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/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 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

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 30- 60% 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

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

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.

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.

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.

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.

Total Population

Total Population

Total Population

Total Population

Arena Size

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

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

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.

Maximum Speed

Maximum Speed

Maximum Speed

Maximum Speed

Infection Radius

Infection Radius

Infection Radius

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).

Probability of Infection

Probability of Infection

Probability of Infection

Probability of Infection

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

Recovery/ Removal Cycle

Recovery/ Removal Cycle

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.

Initial Position of Infectives Averages of 100 runs

Logistic Fitting Initial Infective Centered in Arena

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

2 Regions: Wall Gap Averages of 100 runs

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

2 Regions: Wall Thickness Averages of 100 runs

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

4 Regions: Wall Gap Averages of 100 runs

4 Regions: Wall Thickness Thickness of 30 Thickness of 10 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 http://www.answersingenesis.org/articles/am/v2/n3/antibiotic-resistance-of-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/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