Stochastic Spatial Dynamics of Epidemic Models

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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.

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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://commons.wikimedia.org/wiki/Image: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
• Classifications and Equations
• First Model: Simple Square Region
• Assumptions
• The Effect of Changing Variables
• Logistic Fitting
• Comparison to SIR
• Conclusions
• Second Model: Wall Obstructions
• 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:
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:
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
Construct a square region.

Insert 1 Infective randomly.

Individuals move randomly.

The Infectives infect Susceptibles on contact.

Infectives are changed to Recovered Individuals after a set time.

Making Our Model
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
We Change the Following:
• Total population
• Arena size
• Maximum speed of individuals
• Probability of infection on contact (infectivity)
• The time gap between infection and recovery
• The initial position of the infected population
Initial Position of Infectives

Averages of 100 runs

Logistic Fitting

Initial Infective Centered in Arena

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
• The rate of infection grows with:
• 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.
• 2 Regions
• 4 Regions
2 Regions: Wall Gap

Gap of 110

Gap of 20

Gap of 60

2 Regions: Wall Gap

Averages of 100 runs

2 Regions: Wall Thickness

Thickness of 10

Thickness of 40

Thickness of 70

2 Regions: Wall Thickness

Averages of 100 runs

4 Regions: Wall Gap

Gap of 80

Gap of 50

Gap of 20

4 Regions: Wall Gap

Averages of 100 runs

4 Regions: Wall Thickness

Thickness of 10

Thickness of 30

Thickness of 50

4 Regions: Wall Thickness

Averages of 100 runs

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
• Without Obstructions
• With Obstruction
• Our model compared to the SIR model
Possible Future Work
• 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
• 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/sanofi-pasteur/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