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.