Title: Computational Methods for Design Lecture 2
1Computational Methods for Design Lecture 2
Some Simple ApplicationsJohn A. BurnsCenter
for Optimal Design And ControlInterdisciplinary
Center for Applied MathematicsVirginia
Polytechnic Institute and State
UniversityBlacksburg, Virginia 24061-0531
- A Short Course in Applied Mathematics
- 2 February 2004 7 February 2004
- N8M8T Series Two Course
- Canisius College, Buffalo, NY
2Todays Topics
3A Falling Object
Newtons Second Law
WARNING !! THIS IS A SPECIAL CASE !!
4A Falling Object (constant mass)
5A Falling Object Problems?
6A Falling Object Problems?
7Terminal Velocity
FOR A FALLING OBJECT
8Terminal Velocity
9Comments About Modeling
Newtons Second Law IS Fundamental
- TWO PROBLEMS
- FINDING ALL THE FORCES (OF IMPORTANCE)
- KNOWING HOW MASS DEPENDS ON VELOCITY
ASSUMING CONSTANT MASS
CORRECTION FOR AIR RESISTANCE
THE MODEL FOR AIR RESISTANCE IS AN
APPROXIMATION TO REALITY
10More Fundamental Physics
- ? HOW DOES THE MASS DEPENDS ON VELOCITY ?
FOLLOWS FORM EINSTEINS FAMOUS ASSUMPTION
11More Fundamental Physics
EINSTEINS CORRECTED FORMULA
12Comments About Mathematics
MATHEMATICIANS MUST DEVELOP NEW MATHEMATICS TO
DEAL WITH THE MORE COMPLEX PROBLEMS AND MODELS
13Comments About Modeling
MORE ACCURATE MORE COMPLEX MORE DIFFICULT
14Population Dynamics
- Use growth of protozoa as example
- A population could be
- Bacteria, Viruses
- Cells (Cancer, T-cells )
- People, Fish, Cows Fish
- Things that live and die
ASSUME PLENTY OF FOOD AND SPACE
15Population Dynamics
Change in cell population
Thomas R. Malthus (1766-1834)
16Population Dynamics
17Population After 5 Days
18Population After 7.5 Days
19Population After 10 days
NOT WHAT REALLY HAPPENS
20Improved Model
Malthus ASSUMED PLENTY OF FOOD AND SPACE
COMPETITION FOR FOOD AND SPACE
21Logistics Equation
22A Comparison First 5 Days
23A Comparison First 7.5 Days
24A Comparison First 10 Days
25Logistic Equation 15 Days
26Modeling in Biology
Logistic LAW of population growth
Malthusian LAW of population growth
? WHAT HAVE WE LEFT OUT ? ? WHAT IS THE CORRECT
LAW ?
NEED A NEWTON OR EINSTEIN FOR SYSTEM BIOLOGY
MORE ACCURATE MORE COMPLEX MORE DIFFICULT
27A Smallpox Inoculation Problem
- Basic issue Compute the gain in life expectancy
if smallpox eliminated as a cause if death? - Very timely problem
- What if smallpox is injected into a large city?
- How does age impact the problem?
28A Smallpox Inoculation Model
- Typical epidemiological model
- Contains age dependent coefficients
- Model applied to Paris
- Not funded by Dept. of Homeland Defense
- Work was done in 1760 and published in 1766 by
Daniel Bernoulli, Essai dune nouvelle analyse
de la mortalité causée par la petite vérole,
Mém. Math. Phys. Acad. Roy. Sci., Paris, (1766),1.
29Predator - Prey Models
- Vito Volterra Model (1925)
- Alfred Lotka Model (1926)
- THINK OF SHARKS AND SHARK FOOD
30Numerical Issues LV Model
- Numerical Schemes?
- Explicit Euler?
- Implicit Euler?
- ODE23?
- ODE45?
- ??????
31Symplectic Methods
Explicit Euler
Symplectic
Implicit Euler
gt
gt
32Epidemic Models
33Epidemic Models
- SIR Models (Kermak McKendrick, 1927)
- Susceptible Infected Recovered/Removed
34SIR Models
NOT ISOLATED
35SIR Model
S(t) I(t) N 1
I(t)
S(t)
36Epidemic Models (SARS)
- SEIJR Susceptibles Exposed - Infected -
Removed
Model of SARS Outbreak in Canada by Chowell,
Fenimore, Castillo-Garsow Castillo-Chavez (J.
Theo. Bio.)
- Multiple cities (patch models)
- Hyman
- Mass transportation
- Castillo-Chavez, Song, Zhang
- Delays
- Banks, Cushing, May, Levin
- Migration Spatial effects
- Aronson, Diekmann, Hadeler, Kendall, Murray, Wu
37Spatial Model I
- OCallaghan and Murray (J. Mathematical Biology
2002) - Spatial Epidemic Model
- Partial-Integro-Differential
Equations
NON-NORMAL
38Spatial Model II
WHERE A generates a delay semi-group
39Remarks
- LOTS OF SIMPLE APPLICATIONS
- OPPORTUNITIES FOR MATHEMATICIANS TO GET INVOLVED
WITH MODELING - JOB SECURITY FOR APPLIED MATHEMATICIANS
- NEW MODELS NEED TO BE DEVELOPED TOGETHER
MATHEMATICIANS WITH - PHYSICS, CHEMISTRY, BIOLOGY
- FLUID DYNAMICS, STRUCTURAL DYNAMICS
- GOOD COMPUTATIONAL MATHEMATICS WILL BE THE KEY TO
FUTURE BREAKTHROUGHS - APPROXIMATIONS MUST BE DONE RIGHT