Course description: Mathematical and quantitative modeling of biological phenomena has become a necessary approach at studying living organisms. Mathematical biologists work towards identifying unified frameworks that describe biological phenomena in different levels of complexity. It ranges from small entities such as viruses and bacteria, cells in tissues and, ecological models where animals compete for resources in a habitat. This course introduces basic concepts of mathematical biology with modern applications in mind. We use dynamical systems tools to model mechanisms of cell growth and cell- cell interaction as well as complex evolution of species. We discuss subjects such as microbial growth, cancer evolution and virus dynamics and epidemiology.

Course time: 2020 Winter 2A (2:25-4:15)

Course Location: Kemeny 004

Instructor: Kamran Kaveh

(Office: Kemeny 244, Email:

Prerequisites: Math 3. Familiarity with programming (for example: MATLAB, Python or Mathematica). Basic probability highly recommended.

Grading: Projects (50%), Problem sets (40%), Attendance & Participation (10%)

Projects: Approximately 4 weeks are given to complete the project. The instructor will suggest project ideas in the third week, but you are allowed to propose your own, which has to be approved by the instructor in the fourth week at the latest. Each project presentation is limited to 15 minutes and preferably in the style of TED talks.

Late policy: By “deadline” we mean it. On the condition of accepting the penalty for turning in homework late (that is, 5% each additional day), however, an extension of maximum 2 days will be granted on a case-by-case basis.

Collaboration policy: Collaborations on problem sets are permitted, but you are required to disclose the names of the other students with whom you collaborated and discussed. You are encouraged to work in groups for course projects (but no more than 2 individuals per project).

Textbooks: Mainly notes. Useful references are “Evolutionary Dynamics”, M. A. Nowak (2006). “Mathematical Biology”, J. D. Murray (2003).


Day 1: Introduction & Overview

Day 2: Population dynamics
How to write ODE for biological dynamics
Fixed points and steady states
Example: Logistic growth

Day 3: Population Dynamics II
Natural selection
Replicator equation
Discrete Time models

Day 4: Interacting populations
Cell-cell interaction and signaling
Predator-prey equation. Evolutionary games
Replicator equation revisited
Stability analysis of complex populations
(Problem set 1: Growth models)

Day 5-6: Evolution of antibiotic resistance
Mechanisms of antibiotic resistance
Death-Galaxy experiment
Mutation-selection processes
Mutational pathways

Day 7: Migration and dispersal
Cell motility
Random walks in bacteria
(Problem set 2)

Day 8: Stochastic models of growth and competition
Branching process
Moran process
Fixation probability
Time to fixation

Day 9-10: Growth and competition in tissues and networks
Moran process on networks
death-birth vs birth-death processes
(Problem set 3)

Day 11: Cancer modeling. I
Oncogenes and Tumor suppressor genes
Moran models of cancer initiation

Day 12: Cancer modeling. II
Tumor microenvironment
Spatial models of tumor growth

Day 13: Stem cell dynamics
Self-renewal vs. differentiation
Population dynamics of stem cells
Stem cells in cancer
(Problem set 4)

Day 14: Infectious disease models. I
S-I model
Basic reproduction number
Dynamics of Immune response

Day 15: Infectious disease models. II
Evolution of virulence
HIV dynamics
Drug resistance in HIV
(Problem set 5)

Day 16: Epidemiology and network models
S-I model on networks
Vaccination strategies

Day 17: Viruses meets Cancer cells: Oncolytic viral therapy
Oncolytic virus dynamics
Spatial models of oncolytic viral therapy

Days 18-19: Project Presentations