# MATH 428/528

## Stochastic Elements in Mathematical Biology

 Instructor: Yevgeniy Kovchegov e-mail: kovchegy @math. oregonstate.edu Office: Kidder 368C Office Phone No: 7-1379 Office Hours: MW 1:00pm - 2:00pm in MSLC
Place and time: MWF 2:00pm to 2:50pm, room STAG 261.

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

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.

The course will cover the following topics:

• Discrete time and continuous time Markov chains.
• Mathematical models of genetic drift. Wright-Fisher model and binomial distribution.
• Application: Wright-Fisher model as a Markov chain.
• Moran process (aka 'Moran model') as a model of finite populations.
• Branching processes and their applications in genealogy.
• Birth-and-death processes. Applications in demography, epidemiology, and biology.
• Yule preferential attachment process. Application in bacteria population growth.
• Coalescent processes and their applications in population genetics.
• Other applications
A variety of mathematical techniques will be covered when analyzing these models.

Syllabus:  PDF

Assignments:

Homework #1 (due Friday, May 11):  Assignment 1 (PDF)

Schedule:
Monday, April 2  Review of probability. Conditional probability. Bayes’ Theorem. Lecture 1 slides (PDF)
Wednesday, April 4  Review of probability. Conditional probability. Bayes’ Theorem. Independent events. Lecture 2 slides (PDF)
Friday, April 6  Review of probability. Bayes’ Theorem. Independent events. Examples. Lecture 3 slides (PDF)
Review of combinatorics. Permutations and combinations. Generalized combinations. Binomial theorem. Lecture 4 slides (PDF)
Monday, April 9  Introduction to random variables. Binomial random variable. Expectation of a random variable. Wright-Fisher Model. Lecture 5 slides (PDF)
Wednesday, April 11  Binomial random variable. Expectation of a random variable. Poisson random variable. Geometric random variables. Variance and standard deviation. Lecture 6 slides (PDF)
Friday, April 13  Variance and standard deviation of discrete random variables. Markov and Chebyshev inequalities. Lecture 7 slides (PDF)
Monday, April 16  Introduction into Markov chains. Wright-Fisher model as a Markov chain. Birth-and-death processes. Moran process. Lectures 8-11 slides (PDF)
Wednesday, April 18  Birth-and-death processes. Moran process. Lectures 8-11 slides (PDF)
Wednesday, April 18, 6pm in Kidder 364  Moran process. Fixation times for Moran process. Lectures 8-11 slides (PDF)
Monday, April 30  Moran process. Fixation times for Moran process. Lectures 8-11 slides (PDF)
Wednesday, May 2  Fixation times for Moran process: alternative approach. Lectures 12-15 slides (PDF)
Wednesday, May 2, 6pm in Kidder 364  Fixation times for Moran process: alternative approach. Martingales. Lectures 12-15 slides (PDF)