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Understanding The Dynamics of Drug-Related Crimes In Chicago Over
Time
Joanna Sasara
Understanding The Dynamics Of Drug-Related Crimes In
Chicago Over Time Using Differential
Equations Joanna Sasara
Dominican University
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Background
• Surveys
• Impact of crime
• Theory of Differential
Association
• Flows
• Fitting to data
• Approach2
Model
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Model Development
Criminal
NoncriminalSusceptible α
γη
μC
β
α - demographic and agingβ – demographic, social and
economic conditions, and deterrence
γ - pressure on criminals to withdraw from the criminal category
η - negative effects of deterrence and
social and
economic conditions
μC - social interactionFlows between the groups are proportional to the group they flow out from
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System of Differential Equations
Criminal
NoncriminalSusceptible α
γημC
β
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Basic Properties
Solutions and
Initial time
𝑁+𝑆+𝐶=1 0.6+0.3+0.1=1
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𝑁=1−𝐶−𝑆
Model Analysisη = 0
Criminal
NoncriminalSusceptible α
γμC
β
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Equilibria Points
Second equilibrium point:
First equilibrium point:
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Equilibrium Positivity Local Stability
ALWAYS
ALWAYS
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Model Analysis = 0
, )
Stable if
Oscillation if
()+
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Data
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Chicago Data
Year Population Criminals %C
2005 2,795,807 57,314 2.05
2006 2,775,765 56,310 2.02
2007 2,755,723 54,445 1.98
2008 2,735,681 44,347 1.62
2009 2,715,639 43,176 1.59
2010 2,695,598 43,084 1.60
Census CPD
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Community Areas Population
District Name 2006 2007 2008 2009 2010
16 Dunning 42,025 42,002 41,979 41,956 41,933
25 Montclare 13,114 13,192 13,270 13,348 13,426
25 Hermosa 25,769 25,579 25,389 25,199 25,009
14 Avondale 40,788 40,407 40,026 39,645 39,264
14 Logan Square 77,243 76,331 75,419 74,507 73,595
11&25 Humboldt Park 60,129 59,178 58,227 57,276 56,325
13&14 West Town 83,834 83,234 82,634 82,034 81,434
15&25 Austin 106,120 104,219 102,318 100,417 98,516
11 West Garfield Park 20,008 19,506 19,004 18,502 18,000
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Chicago Districts’ Data 2006
District Criminals Population %C
1 824 40,761 0.632 1,444 37,049 3.8983 2,789 98,413 2.8344 3,033 111,191 2.7285 2,864 96,087 2.9826 2,639 102,649 2.5717 3,989 73,897 5.3988 2,855 238,232 1.1989 2,636 148,352 1.668
10 3,636 141,793 2.23714
Chicago Map 2007
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Low Rate
Medium Rate
High Rate
Year Low Rate Medium Rate High Rate2005 0.006 0.017 0.0722006 0.007 0.017 0.0842007 0.006 0.016 0.0572008 0.006 0.015 0.0422009 0.006 0.017 0.0642010 0.005 0.016 0.067
Crime Level
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Fitting The ModelWhen η=0
Chicago fitting when η= 0 with α= 0.00424, μ= 0.1, γ= 0.2μ, β= 0.015, S(0) = 0.1, C(0) = 0.016 and results of = 1.6%, = 22%, and 17
High Rate fitting when η= 0 with α= 0.015, μ= 0.1, γ= 0.18μ, β= 0.0558, S(0) = 0.1, C(0) = 0.0717 and results of = 6.8%, = 21%, and 18
Fitting The Model When μ=0
Chicago fitting when μ = 0 with α= 0.005, η= 0.006, γ= 0.03, β= 0.051, S(0) = 0.1, C(0) = 0.016 and results of = 1.6%, = 8%,
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High Rate fitting when μ = 0 with α= 0.635, η= 0.504, γ= 0.28345, β= 0.99, S(0) = 0.1, C(0) = 0.0717 and results of = 6.8%, = 3.8%,
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Conclusion
• Summary
• Approach
• Divide population into three groups
• System of Diff. Eq.
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• Case 1, η = 0
• Necessity of susceptible group
• Vanish
• Case 2, μ = 0
• Independence
• Constant appearance
• Insufficiency
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Future Prospects
• Fitting for high crime scenario
• Varying parameters
• Spatial movement between groups
• Different initial conditions
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Acknowledgment
Dr. Marion Weedermann and Dr. Sara Quinn
Chicago Police Department
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References•E. L. Glaeser and B. Sacerdote and J. A. Scheinkman, “Crime And Social Interactions,” The Quarterly Journal of Economics, Vol. 111(2), 1996.• L. Edelstein-Keshet, “Mathematical Model In Biology,” Siam, 1988.• L.J.S. Allen, “An Introduction to Mathematical Biology,” Pearson, Prentice Hall, 2006.• M. Campbell and P. Ormerod, “Social interaction And The Dynamics Of Crime,” Voltera Conlusting Ltd. Research Paper, 1997.• U.S. Department Justice, National Drug Intelligence Center, “National Drug Threat Assessment 2011.”• N. J. Herman, “Deviance: A Symbolic Interactioanist Approach,” Rowman & Littlefield, 1995.
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