# MATH 428/528: Stochastic Elements in Mathematical Biology

## Spring 2021

 Instructor: Yevgeniy Kovchegov e-mail: kovchegy @math. oregonstate.edu Office: Kidder 60 Office Phone No: 7-1379 Office Hours: by appointment, via Zoom
Instructions: MWF 1:00pm to 1:50pm, via Zoom.

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

Schedule:
Monday, March 29  Review of probability. Conditional probability. Bayes’ Theorem. Lectures 1-3 slides (PDF)
Wednesday, March 31  Review of probability. Conditional probability. Bayes’ Theorem. Independent events. Lectures 1-3 slides (PDF)
Friday, April 2  Review of probability. Bayes’ Theorem. Independent events. Examples. Lectures 1-3 slides (PDF)
Monday, April 5  Review of combinatorics. Permutations and combinations. Generalized combinations. Binomial theorem. Lecture 4 slides (PDF)
Wednesday, April 7  Introduction to random variables. Binomial random variable. Expectation of a random variable. Wright-Fisher Model. Lecture 5 slides (PDF)
Friday, April 9  Binomial random variable. Expectation of a random variable. Wright-Fisher Model. Poisson random variable. Geometric random variables. Variance and standard deviation. Lecture 6 slides (PDF)
Monday, April 12  Variance and standard deviation of discrete random variables. Markov and Chebyshev inequalities. Lecture 7 slides (PDF)
Wednesday, April 14  Introduction into Markov chains. Wright-Fisher model as a Markov chain. Birth-and-death processes. Moran process. Lectures 8-11 slides (PDF)
Friday, April 16  Birth-and-death processes. Moran process. Lectures 8-11 slides (PDF)
Monday, April 19  Moran process. Fixation times for Moran process. Lectures 8-11 slides (PDF)
Wednesday, April 21  Moran process. Fixation times for Moran process. Lectures 8-11 slides (PDF)
Friday, April 23  Fixation times for Moran process: alternative approach. Lectures 12-15 slides (PDF)
Monday, April 26  Fixation times for Moran process: alternative approach. Martingales. Lectures 12-15 slides (PDF)
Wednesday, April 28  Martingales. Stopping times. The Optional Stopping Theorem. Lectures 12-15 slides (PDF)
Friday, April 30  The Optional Stopping Theorem. Probability harmonic functions. Lectures 12-15 slides (PDF)
Monday, May 3  Moran model with mutation. Stationary distribution. Lectures 16-20 slides (PDF)
Wednesday, May 5  Detailed balance conditions and reversibility. Stationary distribution for a birth-and-death chain. Lectures 16-20 slides (PDF)
Friday, May 7  Stationary distribution for Moran model with mutation. Lectures 16-20 slides (PDF)