Math 605 - Stochastic Methods for Biology, Fall 2013

Course Summary:  This course is, first and foremost, an introduction to stochastic processes (models that incorporate randomness) that is equivalent in level, and to a large extent content, to Math 632.  Thus, we will cover the following mathematical objects:
However, the applications will be drawn primarily from the biosciences, with a special emphasis on the continuous time Markov chains used to model biochemical and other population processes. Further, as simulation is a large part of how scientists study their models, we will spend considerable time on computational methods.

You may download the Syllabus.

Tuesday and Thursday, 1:00 PM - 2:15 PM. 
Location: Van Vleck B139.

David Anderson
   Office: 617 Van Vleck
   E-mail: anderson at math dot wisc dot edu

Office hours:  Tuesday, 3:30-4:30pm, Friday 1:00-2:00pm, and by appointment.

Textbook: There is no text for the course.  Instead, I will provide lecture notes, which can be found directly below.  Some optional texts, which I am going to put on reserve in the math library, are:

  1. Stochastic modelling for systems biology, by Darren J. Wilkinson.
  2. Introduction to Stochastic processes, 2nd ed., by Gregory F. Lawler.  
  3. Adventures in Stochastic Processes, by Sidney Resnick.

Lecture notes.  I will post the lecture notes as they become available.  Below are current as of April 9th, 2013. 

A note to visitors to this site (i.e. not my students):  Please feel free to peruse these notes.  However, I ask that you please notify me of any typos, or potential improvements, as these are a work in progress and such comments would be extremely useful.
The course notes are available here.

Intended audience: Advanced undergraduate students and/or graduate students in mathematics, physics, computer science, engineering, and related disciplines with an interest in biology, and students in biochemistry, biology, and related disciplines, with an interest in quantitative approaches in biology.

It is important that the student has taken Calculus and an introductory probability course (at the level of Math 331/431 or Stat 309/311).  A basic knowledge of ordinary differential equations and linear algebra is also important.  Having some experience with writing codes would also be useful, though not strictly necessary as this can be learned during the course.

Homework:  Homework assignments will be given throughout the semester.  Every student must turn in an individual assignment.  However, working on the assignment in groups is highly encouraged.

Matlab:  Each homework assignment will involve some use of MATLAB, which is a mathematical software package.  If you do not already have MATLAB (and I'm guessing most, if not all, of you do not have MATLAB on your personal computers) then you can use MATLAB on any Windows machine in a University of Wisconsin computer lab.  A list of the labs, with a map, is here.

Grading: In determining your final grade your work will be weighted in the following manner:

Midterm Exam 1
25 %
Midterm Exam 2
25 %
Final Exam 35 %
15 %

  1. Exam 1 time is Tuesday evening, October 15th from 7:15 - 9:15pm.  The exam will cover material up through and including Section 3.2.  The exam room is 6203 Social Sciences.
  2. Exam 2 date is scheduled for Monday evening, November 25th from 7:15 - 9:15pm.  The exam room is B239 in Van Vleck
  3. Final Exam is set for Monday, December 16th, from 12:25 - 2:25PM.  This can not be changed, and you *must* take the exam at this time.  The exam room is Ingraham 120.

The exams may be evening exams.

Sample MATLAB files:

Homework assignments:  All homework is due at the beginning of class on the due date.
  1. Due Tuesday, Sept. 10th.  Appendix A, #'s 1 - 8.  Appendix B, #'s 1 - 4. 
  2. Due Tuesday, Sept. 17th.  Chapter 2, #'s 1 - 7.
  3. Due Tuesday, Oct. 1st.  Chapter 2, #'s 8 - 13.
  4. Due Tuesday, Oct. 8th.  Chapter 2, #'s 14 - 16.
  5. Due Thursday, Oct. 17th.  Chapter 3, #'s 1 - 7.  I am not providing a MATLAB code for number 7.  You should be able to tweak a code from a previous problem.   Also, note that the death rate is different in the cases X_n = 0 vs X_n > 0!  Turn in your code.
  6. Due Tuesday, Oct. 29th.  Chapter 3, #'s 8 - 13.  I am not providing MATLAB code for the two computer oriented problems.  At this point, you should be able to write these types of scripts.  If you need help, SEE ME.  Here is my MATLAB code for problem number 13.
  7. Due Thursday, November 14th.  Chapter 4, #'s 1 - 10.
  8. Due Tuesday, December 10th.  Chapter 5, #'s 1, 2, 3, 5, 6, 7, 8, 9.  Solutions posted at
Fall 2013 Schedule: This is mainly to be determined and is always subject to change.  The readings refer to what should be read in preparation for that day's class.

Sept. 3rd
Topic:  Introduction to the course (slides) and background material on the theory of probability (slides). 
Readings: Chapter 1, Appendix (background material on differential/difference equations and probability).
Sept. 5th
Topic:  Discrete Time Markov Chains: description and first properties; examples.
Readings: Section 2.1.
Sept. 10th HW 1 Due.
Topic:  Discrete Time Markov Chains: simulation, higher order transition probabilities. 
Readings: Sections 2.2 - 2.3. 
Sept. 12th.
Topic:  Discrete time Markov chains: reducibility, periodicity, recurrence and transience.
Readings: Section 2.4.
Sept. 17th  HW 2 Due.
Topic: Discrete time Markov chains: reducibility, periodicity, recurrence and transience.  Begin stationary distributions.
Readings: Sections 2.4 and 2.5.
Sept. 19th
Topic: Discrete time Markov chains: Invariant/stationary distributions.
Readings: Section 2.5.
Sept. 24th
Topic: Invariant distributions.
Readings: Section 2.5.
Sept. 26th
Topic: Transient behavior.
Readings: Section 2.6.
Oct. 1st HW 3 Due.
Topic: DTMC in biosciences: Genetic models.
Readings: Section 3.1.
Oct. 3rd  Class cancelled (made up during evening exam).
Topic: NA.
Readings: NA.
Oct. 8th  HW 4 Due.
Topic: DTMC in biosciences:  Birth and death processes.
Readings: Section 3.3.
Oct. 10th
Topic: DTMC in biosciences:  Branching processes.
Readings: Section 3.3.
Oct. 15th  Exam 1 (scheduled for evening)  Room = 6203 Social Sciences.
Topic: Tentative; Guest Lecture by Bret Hanlon (Statistics Department) on Parameter Estimation for DTMCs.
Readings: NA.
Oct. 17th HW 5 Due.
Topic: DTMC in biosciences:  Branching processes.
Readings: Section 3.3.
Oct. 22nd
Topic: Finish branching processes.  Begin renewal processes.
Readings: Sections 3.3 and 4.1.
Oct. 24th
Topic: Renewal processes.  Renewal reward processes. 
Readings: Section 4.1.
Oct. 29th  HW 6 Due.
Topic: Point and Poisson processes.
Readings: Section 4.2.
Oct. 31st
Topic: Poisson processes.
Readings: Section 4.2.
Nov. 5th
Topic: Transformations of Poisson processes.
Readings: Section 4.2.
Nov. 7th
Topic: Finish transformation of Poisson processes.  Begin continuous time Markov chains (CTMCs): basic properties.
Readings: Section 4.2. and Sections 5.1.
Nov. 12th
Topic: CTMCs: basic properties and explosions.
Readings: 5.1 - 5.2.
Nov. 14th  HW 7 Due.
 Topic: CTMCs: explosions;  Forward and backward equations, generator matrix.
Readings: Sections 5.2 - 5.3.
Nov. 19th
Topic: The generator revisited.
Readings: Sections 5.5.
Nov. 21st 
Topic:  CTMCs: stationary distributions and limiting behavior.
Readings: Section 5.4.
Nov. 26th Exam 2 (scheduled for Monday evening - the 25th)
Stationary distributions.  Birth and death processes.
Readings: Section 5.4 and 5.6.
Nov. 28th
No class due to Thanksgiving
Dec. 3rd
Topic: Stochastic models of biochemical processes: general model.
Dec. 5th
Topic: Stochastic models of biochemical processes: general models and simulation.
Readings: 6.1 and 6.2.
Dec. 10th HW 8 Due.
Topic: Stochastic models of biochemical processes: first order reaction networks, and the law of large numbers.
Readings: 6.3 - 6.4.
Dec. 12th
Topic: The classical scaling and the law of large numbers.
Readings: 6.4.