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:
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.
Prerequisites: 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.
Midterm Exam
1 |
25 % |
Midterm Exam
2 |
25 % |
Final Exam | 35 % |
Homework |
15 % |
The exams may be evening exams.
Week |
Tuesday |
Thursday |
1 |
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. |
2 |
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. |
3 |
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. |
4 |
Sept. 24th Topic: Invariant distributions. Readings: Section 2.5. |
Sept. 26th Topic: Transient behavior. Readings: Section 2.6. |
5 |
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. |
6 |
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. |
7 |
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. |
8 |
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. |
9 |
Oct.
29th HW 6 Due. Topic: Point and Poisson processes. Readings: Section 4.2. |
Oct. 31st Topic: Poisson processes. Readings: Section 4.2. |
10 |
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. |
11 |
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. |
12 |
Nov. 19th Topic: The generator revisited. Readings: Sections 5.5. |
Nov. 21st
Topic: CTMCs: stationary distributions and limiting behavior. Readings: Section 5.4. |
13 |
Nov. 26th Exam 2 (scheduled for
Monday evening - the 25th) Topic: Stationary distributions. Birth and death processes. Readings: Section 5.4 and 5.6. |
Nov. 28th No class due to Thanksgiving |
14 |
Dec. 3rd Topic: Stochastic models of biochemical processes: general model. Readings: 6.1. |
Dec. 5th Topic: Stochastic models of biochemical processes: general models and simulation. Readings: 6.1 and 6.2. |
15 |
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. |