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Summer
2002 Modules for KUCSEC
L.G. de Pillis and A.E. Radunskaya
Modeling Tumor-Immune Interactions Mathematics
Lesette de Pillis
Harvey Mudd College
depillis@hmc.edu |
Amy Radunskaya
Pomona College
Ami_Radunskaya@pomona.edu |
Nonlinear Ordinary Differential Equations, Bifurcation Analysis
Numerics: Various ODE solvers, explicit, implicit, non-stiff, stiff
Time-line: Week 1: Model development and qualitative analysis Week
2: Numerical issues in solving ODE, introduction of explicit, implicit,
stiff solvers Week 3: Bifurcation analysis, numerical continuation
In this module we introduce the Tumor-Immune system interaction
equations developed by Kuznetsov(1994). This model is of particular
interest because the mathematics are straightforward, yet the dynamics
of the system are quite rich. We will begin by developing each component
of the differential equations from biological principals. We will
then take the students through an elementary qualitative analysis
of the system, finding critical points and stabilities. The computational
component will then be introduced, and this will allow us to familiarize
the students with simple explicit, implicit and specialized stiff
numerical ODE solvers. The students will learn about the benefits
and drawbacks of each category of solver, and will be guided through
a basic stability analysis of each solver. Once we start running
numerical experiments, we will have the students experiment with
parameter changes, and discover the impacts on the dynamics of the
system. The system we will be working with has several bifurcation
points, equilibria appear and disappear and stability properties
change. We will examine the numerical problems encountered in the
analysis of these bifurcations by investigating methods of numerical
continuation. Continuation is also important in the numerical detection
of unstable equilibria and separatrices. We will then guide the
students through a discussion of the biological interpretation of
these parameter changes. For example, this model allows for an explanation
of tumor dormancy, a phenomenon which is still not fully understood
biologically.
Bifurcation
Bifurcation Slides
EqnDev
EqunDev Slides
Exercises
Immunebackground
intro
intro slides
matlabscripts
module1numerics
module1numerics slides
nondimensionalization
nondimensionalization
slides
overview
parameter_estimation
parameter_estimation
slides
projects
qualanalys
qualanalys slides
wrapup
wrapup slides
For more information, please contact
Ignatios Vakalis (614-236-6587)
Principle Investigator
Andrea Karkowski (614-236-6449)
Co-PI
Terry Lahm (614-236-6800)
Co-PI
Margie Gilbert (614-236-6714)
Grant Administrator
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