jenkins +1 more
Physical Review LettersCharacterization of Strange Attractors
4,842 Citations1983Peter Grassberger, Itamar Procaccia
Physical Review LettersGeometry from a Time Series
3,889 Citations1980Norman H. Packard, James P. Crutchfield +2 more
Journal of Statistical PhysicsQuantitative universality for a class of nonlinear transformations
3,581 Citations1978Mitchell J. Feigenbaum
NaturePopulation biology of infectious diseases: Part I
3,243 Citations1979Roy M. Anderson, Robert M. May
MeccanicaLyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory
1,840 Citations1980Giancarlo Benettin, L. Galgani +2 more
ScienceBiological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos
1,708 Citations1974Robert M. May
This paper presents a dynamical regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but boundedpopulation fluctuations, can occur.
Birkhäuser Boston eBooksIterated Maps on the Interval as Dynamical Systems
1,654 Citations2009Pierre Collet, Jean‐Pierre Eckmann
The American NaturalistBifurcations and Dynamic Complexity in Simple Ecological Models
1,241 Citations1976Robert M. May, George Oster
It is shown that as a hump steepens, the dynamics goes from a stable point, to a bifurcating hierarchy of stable cycles of period 2n, into a region of chaotic behavior where the population exhibits an apparently random sequence of "outbreaks" followed by "crashes".
NaturePopulation biology of infectious diseases: Part II
946 Citations1979Robert M. May, Roy M. Anderson
Consideration is given to the relation between the ecology and evolution of the transmission processes and the overall dynamics, and to the mechanisms that can produce cyclic patterns, or multiple stable states, in the levels of infection in the host population.
Journal of EcologyModelling Fluctuating Populations.
874 Citations1983G. Mani, Roger M. Nisbet +1 more
Zeitschrift für Naturforschung AStrange Attractors, Chaotic Behavior, and Information Flow
801 Citations1981Robert Shaw
The observed geometry of strange attractors is seen to arise naturally from a rule allowing joining but not splitting of trajectories in phase space, and their possible relation to 1/f noise is suggested.
ScienceDirectly Transmitted Infections Diseases: Control by Vaccination
649 Citations1982Roy M. Anderson, Robert M. May
Journal of Combinatorial Theory Series AOn finite limit sets for transformations on the unit interval
607 Citations1973N. Metropolis, M. L. Stein +1 more
An infinite sequence of finite or denumerable limit sets is found for a class of many-to-one transformations of the unit interval into itself and the structure and order of occurrence is universal for the class.
Physics ReportsFluctuations and simple chaotic dynamics
483 Citations1982James P. Crutchfield, J. Doyne Farmer +1 more
SIAM Journal on Applied MathematicsStable Orbits and Bifurcation of Maps of the Interval
463 Citations1978David A. Singer
American Journal of EpidemiologyRECURRENT OUTBREAKS OF MEASLES, CHICKENPOX AND MUMPS
462 Citations1973Wayne P. London, James A. Yorke
Computer simulations show that the persistence of the biennial pattern of measles outbreaks implies that the vaccine is not being used uniformly throughout the population, for populations in which most members are vaccinated.
International Journal of EpidemiologyMeasles in England and Wales—I: An Analysis of Factors Underlying Seasonal Patterns
411 Citations1982Paul Fine, JACQUELINE A CLARKSON
Examination of the weekly measles notifications for England and Wales, 1950-1979, reveals a regular biennial pattern of major and minor epidemics before the national immunization programme began in 1968, followed by an annual cycle ofMinor epidemics, suggesting that the national measles vaccination programme has not lowered the total number of individuals susceptible to measles.
Journal of HygieneVaccination against rubella and measles: quantitative investigations of different policies
324 Citations1983Roy M. Anderson, Robert M. May
It is shown that immunization programmes can, under some circumstances, increase the total number of cases among older age groups; the implications for the overall incidence of measles encephalitis and of congenital rubella syndrome are examined.
Zeitschrift für Naturforschung AChaotic Behavior in Simple Reaction Systems
289 Citations1976Otto E. Rössler
Lecture notes in biomathematicsThe Incidence of Infectious Diseases under the Influence of Seasonal Fluctuations
251 Citations1976Klaus Dietz
The present paper is concerned with the effect of seasonal variations of the contact rate on the incidence of infectious diseases and the theory of nonlinear oscillations according to which the biennial cycles of measles epidemics could be understood as subharmonic parametric resonance.
Physical Review LettersOne-Dimensional Dynamics in a Multicomponent Chemical Reaction
220 Citations1982Reuben H. Simoyi, Alan Wolf +1 more
American Journal of EpidemiologyRECURRENT OUTBREAKS OF MEASLES, CHICKENPOX AND MUMPS
188 Citations1973James A. Yorke, Wayne P. London
The mean monthly contact rates for measles, chickenpox and mumps estimated from the monthly reported cases show systematic differences between the years with many cases and the yearswith few cases, and the clustering of cases within social groups is proposed.
American Journal of EpidemiologySEASONALITY AND THE REQUIREMENTS FOR PERPETUATION AND ERADICATION OF VIRUSES IN POPULATIONS
186 Citations1979James A. Yorke, Neal Nathanson +2 more
It is argued that the population size of 500,000 for measles persistence (described by Bartlett) is primarily a consequence of seasonal variation, and computer modelling can be used to examine the effect of changes in parameters which determine the seasonal cycle of virus perpetuation and fadeout.
Wiley eBooksStatistical physics and chaos in fusion plasmas
156 Citations1984C. W. Horton, L. E. Reichl
Physics Letters APower spectral analysis of a dynamical system
140 Citations1980James P. Crutchfield, Doyne Farmer +4 more
Theoretical Population BiologyOscillatory phenomena in a model of infectious diseases
97 Citations1980Z. Grossman
Simple deterministic models, where the oscillations arise either solely from periodically varying contact rates or from the combined effect of large initial perturbation, small periodic variation of the contact rate, and the destabilizing nature of infectious and latent periods when described as time delays are studied.