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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.
“Transmission” of Incarceration
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.
“Transmission” of Incarceration
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.
“Transmission” of Incarceration
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
SIS Example
spouse
parent-child
sibling
SIS Example
spouse
parent-child
sibling
SIS Example
spouse
parent-child
sibling
friend
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
SIS Example: Month 5
spouse
parent-child
sibling
friend
SIS Example: Month 5
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)
SIS Example: Month 10
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
Birth: 113 Death:194
B4
Birth: 101 Death:174
B1
Birth: 109 Death:187
B3
C2 C3
Birth: 127 Death:205
Birth: 135 Death:211
C4
Birth: 136 Death:216
C1
Birth: 125 Death:208
C6 C8
Birth: 144 Death:214
C5
Birth: 141 Death:228
C7
Birth: 149 Death:241
Birth: 155 Death:246
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
Number of Times Incarcerated
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
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,
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
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
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…
What if…
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