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Mathematical Modelling of The Dynamics of Poverty, Crime and
Imprisonment
Okuh Benjamin Ajokpaoghene, *Omokri Peter Akweni, *Akudo Nkpuruoma Ashinze
Department of Mathematics and Statistics, Delta State Polytechnic, Ogwashi-Uku, Delta State, Nigeria
DOI: https://doi.org/10.51583/IJLTEMAS.2025.1410000007
Received: 16 Sep. 2025; Accepted: 24 Sep. 2025; Published: 28 October 2025
Abstract: This study explores the evolution and application of mathematical modelling to complex social issues such as poverty,
crime, and terrorism. Traditionally rooted in epidemiology, compartmental models have been successfully adopted in criminology
and public health to capture the dynamics of addiction, ideological radicalization, and recidivism. The model was derived from a
five–compartment representation from which a set of five ordinary differential equations (ODEs) was developed to capture the
dynamism of poverty, crime, and imprisonment in a deterministic SCJ₁J₂R compartmental model using the next generation matrix
to obtain the reproduction number R₀. This is used to analyze the local stability of the crime-free equilibrium of the SCJ₁J₂R model.
The crime-free equilibrium and the local stability of the endemic equilibrium show that R₀ is asymptotically stable if R₀ < 1. We
use hypothetical data to simulate the sensitivity of the parameters of the basic reproduction number (R₀) so as to obtain R₀ < 1 for
a crime-free society. The SCJ₁J₂R compartmental model differentiates incarceration based on criminal evidence and accounts for
both natural and crime–induced mortality as well as reintegration processes. This review highlights the growing role of
mathematical approaches in policy-relevant analysis of community safety, systemic intervention, and stability of the models.
Keywords: Reproduction Number (R₀), Crime, Poverty, Incarceration, ODEs, SCJ₁J₂R Model,
I. Introduction
Mathematical modelling of the dynamics of poverty, crime, and imprisonment involves creating systems of equations using ordinary
differential equations (ODEs) to represent these variables and parameters’ interaction and change over time in the enclosed system.
There is a strong relationship between poverty and crime. Some of the reasons of addiction are directly or indirectly related to
poverty such as: ignorance, unhealthy social environment, inability to deal with life and stress, unavailability of social and
psychological help, family and social damages, family troubles and strained relationships, inability to complete education, etc.
(Sakib et al., 2017). Crime is one among the most challenging problems in most developing countries in which poverty,
unemployment, etc. is among the causes. This paper is intended to contribute to the eradication of poverty-related crimes in the
developing countries by proposing a deterministic mathematical model. The model can help to analyze the impact of various
intervention programs such as poverty alleviation programs, law enforcement, and rehabilitation programs (Mataru et al., 2023).
Crime has become an epidemic disease in our society. As such, epidemiologists and scholars are finding possible solutions to curb
the crime menace through the formulation of mathematical modelling. Modelling in criminology and public health has evolved
from the classical epidemic framework into sophisticated multi-layered representation of criminal behaviour, terrorism, and drug-
related activities in our communities (McMillon, 2014; Njagarah, 2013). White and Comiskey (2017) used ordinary differential
equations (ODEs) to embed treatment and relapse, foregrounding the cyclical patterns that the model would generate.
Mushayabasa et al. (2014) extended this paradigm by explicitly incorporating awareness and rehabilitation control, highlighting
how prevention and treatment can reshape the qualitative behaviour of the enclosed system. Parallel development in the
mathematical modelling of terrorism and radicalization recognized that ideology can diffuse through populations with epidemic–
like mechanisms. Considering a unified framework that simultaneously captures drug crime and terrorist activity remains limited.
The research of Malonza and Bonyo (2022) advanced the field with a dual-crime model for illicit trade and armed conflict in East
Africa, which did not fully differentiate incarceration pathways or reintegration processes. Responding to this gap, this paper
proposes an SCJ₁J₂R compartmental model, explicitly separating incarceration with or without crime involvement (J₁ and J₂),
integrating both natural and crime–induced mortality, and allowing recovery and reintegration dynamics that can feed back into the
susceptible class (S) in the enclosed system.
Recent mathematical modelling research has increasingly integrated structural socio-economic determinants. Khan et al. (2023)
constructed a deterministic framework for urban drug trafficking that couples law enforcement with socio-economic stressors, while
Aluko and Olayemi (2022) examined organized crime with recidivism and rehabilitation, emphasizing the “revolving door’’
between incarceration and criminal relapse.
Etim et al. (2024) underscored the effect of poverty and unemployment in operation as upstream drivers of crime, motivating
prevention strategies which act on macro-structural levers rather than purely punitive controls. A complementary methodological
strand stems from the study of criminal hotspots using mathematical models grounded in reaction–diffusion PDEs and self-exciting
processes. The research of Short et al. (2008, 2010) demonstrated how local crime attractiveness, repeat victimization, and offender
movement can produce spatio-temporal clustering (hotspots) and how interventions may dissipate these criminal clusters.
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While these models are not compartmental in the traditional epidemiological sense, they underscore the value of mechanistic
formulations that connect micro-level behaviour to macro-level emergent patterns. Hence, the SCJ₁J₂R model developed in this
study contributes to this trend by offering a unified, policy-relevant structure that can quantify how the differentiated incarceration
paths, mortality channels, and the recovery mechanisms interact to bring the community to safety and stability.
II. Formulation of the SCJ1J2R Model
The model is an SCJ1J2R model which analyses the dynamics of poverty, crime and imprisonment in the enclosed system. The
population is divided into five compartment; the susceptible class(S), the criminal class(C), the jailed class due to crime(J1), the
jailed class without committing crime (J2) and rehabilitation class(R). The total population at time (t) is given by
T(t) = S(t) + C(t) + J1(t) + J2(t) + R(t) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (1)
The system of ODEs governing the model is
����
����
= + ∅�� − (���� + �� + ��)�� - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2(a)
����
����
= ������ − (��3 + �� + �� + ∅)�� - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2(b)
����1
����
= ���� − (�� + ��1)��1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2(c) (2)
����2
����
=∝ �� − (�� + ��2)��2 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2(d)
����
����
= ��1��1 + ��2��2 + ��3�� − (�� + ∅)�� - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2(e)
In the following tables, the variable and parameters of the SCJ1J2R model are interpreted as this:
Table 1: The Model Variables
Variable Descriptions
S(t) Susceptible class
C(t) Criminal class
J1(t) Jailed class due to crime
J2(t) Jailed class without committing crime
R(t) Rehabilitation class
Table 2: The Model Parameters
Parameters Descriptions
�� Recruitment rate
∅ Rate of movement from rehabilitation class to susceptible class
�� Rate of movement from susceptible class to criminal class
�� Rate of movement from susceptible class to jailed class without committing crime
�� Natural death rate
�� Induce death rate due to crime
��1 Rate of movement from jail due to crime to rehabilitation class
��2 Rate of movement from jail without committing crime to rehabilitation class
��3 Rate of movement from criminal class to rehabilitation class
�� Rate of movement from criminal class to jailed class
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Figure 1: The flow chart of the SCJ1J2R model
III. The Model Properties
In this section, we shall discuss the existence and uniqueness of the model solution.
Existence and Uniqueness of Solution
If the system of equations in (2) has solution and it is unique, then we apply the Lipchitz condition to verify the existence and
uniqueness solution of the system.
Theorem 3.1.1
Let D’ denote the region R such that
�� = {(��, ��): |�� − ��0| ≤ ��; |�� − ��0| ≤ ��, (��, �� > 0)}
Where x = x1, x2, x3……..xn, and y = y1, y2, y3……..yn,
Suppose f(t, x) satisfied the Lipchitz condition, then
||��(��1, ��1) − ��(��1, ��2)|| ≤ ��||��1 − ��2||
The pairs (��1, ��1) and (��1, ��2) belong to DI, where k is a positive constant. There is a constant �� > 0 such that there exists a unique
solution which satisfied the requirement that
������
������
, ��, �� = 1,2,3, … … �� be continuous and bounded in DI.
Hence by applying the Lipschitz condition, the system in (2) satisfies the conditions of Theorem 3.1.1, ensuring the existence and
uniqueness of solutions.
IV. Mathematical Analysis of the SCJ1J2R model
In this section, we shall study and analyse the stability status of the equilibrium solutions of the system of equations in (2).
Crime-Free Equilibrium (CFE)
Here, we are interested in a population dynamics of the system where there is crime-free equilibrium (CFE). We assume the absence
of crime in the society, hence C=0, which implies J1 = 0.
S
C
J1
J2
R
��R
��J2
��J1
��3C
��1J1
��S
��SC
��S
(��+∈)��
��C
∅R
π
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Setting all derivatives in (2) to zero:
����
����
=
����
����
=
����1
����
=
����2
����
=
����
����
= 0 (3)
Solving equations (3) and letting ��0 denotes the crime-free equilibrium. Hence
��0 = (
��(��+��2)(��+∅)
(��+∅)(��+��)(��+��2)−∅��2
, 0,0,
����(��+∅)
(��+∅)(��+��)(��+��2)−∅��2
,
��2����
(��+∅)(��+��)(��+��2)−∅��2
) (4)
��ℎ������ �� =
��(��+��2)(��+∅)
(��+∅)(��+��)(��+��2)−∅��2
; ��2 =
����(��+∅)
(��+∅)(��+��)(��+��2)−∅��2
; �� =
��2����
(��+∅)(��+��)(��+��2)−∅��2
Basic Reproduction Number of SCJ1J2R Model
Using the next generation matrix approach, Diekmann and Heesterbeck (2001),
1
0R FV (5)
After computation,
0
3
S
R
(6)
At CFE:
2
0
3 2 2
R
(7)
Local Stability of Crime-Free Equilibrium of the Model
Theorem 3.1.3: The crime-free equilibrium is locally asymptotically stable if R0<1 and asymptotically unstable if R0>1.
Hence;
����(��+��2)(��+∅)
(��3+��+∈+��)(��+∅)(��+��)(��+��2)−∅��2
< 1 (8)
R0 < 1
⋋3= −(�� + ��1) < 0
⋋4= −(�� + ��2) < 0
⋋5= −(�� + ∅) < 0
Therefore for
��(⋋) < 0
Then, R0 =
����(��+��2)(��+∅)
(��3+��+∈+��)(��+∅)(��+��)(��+��2)−∅��2
(9)
By implication
⋋3=
1
��0
< 0, ��0 =
1
⋋3
< 1
Numerical Simulation
Numerical evaluation of the basic reproduction number (��0) using the parameter data supplied in the generated data indicates strong
parameter sensitivity. For the parameter combinations in Table 4, the computed ��0 values lie between 0.07 and 0.59, and all ten
combinations satisfy ��0 < 1. Consequently, under the Table 4 parameter regime the Crime-Free Equilibrium is locally
asymptotically stable and criminality is predicted to die out. By contrast, parameter combinations in Table 5 produce ��0 > 1 (range
1.08–6.52), predicting persistence of criminal behaviour.
Table 3 contains a mix of outcomes (one of ten parameter sets yields ��0 < 1; this highlights that targeted changes in parameters
such as the recruitment rate into the criminal class ( ) or increased rehabilitation (�� terms) can move the system from an
endemic to a crime-free state.
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Table 3
Entry �� µ σ ��1 β �� φ ��2 ∅ �� ��3 R0 1/R0 <1
1 55 0.12 0.2 0.22 0.14 0.04 0.04 0.06 0.22 0.27 0.025 0.9 1.1111
2 55 0.12 0.25 0.22 0.2 0.04 0.06 0.08 0.26 0.29 0.03 1.38 0.7246
3 55 0.12 0.3 0.22 0.26 0.04 0.08 0.1 0.3 0.31 0.035 1.86 0.5376
4 55 0.12 0.35 0.22 0.32 0.04 0.1 0.12 0.34 0.33 0.04 2.34 0.4274
5 55 0.12 0.4 0.22 0.38 0.04 0.12 0.14 0.38 0.35 0.045 2.81 0.3559
6 55 0.12 0.45 0.22 0.44 0.04 0.14 0.16 0.42 0.37 0.05 3.27 0.3058
7 55 0.12 0.5 0.22 0.5 0.04 0.16 0.18 0.46 0.39 0.055 3.7 0.2703
8 55 0.12 0.55 0.22 0.56 0.04 0.18 0.2 0.5 0.41 0.06 4.12 0.2427
9 55 0.12 0.6 0.22 0.62 0.04 0.2 0.22 0.54 0.43 0.065 4.53 0.2208
10 55 0.12 0.65 0.22 0.68 0.04 0.22 0.24 0.58 0.45 0.07 4.91 0.2037
Table 4
Entry �� µ �� ��1 β ∅ φ ��2 δ ρ ��3 R0 1 /R0 <1
1 55 0.11 0.011 0.212 0.19 0.056 0.07 0.1 0.31 0.31 0.04 0.59 1.6949
2 55 0.11 0.0101 0.224 0.26 0.062 0.1 0.14 0.42 0.4 0.05 0.49 2.0408
3 55 0.11 0.0092 0.236 0.33 0.068 0.13 0.18 0.53 0.49 0.06 0.4 2.5
4 55 0.11 0.0083 0.248 0.4 0.074 0.16 0.22 0.64 0.58 0.07 0.33 3.0303
5 55 0.11 0.0074 0.26 0.47 0.08 0.19 0.26 0.75 0.67 0.08 0.27 3.7037
6 55 0.11 0.0065 0.272 0.54 0.086 0.22 0.3 0.86 0.76 0.09 0.22 4.5455
7 55 0.11 0.0056 0.284 0.61 0.092 0.25 0.34 0.97 0.85 0.1 0.17 5.8824
8 55 0.11 0.0047 0.296 0.68 0.098 0.28 0.38 1.08 0.94 0.11 0.14 7.1429
9 55 0.11 0.0038 0.308 0.75 0.104 0.31 0.42 1.19 1.03 0.12 0.1 10.0
10 55 0.11 0.0029 0.32 0.82 0.11 0.34 0.46 1.3 1.12 0.13 0.07 14.2857
Table 5
Entry µ �� ��1 β ∅ φ ��2 δ Ρ ��3 R0
1 110 0.55 0.12 0.09 0.15 0.24 0.025 0.1 0.48 0.4 0.045 1.08
2 110 0.5 0.15 0.11 0.21 0.24 0.035 0.14 0.48 0.58 0.065 1.47
3 110 0.45 0.18 0.13 0.27 0.24 0.045 0.18 0.48 0.76 0.085 1.89
4 110 0.4 0.21 0.15 0.33 0.24 0.055 0.22 0.48 0.94 0.105 2.36
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5 110 0.35 0.24 0.17 0.39 0.24 0.065 0.26 0.48 1.12 0.125 2.87
6 110 0.3 0.27 0.19 0.45 0.24 0.075 0.3 0.48 1.3 0.145 3.45
7 110 0.25 0.3 0.21 0.51 0.24 0.085 0.34 0.48 1.48 0.165 4.08
8 110 0.2 0.33 0.23 0.57 0.24 0.095 0.38 0.48 1.66 0.185 4.8
9 110 0.15 0.36 0.25 0.63 0.24 0.105 0.42 0.48 1.84 0.205 5.6
10 110 0.1 0.39 0.27 0.69 0.24 0.115 0.46 0.48 2.02 0.225 6.52
Empirical Analysis of the Basic Reproduction Number (R₀)
Figure 2: Distribution of R₀ values across parameter
tables
Figure 3: Histogram of R₀ across Tables 3–5
Figure 4: Boxplots of R0 by Table 3, 4 & 5
Figure 5: Sensitivity of R0 to β
Empirical plots of 0R across parameters. Fig 2- Distribution of 0R values across parameter tables, Fig 3- Histogram of 0R across
Tables 3–5, Fig 4- Boxplots of 0R by Table, Fig 5- Sensitivity of 0R to β.
V. Results and Discussion
The four empirical plots collectively highlight the threshold-driven behaviour of the basic reproduction number ( 0R ):
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1. Separation across Parameter Regimes: Both the distribution and histogram demonstrate a clear distinction between parameter
tables. Table 4 values cluster below unity, indicating crime-free stability, while Tables 3 and 5 lie above unity, predicting
persistence of criminal activity.
2. Variability of Outcomes: The boxplots reveal that Table 4 values are tightly bounded and always sub-threshold (
0 1R ),
while Tables 3 and 5 show wide variability with medians above one, underscoring their unstable regimes.
3. Sensitivity to Crime Transmission (β): The sensitivity plot shows divergent effects of β:Table 3 shows a moderate upward
trend with β; Table 4 shows a decreasing trend with β, reflecting stabilizing dynamics; Table 5 shows sharp escalation with β,
underscoring its destabilizing effect.
4. Threshold Property of
0R : Table 4 always produces
0 1R , predicting crime eradication. Table 5 consistently produces
0 1R , predicting persistence. Table 3 straddles the threshold, reflecting sensitivity to parameter tuning.
VI. Conclusion
The combined plots reinforce the threshold nature of R0: sub-threshold values ( 0 1R ) predict eradication of crime, while super-
threshold values ( 0 1R ) predict persistence. Strategic interventions, particularly reducing recruitment (β) and strengthening
rehabilitation (φ, δ2), can transition the system from endemic to crime-free states.
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