A Hybrid Fractional Model for Whooping Cough-Like Infections

Abstract

In this study, the dynamics of whooping cough-like infections are examined. For this, an integer-order model of the disease is transformed into a fractional-order nonlinear epidemic model by replacing the classical derivatives with the Caputo derivatives. The whole population is categorized into four sub-classes, namely, susceptible individuals (S), the exposed population (E), the infected class (I), and the secured compartment (R). Two numerical techniques are designed to find the solutions to the underlying model. A comparison between the two numerical methods is furnished to evaluate the better and more realistic method. The boundedness of the nonlinear fractional model is confirmed. In addition, the existence and uniqueness of the solution to the underlying nonlinear model are proven. The basic reproduction number(R) of the model also worked out for predicting the disease dynamics. The nonlinear epidemic model has two equilibrium states, namely, the cough-free equilibrium state and the cough-existing equilibrium state. The stability of the model at both equilibrium states is examined. The role of R0 in maintaining the stable state is also verified. Two numerical schemes are furnished by using the Grünwald-Letnikov approximation and some other standard results. The boundedness and positivity of the GL-NSFD scheme 2 is ascertained. The simulated graphs are sketched to validate the claims about the numerical scheme. The outcome of the study is summarized in the conclusion section. © 2025 NSP Natural Sciences Publishing Cor.