MATC58 - An Introduction to Mathematical Biology - Winter 2021

• Professor: Yun William Yu
• Contact information: yw.yu@utoronto.ca
• Office hours: Thursdays 4-5pm on Gather.Town (same link as class)

Info

• Class times:
  • Tuesdays, 12-2pm and Thursdays, 3-4pm on Gather.Town
  • The first synchronous class will be on Tuesday, January 12, and the last one on Thursday, April 8. There will be no class during reading week, which is the week of Feb 13 to Feb 19.
• Preliminary outline: http://www.math.toronto.edu/ywyu/MATC58-2021-Winter/#overview
• Course calendar: http://www.math.toronto.edu/ywyu/MATC58-2021-Winter/#schedule
• Syllabus: http://www.math.toronto.edu/ywyu/MATC58-2021-Winter/2021-01-06-Syllabus.pdf
• Quercus: https://q.utoronto.ca/courses/196756
• Piazza: https://piazza.com/utoronto.ca/winter2021/matc58h3slec01
• Lectures: https://youtube.com/playlist?list=PLME5LqfUcYzkQVZGFl9vABhrl51UfrjSU

Preliminary Course Outline (subject to revision)

• Ch 1: Linear difference equations
• Ch 2: Nonlinear difference equations
• Ch 3: Biological examples of difference equations
• Ch 4: Linear differential equations
• Ch 5: Nonlinear ordinary differential equations
• Ch 6: Biological examples of differential equations

Class calendar and schedule

Session 1. Tuesday, January 12

Recommended reading: Allen Sections 1.1-1.2

What is mathematical biology? What are the tools used in modelling, understanding, and explaining biological systems? We will further give a high-level review of the taxonomy of difference equations and differential equations. Then, we will break up into groups to try solving a few introductory problems, and then present them to the rest of class. Furthermore, we will do some problems you should remember how to do from MATB44.

• Vid 0: Notes (Intro to mathematical biology)
• Vid 1.2a: Notes (Classification of difference equations)
• Vid 1.2b: Notes (Extended version of classification of difference equations)

Session 2. Thursday, January 14

Recommended reading: Allen Sections 1.3-1.4

First-order linear difference equations. Higher-order linear difference equations.

• Vid 1.3a: Notes (First order linear difference equations)
• Vid 1.3b: Notes (Extended version of first order linear difference equations)
• Vid 1.3c: Notes (Cobwebbing in Python)
• Vid 1.4a: Notes (Higher order equations)
• Vid 1.4b: Notes (Extended version of higher order equations)

Session 3. Tuesday, January 19

Recommended reading: Allen Section 1.5

Systems of first-order linear difference equations.

• Vid 1.5a: Notes (first-order linear systems)
• Vid 1.5b: Notes (extended version of first-order linear systems)
• Vid X1: Doc link (tutorial on mathematical typesetting and document preparation using HackMD).

Session 4. Thursday, January 21

Recommended reading: Allen Sections 1.6-1.7

Leslie's age-structured model and Perron-Frobenius.

• Vid 1.6-1.7a: Notes (Leslie's age structured model and Perron-Frobenius)
• Vid 1.6-1.7b: Notes (Extended version of Leslie's age structured model and Perron-Frobenius)

Session 5. Tuesday, January 26

Recommended reading: Allen Sections 2.1-2.3

Nonlinear difference equations and local stability of first-order equations

• Vid 2.2a: Notes (Intro to nonlinear difference equations)
• Vid 2.2b: Notes (Extended version of intro to nonlinear difference equations
• Vid 2.3a: Notes (Local stability of first order equations)
• Vid 2.3b: Notes (Extended version of local stability of first order equations

Session 6. Thursday, January 28

Recommended reading: Allen Sections 2.3-2.4

Nonhyperbolic equations and cobwebbing of both nonlinear functions and periodic solutions

• Vid 2.4a: Notes (Nonhyperbolic equations and cobwebbing
• Vid 2.4b: Google Colab (Cobwebbing of nonlinear and periodic functions in Python)

Session 7. Tuesday, February 2

Recommended reading: Allen sections 2.5-2.6

Globally stable equilibria and the approximate logistic difference equation

• Vid 2.5a: Notes (Global stability theorems)
• Vid 2.5b: Notes (Examples of global stability)
• Vid 2.6a: Notes (Approximate logistic equation)

Session 8. Thursday, Feburary 4

Recommended reading: Allen section 2.7

Bifurcation theory

• Vid 2.7a: Notes (Bifurcation theory)
• Vid 2.7b: Google Colab (Plotting some examples of the discrete logistic equation as we change the bifurcation parameter)

Session 9. Tuesday, February 8

Recommended reading: Allen sections 2.7-2.9

Liapunov exponents and stability of first-order systems

• Vid 2.7c: Notes (Liapunov exponents)
• Vid 2.7d: Notes (Liapunov exponents (2nd look with example)
• Vid 2.8a: Notes (stability in first-order systems and theory of jury conditions)
• Vid 2.8b: Notes (extended version of stability in first-order systems and proof of jury condition for n=2)

Session 10. Thursday, February 10

Recommended reading: Allen section 2.10, the news

SIR epidemic model (as a difference equation model). Note that today there's only a single video because we covered only one topic in depth. There's also no separation of theory and application, because the entire lecture is an application of mathematical theory we've already learned in previous videos.

• Vid 2.10: Notes

Session 11. Tuesday, February 23

Recommended reading: Allen sections 3.7 and 3.9

Population genetics and Hardy-Weinberg equilibrium. SIR epidemic model with vaccines, specifically for measles.

• Vid 3.7: Notes
• Vid 3.9: Notes and Google Colab

Session 12. Thursday, February 25

Midterm review

Please look over practice problems on Quercus before class.

Session 13. Tuesday, March 02

Midterm!

Session 14. Thursday, March 04

Recommended reading: Allen sections 4.1-4.4

Review of ODEs, integrating factors, characteristic polynomials, and linear constant coefficient ODEs

• Vid 4.2: Notes (Autonomous, linear, homogeneous definitions)
• Vid 4.3-4.4: Notes (How to solve linear ODEs, especially with constant coefficients)

Session 15. Tuesday, March 08

Recommended reading: Allen sections 4.5, 4.6, 4.7, 4.9

Routh-Hurwitz criteria, first-order linear systems of ODEs, Gershgorin circle theorem

• Vid 4.5a: Notes (Routh-Hurwitz criteria)
• Vid 4.5b: Notes (Routh-Hurwitz criteria, examples)
• Vid 4.6-4.7: Notes (first-order linear systems of ODEs)
• Vid 4.9: Notes (Gershgorin Circle Theorem review)

Session 16. Thursday, March 11

Recommended reading: Allen section 4.8

Phase plane analysis via the eigenvalues, trace, and determinant of a 2x2 matrix.

• Vid 4.8: Notes

Session 17. Tuesday, March 16

Recommended reading: Allen sections 4.10, 4.15

The matrix exponential and a linear pharmacokinetics model.

• Vid 4.X: Notes (matrix exponential lecture)
• Vid 4.10: Notes (linear pharmacokinetics model

Session 18. Thursday, March 18

Recommended reading: Allen sections 5.1-5.4

Nonlinear ODEs, local stability, and population growth models

• Vid 5.1-5.3: Notes (nonlinear ODEs and local stability)
• Vid 5.3b: Notes (local stability of a few simple population growth models)

Session 19. Tuesday, March 23

Recommended reading: Allen sections 5.5-5.6

Linearlization of systems of ODEs. Nonlinear phase-plane analysis using nullclines.

• Vid 5.5a: Notes (theory of linearization of first order systems and local stability analysis)
• Vid 5.5b: Notes (example of linearization of first order systems and local stability analysis)
• Vid 5.6: Notes (phase plane analysis using nullclines)

Session 20. Thursday, March 25

Recommended reading: Allen section 5.7. Strogatz section 7.3.

Periodic solutions and Poincare-Bendixson Theorem

• Vid 5.7: Notes

Session 21. Tuesday, March 29

Recommended reading: Allen sections 5.8, 5.10

Bifurcations (including Hopf bifurcation) and qualitative matrix stability

• Vid 5.8: Notes (bifurcations)
• Vid 5.10: Notes (qualitative matrix stability)

Session 22. Thursday, March 31

Recommended reading: Allen section 5.11

Global stability of systems of differential equations and Liapunov functions

• Vid 5.11: Notes

Session 23. Tuesday, April 5

Recommended reading: Allen section 6.8

Epidemic modelling using compartmentalized ODE systems

• Vid 6.8: Notes

Session 24. Thursday, April 7

Student presentations!

Final Examination. Friday, April 23, 8am

Will be proctored over Zoom and Crowdmark. You are allowed a single double-sided 8.5"x11" handwritten cheat sheet, which you must send to me before class. There will be no other outside materials allowed.