A Hybrid Fractional-Order Differential Model with Graph-Based Intervention Dynamics for Predicting Banditry-Driven Population Displacement in Northern Nigeria

Introduction: The increased prevalence of armed banditry in Northern Nigeria has transformed the region from a localized security concern to one of the gravest humanitarian crises on the continent. The region displaced more than 650,300 people in the Northwest and nearly 601,700 in the North-Central zone by October 2025. The number of civilian deaths in the first half of 2025 alone exceeded the entire death toll recorded in 2024 by 109%. Objectives: In this study, we introduce a hybrid model based on fractional-order differential equations, utilising the Caputo derivative framework. We aim to incorporate graph-based dynamics to evaluate the effects of strategic interventions and to predict displacement patterns in conflict-affected Northern Nigeria. Methods: To achieve this, we categorize the population into three groups within the conflict zone: at-risk citizens R(t), displaced individuals D(t), and intervention agents I(t). These groups interact according to a system of fractional-order differential equations (FODEs), with the fractional order set at \alpha = 0.68. We represent the interactions among intervention actors using a weighted directed graph, G = (V, E, W). This graph includes various social and political entities, such as security agencies, non-governmental organizations (NGOs), state institutions, and community mediators. To ensure the stability of our fractional system, we apply Lyapunov's direct method. For sensitivity analysis, we utilise normalised forward sensitivity indices, and we solve the fractional differential equations numerically using the Adams-Bashforth-Moulton predictor-corrector scheme. Results: The calibration of the model to data from the IOM Displacement Tracking Matrix (2020–2025) has shown the capacity of our fractional model to outpe