An Integrated Mathematical Model for Detecting Known and Unknown Fraud Patterns with Optimal Resource Allocation in the Nigerian Social Security System

Abstract

Social security systems in developing countries face substantial fraud losses threatening system
sustainability, yet resource-constrained agencies can investigate only a small fraction of claims.
We present a comprehensive mathematical framework integrating Bayesian logistic regression for
known fraud patterns, Gaussian mixture model-based anomaly detection for novel schemes, and
optimal resource allocation under capacity constraints. Temporal adaptation through exponential
smoothing maintains detection performance as fraud patterns evolve quarterly. Our greedy allo-
cation policy, derived through marginal benefit analysis, provably minimizes expected cost under
investigation capacity constraints. Empirical validation on 10,000 pension claims from Nigerian state
bureaus (Lagos, Rivers, Kano) collected over 18 months demonstrates operational effectiveness. The
integrated framework achieves area under ROC curve of 0.811 with precision of 84% and recall of
67% on realistic data exhibiting class overlap, label noise (3%), and missing values (5%). Greedy
resource allocation detects 7–11 times more fraud than random selection across all capacity levels,
translating to estimated annual savings of 165 million ($122,000 USD) for Lagos bureau alone and
2 billion ($1.5 million USD) nationally. Sample complexity analysis reveals that 1,000–2,000 inves-
tigated claims suffice for reliable deployment, achievable within 13–18 months for most Nigerian
bureaus. Temporal adaptation with exponential smoothing parameter α = 0.70 maintains stable
performance despite quarterly fraud pattern evolution, preventing the 13-point AUC degradation
observed in static models over 12-month horizons.