Stability analysis and numerical investigation of fractional SIR model for childhood disease transmission with vaccination

This work presents a fractional-order SIR-type framework to investigate the transmission dynamics of childhood infectious diseases under the influence of vaccination. The classical epidemic model is generalized through the Caputo fractional derivative to incorporate memory-dependent effects commonly observed in biological systems. The population is partitioned into susceptible, infected, and recovered compartments, and the impact of key epidemiological parameters such as vaccination rate, transmission coefficient, and recovery rate is examined. The well-posedness of the proposed model is established by proving the existence and uniqueness of solutions using fixed-point arguments in an appropriate Banach space. The local stability of both the disease-free and endemic equilibrium states is analyzed via the Jacobian matrix and the basic reproduction number. For numerical approximation, a fractional Adams-Bashforth-Moulton predictor-corrector scheme is employed, and its performance is assessed. Numerical simulations are carried out for scenarios with and without vaccination to highlight the role of immunization in controlling disease spread. The findings demonstrate that increased vaccination coverage significantly lowers the basic reproduction number and enhances the stability of the disease-free state. Furthermore, comparative analysis indicates that the proposed numerical method yields stable and accurate solutions relative to existing approaches such as the fractional Euler and implicit L₁ schemes. The study emphasizes the importance of incorporating memory effects in epidemic modeling and provides insights relevant to effective disease control strategies.