Comparative Analysis of Classical, Fractional, and Quantum Calculus Models for Measles Transmission Dynamics

Abstract

This study presents three mathematical models of measles transmission that utilize pulse vaccination strategies: classical ordinary differential equations, Caputo fractional derivatives, and Jackson q-derivatives. The basic reproduction number R0 has been derived, and the global stability of equilibria has been investigated using an SIRS model with bilinear incidence. The quantum calculus model facilitates the integration of discrete-pulse vaccination strategies by applying geometric time scales. The existence, uniqueness, positivity, and global asymptotic stability of equilibria have been demonstrated through Lyapunov’s direct method. Numerical results indicate that while equilibria remain model-independent, transient dynamics exhibit significant variation. The fractional-derivative models reduce epidemic peaks by up to 5.6%, whereas the quantum-calculus model results in an increase of 1.1%. These models provide essential tools for enhancing pulse vaccination strategies in measles elimination initiatives.