MA354

Math Modeling Introduction

Outline

A. Three Course Objectives1. Model literacy: understanding a typical model description2. Model Analysis3. Building Models

B. What is a “model”?Models describe relationships among quantities.

C. Building a ModelD. Model Classifications

A. Course Objectives

Interpreting the Mathematical Description of a Model

Implicit and discrete:

System of equations:

Exotic or unfamiliar model: (statistical mechanics)

Course Objectives

• Objective 2: Model Analysis and Validity The second 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 3: Model Construction The third objective is to learn to build models of real-world phenomena by making appropriate simplifying assumptions and identifying key factors.

B. What is a model?

Not the type of model we mean:

Not the type of model we mean:

Also not the type of model we mean:

Describing a relationship among concepts

Also not the type of model we mean:

Describing a relationship among concepts

Also not the type of model we mean:

Describing a relationship among concepts

Also not the type of model we mean:

Describing a relationship among concepts

Fluid Mosaic “Model”

For us, a model is:

• A set of variables {u, v, w, …} – Selected based on those the a phenomenon of interest is

hypothesized to depend on– Together define a system

• A description of the functional quantitative relationship of those variables

Simple example:

Variables : force, mass, accelerationQuantitative relationship is very simple

• force proportional to mass• force proportional to acceleration

“Interesting” examples:• In my opinion, we don’t have a modeling class to study

models like (Studying these equations is important, but when we study them, we are studying physics and/or mathematics.)

• Principles of modeling come into play as the relationships become more interesting:– Antagonistic effects (trade-offs; basic optimization from Cal 1)– Synergistic effects (net effects greater than sum of parts)– Feedback loops

• Negative (antagonistic, permit limiting behavior and oscillations)• Positive (with negative feedback loops, make prediction difficult without

quantitative descriptions)

• New and exotic interactions

“Interesting” examples:• In my opinion, we don’t have a modeling class to study

models like (Studying these equations is important, but when we study them, we are studying physics and/or mathematics.)

• Principles of modeling come into play as the relationships become more interesting:– Antagonistic effects (trade-offs; basic optimization from Cal 1)– Synergistic effects (net effects greater than sum of parts)– Feedback loops

• Negative (antagonistic, permit limiting behavior and oscillations)• Positive (with negative feedback loops, make prediction difficult without

quantitative descriptions)

• New and exotic interactions

C. Building Models

Model Construction..

• A modeler must first select a number of variables, and then determine and describe their relationship.

• Note: pragmatically, simplicity and computational efficiency often trump accuracy.

(A mathematical model describes a system with variables {u, v, w, …} by describing the functional relationship of those variables.)

Model Construction..

• A modeler must first select a number of variables, and then determine and describe their relationship.

• Note: pragmatically, simplicity and computational efficiency often trump accuracy.

(A mathematical model describes a system with variables {u, v, w, …} by describing the functional relationship of those variables.)

Model Construction..

• A modeler must first select a number of variables, and then determine and describe their relationship.

• Note: pragmatically, simplicity and computational efficiency often trump accuracy.

(A mathematical model describes a system with variables {u, v, w, …} by describing the functional relationship of those variables.)

The value of a model is in its ability to make an accurate or useful set of predictions, not realism in all possible aspects.

Principles of Model Design

• 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.– This is a feature, not a problem.

• 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 – does the model allow investigation

of a question of interest?– New phenomena predicted that motivate further experiments.

C. 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