Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator - Yasir A. Madani; Zeeshan Ali; Mohammed Rabih; Amer Alsulami; Nidal H. E. Eljaneid; Khaled Aldwoah; Blgys Muflh (Fractal and Fractional)
This paper presents a new SIRS model for recurrent childhood diseases under the Caputo fractional difference operator. The existence theory is established using Brouwer’s fixed-point theorem and the Banach contraction principle, providing a comprehensive mathematical foundation for the model. Ulam stability is demonstrated using nonlinear functional analysis. Sensitivity analysis is conducted based on the variation of each parameter, and the basic reproduction number <semantics>(R0)</semantics> is introduced to assess local stability at two equilibrium points. The stability analysis indicates that the disease-free equilibrium point is stable when <semantics>R0<1</semantics>, while the endemic equilibrium point is stable when <semantics>R0>1</semantics> and otherwise unstable. Numerical simulations demonstrate the model’s effectiveness in capturing realistic scenarios, particularly the recurrent patterns observed in some childhood diseases.