Sensitivity Analysis, Parameter Estimation, and Disease Prediction for a Generalized Fractional-Order Disease Model with Adaptive Immune Response

Aims/Objectives: Predicting how infectious diseases spread is challenging because biological systems carry memory—today’s infection rate depends on the entire history of past exposure, not just the current moment. This paper presents a disease prediction framework using Caputo fractional calculus to capture memory effects, with explicit humoral and cellular immune responses (the SEIR-HCI model). Methodology: Sensitivity analysis identifies two parameters that drive epidemic risk most strongly: the transmission rate β (sensitivity index +1) and the recovery rate \(γ\) (index ≈ −0.95). The model is fitted to real SARS-CoV-2 daily case data from India and South Africa using weighted least squares, with uncertainty quantification via parametric bootstrap. Results: The proposed model reduces prediction error (RMSE) by roughly 40–41% over a standard integer-order model, with R2 values of 0.957 and 0.946 for India and South Africa, respectively. The estimated memory index \(α̂\) ≈ 0.887–0.912, significantly below one, confirms that real epidemic data carry substantial memory. All computations use the Adams–Bashforth–Moulton predictor-corrector scheme, convergent at order min(2, 1+α). Conclusion: Intervention strategies targeting β (e.g., non-pharmaceutical interventions) and γ (e.g., clinical management) yield the highest returns for epidemic control. The immune compartments H (humoral) and C (cellular) are population-level phenomenological proxies; direct validation against immunological time-series data constitutes a stated limitation of the study.