The many guises of R0 (a didactic note)

Cushing, J. M.; Diekmann, Odo

doi:10.1016/j.jtbi.2016.06.017

Cited by 62 publications

(67 citation statements)

References 29 publications

(44 reference statements)

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“…As mentioned in ( Cushing and Diekmann, 2016 ) and other literature, independent of the formula of the basic reproduction number, the critical condition that it is equal to unity is mathematically identical as long as it is well-defined.…”

Section: Comparison Of the Basic Reproduction Numbersmentioning

confidence: 91%

“…We discuss in this section how the different basic reproduction numbers can be mathematically derived for the model (2) . Subsequently, going by their meanings from the perspective of modeling, we examine how they are different and what nature they have in common (for such possibly different formulas for basic reproduction number, see the arguments in Brauer et al, 2016;Cushing and Diekmann, 2016;van den Driessche, 2017 ).…”

Section: The Basic Reproduction Numbersmentioning

confidence: 99%

“…Conversely, the expected number of infective visitors that a single infective resident can produce, assuming every contact to the visitor is always to the susceptible, is R v r which is expressed by the product of the expected duration of the infectivity of each infective resident 1/ ρ, the resident-visitor transmission coefficient β vr , and the visitor subpopulation size N v . The basic reproduction number R 0 | r can be translated based on its conceptual definition as similarly argued in ( Cushing and Diekmann, 2016 ): The formula (8) can be rewritten as…”

Section: The Basic Reproduction Numbers In Terms Of Each Subpopulationmentioning

confidence: 99%

“…Following this biological definition, a mathematical theory is used to derive the basic reproduction number as the spectral radius of a specific matrix which is called the "next generation matrix" (NGM) for a system of ordinary differential equations governing epidemic dynamics (see Cushing and Diekmann, 2016;Diekmann et al, 2013 for a complete reference, or see van den Driessche, 2017 for a recent review). In the frequently referred paper by van den Driessche and Watmough (2002) , very helpful results were obtained for disease control having investigated the actual definition of R 0 based on a compartmental system of ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

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A model for epidemic dynamics in a community with visitor subpopulation

Dansu

Seno

2019

Journal of Theoretical Biology

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a b s t r a c tWith a five dimensional system of ordinary differential equations based on the SIR and SIS models, we consider the dynamics of epidemics in a community which consists of residents and short-stay visitors. Taking different viewpoints to consider public health policies to control the disease, we derive different basic reproduction numbers and clarify their common/different mathematical natures so as to understand their meanings in the dynamics of the epidemic. From our analyses, the short-stay visitor subpopulation could become significant in determining the fate of diseases in the community. Furthermore, our arguments demonstrate that it is necessary to choose one variant of basic reproduction number in order to formulate appropriate public health policies.

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Section: Comparison Of the Basic Reproduction Numbersmentioning

confidence: 91%

Section: The Basic Reproduction Numbersmentioning

confidence: 99%

Section: The Basic Reproduction Numbers In Terms Of Each Subpopulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A model for epidemic dynamics in a community with visitor subpopulation

Dansu

Seno

2019

Journal of Theoretical Biology

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“…Our model, an extension of a discrete‐time anthrax model of van den Driessche and Yakubu in herbivore populations, includes B. anthracis shedding in both human and animal populations. We use the next‐generation method for computing the basic reproduction number,

R_{0}

, in discrete‐time compartmental epidemic models, developed by Allen and van den Driessche (), Cushing and Yicang (), and Li and Schneider (), to establish verifiable conditions for anthrax invasion or extinction (noninvasion) in the human and herbivore populations (Cushing & Diekmann, ; Diekmann, Heesterbeek, & Metz, ).…”

Section: Introductionmentioning

confidence: 99%

A discrete‐time anthrax model in human and herbivore populations

Yakubu

Ziyadi

2018

Natural Resource Modeling

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We introduce a discrete‐time anthrax disease model in human and herbivore populations, where the herbivore recruitment function is the classic Beverton–Holt model while that of humans is a constant recruitment function. We use the next generation method to compute the basic reproduction numbers for the submodels consisting of only herbivores ( scriptR 0 A ) and only humans ( scriptR 0 H ). In addition, we compute it for the full human–herbivore model ( scriptR 0 A H ). When R 0 A < 1 or R 0 H < 1 or R 0 A H < 1, the number of anthrax infections decreases and the disease does not invade the corresponding population. Whereas, the number of anthrax infections increases and the disease invades the corresponding population when R 0 A > 1 or R 0 H > 1 or R 0 A H > 1. Our simulations show that it is possible for the anthrax disease to invade a human–herbivore population where the disease does not invade either herbivores or humans in isolation. Recommendations for resource managers To help guide anthrax disease prevention and control strategies, we provide basic reproduction numbers for the herbivore only population ( scriptR 0 A ), human‐only population ( scriptR 0 A ) and human–herbivore population ( scriptR 0 A H ). R 0 < 1 implies the number of anthrax infections deceases and anthrax does not invade the population, while R 0 > 1 implies the number of anthrax cases increases and anthrax invasion occurs. It is possible for anthrax to invade a population consisting of herbivores and humans where the disease does not invade either the herbivores or humans in isolation. Thus, more anthrax surveillance is required in mixed populations of herbivores and humans. Our human–herbivore model can be used to predict the number of anthrax‐infected humans and animals given the number of known anthrax infections in populations.

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A practical approach to R₀ in continuous‐time ecological models

Barril¹,

Calsina²,

Ripoll

2017

Math Methods in App Sciences

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In this paper we study the asymptotic behaviour of the solutions in linear models of population dynamics by means of the basic reproduction number R 0 . Our aim is to give a practical approach to the computation of the basic reproduction number in continuous-time population models structured by age and/or space. The procedure is different depending on whether the density of newborns per time unit and the density of population belong to the same functional space or not. Three infinite-dimensional examples are illustrated: a transport model for a cell population, a model of spatial diffusion of individuals in a habitat, and a model of migration of individuals between age-structured local populations. For each model, we have highlighted the possible advantages of computing R 0 instead of the Malthusian parameter.

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The many guises of R0 (a didactic note)

Cited by 62 publications

References 29 publications

A model for epidemic dynamics in a community with visitor subpopulation

A model for epidemic dynamics in a community with visitor subpopulation

A discrete‐time anthrax model in human and herbivore populations

A practical approach to R₀ in continuous‐time ecological models

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The many guises of R0 (a didactic note)

Cited by 62 publications

References 29 publications

A model for epidemic dynamics in a community with visitor subpopulation

A model for epidemic dynamics in a community with visitor subpopulation

A discrete‐time anthrax model in human and herbivore populations

A practical approach to R0 in continuous‐time ecological models

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Product

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A practical approach to R₀ in continuous‐time ecological models