Oscillatory phenomena in a model of infectious diseases

Grossman, Z.

doi:10.1016/0040-5809(80)90050-7

Cited by 87 publications

(60 citation statements)

References 14 publications

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“…A number of different IPDs are used in the literature, including the exponential distribution (in the so-called general stochastic epidemic and implicitly in most deterministic models), the gamma distribution and an almost surely constant infectious period. It has long been recognised that the exponential distribution, though mathematically convenient as it means the epidemic process is Markov, does not provide a realistic model of the infectiousness of real diseases (see, for example, Grossman [28] and Keeling and Grenfell [29]). A fixed infectious period also offers some mathematical advantages and is usually more realistic than an exponentially distributed one, but it still eliminates a potentially important source of randomness in an epidemic model.…”

Section: The Effect Of the Infectious Period Distributionmentioning

confidence: 99%

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

Ball

Sirl

Trapman

2010

Mathematical Biosciences

134

148

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This paper is concerned with a stochastic SIR (susceptible → infective → removed) model for the spread of an epidemic amongst a population of individuals, with a random network of social contacts, that is also partitioned into households. The behaviour of the model as the population size tends to infinity in an appropriate fashion is investigated. A threshold parameter which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak is obtained, as are the probability that a major outbreak occurs and the expected proportion of the population that are ultimately infected by such an outbreak, together with methods for calculating these quantities. Monte Carlo simulations demonstrate that these asymptotic quantities accurately reflect the behaviour of finite populations, even for only moderately sized finite populations. The model is compared and contrasted with related models previously studied in the literature. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.

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Section: The Effect Of the Infectious Period Distributionmentioning

confidence: 99%

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

Ball

Sirl

Trapman

2010

Mathematical Biosciences

134

148

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“…The predicted damping time for infections such as measles and pertussis is long, however, and a variety of processes such as seasonality in transmission and stochastic demographic effects (the inclusion of chance elements in the growth and decay of the susceptible and infectious populations) can perpetuate indefinitely the otherwise damped cycles Bartlett, 1956; (Grossman, 1980;Schwartz & Smith, 1984).…”

Section: Theoretical Predictions and Observed Trendsmentioning

confidence: 99%

“…First, there is considerable mathematical interest in models of recurrent epidemic behaviour as a consequence of their non-linear dynamical properties (see Dietz, 1976;Nussbaum, 1977;Busenberg & Cooke, 1978;Green, 1978;Smith, 1978;Yorke et al 1979;Grossman, 1980;Gripenberg, 1980;Stech & Williams, 1981; Hethcote, Stech & van den Driessche, 1981Schwartz & Smith, 1983;Aron & Schwartz, 1984). The mass-action non-linearity, combined with incubation delays in the course of infection, may induce simple cycles or two-point and higher-order cycles (moving into chaos), depending on parameter values and precise model structure.…”

Section: Introductionmentioning

confidence: 99%

Oscillatory fluctuations in the incidence of infectious disease and the impact of vaccination: time series analysis

Anderson

Grenfell

May

1984

J. Hyg.

156

133

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SUMMARYThis paper uses the techniques of time series analysis (autocorrelation and spectral analysis) to examine oscillatory secular trends in the incidence of infectious diseases and the impact of mass vaccination programmes on these well-documented phenomena. We focus on three common childhood diseases: pertussis and mumps (using published disease-incidence data for England and Wales) and measles (using data from England and Wales, Scotland, North America and France). Our analysis indicates highly statistically significant seasonal and longer-term cycles in disease incidence in the prevaccination era. In general, the longer-term fluctuations (a 2-year period for measles, 3-year periods for pertussis and mumps) account for most of the cyclical variability in these data, particularly in the highly regular measles series for England and Wales. After vaccination, the periods of the longer-term oscillations tend to increase, an observation which corroborates theoretical predictions. Mass immunization against measles (which reduces epidemic fluctuations) magnifies the relative importance of the seasonal cycles. By contrast, we show that high levels of vaccination against whooping cough in England and Wales appear to have suppressed the annual cycle.

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“…Equilibrium stability analyses have been conducted on 'unforced' models that assume constant contact rates [6,7,[29][30][31][32], and bifurcation analyses have been conducted on 'forced' models in which contact rates vary seasonally [6][7][8][9][10][11][12]33]. Lloyd [7] found that the biennial pattern observed in the SI 1 R model is reproduced by the SI n R model but with much weaker seasonality.…”

Section: Dynamics Of Epidemic Models With Erlang-distributed Stage Dumentioning

confidence: 99%

Effects of the infectious period distribution on predicted transitions in childhood disease dynamics

Krylova

Earn

2013

J. R. Soc. Interface.

107

140

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The population dynamics of infectious diseases occasionally undergo rapid qualitative changes, such as transitions from annual to biennial cycles or to irregular dynamics. Previous work, based on the standard seasonally forced 'susceptible-exposed-infectious -removed' (SEIR) model has found that transitions in the dynamics of many childhood diseases result from bifurcations induced by slow changes in birth and vaccination rates. However, the standard SEIR formulation assumes that the stage durations (latent and infectious periods) are exponentially distributed, whereas real distributions are narrower and centred around the mean. Much recent work has indicated that realistically distributed stage durations strongly affect the dynamical structure of seasonally forced epidemic models. We investigate whether inferences drawn from previous analyses of transitions in patterns of measles dynamics are robust to the shapes of the stage duration distributions. As an illustrative example, we analyse measles dynamics in New York City from 1928 to 1972. We find that with a fixed mean infectious period in the susceptible -infectious -removed (SIR) model, the dynamical structure and predicted transitions vary substantially as a function of the shape of the infectious period distribution. By contrast, with fixed mean latent and infectious periods in the SEIR model, the shapes of the stage duration distributions have a less dramatic effect on model dynamical structure and predicted transitions. All these results can be understood more easily by considering the distribution of the disease generation time as opposed to the distributions of individual disease stages. Numerical bifurcation analysis reveals that for a given mean generation time the dynamics of the SIR and SEIR models for measles are nearly equivalent and are insensitive to the shapes of the disease stage distributions.

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Oscillatory phenomena in a model of infectious diseases

Cited by 87 publications

References 14 publications

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

Oscillatory fluctuations in the incidence of infectious disease and the impact of vaccination: time series analysis

Effects of the infectious period distribution on predicted transitions in childhood disease dynamics

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