A fractional-order epidemic model with vaccination and hospitalization: Dynamics and actuarial applications

Abstract

In this work, we develop and analyze a Caputo fractional-order compartmental model that extends the classical SIS framework by incorporating vaccination, hospitalization, and disease-induced mortality. The proposed model, denoted as $\mathcal{SVIHD}$, accounts for memory effects inherent in epidemic processes through the fractional derivative formulation. We establish fundamental properties of the system including non-negativity, boundedness, and the existence of unique solutions. The disease-free and endemic equilibria are computed, and the vaccination-adjusted basic reproduction number \(\mathcal{R}_0^{\nu,\kappa}\) is derived via the next-generation matrix method. Comprehensive stability analysis is performed using Matignon's criterion, Lyapunov functions, and the fractional LaSalle invariance principle, demonstrating that the disease-free equilibrium is globally asymptotically stable when \(\mathcal{R}_0^{\nu,\kappa} \leq 1\), while the endemic equilibrium is globally asymptotically stable when \(\mathcal{R}_0^{\nu,\kappa} > 1\). Sensitivity analysis identifies key parameters influencing disease transmission. Furthermore, we apply the model to health insurance pricing, where the fractional-order dynamics inform the calculation of fair premiums and solvency reserves. The equivalence principle yields zero-profit premiums, while an admissibility condition ensures non-negative reserves throughout the coverage horizon. Numerical simulations using the Adams-Bashforth-Moulton predictor-corrector scheme for \(\kappa \in \{1.0, 0.9, 0.8\}\) validate the theoretical findings and demonstrate that stronger memory effects (smaller \(\kappa\)) lead to slower epidemic decay and consequently higher insurance premiums. The results establish a direct quantitative link between epidemic memory strength and actuarial requirements, providing a novel framework for pandemic risk management in insurance applications.