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: firstname.lastname@example.org)
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 Chemotaxis (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