Equilibria and Stability Analysis of a Compartmental Model for Crime Dynamics with Recidivism and Corruption

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.