Dynamic Analysis of a Modified SETPR Terrorism Model Incorporating Recidivism of Imprisoned Terrorists

Abstract

Terrorism poses a serious threat to national security, particularly when former terrorist inmates return to active duty due to the failure of deradicalization programs. This study modifies the $SETPR$ model by adding a direct recidivism path from the imprisoned compartment (\(P\)) to the active terrorist compartment (\(T\)) via the parameter \(k\). This model consists of five subpopulations: the susceptible (\(S\)), the exposed (\(E\)), active terrorists (\(T\)), imprisoned (\(P\)), and subpopulation that has ceased terrorist activities (\(R\)). The basic reproduction number is given by \(\mathcal{R}_0,\) which is derived using the next-generation matrix method and serves as a threshold parameter for the spread of terrorism. Local stability analysis shows that the terrorism-free equilibrium is locally asymptotically stable if \(\mathcal{R}_0 < 1\), while the endemic equilibrium is locally asymptotically stable if \(\mathcal{R}_0 > 1\). Numerical simulations using Matlab R2013a and Maple 11 show that when \(\mathcal{R}_0 = 0.0039 < 1\), the system converges to the terrorism-free equilibrium, meaning terrorism will become extinct. Conversely, with \(\mathcal{R}_0 = 5.8535 > 1\), the system is stable at the endemic equilibrium, indicating that terrorism persists at a positive level. These results show that it is important to control key parameters through integrated intervention strategies, so that the \(\mathcal{R}_0\) value can be reduced below unity.