The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations

Chalub, Fabio A. C. C.; Souza, Max O.

doi:10.1007/s00285-013-0657-7

Cited by 30 publications

(53 citation statements)

References 63 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As a consequence it does not reach the value of unity at the time, T, by which the expected trajectory achieves fixation. (Chalub and Souza 2013). It is plausible that in all such systems, the ideas and methods we have presented in this work, associated with the expected trajectory and fictitious selection, could shed new insights into the phenomena that these complex systems exhibit.…”

Section: Other Applicationsmentioning

confidence: 88%

The Characteristic Trajectory of a Fixing Allele: A Consequence of Fictitious Selection That Arises from Conditioning

et al. 2013

View full text Add to dashboard Cite

This work is concerned with the historical progression, to fixation, of an allele in a finite population. This progression is characterized by the average frequency trajectory of alleles that achieve fixation before a given time, T. Under a diffusion analysis, the average trajectory, conditional on fixation by time T, is shown to be equivalent to the average trajectory in an unconditioned problem involving additional selection. We call this additional selection "fictitious selection"; it plays the role of a selective force in the unconditioned problem but does not exist in reality. It is a consequence of conditioning on fixation. The fictitious selection is frequency dependent and can be very large compared with any real selection that is acting. We derive an approximation for the characteristic trajectory of a fixing allele, when subject to real additive selection, from an unconditioned problem, where the total selection is a combination of real and fictitious selection. Trying to reproduce the characteristic trajectory from the action of additive selection, in an infinite population, can lead to estimates of the strength of the selection that deviate from the real selection by .1000% or have the opposite sign. Strong evolutionary forces may be invoked in problems where conditioning has been carried out, but these forces may largely be an outcome of the conditioning and hence may not have a real existence. The work presented here clarifies these issues and provides two useful tools for future analyses: the characteristic trajectory of a fixing allele and the force that primarily drives this, namely fictitious selection. These should prove useful in a number of areas of interest including coalescence with selection, experimental evolution, time series analyses of ancient DNA, game theory in finite populations, and the historical dynamics of selected alleles in wild populations.T HE phenomenon of fixation, where an allele becomes fully established in a population, was first studied by Fisher (1922) and Haldane (1927). Building on results previously obtained by Fisher (1922), Haldane (1927) showed that the probability of fixation of a new beneficial allele in a large population is 2s, where s is the allele's selective advantage (i.e., its selection coefficient). For typical values of s, the probability of fixation of a beneficial allele is very small. Hence even in a large population, the majority of positively selected alleles are unlikely to fix. The probability of fixation of neutral and deleterious mutations is nonzero but lies beyond the reach of Haldane's result; they are even less likely to fix than beneficial mutations (Kimura 1962). However, despite the improbability of any new allele fixing, over time there is repeated production of new alleles in a population, so that eventually fixation will occur. Apart from the long-term changes in a population that fixation entails, we note that alleles "on the way" to fixation have intermediate and high frequencies for extended periods of time, and the ass...

show abstract

Section: Other Applicationsmentioning

confidence: 88%

The Characteristic Trajectory of a Fixing Allele: A Consequence of Fictitious Selection That Arises from Conditioning

et al. 2013

View full text Add to dashboard Cite

show abstract

“…van Kampen, 2001), but following a more analytical route as pursued by the authors in Chalub and Souza (2009a,b) (and also more recently in Chalub and Souza (2013)) leads to a continuous equation for the probability density, that we term SIS-PDE, together with a boundary condition at one end, and with a conservation law.…”

Section: Resultsmentioning

confidence: 99%

Discrete and continuous SIS epidemic models: A unifying approach

Chalub

Souza

2014

Ecological Complexity

Self Cite

View full text Add to dashboard Cite

The Susceptible-Infective-Susceptible (SIS) epidemiological scheme is the simplest description of the dynamics of a disease that is contact-transmitted, and that does not lead to immunity. Two by now classical approaches to such a description are: (i) the use of a mass-action compartamental model that leads to a single ordinary differential equation (SIS-ODE); (ii) the use of a discrete-time Markov chain model (SIS-DTMC). While the former can be seen as a mean-field approximation of the latter under certain conditions, it is also known that their dynamics can be significantly different, if the basic reproduction number is greater than one. The goal of this work is to introduce a continuous model, based on a partial differential equation (SIS-PDE), that retains the finite populations effects present in the SIS-DTMC model, and that allows the use of analytical techniques for its study. In particular, it will reduce itself to the SIS-ODE model in many circumstances. This is accomplished by deriving a diffusion-drift approximation to the probability density of the SIS-DTMC model. Such a diffusion is degenerated at the origin, and must conserve probability. These two features then lead to an interesting consequence: the biologically correct solution is a measure solution. We then provide a convenient representation of such a measure solution that allows the use of classical techniques for its computation, and that also provides a tool for obtaining information about several dynamical features of the model. In particular, we show that the SIS-ODE gives the most likely state, conditional on non-absorption. As a further application of * Corresponding Author Preprint submitted to ElsevierOctober 12, 2013 such representation, we show how to define the disease-outbreak probability in terms of the SIS-PDE model, and show that this definition can be used both for certain and uncertain initial presence of infected individuals. As a final application, we compute an approximation for the extinction time of the disease. In addition, we present many numerical examples that confirm the good approximation of the SIS-DTMC by the SIS-PDE.

show abstract

“…As noted, if selection is also included, then the coefficients of V are at least quadratic polynomials, and can be quite complicated, see [8]. So long as the second order part remains L Kim , then a result of Ethier, using the Trotter product formula, makes it possible to again define a strongly continuous semi-group, see [15].…”

Section: Introductionmentioning

confidence: 99%

“…We use an induction over the maximal codimension of bP, which allows us to construct an approximate solution operator for the Cauchy problem of the form 8) with Q t i the "interior" and Q t b the "boundary" contributions, respectively. We then analyze the operator:…”

Section: Construction Of the Resolventmentioning

confidence: 99%

Degenerate Diffusion Operators Arising in Population Biology

Epstein¹,

Mazzeo²

2013

View full text Add to dashboard Cite

The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations

Cited by 30 publications

References 63 publications

The Characteristic Trajectory of a Fixing Allele: A Consequence of Fictitious Selection That Arises from Conditioning

The Characteristic Trajectory of a Fixing Allele: A Consequence of Fictitious Selection That Arises from Conditioning

Discrete and continuous SIS epidemic models: A unifying approach

Degenerate Diffusion Operators Arising in Population Biology

Contact Info

Product

Resources

About