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

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 First Model: Simple Square Region
Classifications and Equations
First Model: Simple Square Region
Assumptions
The Effect of Changing Variables
Logistic Fitting
Comparison to SIR
Conclusions
Second Model: Wall Obstructions

5. History Epidemics in History: Modeling Epidemics:
Black Death/ Black Plague
Avian Flu
HIV/AIDS
Modeling Epidemics:
Kermack and McKendrick, early 1900’s
SIR model
Black Death/ Black Plague: occurred in the 1340s and is considered to be the deadliest pandemic in history. It spread from Central Asia to Europe with a total of 75 million deaths, which killed % of Europe’s population.
Avian Flu: occurred from 1918 to 1920 during World War 1. More people were hospitalized from the epidemic than wounds. There were 20 to 100 million deaths, according to the World Health Organization.
HIV/AIDS: first identified in 1981 and its rate is steadily increasing today. As of 2006, it is estimated more than 25 million people have died from AIDS. This is one of the most destructive pandemics recorded.
Model: General Epidemic 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. Making Our Model Construct a square region. Add n-1 Susceptibles.
Insert 1 Infective randomly.
Individuals move randomly.
1)…of width w.
2)…whereas n is the total population.
3)One Infective is placed randomly into the plane.
4) Each individual moves randomly up to a maximum speed and distance.
5) Susceptibles become Infectives when they come in contact.
6) Infectives become Recovered Individuals after the recovery cycle starts.
The Infectives infect Susceptibles on contact.
Infectives are changed to Recovered Individuals after a set time.

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
Infectious: as it will turn red.
Degree: regardless of age, gender, or ethnicity.

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
Infectious: as it will turn red.
Degree: regardless of age, gender, or ethnicity.

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
Arena: the area available for motion
Radius: which is the distance necessary for contact.
Cycle: the point when the people begin to recover.

13. Total Population

14. Total Population

15. Total Population

16. Total Population

17. Arena Size

18. Arena Size
Less people are becoming infected as the arena size increases.

19. Arena Size
The area available for motion directly affects the people in the space.

20. Arena Size
Meaning, the smaller the arena size the more people will be infected and at a quicker rate, depending on the concentration of people.

21. Maximum Speed

22. Maximum Speed

23. Maximum Speed

24. Maximum Speed

25. Infection Radius

26. Infection Radius

27. Infection Radius

28. Infection Radius
If the epidemic is able to be contracted from a distance of 8, then more people will become infected quicker as oppose to a distance of 3.
(They are also allowed to recover quicker).

29. Probability of Infection

30. Probability of Infection

31. Probability of Infection

32. Probability of Infection

33. Recovery/ Removal Cycle
In this graph, fewer people are being infected because individuals are recovering faster.

34. Recovery/ Removal Cycle

35. Recovery/ Removal Cycle

36. Recovery/ Removal Cycle
More people become infected due to the time gap between the infection and removal. Thus, fewer people are being infected because individuals are recovering faster.

37. Initial Position of Infectives
Averages of 100 runs

38. Logistic Fitting
Initial Infective Centered in Arena

39. An average of 105 program runs
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 60
Gap of 110
Gap of 20

45. 2 Regions: Wall Gap
Averages of 100 runs

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

47. 2 Regions: Wall Thickness
Averages of 100 runs

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

49. 4 Regions: Wall Gap
Averages of 100 runs

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

51. 4 Regions: Wall Thickness
Averages of 100 runs

52. 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

53. 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

54. Summary History The SIR model Our Model
Without Obstructions
With Obstruction
Our model compared to the SIR model

55. 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

56. Bibliography
Bongaarts, John, Thomas Buettner, Gerhard Heilig, and Francois Pelletier. "Has the HIV epidemic peaked?" Population and Development Review 34(2): (2008).
Capasso, Vincenzo. Mathematical Structures of Epidemic Systems. New York, NY: Springer-Verlag (1993).
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).

57. Acknowledgements Dr. Russell Herman Mr. David Glasier
Mr. and Mrs. Cavender
SVSM Staff
Joanna Sanborn
Dr. Linda Purnell
Our parents
All supporters