Math 865


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Course: Mathematical Modeling of Biological Processes


References: (1) Models of Cellular Regulation by Baltazar Aguda and Avner Friedman

bullet Policy: Late Homework and project will be no credit.
bulletTentative Schedule: updated regularly


Tuesday: EA 295 Thursday: EA 295
Mar 30

Chemical Kinetics

Mass action law

Michaelis-Menten and Hill type kinetics

Basic ODE thoery


uniqueness by successive iterations



Apr 1

Stability of steady state for one ODE

Phase portraits in the plane

Nullclines and Bistability


Apr 6

Bifurcation diagram

Bistability and hystereris

Hopf bifurcation

Singular perturbation

Apr 8

Euler Method: Convergence and Stability

Computation: XPPAUT introduction



Apr 13

Virus dynamics

Ref: Leenheer & Smith, Virus Dynamics: A Global Analysis

SIAM J. Appl. Math Vol 63, No.4 pp 1313-1327, 2003

Basic reproduction number


Apr 15

HW: (1) pick f_1 and f_2 you like to generate similar figure as Fig.3.5 in p.27 (handout) and describe the behavior of equalibrium points

Solve the virus model with N = 15 and N = 30. Discuss the biological meanings.

Computation: XPPAUT introduction II

Numerical experiments of virus dynamics


Apr 20

Epidemiological models

SIR model

SIER model

Reference book: Mathematical epidemiology by Fred Brauer, Pauline Van den Driessche, Jianhong Wu and Linda J. S. Allen, Springer, 2008

Apr 22

Computation: XPPAUT introduction III



Apr 27

Cell cycle

The Goldbetter model

Reaction Diffusion equations

Hyperbolic systems

Apr 29

Simulation of the Goldbetter model

Computation: XPPAUT introduction IV




May 4

System of Diffusion Reaction Model

Tumor model

May 6

Simulation for parabolic and hyperbolic equations

Matlab pdepe help



May 11

Tumor model with several cell types

May 13

Simulations for viral therapy of tumor



May 18

Tumor model with several cell types

A model of radially symmetric tumor and it stationary solution

Stability/instability of the stationary solution

Cell differentiation; the Yates-Callard-Stark (YCS) model of Th0 differentiation into Th1 and Th2

Cell differentiation (continued), asymptotic behaviour

May 20

Numerical simulation for cell differentiation


May 25 Project I: first reference: Dictyostelium discoideum: cellular self-organization in an excitable biological medium by Thomas Hofer, Jonathan A. Sherratt and Philip K. Maini, Proc R. Soc Lon. B (1995) 259, 249-257

second reference: Cellular pattern formation during Dictyostelium aggregation by Thomas Hofer, Jonathan A. Sherratt and Philip K. Maini, Physica D 85 (1995) 425-444
May 27 Project II: Malaria model with periodic mosquito birth and death rates. Bassidy Dembele, Avner Friedman and Abdul-Aziz Yakubu, Journal of Biological Dynamics, Vol 3, No. 4, July 2009, 430-445

Jun 1 Project III: Reference I, A mathematical model of tumor-immune evasion and siRNA treatment. J.C. Arciero, T.L. Jackson, and D.E. Kirschner. Discrete and Continuous Dynamical Systems-Series B, Vol 4, No. 1, Feb 2004, 39-58

Project III: Reference II, Modeling immunotherpy of the tumor--immune iteraction by Denise Kirschner and John Carl Panetta, J. Math. Biol. (1998)37: 235-252

Jun 3 Project IV: Reference I, A Mathematical Model for Collagen Fibre Formation during Foetal and Adult Dermal Wound Healing. Paul D. Dale,; Jonathan A. Sherratt and; Philip K. Maini, Proc. R. Soc. Lond. B (1996) 263, 653-660
Jun 8

Project V: Leukocytes. James Keener, James Sneyd. Mathematical Physiology. Vol II: Systems Physiology, Chapter 16, 495-507.

Jun 10

Project VI: Respiration. James Keener, James Sneyd. Mathematical Physiology. Vol II: Systems Physiology, Chapter 17, 516-527.