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Equilibria and Stability Analysis of a Compartmental Model for Crime Dynamics with Recidivism and Corruption
Research Article | Published: 15 January 2026
ICCK Journal of Applied Mathematics Volume 2, Issue 1, 2026: 1-43
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ICCK Journal of Applied Mathematics, Volume 2, Issue 1, 2026: 1-43

Open Access | Research Article | 15 January 2026
Equilibria and Stability Analysis of a Compartmental Model for Crime Dynamics with Recidivism and Corruption
by
Sarita Pippal Sarita Pippal 1 *
Ajay Ranga Ajay Ranga 2
1 Department of Mathematics, Panjab University, Chandigarh 160014, India
2 J.C. Bose University of Science & Technology, YMCA, Faridabad, Haryana 121006, India
* Corresponding Author: Sarita Pippal, [email protected]
DOI: 10.62762/JAM.2025.240326
ARK: ark:/57805/jam.2025.240326
Received: 25 October 2025, Accepted: 09 December 2025, Published: 15 January 2026  
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Abstract
A nonlinear compartmental model is developed to analyze crime dynamics in a structured society. The population is stratified into eight compartments: \(S(t)\) (susceptible), \(E(t)\) (exposed), \(C(t)\) (active criminals), \(C_v(t)\) (convicted criminals), \(P_h(t)\) (passive-honest), \(P_c(t)\) (committed-honest), \(J_h(t)\) (honest judges), and \(J_c(t)\) (corrupt judges). The model incorporates nonlinear mechanisms such as institutional corruption (\(\kappa_1\)), judicial correction (\(\kappa_2\)), recidivism feedback (\(\rho_1,\rho_2\)), exposure intensity (\(\eta_1\)), and rehabilitation (\(r_2\)), providing a realistic portrayal of crime--justice interactions. Solutions remain positive and bounded within a feasible domain \(\mathcal{D}\). Linear stability analysis of the crime--free equilibrium \(Z_0\) is performed via the Jacobian \(J(Z_0)\). Numerical simulations explore the long--term dynamics under variations of key parameters (\(\beta_1,\eta_1,\rho_1,\rho_2,r_2,\kappa_1,\kappa_2\)). Results show that strong judicial recruitment (\(a_1\)) and honest reinforcement (\(\beta_4\)) suppress criminal activity, whereas increased corruption (\(\kappa_1\)) and recidivism (\(\rho_1,\rho_2\)) promote its growth. Bifurcation curves, contour maps, and stability basins highlight critical thresholds and equilibrium structures. The analysis demonstrates that proactive policy measures---reducing corruption, discouraging recidivism, and enhancing judicial integrity---can significantly lower crime levels and foster honest societal behavior, offering valuable guidance for designing effective crime--prevention strategies across diverse socio--political contexts.
Graphical Abstract
Equilibria and Stability Analysis of a Compartmental Model for Crime Dynamics with Recidivism and Corruption
Keywords
dynamical systems
nonlinear system
differential equations
stability
bifurcation
Data Availability Statement
Data will be made available on request.
Funding
This work was supported without any funding.
Conflicts of Interest
The authors declare no conflicts of interest.
AI Use Statement
The authors declare that no generative AI was used in the preparation of this manuscript.
Ethical Approval and Consent to Participate
Not applicable. This is a theoretical mathematical modeling study using only publicly available aggregated data; no human participants were involved.
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Cite This Article
APA Style
Pippal, S., & Ranga, A. (2026). Equilibria and Stability Analysis of a Compartmental Model for Crime Dynamics with Recidivism and Corruption. ICCK Journal of Applied Mathematics, 2(1), 1–43. https://doi.org/10.62762/JAM.2025.240326
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TY  - JOUR
AU  - Pippal, Sarita
AU  - Ranga, Ajay
PY  - 2026
DA  - 2026/01/15
TI  - Equilibria and Stability Analysis of a Compartmental Model for Crime Dynamics with Recidivism and Corruption
JO  - ICCK Journal of Applied Mathematics
T2  - ICCK Journal of Applied Mathematics
JF  - ICCK Journal of Applied Mathematics
VL  - 2
IS  - 1
SP  - 1
EP  - 43
DO  - 10.62762/JAM.2025.240326
UR  - https://www.icck.org/article/abs/JAM.2025.240326
KW  - dynamical systems
KW  - nonlinear system
KW  - differential equations
KW  - stability
KW  - bifurcation
AB  - A nonlinear compartmental model is developed to analyze crime dynamics in a structured society. The population is stratified into eight compartments: \(S(t)\) (susceptible), \(E(t)\) (exposed), \(C(t)\) (active criminals), \(C_v(t)\) (convicted criminals), \(P_h(t)\) (passive-honest), \(P_c(t)\) (committed-honest), \(J_h(t)\) (honest judges), and \(J_c(t)\) (corrupt judges). The model incorporates nonlinear mechanisms such as institutional corruption (\(\kappa_1\)), judicial correction (\(\kappa_2\)), recidivism feedback (\(\rho_1,\rho_2\)), exposure intensity (\(\eta_1\)), and rehabilitation (\(r_2\)), providing a realistic portrayal of crime--justice interactions. Solutions remain positive and bounded within a feasible domain \(\mathcal{D}\). Linear stability analysis of the crime--free equilibrium \(Z_0\) is performed via the Jacobian \(J(Z_0)\). Numerical simulations explore the long--term dynamics under variations of key parameters (\(\beta_1,\eta_1,\rho_1,\rho_2,r_2,\kappa_1,\kappa_2\)). Results show that strong judicial recruitment (\(a_1\)) and honest reinforcement (\(\beta_4\)) suppress criminal activity, whereas increased corruption (\(\kappa_1\)) and recidivism (\(\rho_1,\rho_2\)) promote its growth. Bifurcation curves, contour maps, and stability basins highlight critical thresholds and equilibrium structures. The analysis demonstrates that proactive policy measures---reducing corruption, discouraging recidivism, and enhancing judicial integrity---can significantly lower crime levels and foster honest societal behavior, offering valuable guidance for designing effective crime--prevention strategies across diverse socio--political contexts.
SN  - 3068-5656
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  -
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@article{Pippal2026Equilibria,
  author = {Sarita Pippal and Ajay Ranga},
  title = {Equilibria and Stability Analysis of a Compartmental Model for Crime Dynamics with Recidivism and Corruption},
  journal = {ICCK Journal of Applied Mathematics},
  year = {2026},
  volume = {2},
  number = {1},
  pages = {1-43},
  doi = {10.62762/JAM.2025.240326},
  url = {https://www.icck.org/article/abs/JAM.2025.240326},
  abstract = {A nonlinear compartmental model is developed to analyze crime dynamics in a structured society. The population is stratified into eight compartments: \(S(t)\) (susceptible), \(E(t)\) (exposed), \(C(t)\) (active criminals), \(C\_v(t)\) (convicted criminals), \(P\_h(t)\) (passive-honest), \(P\_c(t)\) (committed-honest), \(J\_h(t)\) (honest judges), and \(J\_c(t)\) (corrupt judges). The model incorporates nonlinear mechanisms such as institutional corruption (\(\kappa\_1\)), judicial correction (\(\kappa\_2\)), recidivism feedback (\(\rho\_1,\rho\_2\)), exposure intensity (\(\eta\_1\)), and rehabilitation (\(r\_2\)), providing a realistic portrayal of crime--justice interactions. Solutions remain positive and bounded within a feasible domain \(\mathcal{D}\). Linear stability analysis of the crime--free equilibrium \(Z\_0\) is performed via the Jacobian \(J(Z\_0)\). Numerical simulations explore the long--term dynamics under variations of key parameters (\(\beta\_1,\eta\_1,\rho\_1,\rho\_2,r\_2,\kappa\_1,\kappa\_2\)). Results show that strong judicial recruitment (\(a\_1\)) and honest reinforcement (\(\beta\_4\)) suppress criminal activity, whereas increased corruption (\(\kappa\_1\)) and recidivism (\(\rho\_1,\rho\_2\)) promote its growth. Bifurcation curves, contour maps, and stability basins highlight critical thresholds and equilibrium structures. The analysis demonstrates that proactive policy measures---reducing corruption, discouraging recidivism, and enhancing judicial integrity---can significantly lower crime levels and foster honest societal behavior, offering valuable guidance for designing effective crime--prevention strategies across diverse socio--political contexts.},
  keywords = {dynamical systems, nonlinear system, differential equations, stability, bifurcation},
  issn = {3068-5656},
  publisher = {Institute of Central Computation and Knowledge}
}
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