Mathematical Modeling
UAB/EdGrid
Graduate Assistants:
Clinton Curry
Brent Shivers
John C. Mayer
UAB/Mathematics
Mathematical Modeling
• Creates a mathematical representation of some
phenomenon to better understand it.
• Matches observation with symbolic representation.
• Informs theory and explanation.
The success of a mathematical model depends on how easily
it can be used, and how accurately it predicts and how well it
explains the phenomenon being studied.
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Mathematical Modeling
• A mathematical model is central to most
computational scientific research.
• Other terms often used in connection with
mathematical modeling are
–
–
–
–
Computer modeling
Computer simulation
Computational mathematics
Scientific Computation
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Mathematical Modeling and the
Scientific Method
• How do we incorporate mathematical
modeling/computational science in the
scientific method?
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Mathematical Modeling
Problem-Solving Steps
• Identify problem area
• Conduct background
research
• State project goal
• Define relationships
• Develop mathematical
model
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• Develop
computational
algorithm
• Perform test
calculations
• Interpret results
• Communicate results
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Syllabus: MA
261/419/519
Spring, 2006
Syllabus: MA 261/419/519
Introduction to Mathematical Modeling
• Prerequisite: Calculus 1
• Goals
• Learn to build models
• Understand mathematics behind models
• Improve understanding of mathematical concepts
• Communicate mathematics
• Models may be mathematical equations,
spreadsheets, or computer simulations.
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Contact Information
• Instructor: Dr. John Mayer - CH 490A
– mayer@math.uab.edu - 934-2154
– ??
• Assistant: Mr. Robert Cusimano - CH 478B
– rob5236@uab.edu - 934-2154
– ??MW 4:00 – 5:30 PM (computer lab CB 112)
• Assistant: Ms. Jeanine Sedjro – ??
– ??
– ??MW 6:45 – 8:00 PM (computer lab CB 112)
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Class Meetings
• Lectures.
– Monday and
Wednesday
– 5:30 – 6:45 PM
– CB15 112.
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• Computer lab.
– Monday and
Wednesday
– 4:00 – 5:30 PM
– 6:45 – 8:00 PM
– CB15 112.
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Software Tools
• Spreadsheet:
• Microsoft EXCEL
• Compartmental Analysis and System
Dynamics programming:
• Isee Systems STELLA
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Computer Lab
• The only computer labs that have STELLA
installed are CB15 112 and EDUC 149A.
• Always bring a formatted 3.5” diskette to the
lab.
• Label disk: “Math Modeling, Spring, 2006,
Your Name,
Math Phone Number: 934-2154”
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Assignments
• One or two assignments every week.
• First few assignments have two due dates.
– Returned next class.
– Re-Graded for improved grade.
• Written work at most two authors.
– You may work together with a partner on the
computer.
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Midterm Tests
• Two tests, one about every 5 weeks.
• Given in the computer lab.
• Basic building blocks relevant to the kinds of
models we are constructing.
• Mathematics and logic behind the computer
models.
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System Dynamics Stories and
Projects
• System Dynamics Stories.
– Scenarios describing realistic situations to be modeled.
– Entirely independent work – no partners.
– Construct model and write 5-10 page technical paper
(template provided).
• Project due date to be announced.
– Begin work about middle of term.
– Preliminary evaluation three weeks prior to due date.
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Grading
MA 261
MA 419
MA 519
Assignments
40%
30%
30%
Tests (2)
30%
30%
25%
Model 1
10%
10%
10%
Model 2
na
na
10%
20%
20%
15%
na
10%
10%
Paper
Lesson Plan
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A Course
Sampler
Topics for Tools
• Spreadsheet Models
– Data analysis: curve
fitting.
– Recurrence Relations
and difference
equations.
– Cellular Automata:
nearest-neighbor
averaging.
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• Compartmental
Models
– Introduction to
System Dynamics.
– Applications of
System Dynamics.
– System Dynamics
Stories (guided
projects).
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Spreadsheet Models: Excel
• Curve fitting – introduction to
(linear) regression
• Difference Equations: modeling
growth
• Nearest-neighbor averaging
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Morteville
by Doug Childers
• Anthrax detected in
Morteville
• Is terrorism the
source?
• Infer geographic
distribution from
measures at several
sample sites
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• Build nearest
neighbor averaging
automaton in Excel
• Form hypothesis
• Get more data and
compare
• Revise hypothesis
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Morteville View 1
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Anthrax Distribution 1
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Compartmental Modeling
• How to Build a
Stella Model
• Simple Population
Models
• Generic Processes
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• Advanced
Population Models
• Drug Assimilation
• Epidemiology
• System Stories
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Population Model
•Stella Model
•Equations
•Graphical Output
Population(t) = Population(t - dt) + (Births Deaths) * dt
INIT Population = 100000 {people}
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INFLOWS:
Births =
Population*Fraction_That_Are_Female*Reprodu
cing_Females*Births_per_Rep_Female
{people/year}
OUTFLOWS:
Deaths = Population*Death_Fraction
{people/year}
Births_per_Rep_Female = 66/1000 {people/1000
rep females/year}
Fraction_That_Are_Female = 0.5
{females/people}
Reproducing_Females = .45 {rep females/female}
Death_Fraction = GRAPH(Population)
(120000, 0.01), (125000, 0.011), (130000, 0.013),
(135000, 0.015), (140000, 0.017), (145000,
0.019), (150000, 0.02)
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Generic Processes
T yp ewrite rs i n ware ho use
• Linear model with
external resource
• Exponential
growth or decay
model
Emp lo ye es
T yp ewrite rs ma de pe r e mpl o yee
T em perature
Cool ing
cool i ng rate
Weight
• Convergence
model
weight gained
weight gain
per month
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T yp ewrite rs ma de
p er we ek
weight lost
weight loss rate
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Calculus Example
STELLA Diagram
Stock X
Flow 1
Flow is
proportional to X.
STELLA Equations:
Constant a
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Stock_X(t) = Stock_X(t - dt) + (Flow_1) * dt
INIT Stock_X = 100
Flow_1 = Constant_a*Stock_X
Constant_a = .1
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Connecting the Discrete to the
Continuous
• Stock_X(t) = Stock_X(t - dt) + (Flow_1) * dt
• (Stock_X(t) - Stock_X(t - dt))/dt = Flow_1
• Flow_1 = Constant_a*Stock_X(t – dt)
• (X(t) - X(t - dt))/dt = a X(t - dt)
• Let dt go to 0
• dX/dt = a X(t) (a Differential Equation)
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Solution to DE
•
•
•
•
•
•
dX/dt = a X(t)
dX/X(t) = a dt
Integrate
log (X(t)) = at + C
X(t) = exp(C) exp(at)
X(t) = X(0) exp(at)
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System
Dynamics
Stories and
Projects
System Dynamics Stories and
Projects
• Scenarios describing realistic
situations to be modeled.
• Fully independent work.
• Construct model.
• Write 5-10 page technical paper
(template provided).
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Modeling a
Dam 2
Boysen Dam
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• Boysen Dam has several
purposes: It "provides
regulation of the streamflow for
power generation, irrigation,
flood control, sediment
retention, fish propagation, and
recreation development." The
United States Bureau of
Reclamation, the government
agency that runs the dam,
would like to have some way of
predicting how much power
will be generated by this dam
under certain conditions.
– Clinton Curry
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Powe r Ge ne …
Powe r Ge ne ra ted
h ei gh t o f
h ea dwate r
Dam Ca pa ci ty
T urb in e Flo w
~
Spi l l m od ifi er
~
~
h ei gh t o f
tai l water
h ei gh t o f
h ea dwate r
Perce nt ful l
Ri ver flo w
Ini ti al val u es
h ei gh t o f
tai l water
~
T urb in e Flo w
T ai l water
?
Wa ter Vo l ume
~
T ai l water
El im in ati on
Spi l l wa y Fl o w
?
T urb in e Cap acity
Ri ver flo w
T arg et Wate r
~
Vol u me Perce nt
T en Ye ar Avera ge
T arg et Wate r Hei g ht T urb in e mu l ti p li e r~
~
Spi l l wa y Rel ea se Cap acity
Cal cu l ati ng Sp il l …
Spi l l m od ifi er
Ini ti al Wate r Hei g ht
Ma xi mu m He i gh t
~
~
Ini ti al Wate r
Vol u me Perce nt
Mi n im um Hei g ht
Mi n im um Hea d
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Dam Ca pa ci ty
~
Wa ter Vo l ume
T arg et Wate r
Vol u me Perce nt
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Boysen Dam
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Tot a l o rigina l
Modeling a Smallpox Epidemic
• One infected
terrorist comes
to town
• How does the
system handle
the epidemic
under different
assumptions?
– Alicia Wilson
immu n e su p pr e ssed
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p op u la tion
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Tot a l d e at h s
h ave sma llpo x
o ld er vac c in a te d
d ea t hs
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h ave sma llpo x
Rec e n tly
Tot a l p e op le
va c c in at e d
wit h sma llp o x
Tot a l n e ve r
in fe c te d p e op le
o ld er
va c c in at e d
immu n e su p pr e ssed
d ea t hs
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immu n e su p pr e ssed
immu n e
h ave sma llpo x
su p p re sse d
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