Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data

Abstract

This paper is all about the development and analysis of an epidemiological model related to the disease of diarrhea that occurred in Ghana during 2008–2015. Using real statistical data, three new fractional-order mathematical models have been developed on the basis of having information about existing classical model . The new models are formulated with Caputo, Caputo–Fabrizio–Caputo and the Atangana–Baleanu–Caputo fractional-order approaches while taking care of the dimensional analysis during the process of fractionalization. Besides, existence and uniqueness for the solutions of the fractional-order models under each case are proved with the help of fixed point theory whereas positivity and boundedness of models’ solution are also investigated. Steady-states (disease-free and endemic equilibria) points of the model and sensitivity of the basic reproductive number (R0) are also explored. While many of the model’s parameters are fixed, the transmission rate (β) of the disease has been estimated and so is the case with orders of the fractional models. Using minimum distance approach, it has been found that the diarrhea model under investigation estimates the real statistical data well enough when considered with the Atangana–Baleanu–Caputo fractional order operator which has non-local and non-singular kernel. Thus, this fractional-order operator of Atangana–Baleanu in the present research study for the diarrhea model outperforms those having index law, power law and stretched exponential kernels.