Application of the Lambert <i>W</i> Function to the SIR Epidemic Model

Wang, Frank

doi:10.4169/074683410x480276

Cited by 14 publications

(12 citation statements)

References 5 publications

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“…(4) that precludes the need to estimate S MSY via numerical methods or Hilborn ’s ( 1985 ) linear approximation. This function has been used for explicit solutions to Roger’s random predator equation in ecology ( McCoy & Bolker, 2008 ) and susceptible-infected-removed (SIR) models in epidemiology ( Reluga, 2004 ; Wang, 2010 ). Specifically, W ( z ) is defined as the function that satisfies for any complex number z (Lambert 1758 and Euler 1783 as cited in Corless et al, 1996 ).…”

Section: Methodsmentioning

confidence: 99%

An explicit solution for calculating optimum spawning stock size from Ricker’s stock recruitment model

Scheuerell

2016

PeerJ

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Stock-recruitment models have been used for decades in fisheries management as a means of formalizing the expected number of offspring that recruit to a fishery based on the number of parents. In particular, Ricker’s stock recruitment model is widely used due to its flexibility and ease with which the parameters can be estimated. After model fitting, the spawning stock size that produces the maximum sustainable yield (SMSY) to a fishery, and the harvest corresponding to it (UMSY), are two of the most common biological reference points of interest to fisheries managers. However, to date there has been no explicit solution for either reference point because of the transcendental nature of the equation needed to solve for them. Therefore, numerical or statistical approximations have been used for more than 30 years. Here I provide explicit formulae for calculating both SMSY and UMSY in terms of the productivity and density-dependent parameters of Ricker’s model.

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Section: Methodsmentioning

confidence: 99%

An explicit solution for calculating optimum spawning stock size from Ricker’s stock recruitment model

Scheuerell

2016

PeerJ

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“…Expression (7) can be used to easily check, for some given values of the reproduction number, initial conditions and the threshold M , if I max will exceed or not the threshold M . Furthermore, expression (7) can be used to estimate conditions on the parameters or on the initial conditions, ensuring I max ≤ M .…”

Section: Controlling the Maximum Number Of Infectedmentioning

confidence: 99%

“…Previous works have considered the Lambert W function in the context of epidemiological models. In [6,7,8] for example, the Lambert W function is used to express the final sizes in different epidemiological models and in [9], the Lambert W function is used to manipulate and analyze an epidemiological model with piece-wise smooth incidence rate. Additional details and applications of the Lambert W function can be found in [5,10].…”

Section: Introductionmentioning

confidence: 99%

Analysis of infected population threshold exceedance in an SIR epidemiological model

Báez-Sánchez¹,

Bobko²

2020

Preprint

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We consider an epidemiological SIR model and a positive threshold M . Using a parametric expression for the solution curve of the SIR model and the Lambert W function, we establish necessary and sufficient conditions on the basic reproduction number R 0 to ensure that the infected population does not exceed the threshold M .We also propose and analyze different measures to quantify a possible threshold exceedance.

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“…The infection dynamics cannot be solved explicitly as a function of time. However, Wang (2010) has shown that the asymptotic values (i.e. the values of S, I and R after the epidemic is over) have an explicit solution with the Lambert W function.…”

Section: T H E S I R E P I D E M I C M O D E Lmentioning

confidence: 99%

“…The key is again to eliminate time by dividing the first equation in (14) by the last. Wang derives simple, explicit, asymptotic formulas for S and R , which provide mathematical insight into the threshold nature of epidemics (Wang ).…”

Section: The Lambert W Function In Ecology and Evolutionary Biologymentioning

confidence: 99%

The Lambert W function in ecological and evolutionary models

Lehtonen

2016

Methods Ecol Evol

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Summary The Lambert W function is a mathematical function with a long history, but which was named and rigorously defined relatively recently. It is closely related to the logarithmic function and arises from many models in the natural sciences, including a surprising number of problems in ecology and evolution. I describe the basic properties of the function and present examples of its application to models of ecological and evolutionary processes. The Lambert W function makes it possible to solve explicitly several models where this is not possible with elementary functions. I present examples of such models from existing literature, as well as novel models. Solving models explicitly with the Lambert W function can provide deeper insight and a new point of view on a biological problem. Explicit solutions with the Lambert W function are easily amenable to further mathematical operations, such as differentiation and integration. These advantages apply to a wide range of models, from the marginal value theorem to population growth rates and disease epidemics.

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Application of the Lambert W Function to the SIR Epidemic Model

Cited by 14 publications

References 5 publications

An explicit solution for calculating optimum spawning stock size from Ricker’s stock recruitment model

An explicit solution for calculating optimum spawning stock size from Ricker’s stock recruitment model

Analysis of infected population threshold exceedance in an SIR epidemiological model

The Lambert W function in ecological and evolutionary models

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