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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