computational methods for design lecture 2 – some “ simple ” applications john a. burns c...
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Computational Methods for Design Lecture 2 – Some “Simple” Applications
John A. Burns
Center for Optimal Design And Control
Interdisciplinary Center for Applied MathematicsVirginia Polytechnic Institute and State University
Blacksburg, Virginia 24061-0531
A Short Course in Applied Mathematics
2 February 2004 – 7 February 2004
N∞M∞T Series Two Course
Canisius College, Buffalo, NY
Today’s Topics
Lecture 2 – Some “Simple” Applications A Falling Object: Does F=ma ? Population Dynamics System Biology A Smallpox Inoculation Problem Predator - Prey Models A Return to Epidemic Models
A Falling Object
( ) ( )F t ma t“Newton’s Second Law”
WARNING!! THIS IS A SPECIAL CASE !!
( ) ( ) ( )d ddt dtF t p t mv t
IF m(t) = m is constant, then
( ) ( )F t ma t
( ) ( )mg F t ma t
ASSUME the only force acting onthe body is due to gravity …
. y(t)
A Falling Object (constant mass)
( ) ( ) ( ) ( )d ddt dtmg m t v t m y t my t
. y(t)
( )y t gODE
0 0(0) (0)y h y v INITIAL VALUES
2( ) / 2y t gt at b GENERAL SOLUTION
20 0( ) / 2y t gt v t h
A Falling Object: Problems?
0 5 10 15 20 250
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
(0) 10,000 (0) 0y y
( )y t
A Falling Object: Problems?
0 5 10 15 20 25-900
-800
-700
-600
-500
-400
-300
-200
-100
0
( ) ( )v t y t
(0) 10,000 (0) 0y y
800 ft/sec 445 m/hr
Terminal Velocity
( ) ( ) ( ) ( ) ( )g dampmy t F t F t mg y t y t AIR RESISTANCE( ) ( )v t y t
( ) 0v t FOR A FALLING OBJECT
( ) ( ) ( )mv t mg v t v t
2( ) ( )mv t mg v t
2 /
2 /
1( )
1
t g mmg
t g m
ev t
e
Comments About Modeling
( ) ( )ddt mv t F t
Newton’s Second Law IS Fundamental
TWO PROBLEMS1. FINDING ALL THE FORCES (OF IMPORTANCE)2. KNOWING HOW MASS DEPENDS ON VELOCITY
ASSUMING CONSTANT MASS
( )mv t mg( ) ( ) ( )mv t mg v t v t “CORRECTION” FOR AIR RESISTANCE
THE “MODEL” FOR AIR RESISTANCEIS AN APPROXIMATION TO REALITY
More Fundamental Physics
? HOW DOES THE MASS DEPENDS ON VELOCITY ?
186,000 mi/secc
FOLLOWS FORM EINSTEIN’S FAMOUS ASSUMPTION
2E mc
( ) ( ) ( )dE t F t v t
dt
2 ( ) ( ) ( )d d
mc t mv t v tdt dt
2 ( ) ( ) ( ) ( )d d
c m t m t mv t mv tdt dt
More Fundamental Physics
EINSTEIN’S CORRECTED FORMULA
22 2 2 2( ) ( ) ( ) ( )c m t mv t C m t v t C 2 2
0 0c m C C
22 2 2 20( ) ( )c m t mv t c m
22 2
02
( )( ) 1
v tm t m
c
0
2 2( )
1 ( ) /
mm t
v t c
Comments About Mathematics
0
2 2( )
1 ( ) /
mm t
v t c
ONLY IMPORTANT WHEN STILL DOESN’T HELP WITH MODELING FORCES SCIENTISTS AND ENGINEERS MUST FIND THE
“IMPORTANT” RELATIONSHIPS
v c
( ) ( ) ( )dampF t y t y t
MATHEMATICIANS MUST DEVELOP NEW MATHEMATICS TO DEAL WITH THE MORE
COMPLEX PROBLEMS AND MODELS
Comments About Modeling
MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT
)()()( tytym
gty (0) 10,000 (0) 0y y
)(ty)()( tytv
4465
Population 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
)(
)(
)(
time
td
tb
tp
t sec, hrs, days, years ….
Number of cells at time t
Probability that a cell divides in unit time at time t
Probability that a cell dies in unit time at time t
Population Dynamics
ttptb )()(Number of new cells on ttt ,
ttptd )()(Number of cell deaths on ttt ,
ttptbttptbtpttp )()()()()()(Change in cell population
)()()()()()()(
tptrtptdtbt
tpttp
TAKE LIMIT AS 0t
Malthusian LAWof population growth
)()()( tptrtpdt
d
Thomas R. Malthus (1766-1834)
Population Dynamics
000
0
0
)(
)(
)(
dbrtr
dtd
btb
)()( 0 tprtpdt
d
)(
)(
)(
tr
td
tb BIRTH RATE
DEATH RATE
REPRODUCTIVE RATE
00)( petp tr
ASSUMECONSTANT
DO AN EXPERIMENT
10)0( 0 pp
2510)1( 10 rep
5.210
250 re
)5.2ln(ln 00 rer
9163.00 r
Population After 5 Days
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
100
200
300
400
500
600
700
800
900
1000
tetp 9163.10)(
Population After 7.5 Days
0 1 2 3 4 5 6 7 80
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
tetp 9163.10)(
Population After 10 days
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10x 10
4
tetp 9163.10)(
NOT WHAT REALLY
HAPPENS
Improved Model
COMPETITION FOR FOOD AND SPACE
MalthusASSUMED
PLENTY OF FOOD AND SPACE
Pierre-Fancois Verhulst (1804-1849)
)()( ,)( 1010 tpddtdp(t)bbtb
)(1
1)(1)( 00
110 tp
Krtp
r
dbrtr
)()()()( 1100 tpdbdbtdtbtr
Logistics Equation
)()()( tptrtpdt
d
11
0
db
rK
)(
11)( 0 tp
Krtr
11
0
db
rK
CARRYING CAPACITY
NATURAL REPRODUCTIVE RATE
)()(1
1)( 0 tptpK
rtpdt
d
LE
IS THIS A BETTER(MORE ACCURATE)
MODEL? ?
A Comparison: First 5 Days
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
100
200
300
400
500
600
700
800
900
1000
tetp 9163.10)(
Malthusian LAWof population growth
)()( 0 tprtpdt
d )()(
11)( 0 tptp
Krtp
dt
d
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
100
200
300
400
500
600
700
800
900
1000
)(tp
Logistic LAWof population growth
A Comparison: First 7.5 Days
0 1 2 3 4 5 6 7 80
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Malthusian LAWof population growth
0 1 2 3 4 5 6 7 80
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Logistic LAWof population growth
)()( 0 tprtpdt
d )()(
11)( 0 tptp
Krtp
dt
d
10,000 5,000
A Comparison: First 10 Days
0 1 2 3 4 5 6 7 8 9 100
1000
2000
3000
4000
5000
6000
7000
8000
9000
Logistic LAWof population growth
)()( 0 tprtpdt
d )()(
11)( 0 tptp
Krtp
dt
d
Malthusian LAWof population growth
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10x 10
4
9,000100,000
Logistic Equation: 15 Days
0 5 10 150
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
)(tp
)()(1
1)( 0 tptpK
rtpdt
d
K
Modeling in Biology
Malthusian LAWof population growth
Logistic LAWof population growth
)()( 0 tprtpdt
d )()(
11)( 0 tptp
Krtp
dt
d
MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT
? WHAT HAVE WE LEFT OUT ?? WHAT IS THE CORRECT LAW ?
NEED A NEWTON OR EINSTEINFOR SYSTEM BIOLOGY
))(),(,()( tptptftpdt
d
A 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? a = AGE
)(
)(
)(1)(
)(
)(
)(
ay
ax
asac
a
as
a
Fraction of susceptibles who survive & become immune
Death rate at age a due to all causes
Rate at which susceptibles become infected
Fraction that dies due to the infection
Probability that a newborn is alive and susceptible at age a
Probability that a newborn is alive and immune at age a
A Smallpox Inoculation Model
0)0( ),()()()()(1)(
1)0( ),()()()(
yayaaxaacayda
d
xaxaaaxda
d
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 d’une nouvelle analyse de la mortalité causée par la petite vérole”, Mém. Math. Phys. Acad. Roy. Sci., Paris, (1766),1.
Predator - Prey Models
)()()(
)()()(
tdyctytydt
d
tbyatxtxdt
d
Vito Volterra Model (1925) Alfred Lotka Model (1926)
THINK OF SHARKS AND SHARK FOOD
dcba
ty
tx
,,,
)(
)(
Numbers of predators
Numbers of prey
Parameters
2)(tx
Numerical Issues: LV Model
)()()(
)()()(
tdyctytydt
d
tbyatxtxdt
d
Numerical Schemes? Explicit Euler? Implicit Euler? ODE23? ODE45? ??????
a/b
c/d
o
>
>
>
y
xo
Epidemic Models SIR Models (Kermak – McKendrick, 1927)
Susceptible – Infected – Recovered/Removed
( ) ( ) ( )dS t S t I t
dt
( ) ( ) ( ) ( )dI t S t I t I t
dt
( ) ( )dR t I t
dt
( ) ( ) ( ) constantS t I t R t N
SIR Models
)()()()()()(
)()()(
tStItItItStIdt
d
tItStSdt
d
constant)()()(
)()(
NtRtItS
tItRdt
d
0)()()(
then,)( If
tStItIdt
d
tS
NOT ISOLATED
NSI ee 0 ,0
Equilibrium
Epidemic Models (SARS) SEIJR: Susceptibles – Exposed - Infected - Removed
)()()(
)()()(
222
111
tPtSrtSdt
d
tPtSrtSdt
d
)()()(
)()()()(
)()()()(
21
2
1
tJtItRdt
d
tJtItJdt
d
tItkEtIdt
d
)()()()()()( 2211 tkEtPtSrtPtSrtEdt
d
)(/))()()(()( tNtlJtqEtItP
Model of SARS Outbreak in Canada
byChowell, 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 …
Spatial Model I O’Callaghan and Murray (J. Mathematical Biology 2002)
Spatial Epidemic Model Partial-Integro-Differential Equations
NON-NORMAL
DELAY = latent
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Remarks
GOOD COMPUTATIONAL MATHEMATICS WILL BE THE KEY TO FUTURE BREAKTHROUGHSAPPROXIMATIONS MUST BE DONE RIGHT
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 …