Mathematical modeling of childhood infectious diseases using fractional derivatives and control strategies

Abstract

Childhood infectious diseases continue to pose significant global health challenges, particularly in regions with limited access to healthcare and vaccination services. To capture the memory-dependent nature of disease transmission, this study develops a fractional-order epidemiological model based on the Caputo–Fabrizio derivative. The framework incorporates two realistic intervention strategies—vaccination and isolation/treatment—and analyzes their combined effectiveness in reducing infection levels. Analytical solutions are obtained using the Laplace–Adomian Decomposition Method, while numerical approximations are generated through an Euler-based scheme. The epidemic threshold is derived, and the influence of key parameters on disease persistence is examined. Both disease-free and endemic equilibria are established, and their local and global stability properties are rigorously investigated. Additionally, the Sumudu transform is employed to further assess the system’s dynamic behavior, benefiting from its suitability for fractional operators with non-singular kernels. Fundamental mathematical properties of the model, including existence, uniqueness, positivity, and boundedness, are proven. Optimal control theory, via Pontryagin’s Maximum Principle, is applied to determine cost-effective strategies that minimize infections and maximize recovery. Computational simulations support the theoretical findings, demonstrating that integrated control measures substantially mitigate the spread of childhood infectious diseases.