Caputo-Fabrizio Fractional Order Derivative Mathematical Modeling and Optimal Control, Cost-Effectiveness Analysis of Diphtheria Transmission Dynamics in Nigeria

Abstract

Diphtheria, a potentially fatal vaccine-preventable disease caused by Corynebacterium diphtheriae, has seen a resurgence in several regions, including sub-Saharan Africa. This study presents a comprehensive mathematical model for diphtheria transmission dynamics that incorporates environmental bacterial persistence, public awareness campaigns, and vaccine hesitancy. The model subdivides the human population into epidemiologically relevant classes, including susceptible, exposed, symptomatic infectious, asymptomatic infectious, vaccinated, and hesitant individuals, with an additional compartment representing environmental bacterial load. Analytical results establish the model’s positivity, boundedness, and stability properties, with the basic reproduction number ( ) derived to determine disease threshold conditions. A global sensitivity analysis using the Partial Rank Correlation Coefficient (PRCC) identified the transmission rate , exposure progression rate , and vaccination rate as the most influential parameters affecting . The optimal control analysis, formulated via Pontryagin’s Maximum Principle, evaluated time-dependent interventions representing vaccination, treatment, and public enlightenment efforts. Numerical simulations using parameterized incidence data from Bauchi State, Northeast Nigeria, revealed that combined control strategies significantly reduce infection prevalence and environmental contamination compared to single interventions. The cost-effectiveness analysis (CEA) demonstrated that the combined vaccination and public enlightenment campaign strategy produced the lowest Incremental Cost-Effectiveness Ratio (ICER) and was classified as “very cost-effective” according to WHO-CHOICE thresholds. The findings underscore that effective control requires a multi-pronged, dynamically-adapted strategy targeting all transmission pathways. High vaccination coverage remains the cornerstone, but its impact is significantly amplified when synergistically combined with public health campaigns and environmental decontamination. The results offer evidence-based, cost-effective guidance for public health policymakers to design efficient diphtheria outbreak response and elimination programs. Moreover, numerical solutions obtained for different fractional orders highlight the Caputo-Fabrizio fractional derivative's capability to simulate memory effects, thereby offering a more nuanced representation of the real-life problem under study.