Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator

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 (ℛ0) is introduced to assess local stability at two equilibrium points. The stability analysis indicates that the disease-free equilibrium point is stable when ℛ0<1, while the endemic equilibrium point is stable when ℛ0>1 and otherwise unstable. Numerical simulations demonstrate the model’s effectiveness in capturing realistic scenarios, particularly the recurrent patterns observed in some childhood diseases.