1. MA354 An Introduction to Math Models (more or less corresponding to 1.0 in your book)

2. 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.

3. 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.

4. 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.

5. 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.

6. Objective 2: Model Construction The second objective is to build models of real-world phenomena by making appropriate simplifying assumptions and identifying key factors.

7. 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.

8. MA354 (Part 1) Classifying Models

9. By application (ecological, epidemiological,etc) Discrete or continuous? Stochastic or deterministic? Simple or Sophisticated Validated, Hypothetical or Invalidated

10. DISCRETE OR CONTINUOUS?

11. Discrete verses Continuous Discrete: –Values are separate and distinct (definition)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)

12. Modeling Approaches Continuous Verses Discrete Continuous Approaches (differential equations) Discrete Approaches (lattices)

13. Modeling Approaches Continuous Verses Discrete Continuous Approaches (smooth equations) Discrete Approaches (discrete representation)

14. Continuous Models Good models for HUGE populations (10 23 ), 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

15. 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

16. 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.

17. 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:

18. 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 Approaches Deterministic Verses Stochastic

19. Stochastic Models Accounts for random, probabilistic phenomena by considering specific possibilities. In practice, the generation of random numbers is required. Different result each time.

20. Deterministic Models One result. Thus, analytic results possible. In a process with a probabilistic component, represents average result.

21. 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.

22. 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.

23. 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.

24. Spatially Explicit Models Spatial variables (x,y) or (r,  ) Generally, much more sophisticated. Generally, much more complex! ODE: no spatial variables PDE: spatial variables

25. MA354 (Part 2) Models Describe Relationships Between Variables

26. 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

27. A Relationship Between Two Quantities Points to an interaction –May be direct –May be indirect In my opinion, a good model correctly describes their interaction Example: oranges and soap bubbles both form spheres, but for different reasons

28. Example: Hooke’s Law An ideal spring. F=-kx x = displacement(variable) k = spring constant(parameter) F = resulting force vector

29. Other Examples Circumference of a circle is proportional to r Weight is proportional to mass and the gravitational constant Etc.