DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020046 DYNAMICAL SYSTEMS SERIES S Volume 13, Number 3, March 2020 pp. 805–821 MATHEMATICAL MODELING APPROACH TO THE FRACTIONAL BERGMAN’S MODEL Victor Fabian Morales-Delgado Facultad de Matem´ aticas. Universidad Aut´onoma de Guerrero Av. L´azaro C´ardenas S/N, Cd. Universitaria Chilpancingo, Guerrero, M´ exico Jos´ e Francisco G´ omez-Aguilar * CONACyT-Tecnol´ogico Nacional de M´ exico/CENIDET Interior Internado Palmira S/N, Col. Palmira C.P. 62490, Cuernavaca Morelos, M´ exico Marco Antonio Taneco-Hern´ andez * Facultad de Matem´ aticas. Universidad Aut´onoma de Guerrero Av. L´azaro C´ardenas S/N, Cd. Universitaria Chilpancingo, Guerrero, M´ exico Abstract. This paper presents the solution for a fractional Bergman’s min- imal blood glucose-insulin model expressed by Atangana-Baleanu-Caputo frac- tional order derivative and fractional conformable derivative in Liouville-Caputo sense. Applying homotopy analysis method and Laplace transform with homo- topy polynomial we obtain analytical approximate solutions for both deriva- tives. Finally, some numerical simulations are carried out for illustrating the results obtained. In addition, the calculations involved in the modified homo- topy analysis transform method are simple and straightforward. 1. Introduction. Bergman’s minimal model consider a body as a compartment with a basal concentration of glucose and insulin. The minimal model has two variations, the first describes how blood glucose concentration reacts with blood insulin concentration, and the second describes how blood insulin concentration reacts with blood glucose concentration. The two models take glucose and insulin data as an input, and have mostly been used to interpret the kinetics during the glucose tolerance test [14]- [13]. Fractional Calculus (FC) describes the evolution of biological models with mem- ory, this evolution is presented in the fractional exponent of the derivative. It implies the next state of a fractional system depends not only upon its current state but also upon all of its historical states. The fractional orders of differentiation highlight the intermediate behaviours that cannot be modeled by ordinary equations [28]- [26]. Atangana and Baleanu suggested a derivative in the Liouville-Caputo sense with non-singular and non-local kernel based in the generalized Mittag-Leffler law [10]. 2010 Mathematics Subject Classification. Primary: 34A34, 65M12; Secondary: 26A33, 34A08, 65C20, 65P20. Key words and phrases. Bergman’s model, fractional conformable derivative, Atangana- Baleanu fractional derivative, Laplace transform, modified homotopy analysis transform method. * Corresponding authors: J. F. G´omez-Aguilar and M. A. Taneco-Hern´andez. 805