MA354 Mathematical Modeling T H 2:45 pm– 4:00 pm Dr. Audi Byrne

Your Instructor Instructor: Dr. Audi Byrne

Dr. Audi Byrne PhD in mathematics from the University of Notre Dame

Dr. Audi Byrne Research area in biomathematics. (Dynamical systems and modeling. ) Cellular automata Multi-cellular Systems Stochastic Processes

Contacting Your Instructor Office: ILB 452 Office Hours: 10:00am-11:00am daily And by appointment.

Course Information Course webpage Google ‘Byrne South Alabama’ Eventually, stuff on Ecompanion.

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 expts.

Types of Models Discrete or Continuous Stochastic or Deterministic Simple or Sophisticated Good or bad (elegant or sloppy) Validated or Invalidated

Continuous or Discrete

Modeling Approaches Continuous Approaches (PDEs) Discrete Approaches (lattices)

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

Continuous Models Biological applications: Cells/Molecules = density field. 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

Stochastic vs Deterministic

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 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, more sophisticated. Generally, more complex! ODE: no spatial variables PDE: spatial variables

Other Ways that Model Differ What is being described? –The largest expected diameter of a tumor. –The diameter of the tumor over time. –The shape of the tumor over time.

Objective 1: Model Analysis and Validity The first objective is to study the behavior of mathematical models of real-world problems 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 2: Model Construction The second objective is to model real-world observations by making appropriate simplifying assumptions and identifying key factors.

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. Regarding Accuracy: simplicity and computational efficiency may trump accuracy.

Functional Relationships Among Variables x,y No Relationship –Or effectively no relationship. –No need 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

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

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