MA354
Mathematical Modeling
T H 2:45 pm– 4:00 pm
Dr. Audi Byrne
Your Instructor
• Instructor: Dr. Audi Byrne
Dr. Audi Byrne
PhD in mathematics from the University of Notre Dame
Dr. Audi Byrne
Research area in biomathematics.
(Dynamical systems and modeling. )
Cellular automata
Multi-cellular Systems
Stochastic Processes
Contacting Your Instructor
• Office: ILB 452
• Office Hours: 10:00am-11:00am daily
And by appointment.
• E-mail: abyrne@jaguar1.usouthal.edu
Course Information
• Course webpage
• Google ‘Byrne South Alabama’
• Eventually, stuff on Ecompanion.
Mathematical Modeling
• Model design:
– Models are extreme simplifications!
– A model should be designed to address a particular question; for a focused
application.
– The model should focus on the smallest subset of attributes to answer the
question.
• Model validation:
– Does the model reproduce relevant behavior?  Necessary but not
sufficient.
– New predictions are empirically confirmed.  Better!
• Model value:
– Better understanding of known phenomena.
– New phenomena predicted that motivates further expts.
Types of Models
• Discrete or Continuous
• Stochastic or Deterministic
• Simple or Sophisticated
• Good or bad (elegant or sloppy)
• Validated or Invalidated
Continuous or Discrete
Modeling Approaches
• Continuous Approaches (PDEs)
• Discrete Approaches (lattices)
Continuous Models
• Good models for HUGE populations (1023),
where “average” behavior is an appropriate
description.
• Usually: ODEs, PDEs
• Typically describe “fields” and long-range
effects
• Large-scale events
– Diffusion: Fick’s Law
– Fluids: Navier-Stokes Equation
Continuous Models
http://math.uc.edu/~srdjan/movie2.gif
Rotating
Vortices
Biological applications:
Cells/Molecules = density field.
http://www.eng.vt.edu/fluids/msc/gallery/gall.htm
Discrete Models
• E.g., cellular automata.
• Typically describe micro-scale events and short-range
interactions
• “Local rules” define particle behavior
• Space is discrete => space is a grid.
• Time is discrete => “simulations” and “timesteps”
• Good models when a small number of elements can
have a large, stochastic effect on entire system.
Hybrid Models
• Mix of discrete and continuous components
• Very powerful, custom-fit for each application
• Example: Modeling Tumor Growth
– Discrete model of the biological cells
– Continuum model for diffusion of nutrients and
oxygen
– Yi Jiang and colleagues
Stochastic vs Deterministic
Stochastic Models
• Accounts for random, probabilistic phenomena
by considering specific possibilities.
• In practice, the generation of random numbers
is required.
• Different result each time.
Deterministic Models
• One result.
• Thus, analytic results possible.
• In a process with a probabilistic component,
represents average result.
Stochastic vs Deterministic
• Averaging over possibilities  deterministic
• Considering specific possibilities  stochastic
• Example: Random Motion of a Particle
– Deterministic: The particle position is given by a
field describing the set of likely positions.
– Stochastic: A particular path if generated.
Other Ways that Model Differ
• What are the variables?
– A simple model for tumor growth depends upon
time.
– A less simple model for tumor growth depends
upon time and average oxygen levels.
– A complex model for tumor growth depends upon
time and oxygen levels that vary over space.
Spatially Explicit Models
• Spatial variables (x,y) or (r,)
• Generally, more sophisticated.
• Generally, more complex!
• ODE: no spatial variables
• PDE: spatial variables
Other Ways that Model Differ
• What is being described?
– The largest expected diameter of a tumor.
– The diameter of the tumor over time.
– The shape of the tumor over time.
Objective 1: Model Analysis and Validity
The first objective is to study the behavior
of mathematical models of real-world
problems analytically and numerically. The
mathematical conclusions thus drawn are
interpreted in terms of the real-world problem
that was modeled, thereby ascertaining the
validity of the model.
Objective 2: Model Construction
The second objective is to model real-world
observations by making appropriate
simplifying assumptions and identifying key
factors.
Model Construction..
• A model describes a system with variables {u,
v, w, …} by describing the functional
relationship of those variables.
• A modeler must determine and “accurately”
describe their relationship.
• Regarding Accuracy: simplicity and
computational efficiency may trump accuracy.
Functional Relationships
Among Variables x,y
• No Relationship
– Or effectively no relationship.
– No need to use x in describing y.
• Proportional Relationship
– Or approximately proportional.
– x = k*y
• Inversely proportional relationship
– x=k/y
• More complex relationship
–
–
–
–
Non-linearity of relationship often critical
Exponential
Sigmoidal
Arbitrary functions
Hooke’s Law
• An ideal spring.
• F=-kx
x = displacement
k = spring constant
F = resulting force vector
(variable)
(parameter)
Other Examples
• Circumference of a circle is proportional to r
• Weight is proportional to mass and the
gravitational constant