Transcript Jones_Smith

Stochastic Spatial Dynamics of
Epidemic Models
Mathematical Modeling
Nathan Jones and Shannon Smith
Raleigh Latin School and KIPP: Pride High School
2008
Spatial Motion and Contact in
Epidemic Models
http://www.answersingenesis.org/articles/
am/v2/n3/antibiotic-resistance-of-bacteria
http://commons.wikimedia.org/wiki/Im
age:Couple_of_Bacteria.jpg
Problem
If we create a model in which individuals move
randomly in a restricted area, how will it
compare with the General Epidemic Model?
Outline


History
The SIR Model
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First Model: Simple Square Region
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
Classifications and Equations
Assumptions
The Effect of Changing Variables
Logistic Fitting
Comparison to SIR
Conclusions
Second Model: Wall Obstructions
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
The Effect of Changing Variables
Conclusions
History

Epidemics in History:
Black Death/ Black Plague
 Avian Flu
 HIV/AIDS


Modeling Epidemics:
Kermack and McKendrick, early 1900’s
 SIR model

The SIR Model: Equations

Susceptibles:
α is known as the transmittivity constant
 The change in the number of Susceptibles
is related to the number of Infectives and
Susceptibles:

dS
 IS
dt
The SIR Model: Equations

Infectives:


β is the rate of recovery
The number of Infectives mirrors the number of
Susceptibles, but at the same time is decreased
as people recover:
dI
 IS  I
dt
The SIR Model: Equations

Recovered Individuals


β is the rate of recovery
The number of Recovered Individuals is
increased by the same amount it removes from
the Infectives
dR
 I
dt
Making Our Model



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
Construct a square region.
Add n-1 Susceptibles.
Insert 1 Infective
randomly.
Individuals move
randomly.
The Infectives infect Susceptibles on contact.
Infectives are changed to Recovered
Individuals after a set time.
Original Assumptions of First Model
The disease is communicated solely
through person to person contact
 The motion of individuals is effectively
unpredictable
 Recovered Individuals cannot become
re-infected or infect others
 Any infected individual immediately
becomes infectious
 There is only one initial infective

Original Assumptions of First Model
The disease does not mutate
 The total population remains constant
 All individuals possess the same
constant mobility
 The disease affects all individuals to the
same degree
 Only the boundary of the limited region
inhibits the motion of the individuals
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We Change the Following:
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Total population
Arena size
Maximum speed of individuals
Infection Radius
Probability of infection on contact (infectivity)
The time gap between infection and recovery
The initial position of the infected population
Total Population
Total Population
Total Population
Total Population
Arena Size
Arena Size
Arena Size
Arena Size
Infected People
Infective Trend According to Arena Size
350
300
250
200
150
100
50
0
Arena Size of 100
Arena Size of 200
Arena Size of 300
0
200
400
Time
600
Maximum Speed
Maximum Speed
Maximum Speed
Maximum Speed
Infection Radius
Infection Radius
Infection Radius
Infection Radius
Infected Individuals
Infective Trend According to the Size of the
Infection Radius
300
250
200
Infection Radius of 3
150
Infection Radius of 5
100
Infection Radius of 8
50
0
0
200
400
Time
600
800
Probability of Infection
Probability of Infection
Probability of Infection
Probability of Infection
Recovery/ Removal Cycle
Recovery/ Removal Cycle
Recovery/ Removal Cycle
Recovery/ Removal Cycle
Infected Individuals
Infective Trend According to the Recovery Cycle
300
250
200
Recovery Cycle of 50
150
Recovery Cycle of 100
100
Recovery Cycle of 200
50
0
0
200
400
Time
600
800
Initial Position of Infectives
Initial
Initial
Infective
Infective
Placed
Placed
inon
Corner
Side
of
ofArena
Arena
Initial
Infective
Centered
in Arena
Averages of 100 runs
350
300
Individuals
250
Susceptibles
Infectives
Removed
200
150
100
50
0
0
100
200
300
400
Time
500
600
700
800
Logistic Fitting
Initial Infective Centered in Arena
Population
Population
Susceptible
Population
Removed Population
350
350
300
250
200
150
100
50
0
0
200
400
600
350
350
300
250
200
150
100
50
0
800
Tim e
Time
 297
.300
295
.370
S (tt))  297
.300
2.680
.023
t 155
250.000
.000))))
( 0(.0023
( t( 
11ee
Comparison to SIR
An average of 105 program runs
The Discrepancy
Why is there a
discrepancy?
 The Infectives
tend to isolate
each other from
Susceptibles

A Partial Solution
Average of 100 runs
Conclusions for the First Model
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The rate of infection grows with:
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The population density
The rate of transportation
The radius of infectious contact
The probability of infection from contact
The rate of infection decreases when
individuals recover more quickly
 The position of the initial infected can
significantly affect the data
 Our model does not match the SIR, primarily
due to spatial dynamics, but is still similar
Second Model: Wall Obstructions
The movement of the individuals is now
affected by walls in the arena.
 4 Regions
 2 Regions

2 Regions: Wall Gap
Gap of 60
110
20
2 Regions: Wall Gap
Gap
Between
Walls
1/8of
of
Arena
Gap
Gap
Between
Between
Walls
Walls
isisis
11/16
3/8
ofArena
Arena
Averages of 100 runs
350
350
300
300
Individuals
Individuals
250
250
Susceptibles
Susceptibles
Infectives
Infectives
Removed
Removed
200
200
150
150
100
100
50
50
00
00
200
200
400
400
600
600
Time
Time
800800
10001000
1200 1200
2 Regions: Wall Thickness
Thickness of 40
10
70
2 Regions: Wall Thickness
10
Wall Thickness of 70
40
Averages of 100 runs
350
300
Individuals
250
Susceptibles
Infectives
Removed
200
150
100
50
0
0
200
400
600
Time
800
1000
1200
4 Regions: Wall Gap
Gap of 20
80
50
4 Regions: Wall Gap
No
Walls
(wall
gapof
80)
Wall
Gaps
are
50
ofArena)
Arena)
Wall
Gaps
are
20 (5/16
(1/8
350
300
Individuals
250
Susceptibles
Infectives
Removed
200
150
100
50
0
0
200
400
600
800
Time
Averages of 100 runs
1000
1200
4 Regions: Wall Thickness
Thickness of 30
10
50
4 Regions: Wall Thickness
10
Wall Thickness of 30
50
350
350
Averages of 100 runs
300
300
Individuals
Individuals
250
250
Susceptibles
Susceptibles
Infectives
Infectives
Removed
Removed
200
200
150
150
100
100
50
50
0
0
0
0
200
200
400
400
600
600
Time
Time
800
800
1000
1000
1200
1200
Conclusions for Second Model

2 Regions and 4 Regions:
Shrinking the gap lowers the final number of
removed individuals
 Increasing the thickness generally lowers
the final number of removed individuals

What we learned
The effects of varying parameters on our
simulated epidemic
 The effects of obstruction on the spread
of epidemics
 How spatial dynamics can affect the
spread of an epidemic
 How simulation and modeling can be
used to repeat and examine events

Summary
History
 The SIR model
 Our Model
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Without Obstructions
 With Obstruction
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Our model compared to the SIR model
Possible Future Work
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Change assumptions
Reconstruct the single run tests using the
averaging program
Find further logistic curves for our data sets
Make more complex arenas
Find constants to account for spatial dynamics
Examine data for ratios and critical points
Compare our model to other epidemic models
Compare our simulated epidemics to real data
Bibliography
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http://www.answersingenesis.org/articles/am/v2/n3/antibiotic-resistanceof-bacteria
Bongaarts, John, Thomas Buettner, Gerhard Heilig, and Francois
Pelletier. "Has the HIV epidemic peaked?" Population and Development
Review 34(2): 199-224 (2008).
Capasso, Vincenzo. Mathematical Structures of Epidemic Systems. New
York, NY: Springer-Verlag (1993).
http://commons.wikimedia.org/wiki/Image:Couple_of_Bacteria.jpg
http://www.epidemic.org/theFacts/theEpidemic/
http://mvhs1.mbhs.edu/mvhsproj/epidemic/epidemic.html
http://www.sanofipasteur.us/sanofipasteur/front/index.jsp?codePage=VP_PD_Tuberculosis&codeRubrique
=19&lang=EN&siteCode=AVP_US
Smith, David and Moore, Lang. “The SIR Model for Spread of Disease”
Journal of Online Mathematics and its Applications: 3-6 (2008).
Mollison, Denis, ed. Epidemic Models: Their Structure and Relation to
Data. New York, NY: Cambridge University (1995).
Acknowledgements
Dr. Russell Herman
Mr. David Glasier
Mr. and Mrs. Cavender
SVSM Staff
Joanna Sanborn
Dr. Linda Purnell
Our parents
All supporters