On the formulation of epidemic models (an appraisal of Kermack and McKendrick)

Breda, Dimitri; Diekmann, Odo; Graaf, Wieke Freudenburg-de; Pugliese, Andrea; Vermiglio, Rossana

doi:10.1080/17513758.2012.716454

Cited by 115 publications

(127 citation statements)

References 47 publications

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“…So the information is problem specific and it cannot be deduced from just Equation (2). In [13, Section X 3…”

Section: Resultsmentioning

confidence: 99%

“…For differential delay equations the rule specifies the derivative at the current time point [13, 17, 19, 28], whilst for renewal equations the rule specifies the value of the function itself in the current time point [3, 11, 14, 20]. Many physiologically structured population models lead to systems of equations that have differential delay as well as renewal components [15].…”

Section: Introductionmentioning

confidence: 99%

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A didactical note on the advantage of using two parameters in Hopf bifurcation studies

Diekmann

Korvasová

2013

Journal of Biological Dynamics

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In order to maximize the information that a linearized stability analysis provides, one should work with two free parameters rather than one. Moreover, it is recommended to first consider coefficients in the characteristic equation as parameters and in a second step (try to) invert the map that defines the coefficients in terms of the parameters as they occur in the original equation. Our aim is to substantiate these claims by way of a delay equation example taken from the literature.AMS Subject Classification: 34K18; 34K20; 34K60; 92C99

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“…So the information is problem specific and it cannot be deduced from just Equation (2). In [13, Section X 3…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A didactical note on the advantage of using two parameters in Hopf bifurcation studies

Diekmann

Korvasová

2013

Journal of Biological Dynamics

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“…There is a need to study this process, both in terms of the underlying biology and in terms of its dynamic consequences (Breda et al, 2012). …”

Section: Introductionmentioning

confidence: 99%

Nine challenges for deterministic epidemic models

et al. 2015

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Deterministic models have a long history of being applied to the study of infectious disease epidemiology. We highlight and discuss nine challenges in this area. The first two concern the endemic equilibrium and its stability. We indicate the need for models that describe multi-strain infections, infections with time-varying infectivity, and those where super infection is possible. We then consider the need for advances in spatial epidemic models, and draw attention to the lack of models that explore the relationship between communicable and non-communicable diseases. The final two challenges concern the uses and limitations of deterministic models as approximations to stochastic systems.

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“…Individuals are recruited (by birth or immigration) into the susceptible category at constant rate μ > 0 and die at percapita rate μ, and following infection are assumed to become permanently immune. The deterministic version of the model is ds dt = μ − βsi − μs, (1) di dt = βsi − γ i − μi, (2) dr dt…”

Section: Introductionmentioning

confidence: 99%

“…Age-structured models in which the recovery rate is allowed to depend upon both age and time since infection are described and studied in [8,13,15,16]. Closer to the spirit of the current work is the recent paper [1], in which the authors study a model based upon that of [18,19], in that an individual's expected infectivity is allowed to be a general function of time since infection, but allowing a general lifetime distribution.…”

Section: Introductionmentioning

confidence: 99%

Generality of endemic prevalence formulae

Clancy

2015

Mathematical Biosciences

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In simple infection models, the susceptible proportion s(*) in endemic equilibrium is related to the basic reproduction number R0 by s(*)=1/R0. We investigate the extent to which this relationship remains valid under more realistic modelling assumptions. In particular, we relax the biologically implausible assumptions that individuals' lifetimes and infectious periods follow exponential distributions; allow a general recruitment process; allow for multiple stages of infection; and consider extension to a multigroup model in which the groups may represent, for instance, spatial heterogeneity, or the existence of super-spreaders. For a homogeneous population, we find that: (i) the susceptible proportion is s(*)=1/R0(e), where R0(e) is a modified reproduction number, equal to R0 only in certain circumstances; (ii) the proportions of the population in each stage of infection are proportional to the expected time spent by an infected individual in that stage before recovery or death. We demonstrate robustness of the formula s(*)=1/R0 for many human infections by noting conditions under which R0(e) is approximately equal to R0, while pointing out other circumstances under which this approximation fails. For heterogeneous populations, the formula s(*)=1/R0 does not hold in general, but we are able to exhibit symmetry conditions under which it is valid.

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On the formulation of epidemic models (an appraisal of Kermack and McKendrick)

Abstract: The aim of this paper is to show that a large class of epidemic models, with both demography and non-permanent immunity incorporated in a rather general manner, can be mathematically formulated as a scalar renewal equation for the force of infection.

Cited by 115 publications

References 47 publications

A didactical note on the advantage of using two parameters in Hopf bifurcation studies

A didactical note on the advantage of using two parameters in Hopf bifurcation studies

Nine challenges for deterministic epidemic models

Generality of endemic prevalence formulae

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