Modelling the transmission behavior of measles disease considering contaminated environment through a fractal-fractional Mittag-Leffler kernel

Abstract

This work utilises a This work utilises a fractal-fractional operator to examine the dynamics of transmission of measles disease. The existence and uniqueness of the measles model have been thoroughly examined in the context of the fixed point theorem, specifically utilising the Atangana-Baleanu fractal and fractional operators. The model has been demonstrated to possess both \textcolor{blue}{Hyers-Ulam} stability and \textcolor{blue}{Hyers-Ulam Rassias} stability. Furthermore, a qualitative analysis of the model was performed, including examination of key parameters such as the fundamental reproduction number, the \textcolor{blue}{measles}-free and measles-present equilibria, and assessment of global stability. This research has shown that the transmission of \textcolor{blue}{measles} disease is affected by natural phenomena, as changes in the fractal-fractional order lead to changes in the disease dynamics. Furthermore, environmental contamination has been shown to play a significant role in the transmission of the \textcolor{blue}{measles} disease. } fractal-fractional operator to examine the dynamics of transmission of Measles disease. The existence and uniqueness of the Measles model have been thoroughly examined in the context of the fixed point theorem, specifically utilising the Atangana-Baleanu fractal and fractional operators. The model has been demonstrated to possess both hyers ulam stability and hyers lam rassias stability. Furthermore, a qualitative analysis of the model was performed, including examination of key parameters such as the fundamental reproduction number, the Measles-free and Measles-present equilibria, and assessment of global stability. This research has shown that the transmission of Measles disease is affected by natural phenomena, as changes in the fractal-fractional order lead to changes in the disease dynamics. Furthermore, environmental contamination has been shown to play a significant role in the transmission of the Measles disease.