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 Schedule Text books Matlab programs Requirements Links

Mathematical Modeling in Biology and Medicine

Fall 2012

Lectures: Wednesdays,
11:00-13:00Ziskind 1
Instructor: Vered Rom-Kedar

Teaching assistant:   Mary Kloc email: marykloc at gmail.com

Announcements: Notice we are back to the OLD TIME
Go to MAAA seminars on Tuesdays @ 11:00

Various strategies of modeling of biological and medical problems and various mathematical and computational tools for analyzing these models will be presented.

Mass-action kinetics  will be first discussed in various biological contexts. Modeling assumptions, scaling, and some qualitative studies of the dynamics will be presented for these problems. We will then use some of these ideas to discuss population dynamics with applications to ecology, cells dynamics, epidemiology and the immune system. These and other methodologies will be utilized to describe some of the current mathematical approaches to cancer modeling.

To participate students must have adequate mathematical background :
Calculus 1,2,3, Linear Algebra, Ordinary differential equations and basic partial differential equation undergraduate courses
Basic knowledge and programming in Matlab will be also assumed.

 Date Topics Homework Reading 31/10/12 1. Introduction: motivation, modeling & examples - single species models. [EK] pg 152-154  ex 5,7 [EK] Ch 4.1-4.6, [M] Ch 1 7/11/12 2. Modeling: non-dimensionalization, pi theorem, one dimensional models [F] Ch 2 and appendix 2 The Buckingham Pi Theorem 14/11/12 3. Population dynamics -  axiomatic models,  bifurcations, ecology, infections Homework 2 [F] Ch 2 and appendix 2 [p3] [p4] 21/11/12 4. 2d and nd models, Stability, bifurcations, structural stability and robustness [EK] pg 154-157 ex 13, 14, 25, 26 28/11/12 5. Population dynamics - interactions, innate immune system Find a research paper on Math-Biology/Medicine that interests you and uses low dimensional ODE model. Write the ODE model and explain the main assumptions and the main results of the paper. [submit after completing next week assignment] [M] Ch 3 [p5] [p6] 5/12/12 6. Population dynamics  -  difference equations, chaos [continuation] Which analytical/computational tools are utilized in the paper? Discuss the robustness of these to inaccuracies in the model (please attach the paper you considered). [M] Ch 2.2-2.4 Any of the DS books, most extended exposition Devaney  (D). 12/12/12 7. Population dynamics - spatial effects - partial differential equations approach Find a research paper on Math-Biology/Medicine that interests you and uses a PDE model.  Follow the same instructions as above (if possible, find a paper on the same subject as before and compare the results). A two-weeks assignment. [EK] 19/12/12 8.  Population dynamics - PDEs - Turing instability. Stochastic models (small populations and/or mutations). [WE] 26/12/12 9.   Population dynamics -stochastic models Find a research paper on Math-Biology/Medicine that interests you and uses a stochastic model.  Follow the same instructions as above (if possible, find a paper on the same subject as before and compare the results). A two-weeks assignment. [WE],[p7] and references therein 2/1/13 10. Mass action kinetics & robustness of networks (one hour lecture) [M] Ch 5, [p8] , [p2] 9/1/13 11. Mass action kinetics 16/1/13 12. Cancer and the innate immune system Start working on the  final assignment, see instructions below [p1], [p3-p6] 23/1/13 13. Cancer models - review. [p1], [p9],[p10] 30/1/13 14. Optimal Control in math biology models (lecture by Shalev Itzkovitz) Exam  13/2/13 Final assignment

Math Biology books and lecture notes:

Relevant Journal papers:
Dynamical systems books:
• (W) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Stephen Wiggins, 1990. (Texts in Applied Mathematics, Vol 2).
• (PR) Introduction to Dynamics. Ian Percival and Derek Richards 1982 Cambridge University Press.
• (Ott) Chaos in dynamical systems. Edward Ott. 1993 Cambridge University Press.
• (St) Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering by Steven H. Strogatz
• (Sc) Deterministic Chaos: An Introduction; Heinz Georg Schuster, [VCH, 2nd edition, 1989]
• (JS) Classical Dynamics, a contemporary approach. Jorge V. Jose and Eugene J. Saletan. 1993 Cambridge University Press.
• (H)  Chaos near resonance. G. Haller, Applied Mathematical Sciences, 138. Springer-Verlag, New York 1999
• (D) An Introduction to Chaotic dynamical systems. R. L. Devaney, 2nd Edition, 1989.
• (LS) Mathematics applied to deterministic problems in the natural sciences, C.C. Lin and L.A. Segel, McMillan Publishing Co., 1974.
• (BO) Advanced mathematical methods for scientists and engineers,  C.M. Bender and S.A. Orszag, McGraw-Hill book company, 1978

Homework assignments will be given every week  (no late submissions).
There will be an exam.
Grade:  60% homework (best 80%) +  40% exam.

Matlab Programs

rk2  rk4
logistic map  Euler scheme circle map  lorenz lorenz w manifolds  henon pendulum standard map  pendulum w forcing