ÊEiman ÊSaowaluck ChasreechaiThanin SitthiwiratthamM.A. El-ShorbagyMuhammad SohailUbaid UllahMati Ur Rahman2025-03-102025-03-102022AIMS Mathematics2473698810.3934/math.20227922-s2.0-85131298098https://repository.dusit.ac.th//handle/123456789/4636Qualitative theory, together with approximate solutions to a dynamic system, are investigated. The proposed mathematical model is composed of protected, susceptible, infected and treated classes. The adopted model expresses the mechanism of disease due to Typhoid fever. A modified type Caputo-Fabrizio fractional derivative (CFFD) is considered for the intended results. With the help of fixed point theory, some sufficient conditions for the existence of approximate solutions are developed. Also, to compute an approximate solution with respect to each compartment, we utilize the Laplace Transform and the Adomian decomposition method (ADM). A graphical presentation corresponding to some fundamental data is given. © 2022 the Author(s), licensee AIMS Press.All Open Access; Gold Open AccessADMapproximate solutionCFFDdynamical systemArticleScopus