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 First Model: Simple Square Region Classifications and Equations 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: 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 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 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 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( 11ee 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. 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 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-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