MA354

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MA354

• Mathematical Modeling
• T H 245 pm 400 pm
• Dr. Audi Byrne

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Your Instructor

• Instructor Dr. Audi Byrne

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Dr. Audi ByrnePhD in mathematics from the
University of Notre Dame

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Dr. Audi ByrneResearch area in
biomathematics.(Dynamical systems and modeling. )

Cellular automata
Stochastic Processes
Multi-cellular Systems
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Contacting Your Instructor

• Office ILB 452
• Office Hours 1000am-1100am daily
• And by appointment.
• E-mail abyrne_at_jaguar1.usouthal.edu

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Course Information

• Course webpage
• www.southalabama.edu
• ?Math and Statistics
• ?Faculty and Staff
• ?Audi Byrne
• ? link to personal homepage
• ? teaching
• ? MA 354
• http//www.southalabama.edu/mathstat/personal_page
  s/byrne/MA354.htm

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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.

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Types of Models

• Discrete or Continuous
• Stochastic or Deterministic
• Simple or Sophisticated
• Good or bad (elegant or sloppy)
• Validated or Invalidated

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Modeling Approaches

• Continuous Approaches (PDEs)
• Discrete Approaches (lattices)

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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 Ficks Law
• Fluids Navier-Stokes Equation

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Continuous Models

• http//math.uc.edu/srdjan/movie2.gif
• Biological applications
• Cells/Molecules density field.

Rotating Vortices
http//www.eng.vt.edu/fluids/msc/gallery/gall.htm
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Discrete Models

• E.g., cellular automata.
• Typically describe micro-scale events and
  short-range interactions
• Local rules define particle behavior
• Space is discrete gt space is a grid.
• Time is discrete gt simulations and timesteps
  
• Good models when a small number of elements can
  have a large, stochastic effect on entire system.

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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

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Stochastic Models

• Accounts for random, probabilistic phenomena by
  considering specific possibilities.
• In practice, the generation of random numbers is
  required.
• Different result each time.

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Deterministic Models

• One result.
• Thus, analytic results possible.
• In a process with a probabilistic component,
  represents average result.

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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.

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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.

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Spatially Explicit Models

• Spatial variables (x,y) or (r,?)
• Generally, much more sophisticated.
• Generally, much more complex!
• ODE no spatial variables
• PDE spatial variables
• (most general difference)

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Other Ways that Model Differ

• What is being described?
• The domain of the function.
• The largest expected diameter of a tumor.
• The diameter of the tumor over time.
• The shape of the tumor over time.

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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.

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Objective 2 Model Construction

• The second objective is to model real-world
  observations by making appropriate simplifying
  assumptions and identifying key factors.

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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.
• Accuracy simplicity and computational
  efficiency may trump accuracy.

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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 ky
• Inversely proportional relationship
• xk/y
• More complex relationship
• Non-linearity of relationship often critical
• Exponential
• Sigmoidal
• Arbitrary functions

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Hookes Law

• An ideal spring.
• F-kx
• x displacement (variable)
• k spring constant (parameter)
• F resulting force vector

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Other Examples

• Circumference of a circle is proportional to r
• Weight is proportional to mass and the
  gravitational constant