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MA354
An Introduction to Math Models
(more or less corresponding to
1.0 in your book)
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 experiments.
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 experiments.
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 experiments.
Objective 1: Model Analysis and Validity
The first objective is to study mathematical models 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 build models of real-world phenomena 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.
• Note: pragmatically, simplicity and computational efficiency often trump accuracy.
MA354
(Part 1)
Classifying Models
Classifying Models
• By application (ecological, epidemiological,etc)
• Discrete or continuous?
• Stochastic or deterministic?
• Simple or Sophisticated
• Validated, Hypothetical or Invalidated
DISCRETE OR CONTINUOUS?
Discrete verses Continuous
• Discrete:– Values are separate and distinct (definition)– Either limited range of values (e.g., measurements
taken to nearest quarter inch)– Or measurements taken at discrete time points (e.g.,
every year or once a day, etc.)
• Continuous– Values taken from the continuum (real line)– Instantaneous, continuous measurement (in theory)
Modeling ApproachesContinuous Verses Discrete
• Continuous Approaches (differential equations)
• Discrete Approaches (lattices)
Modeling ApproachesContinuous Verses Discrete
• Continuous Approaches
(smooth equations)
• Discrete Approaches
(discrete representation)
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
Biological applications:
Cells/Molecules = density field.
http://www.eng.vt.edu/fluids/msc/gallery/gall.htm
Rotating Vortices
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:
• Deterministic Approaches– Solution is always the same and represents the average
behavior of a system.
• Stochastic Approaches– A random number generator is used.– Solution is a little different every time you run a simulation.
• Examples: Compare particle diffusion, hurricane paths.
Modeling ApproachesDeterministic Verses Stochastic
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 is being described?
• What question is the model trying to investigate?
• Example: An epidemiology model that describes the spread of a disease throughout a region, verses one that tries to describe the course of a disease in one patient.
Increasing the Number of Variables Increases the Complexity
• 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, much more sophisticated.
• Generally, much more complex!
• ODE: no spatial variables
• PDE: spatial variables
MA354
(Part 2)
Models Describe Relationships
Between Variables
Functional Relationships Among Variables x,y
• No Relationship– Or effectively no relationship.– No need (and not useful) 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
A Relationship Between Two Quantities
• Points to an interaction– May be direct– May be indirect
• In my opinion, a good model correctly describes their interactionExample: oranges and soap bubbles both form
spheres, but for different reasons
Example: Hooke’s Law
• An ideal spring.
• F=-kxx = displacement (variable)
k = spring constant (parameter)
F = resulting force vector
Other Examples
• Circumference of a circle is proportional to r
• Weight is proportional to mass and the gravitational constant
• Etc.