modeling incarceration as an epidemic · recidivism 0.00 0.25 0.50 0.75 1.00 3 6 9 12 15 18 21 24...

Modeling Incarceration as an Epidemic Kristian Lum1, Samarth Swarup1, Stephen Eubank1, Jim Hawdon2

SAMSI Transition Workshop May 7, 2014

1Network Dynamics and Simulation Science Laboratory, VBI, VT 2Department of Sociology and Center for Peace Studies and Violence Prevention, VT

Incarceration rates in the US

According to data from the Bureau of Justice Statistics, the per capita rate of incarceration nearly quadrupled between 1978 and 2011 from 137 to 511 persons per

100,000. As of 2011, the incarceration rate of black males was 3,023 per 100,000, whereas non-Hispanic white males were incarcerated at the much lower rate of 478

per 100,000

0.000

0.005

0.010

0.015

0.020

1980 1990 2000 2010years

Prop

ortio

n In

carc

erat

ed

race

black

white

California Incarceration Rates

# incarcerated by race taken from National Prisoner Statistics data set from the Inter-University Consortium for Political and Social Research

Total # by race taken from data released by the California Department of Finance

Proportion incarcerated =

“Transmission” of Incarceration

Incarcerated

Direct influence: •  Exposure to criminal norms •  Involvement in criminal subculture

Robert Agnew. An empirical test of general strain theory. Criminology, 30(4):475–500, 1992. Edwin H Sutherland and DR Cressey. Criminology (9th edn). Lippincott, Philadelphia, PA, 1974.


Incarcerated

Demographic influence: •  Decreased household income due to the inmate's inability to work while incarcerated •  Inability of the inmate to contribute to child care responsibilities, put the remaining

family members at increased risk of work-family conflicts •  From a long term perspective, incarceration decreases one’s expected earnings and

opportunities for education. Intergenerational mobility data suggests that lower parental earnings and education tend to result in lower earnings and educational attainment for children.

Joyce A Arditti, Jennifer Lambert-Shute, and Karen Joest. Saturday morning at the jail: Implications of incarceration for families and children*. Family Relations, 52(3):195–204, 2003. Olga Grinstead, Bonnie Faigeles, Carrie Bancroft, and Barry Zack. The financial cost of maintaining relationships with incarcerated african american men: A survey of women prison visitors. Journal of African American Studies, 6(1):59–69, 2001. Bhashkar Mazumder. Upward intergenerational economic mobility in the United States. Economic Mobility Project, Pew Charitable Trusts, 2008.


Incarcerated

Official bias: •  the police and courts pay more attention to the inmate’s family and friends thereby

increasing the probability they will be caught, prosecuted and imprisoned

Sytske Besemer, David P Farrington, and Catrien CJH Bijleveld. Official bias in intergenerational transmission of criminal behaviour. British Journal of Criminology, 53(3):438–455, 2013. David P Farrington. Predicting adult official and self-reported violence. In G-F. Pinard and L. Pagani, editors, Clinical Assessment of Dangerousness, pages 66–88. Cambridge University Press, Cambridge, 2001. Donald James West and David P Farrington. Who becomes delinquent? Second report of the Cambridge Study in Delinquent Development. Heinemann Educational, 1973.


Incarcerated

Bottom line: •  Regardless of the mechanisms involved, the incarceration of one family member

undoubtedly increases the likelihood of other family members being incarcerated •  This suggests that models of contagion may aptly characterize incarceration.

Christopher Wildeman. Paternal incarceration and children’s physically aggres- sive behaviors: evidence from the fragile families and child wellbeing study. Social Forces, 89(1):285–309, 2010. Sara Wakefield and Christopher Wildeman. Mass imprisonment and racial disparities in childhood behavioral problems. Criminology & Public Policy, 10(3):793–817, 2011. Terence P Thornberry. The apple doesn’t fall far from the tree (or does it?): Intergenerational patterns of antisocial behavior—the American Society of Crim- inology 2008 Sutherland Address. Criminology, 47(2):297–325, 2009.

Feedback loop of Incarceration

Incarcerated

Feedback loop of Incarceration

Incarcerated

Incarcerated

Susceptible Infected / Incarcerated

Incarcerated with probability p

Released after s months

SIS Model

Basic idea: •  Each iteration (month), each infected agent independently infects each of its

“neighbors” with probability p. •  At the end of an agent’s infectious period, it recovers and returns to the

“susceptible” state. It may subsequently be re-infected by its infected neighbors neighbors.

SIS Example

SIS Example

spouse

parent-child

sibling

SIS Example

spouse

parent-child

sibling

friend

SIS Example

spouse

parent-child

sibling

friend

Sentence: 6 months

SIS Example

spouse

parent-child

sibling

friend

Sentence: 6 months

During each iteration (year, month, week, minute….), those who are connected with her are at increased risk of incarceration themselves. So, if she has probability p of transmitting to her son in each iteration (here, month), then the probability that she transmits to him over the course of her sentence (s iterations) is psentence = 1-(1-p)s.

0 10 20 30 40 500.

00.

10.

20.

30.

4

Transmission Probability Over Whole Sentence

Sentence Length (months)

Mon

thly

Pro

babi

lity

of T

rans

mis

sion

SIS Example

spouse

parent-child

sibling

friend

The monthly probability of transmission may not also be constant across her contacts. For example, her children may be more effected by her absence than her friends. The probability of transmission may also vary based on personal characteristics of her contacts: males may be more susceptible than females.

pchild pchild pchild

pfriend

pfriend pspouse

Pr(i à j) = f(relationship(i,j), Xi, Xj)

SIS Example: Month 1

spouse

parent-child

sibling

friend


spouse

parent-child

sibling

friend


Some agents are now connected to two inmates—under this model, their probability of incarceration increases.

0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

# Incarcerated Contacts

Prob

abilit

y of

Tra

nsm

issi

on

spouse

parent-child

sibling

friend

P(incarceration) = 1-(1-pparent)(1-psibling)


spouse

parent-child

sibling

friend

And so on…

SIS model components

Agent-based modeling approach: •  Multi-generational family and friend “influence network”.

•  In the analogous disease model, this would be the “contact network”. •  Sentence Lengths •  Transmission probabilities

ODE approach: •  Compartments* •  Transmission probabilities, mixing rates •  Sentence Lengths

* We cannot compartmentalize this population because an agent is simultaneously several roles… mother, daughter, sister, friend, etc…

Synthetic Population

We simulate an evolving population of agents: •  Birth rates are derived from data from the CDC’s National Vital Statistics Report •  Death rates from the Social Security Administration’s life tables •  Mate selection informed by US Census data •  Friendship networks based on information from Add Health and the General Social

Survey •  All family relationships are stored

A2

Birth: 79 Death:154

Birth: 75 Death:163

A1

Birth: 107 Death:177

B2


B4


B1


B3

C2 C3



C4


C1


C6 C8


C5


C7



4 children… 3 children… 2 children…

Synthetic Population

We simulate an evolving population of agents: •  Birth rates are derived from data from the CDC’s National Vital Statistics Report •  Death rates from the Social Security Administration’s life tables •  Mate selection informed by US Census data •  Friendship networks based on information from Add Health and the General Social

Survey •  All family relationships are stored

•  Bureau of Justice Statistics lists mean and median sentences by race for various crimes. We use the drug possession statistics.

Sentencing

White Black

Mean 14 17

Median 10 12

•  Dellaire (2007) gives the percent of people whose family members are in jail by relation.

Transmission Probabilities

Rate per month calibrated to s = 14 month sentences

pmonth

= 1� (1� psentence

)1s

women men

mother 0.001 0.003

father 0.010 0.010

sister 0.007 0.004

brother 0.030 0.027

spouse 0.004 0.001

adult child 0.015 0.006

Monthly transmission rates women men

mother 0.012 0.048

father 0.147 0.148

sister 0.107 0.059

brother 0.377 0.349

spouse 0.059 0.011

adult child 0.213 0.085

Rates given in survey

•  Initialize some percent of individuals as incarcerated (1%). •  In each month an agent is incarcerated, it infects its family

members and friends (friends treated as siblings) according to the probabilities listed in the table.

•  The duration of incarceration is a random draw from the respective distribution.

•  We run this for 50 years (600 months) under the Black sentence distribution and the White sentence distribution

Overview

Results

WhiteBlack

0.01

0.02

0.03

0 10 20 30 40 50time

Prop

ortio

n In

carc

erat

ed

a) Mean Incarceration Rate Over Time

p−value = .05

−150

−100

−50

0

0 10 20 30 40 50time (years)

log

p−va

lue

Results

Recidivism

0.00

0.25

0.50

0.75

1.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Simulation Results

0.00

0.25

0.50

0.75

1.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CaliforniaFlorida

New Zealand

Pennsylvania

Recidivism Rate by Number of Times Incarcerated

Number of Times Incarcerated

Rec

idiv

ism

Rat

e

Results: Recidivism

0.00

0.25

0.50

0.75

1.00

(18,

24]

(24,

29]

(29,

34]

(34,

39]

(39,

44]

(44,

49]

(49,

54]

(54,

59]

(59,

64]

(64,

69]

(69,

74]

(74,

79]

(79,

120]

Simulation Results

0.00

0.25

0.50

0.75

1.00

[18,

19]

[20,

24]

[25,

29]

[30,

34]

[35,

39]

[40,

44]

[45,

49]

[50,

54]

[55,

59]

60+

CaliforniaArizona

Iowa

Texas

Recidivism Rate by Age at Release

Age at Release

Rec

idiv

ism

Rat

e

Recidivism

0.00

0.25

0.50

0.75

1.00

3 6 9 12 15 18 21 24 27 30 33 36

Simulation Results

0.00

0.25

0.50

0.75

1.00

3 6 9 12 15 18 21 24 27 30 33 36

CaliforniaArizona

Florida

Massachusetts

Recidivism Rate by Months Since Release

Months Since Release

Rec

idiv

ism

Rat

e

Recidivism

0.00

0.25

0.50

0.75

1.00

[0,6

)[6

,12)

[12,

18)

[18,

24)

[24,

30)

[30,

36)

[36,

48)

[48,

60)

[60,

72)

[72,

84)

[84,

96)

[96,

600)

Simulation Results

0.000.250.500.751.00

[0,6

](6

,12]

(12,

18]

(18,

24]

(24,

36]

(36,

48]

(48,

60]

(60,

120]

(120

,180

]18

0+

CaliforniaIndiana

Delaware

Florida

Recidivism Rate by Length of Sentence

Length of Sentence

Rec

idiv

ism

Rat

e

ODE Approach

Under assumptions of random mixing and homogeneity of transmission rate, the SIS model can be written as a set of ordinary differential equations. This gives a closed form solution for the steady state prevalence, I.

Let s be the sentence length and p be the transmission rate, then…

I = 0 if s < 1/p and I = 1-1/ps otherwise.

We estimate p using our model

p = 0.0612

Non-Contagious Model Results

WhiteBlack

0.01

0.02

0.03

0 10 20 30 40 50time

Prop

ortio

n In

carc

erat

ed

a) Mean Incarceration Rate Over Time

p−value = .05

−150

−100

−50

0

0 10 20 30 40 50time (years)

log

p−va

lue

WhiteBlack

0.010

0.015

0.020

0.025

0.030

0.035

0 10 20 30 40 50x

Prop

ortio

n In

carc

erat

ed

c) Mean Incarceration Rate Over Time

p−value = .05

−150

−100

−50

0

0 10 20 30 40 50time (years

log

p−va

lue

Non-Contagious Model Results

0.00

0.25

0.50

0.75

1.00

1 2 3 4 5 6 7 8 9 10


Rec

idiv

ism

Rat

e

0.00

0.25

0.50

0.75

1.00

(18,

19]

(19,

24]

(24,

29]

(29,

34]

(34,

39]

(39,

44]

(44,

49]

(49,

54]

(54,

59]

(59,

64]

(64,

69]

(69,

74]

(74,

79]

80+

Age at First Release

Rec

idiv

ism

Rat

e

0.00

0.25

0.50

0.75

1.00

3 6 9 12 15 18 21 24 27 30 33 36


Rec

idiv

ism

Rat

e

0.00

0.25

0.50

0.75

1.00

(0,6

](6

,12]

(12,

18]

(18,

24]

(24,

30]

(30,

36]

(36,

42]

(42,

48]

(48,

54]

(54,

60]

(60,

66]

(66,

72]

(72,

78]

(78,

84]

(84,

90]

90+

Length of Sentence (Months)

Rec

idiv

ism

Rat

e

Non−Contagious Simulation Results

Non-Contagious Model Results (Comparison)

0.00

0.25

0.50

0.75

1.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Simulation Results

0.00

0.25

0.50

0.75

1.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CaliforniaFlorida

New Zealand

Pennsylvania

Recidivism Rate by Number of Times Incarcerated


Rec

idiv

ism

Rat

e

0.00

0.25

0.50

0.75

1.00

(18,

24]

(24,

29]

(29,

34]

(34,

39]

(39,

44]

(44,

49]

(49,

54]

(54,

59]

(59,

64]

(64,

69]

(69,

74]

(74,

79]

(79,

120]

Simulation Results

0.00

0.25

0.50

0.75

1.00

[18,

19]

[20,

24]

[25,

29]

[30,

34]

[35,

39]

[40,

44]

[45,

49]

[50,

54]

[55,

59]

60+

CaliforniaArizona

Iowa

Texas

Recidivism Rate by Age at Release

Age at Release

Rec

idiv

ism

Rat

e

0.00

0.25

0.50

0.75

1.00

3 6 9 12 15 18 21 24 27 30 33 36

Simulation Results

0.00

0.25

0.50

0.75

1.00

3 6 9 12 15 18 21 24 27 30 33 36

CaliforniaArizona

Florida

Massachusetts

Recidivism Rate by Months Since Release


Rec

idiv

ism

Rat

e

0.00

0.25

0.50

0.75

1.00

[0,6

)[6

,12)

[12,

18)

[18,

24)

[24,

30)

[30,

36)

[36,

48)

[48,

60)

[60,

72)

[72,

84)

[84,

96)

[96,

600)

Simulation Results

0.000.250.500.751.00

[0,6

](6

,12]

(12,

18]

(18,

24]

(24,

36]

(36,

48]

(48,

60]

(60,

120]

(120

,180

]18

0+

CaliforniaIndiana

Delaware

Florida

Recidivism Rate by Length of Sentence

Length of Sentence

Rec

idiv

ism

Rat

e

Conclusions

•  We have shown that under a reasonable set of parameters, we are able to reproduce many of the facets of the incarceration epidemic in the United States.

•  We have shown that it is plausible that differential sentencing plays an important, causal role in producing incarceration rates as are seen today.

•  The effects of sentencing policies go far beyond the individual inmate and even beyond their immediate family– they have the potential to affect whole communities.

•  We have demonstrated the utility of a synthetic information approach. This would not have been possible using a differential equation model given the available data.

•  This does not mean that differential sentencing is the whole story– incarceration is complex.

•  This does mean that serious thought should be given to sentencing policy as it pertains to individual-, family-, and community-wide effects.

What if…

modeling incarceration as an epidemic · recidivism 0.00 0.25 0.50 0.75 1.00 3 6 9 12 15 18 21 24...

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