A central question in gang-intervention modeling is whether reducing recruitment below an invasion threshold is enough to eliminate gang activity. Here, we show that relapse from a reformed class can make it insufficient. We developed a deterministic compartmental SGR model with susceptible, active gang, and reformed classes, in which recruitment was reduced by perceived-risk deterrence and by an advocacy interaction between reformed and susceptible individuals. We established positive invariance and an invasion threshold for a weighted-contact formulation, and showed that its reciprocity-consistent proportional-mixing reduction exhibits backward bifurcation when relapse is sufficiently strong. The backward bifurcation generates a subthreshold positive gang equilibrium even when the simplified reproduction number is below one; for the worked parameter regime, this branch turns at a saddle-node and yields bistability, so the long-term outcome is set by basin membership rather than by the threshold alone. In dimensionless form, the backward-bifurcation condition reduces, under typical parameter regimes, to the requirement that the dimensionless relapse rate exceed the natural outflow rate of the active gang class. A worked instance at $\mathcal R_0^{\rm{simp}} = 0.5$ yields gang-free and high-gang attractors separated by a low-prevalence saddle with a substantial reformed-class reservoir; numerical continuation places the saddle-node at $\mathcal R_0^{\rm{simp}}\approx 0.08755$, so the bistable interval spans approximately the upper 91% of the unit interval. Numerical bifurcation diagrams and a baseline-centered PRCC sensitivity analysis identify relapse and reformation as the dominant opposing drivers of long-term gang prevalence in the sampled regime, with recruitment a smaller positive driver. These results suggest that gang reduction may require coordinated strategies that reduce recruitment, prevent relapse, strengthen exit pathways, and sustain reformed-led prevention efforts.
Citation: Theophilus Kwofie, Lucero Rodriguez Rodriguez, Xia Wang, James M. Hyman, Yun Kang. Relapse-driven bistability in a nonlinear model of gang dynamics with deterrence and reformed-led advocacy[J]. Mathematical Biosciences and Engineering, 2026, 23(9): 2807-2859. doi: 10.3934/mbe.2026100
A central question in gang-intervention modeling is whether reducing recruitment below an invasion threshold is enough to eliminate gang activity. Here, we show that relapse from a reformed class can make it insufficient. We developed a deterministic compartmental SGR model with susceptible, active gang, and reformed classes, in which recruitment was reduced by perceived-risk deterrence and by an advocacy interaction between reformed and susceptible individuals. We established positive invariance and an invasion threshold for a weighted-contact formulation, and showed that its reciprocity-consistent proportional-mixing reduction exhibits backward bifurcation when relapse is sufficiently strong. The backward bifurcation generates a subthreshold positive gang equilibrium even when the simplified reproduction number is below one; for the worked parameter regime, this branch turns at a saddle-node and yields bistability, so the long-term outcome is set by basin membership rather than by the threshold alone. In dimensionless form, the backward-bifurcation condition reduces, under typical parameter regimes, to the requirement that the dimensionless relapse rate exceed the natural outflow rate of the active gang class. A worked instance at $\mathcal R_0^{\rm{simp}} = 0.5$ yields gang-free and high-gang attractors separated by a low-prevalence saddle with a substantial reformed-class reservoir; numerical continuation places the saddle-node at $\mathcal R_0^{\rm{simp}}\approx 0.08755$, so the bistable interval spans approximately the upper 91% of the unit interval. Numerical bifurcation diagrams and a baseline-centered PRCC sensitivity analysis identify relapse and reformation as the dominant opposing drivers of long-term gang prevalence in the sampled regime, with recruitment a smaller positive driver. These results suggest that gang reduction may require coordinated strategies that reduce recruitment, prevent relapse, strengthen exit pathways, and sustain reformed-led prevention efforts.
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Theophilus Kwofie, Lucero Rodriguez Rodriguez, Xia Wang, James M. Hyman, Yun Kang. Relapse-driven bistability in a nonlinear model of gang dynamics with deterrence and reformed-led advocacy[J]. Mathematical Biosciences and Engineering, 2026, 23(9): 2807-2859. doi: 10.3934/mbe.2026100
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