TY - JOUR

T1 - Numerical method for multi-alleles genetic drift problem

AU - Xu, Shixin

AU - Chen, Xinfu

AU - Liu, Chun

AU - Yue, Xingye

N1 - Funding Information:
\ast Received by the editors September 10, 2018; accepted for publication (in revised form) May 9, 2019; published electronically July 25, 2019. https://doi.org/10.1137/18M1211581 \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The work of the authors was supported by the National Science Foundation grant DMS-1516344 and by the National Natural Science Foundation of China grants 11271281 and 91230106. \dagger School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu, 215006, China, and Zu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University, No. 8 Duke Avenue, Kunshan, Jiangsu, 215316, China (xsxztr@hotmail.com). \ddagger Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260 (xinfu@pitt.edu). \S Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616 (cliu124@iit.edu). \P School of Mathematical Sciences, Soochow University, Suzhou, 215006 Jiangsu, China (xyyue@ suda.edu.cn).
Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics.

PY - 2019

Y1 - 2019

N2 - Genetic drift describes random fluctuations in the number of genes variants in a population. One of the most popular models is the Wright-Fisher model. The diffusion limit of this model is a degenerate diffusion-convection equation. Due to the degeneration and convection, Dirac singularities will always develop at the boundaries as time evolves, i.e., the fixation phenomenon occurs. Theoretical analysis has proven that the weak solution of this equation, regarded as measure, conserves total probability and expectations. In the current work, we propose a scheme for 3-alleles model with absolute stability and generalize it to N-alleles case (N > 3). Our method can conserve not only total probability and expectations, but also positivity. We also prove that the discrete solution converges to a measure as the mesh size tends to zero, which is the exact measure solution of the original problem. The simulations illustrate that the probability density decays to zero first on the inner nodes, then also on the edge nodes except at the three vertex nodes, on which the density finally concentrates. The results correctly predict the fixation probability and are consistent with theoretical ones and with direct Monte Carlo simulations.

AB - Genetic drift describes random fluctuations in the number of genes variants in a population. One of the most popular models is the Wright-Fisher model. The diffusion limit of this model is a degenerate diffusion-convection equation. Due to the degeneration and convection, Dirac singularities will always develop at the boundaries as time evolves, i.e., the fixation phenomenon occurs. Theoretical analysis has proven that the weak solution of this equation, regarded as measure, conserves total probability and expectations. In the current work, we propose a scheme for 3-alleles model with absolute stability and generalize it to N-alleles case (N > 3). Our method can conserve not only total probability and expectations, but also positivity. We also prove that the discrete solution converges to a measure as the mesh size tends to zero, which is the exact measure solution of the original problem. The simulations illustrate that the probability density decays to zero first on the inner nodes, then also on the edge nodes except at the three vertex nodes, on which the density finally concentrates. The results correctly predict the fixation probability and are consistent with theoretical ones and with direct Monte Carlo simulations.

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U2 - 10.1137/18M1211581

DO - 10.1137/18M1211581

M3 - Article

AN - SCOPUS:85072115921

SN - 0036-1429

VL - 57

SP - 1770

EP - 1788

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

IS - 4

ER -