Analysis and Numerical Investigation of a Fractional Model for Community- and Hospital-Acquired Methicillin-Resistant Staphylococcus aureus in Hospital Settings

Abstract

In this work, we analyze a fractional model describing the transmission dynamics of community-acquired and hospital-acquired methicillin-resistant Staphylococcus aureus (CA-MRSA and HA-MRSA) in a hospital setting. The model replaces the classical time derivative by a Caputo fractional derivative to represent memory effects in the transmission process. We establish existence, uniqueness, and Hyers–Ulam stability under explicit sufficient conditions using fixed-point arguments. We then apply the Laplace residual power series method (LRPSM) to construct approximate solutions and assess their accuracy against RK4 and Adams–Bashforth–Moulton (ABM) reference solutions. For the baseline parameter set, the 30th-order LRPS solution has error below 3 × 10⁻⁶ for η ≥ 0.75 and does not exceed 2.1 × 10⁻³ over the reported test interval; for the second parameter set, the corresponding error is at most 1 × 10⁻⁵ at the tabulated times. Long-time simulations further show that decreasing the fractional order slows the decay toward the disease-free state, reflecting the memory effect. These results provide a quantitative assessment of the LRPSM approximation and clarify the range in which the analytical fixed-point results apply. For η < 1 the reference is itself a numerical ABM solution with a resolution of about 2 × 10⁻⁵, so errors of that size or smaller measure agreement with the reference.