MA354
Math Modeling Introduction
Outline
A. Three Course Objectives
1. Model literacy: understanding a typical model description
2. Model Analysis
3. Building Models
B. What is a “model”?
Models describe relationships among quantities.
C. Building a Model
D. 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:
Fluid Mosaic
“Model”
Describing a
relationship
among
concepts
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:
𝐹 = 𝑚𝑎
Variables : force, mass, acceleration
Quantitative 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 𝐹 = 𝑚𝑎
(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 𝐹 = 𝑚𝑎
(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.
The value of a model is in its
ability
to make an
accurate
or
(A mathematical model describes
a system
with
variables
useful set of predictions, not
in all possible
{u, v, w, …} by describing the realism
functional
relationship of
aspects.
variables.)
those
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 Approaches
Continuous Verses Discrete
• Continuous Approaches (differential equations)
• Discrete Approaches (lattices)
Modeling Approaches
Continuous 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
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:
Modeling Approaches
Deterministic Verses Stochastic
• 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.
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