Mathematical Modeling and Analysis of the Spread of Varicella (Chickenpox) Disease with the Caputo Operator

Abstract

In this study, we present a mathematical model for studying the dynamics of varicella disease transmission incorporating Caputo's fractional derivative to account for memory effects in disease spread. The aim of this research is to construct a robust mathematical framework that combines fractional calculus with disease modeling and explore fractional order parameters impacts on diseases propagations. Thus, the significant contribution to knowledge is the improved understanding of infectious diseases? dynamism as influenced by memory effects thus providing profound insights which can be used as cornerstones in developing preventive measures. The technique of this study relies on integration a system of differential equations which are subjected to the Caputo fractional operator. A variety of data sources were used, including existing epidemiological varicella literature for model calibration. We begin by analyzing the existences and uniqueness of the solutions to this model, utilizing the Banach Fixed-point theorem, the linear stability analysis of the model, using Lyapunov Functional approach. The analysis also includes key epidemiological parameters ($R_0$: basic reproduction number) and dynamic response of the disease-free equilibrium. The effect of the fractional order parameter is investigated through numerical simulations performed by appropriate computational tools especially concerning an infection peak and asymptotic behavior. The results indicate that the varicella transmission dynamics are related to order parameter of fractional order $\alpha$. The results of this study demonstrated that to better describe epidemiological data, it is essential to introduce fractional order operators within infectious disease models able to represent nonlocal and complex phenomena.