MATC58 - An Introduction to Mathematical Biology - Winter 2023

• Professor: Yun William Yu
• Contact information: yw.yu@utoronto.ca
• Office hours: Thursdays 1-2pm in IC 343

Info

• Class times:
  • Mondays, 11-12pm in HL B108 and Thursdays, 11-1pm in HL B106
  • The first synchronous class will be on Monday, January 9, and the last one on Thursday, April 6. There will be no class during reading week, which is the week of Feb 20 to Feb 24.
• Preliminary outline: https://courses.ywyu.net/MATC58-2023-Winter/#overview
• Course calendar: https://courses.ywyu.net/MATC58-2023-Winter/#schedule
• Syllabus: https://courses.ywyu.net/MATC58-2023-Winter/Syllabus-20230105.pdf
• Quercus: https://q.utoronto.ca/courses/287634
• Piazza: https://piazza.com/utoronto.ca/winter2023/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. Monday, January 9

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 12

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. Monday, January 16

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)

Session 4. Thursday, January 19

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. Monday, January 23

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 26

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. Monday, January 30

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 2

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. Monday, February 6

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 9

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. Monday, February 13

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 16

In-class midterm!

Session 13. Monday, February 27

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 14. Thursday, March 02

Recommended reading: Allen sections 4.5-4.7

Routh-Hurwitz criteria and first-order linear systems of ODEs

• 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)

Session 15. Monday, March 06

Recommended reading: Allen sections 4.8

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

• Vid 4.X: Notes (matrix exponential lecture)
• Vid 4.8: Notes (phase plane analysis)

Session 16. Thursday, March 09

Recommended reading: Allen section 4.9, 4.15

Gershgorin circle theorem. A linear pharmacokinetics model.

• Vid 4.9: Notes (Gershgorin Circle Theorem review)
• Vid 4.10: Notes (linear pharmacokinetics model

Session 17. Monday, March 13

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 18. Thursday, March 16

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 19. Monday, March 20

Recommended reading: Allen section 5.7. Strogatz section 7.3.

Periodic solutions and Poincare-Bendixson Theorem

• Vid 5.7: Notes

Session 20. Thursday, March 23

Recommended reading: Allen section 5.8

Bifurcations (including Hopf bifurcation)

• Vid 5.8: Notes (bifurcations)

Session 21. Monday, March 27

Recommended reading: Allen sections 5.10

Qualitative matrix stability

• Vid 5.10: Notes (qualitative matrix stability)

Session 22. Thursday, March 30

Recommended reading: Allen section 5.11

Global stability of systems of differential equations and Liapunov functions

• Vid 5.11: Notes

Session 23. Monday, April 3

Recommended reading: Allen section 6.8

Epidemic modelling using compartmentalized ODE systems

• Vid 6.8: Notes

Session 24. Thursday, April 6

Student presentations!

Final Examination. Tuesday, April 25, 9-11am.