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- 2. 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. 2
- 3. 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 3
- 4. Mathematical Modeling and the Scientific Method How do we incorporate mathematical modeling/computational science in the scientific method? 4
- 5. Mathematical Modeling Problem-Solving Steps Identify problem area Conduct background research State project goal Define relationships Develop mathematical model 5 • Develop computational algorithm • Perform test calculations • Interpret results • Communicate results
- 7. 7Syllabus: 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.
- 8. 8Contact 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)
- 9. 9 Class Meetings • Lectures. – Monday and Wednesday – 5:30 – 6:45 PM – CB15 112. • Computer lab. – Monday and Wednesday – 4:00 – 5:30 PM – 6:45 – 8:00 PM – CB15 112.
- 10. 10 Software Tools • Spreadsheet: • Microsoft EXCEL • Compartmental Analysis and System Dynamics programming: • Isee Systems STELLA
- 11. 11 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”
- 12. 12 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.
- 13. 13 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.
- 14. 14 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.
- 15. Grading MA 261 MA 419 MA 519 Assignment s 40% 30% 30% Tests (2) 30% 30% 25% Model 1 10% 10% 10% Model 2 na na 10% Paper 20% 20% 15% Lesson Plan na 10% 10% 15
- 16. A Course Sampler
- 17. 17 Topics for Tools • Spreadsheet Models – Data analysis: curve fitting. – Recurrence Relations and difference equations. – Cellular Automata: nearest-neighbor averaging. • Compartmental Models – Introduction to System Dynamics. – Applications of System Dynamics. – System Dynamics Stories (guided projects).
- 18. Spreadsheet Models: Excel Curve fitting – introduction to (linear) regression Difference Equations: modeling growth Nearest-neighbor averaging 18
- 19. Morteville by Doug Childers Anthrax detected in Morteville Is terrorism the source? Infer geographic distribution from measures at several sample sites Build nearest neighbor averaging automaton in Excel Form hypothesis Get more data and compare Revise hypothesis 19
- 20. Morteville View 1 20 0.2 0 2.1 4.1 6.3 8.4 9 9.7 11 12 13 14 14 0.3 0.9 2.2 4 6.3 10 9 9.4 10 12 13 15 14 0 0.9 1.9 3.3 5 6.7 7.4 8.6 10 11 12 13 13 0.3 0.7 1.2 2.4 3.5 4.5 5.3 7.6 10 11 12 13 12 0.3 0.3 0 1.4 2.3 2.6 1.5 6.6 12 10 11 13 11 0.3 0.3 0.3 0.8 1.6 2.2 2.9 5.1 7.2 7.8 8.2 8.6 8.5 0.2 0.2 0.2 0 1.1 1.8 2.6 3.6 4 5.3 5.5 4.9 5.5 0.1 0.1 0.2 0.4 0.9 1.4 2.1 2.9 3.5 4.1 3.6 0 3 0.1 0 0.2 0.4 0.6 0.9 1.6 2.2 3 4.1 4.8 3.4 3.6 0.1 0.1 0.2 0.3 0.3 0 1 1.6 2.2 4.3 8 5.2 4.5 0.2 0.2 0.2 0.3 0.3 0.4 0.7 0.8 0 3 5 4.8 4.6 0.2 0.2 0.3 0.3 0.4 0.5 0.7 0.9 1.2 2.8 4.1 4.5 4.5
- 21. Anthrax Distribution 1 21
- 22. Compartmental Modeling How to Build a Stella Model Simple Population Models Generic Processes Advanced Population Models Drug Assimilation Epidemiology System Stories 22
- 23. Population Model 23 •Stella Model Population(t) = Population(t - dt) + (Births - Deaths) * dt INIT Population = 100000 {people} INFLOWS: Births = Population*Fraction_That_Are_Female*Reproduc ing_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) •Equations •Graphical Output
- 24. Generic Processes Linear model with external resource 24 Typewriters in warehouse Typewriters made per week Typewriters made per employee Employees Temperature Cooling cooling rate Weight weight gained weight lost weight gain per month weight loss rate • Exponential growth or decay model • Convergence model
- 25. 25Calculus Example Stock X Flow 1 Constant a STELLA Equations: Stock_X(t) = Stock_X(t - dt) + (Flow_1) * dt INIT Stock_X = 100 Flow_1 = Constant_a*Stock_X Constant_a = .1 Flow is proportional to X. STELLA Diagram
- 26. 26 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)
- 27. 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) 27
- 28. System Dynamics Stories and Projects
- 29. 29 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).
- 30. Modeling a Dam 2 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 30 Boysen Dam
- 31. 31 Tailwater Elimination Water Volume Dam Capacity Percent full ~ height of headwater ~ Ten Year Average ? River flow ~Turbine multiplier Minimum Height Target Water Height Spillway Release Capacity Turbine Flow Tailwater Spillway Flow Turbine Flow Turbine Capacity Minimum Head ~ height of tailwater ~ Initial Water Volume Percent ~ height of headwater ~ height of tailwater Power Generated Initial Water Height ~ Spill modifier ~ Spill modifier ? River flow Water Volume Dam Capacity ~ Target Water Volume Percent Maximum Height ~ Target Water Volume Percent Initial values Power Gene… Calculating Spill …
- 32. 32Boysen Dam
- 33. Modeling a Smallpox Epidemic One infected terrorist comes to town How does the system handle the epidemic under different assumptions? – Alicia Wilson unvaccinated susceptible Unvaccinated have smallpox unvaccinated getting smallpox Total never infected people exposure probability current population unvaccinated deaths unvaccinated mortality new exposures Total deaths new exposures per unvaccinated sick person unvaccinated probability of catching smallpox unvaccinated survivors unvaccinated survival unvaccinated death rate Recently vaccinated recently vaccinated have smallpox immune suppressed mortality Total people with smallpox exposure probability recently vaccinated getting smallpox Recently vaccinated deaths recently vaccinated mortality recently vaccinated survivors recently vaccinated survival vaccinations Recent vaccination probability of catching smallpox Vaccination rate recently vaccinated death rate exposure probability multiple exposure correction older vaccinated mortality immune suppressed immune suppressed have smallpox immune suppressed deaths immune suppressed getting smallpox Total deaths older vaccinated older vaccinated have smallpox older vaccinated getting smallpox older vaccinated deaths older vaccinated survivors older vaccinated survival older vaccinated death rate older vaccinated susceptibility Revaccinations revaccination rate unvaccinated deaths Recently vaccinated deaths older vaccinated deaths Unvaccinated have smallpox recently vaccinated have smallpox older vaccinated have smallpox Total people with smallpox unvaccinated susceptible Recently vaccinated older vaccinated Total never infected people Total original population new esposures per vaccinated sick person Unvaccinated have smallpox recently vaccinated have smallpox older vaccinated have smallpox new exposures per older vaccinated sick person immune suppressed survivors immune suppressed survival immune suppressed mortality rate immune suppressed probability of catching smallpox immune suppressed immune suppressed have smallpox immune suppressed deaths immune suppressed have smallpox new exposures per immune suppressed sick person 33
Editor's Notes
- Identify problem area. Conduct background research: Find the back-ground information to narrow the focus of the problem. Internet and library research A mentor Textbook and teachers State project goal: Write a reasonable problem definition. What do you want to find out? What do you expect to discover? Define relationships: Focus on how variables are related State governing principles = laws/relationships from physics, biology, engineering, economics, etc. State simplifying assumptions, e.g., no friction system (physics), no immigration of population (economics), constant growth rate (biology), etc. Define input and output variables/parameters Give units Develop mathematical model: Define mathematical relationships between variables. Variables (Output and input) and parameters (constants). Output variables give the model solution . The choice of what to specify as input variables and what to specify as parameters is somewhat arbitrary and often model dependent. Input variables characterize a single physical problem while parameters determine the context or setting of the physical problem. For example, in modeling the decay of a single radioactive material, the initial amount of material and the time interval allowed for decay could be input variables, while the decay constant for the material could be a parameter. The output variable for this model is the amount of material remaining after the specified time interval.