Mathematical Modelling and Analysis of Infiltration of Firearms and Its Insecurity Implications in the Northern Region of Nigeria

Abstract

This research produces and analyzes an innovative nonlinear compartmental model to explore the infiltration of firearms into Nigeria and its resulting insecurity dynamics in Northern Nigeria. The model incorporates four interacting components: susceptible individuals, violent actors, arms and ammunition, and recovered individuals, capturing the bidirectional feedback between arms infiltration and violence recruitment. An effective reproduction number, Re​, was derived to characterize the threshold conditions governing the persistence or elimination of violence. Stability analyses, including Lyapunov-based proofs (), create the conditions for global stability of both the violence-free and endemic equilibria. Normalized sensitivity analysis using Python revealed that arms-related parameters, particularly arms inflow (σ =+0.312), arms to violence activation (α= -0.020), and arms-induced susceptibility (γ=+0.254), are the most influential drivers of violence propagation. Numerical simulations were used to validate the investigative results, which showed how reducing arms availability can substantially lower the long-term levels of violent actors. These findings showed that illicit arms proliferation is the primary catalyst sustaining endemic violence. The necessary interventions, such as addressing arms inflow, distribution networks, and de-escalation processes, are most effective for violence reduction. The simulation results establish that the long-term dynamics of violence in Northern Nigeria are strongly driven by the circulation and availability of illicit firearms. The trajectories for S(t), V(t), and A(t) showed that the reductions in arms-related parameters produce meaningful declines in violence, confirming the sensitivity analysis results. This study provides a quantitative framework and actionable insights to support evidence-based security policy, arms-control strategies, and conflict-mitigation efforts in Northern Nigeria.