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Special Topics in Mathematical Biology with Random Variables
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Instructor: Yevgeniy Kovchegov
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e-mail:
kovchegy
@math.
oregonstate.edu
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Office: Kidder 368C
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Office Phone No: 7-1379
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Office Hours: M 1:30pm - 3:00pm and W 2:00pm - 3:30pm
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Place and time: MWF 10:00am to 10:50am, room WNGR 201.
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Web materials:
Charles M. Grinstead and J. Laurie Snell, Introduction to Probability available as a FREE e-book at http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/amsbook.mac.pdf
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Course description: This course is an introduction to stochastic modeling of biological processes. Stochastic models covered may include Markov processes in both continuous and discrete time, urn models, branching processes, and coalescent processes. Biological applications modeled may include genetic drift, population dynamics, genealogy, demography, and epidemiology. Mathematical results will be qualitatively interpreted and applied to the biological process under investigation.
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The course will cover the following topics:
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Discrete time and continuous time Markov chains.
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Mathematical models of genetic drift. Wright-Fisher model and binomial distribution.
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Application: Wright-Fisher model as a Markov chain.
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Moran process (aka 'Moran model') as a model of finite populations.
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Branching processes and their applications in genealogy.
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Birth-and-death processes. Applications in demography, epidemiology, and biology.
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Yule preferential attachment process. Application in bacteria population growth.
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Coalescent processes and their applications in population genetics.
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Other applications
A variety of mathematical techniques will be covered when analyzing these models.
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Syllabus: PDF
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Homework:
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Homework #1 (due Monday, May 23): Assignment 1 (PDF)
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Schedule:
Wednesday, March 30 Review of probability. Conditional probability. Bayes’ Theorem. Lecture 1 slides (PDF)
Friday, April 1 Review of probability. Conditional probability. Bayes’ Theorem. Independent events. Lecture 2 slides (PDF)
Monday, April 4 Review of probability. Bayes’ Theorem. Independent events. Examples. Lecture 3 slides (PDF)
Wednesday, April 6 Review of combinatorics. Permutations and combinations. Generalized combinations. Binomial theorem. Lecture 4 slides (PDF)
Friday, April 8 Introduction to random variables. Binomial random variable. Expectation of a random variable. Wright-Fisher Model. Lecture 5 slides (PDF)
Monday, April 11 No class.
Wednesday, April 13 Binomial random variable. Expectation of a random variable. Poisson random variable. Geometric random variables. Variance and standard deviation. Lecture 6 slides (PDF)
Friday, April 15 Variance and standard deviation of discrete random variables. Markov and Chebyshev inequalities. Lecture 7 slides (PDF)
Monday, April 18 Introduction into Markov chains. Wright-Fisher model as a Markov chain. Birth-and-death processes. Moran process. Lectures 8-11 slides (PDF)
Friday, April 22 Birth-and-death processes. Moran process. Lectures 8-11 slides (PDF)
Monday, April 25 Moran process. Fixation times for Moran process. Lectures 8-11 slides (PDF)
Wednesday, April 27 Moran process. Fixation times for Moran process. Lectures 8-11 slides (PDF)
Friday, April 29 Fixation times for Moran process: alternative approach. Lectures 12-15 slides (PDF)
Monday, May 2 Fixation times for Moran process: alternative approach. Martingales. Lectures 12-15 slides (PDF)
Wednesday, May 4 Martingales. Stopping times. The Optional Stopping Theorem. Lectures 12-15 slides (PDF)
Friday, May 6 The Optional Stopping Theorem. Probability harmonic functions. Lectures 12-15 slides (PDF)
Monday, May 9 The Optional Stopping Theorem. Probability harmonic functions. Lectures 16-20 slides (PDF)
Wednesday, May 11 Moran model with mutation. Stationary distribution. Lectures 16-20 slides (PDF)
Friday, May 13 Detailed balance conditions and reversibility. Stationary distribution for a birth-and-death chain. Lectures 16-20 slides (PDF)
Monday, May 16 Stationary distribution for Moran model with mutation. Lectures 16-20 slides (PDF)
Wednesday, May 18 Stationary distribution for Moran model with mutation. Lectures 16-20 slides (PDF)
Friday, May 20 Continuous random variables. Discrete time coalescing random walk. Lectures 21-25 slides (PDF)
Monday, May 23 Discrete time coalescing random walk. The coalescent. Lectures 21-25 slides (PDF)
Wednesday, May 25 Discrete time coalescing random walk. The coalescent. Lectures 21-25 slides (PDF)
Friday, May 25 Kingman's coalescent. The Branching processes and their applications in genealogy. Lectures 21-25 slides (PDF)
Wednesday, June 1 A discussion.
Friday, June 3 The Branching processes and their applications in genealogy. Lectures 21-25 slides (PDF)
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