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Stochastic Spatial Dynamics of Epidemic Models

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

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  1. Stochastic Spatial Dynamics of Epidemic Models Mathematical Modeling Nathan Jones and Shannon Smith Raleigh Latin School and KIPP: Pride High School 2008

  2. Spatial Motion and Contact in Epidemic Models http://www.answersingenesis.org/articles/am/v2/n3/antibiotic-resistance-of-bacteria http://commons.wikimedia.org/wiki/Image:Couple_of_Bacteria.jpg

  3. Problem If we create a model in which individuals move randomly in a restricted area, how will it compare with the General Epidemic Model?

  4. 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

  5. History • Epidemics in History: • Black Death/ Black Plague • Avian Flu • HIV/AIDS • Modeling Epidemics: • Kermack and McKendrick, early 1900’s • SIR model

  6. 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:

  7. 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:

  8. 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

  9. 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. Making Our Model

  10. 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

  11. 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

  12. We Change the Following: • 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

  13. Total Population

  14. Total Population

  15. Total Population

  16. Total Population

  17. Arena Size

  18. Arena Size

  19. Arena Size

  20. Arena Size

  21. Maximum Speed

  22. Maximum Speed

  23. Maximum Speed

  24. Maximum Speed

  25. Infection Radius

  26. Infection Radius

  27. Infection Radius

  28. Infection Radius

  29. Probability of Infection

  30. Probability of Infection

  31. Probability of Infection

  32. Probability of Infection

  33. Recovery/ Removal Cycle

  34. Recovery/ Removal Cycle

  35. Recovery/ Removal Cycle

  36. Recovery/ Removal Cycle

  37. Initial Position of Infectives Averages of 100 runs

  38. Logistic Fitting Initial Infective Centered in Arena

  39. Comparison to SIR An average of 105 program runs

  40. The Discrepancy • Why is there a discrepancy? • The Infectives tend to isolate each other from Susceptibles

  41. A Partial Solution Average of 100 runs

  42. 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

  43. Second Model: Wall Obstructions • The movement of the individuals is now affected by walls in the arena. • 2 Regions • 4 Regions

  44. 2 Regions: Wall Gap Gap of 110 Gap of 20 Gap of 60

  45. 2 Regions: Wall Gap Averages of 100 runs

  46. 2 Regions: Wall Thickness Thickness of 10 Thickness of 40 Thickness of 70

  47. 2 Regions: Wall Thickness Averages of 100 runs

  48. 4 Regions: Wall Gap Gap of 80 Gap of 50 Gap of 20

  49. 4 Regions: Wall Gap Averages of 100 runs

  50. 4 Regions: Wall Thickness Thickness of 10 Thickness of 30 Thickness of 50

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