A new fractional model for tuberculosis with relapse via Atangana–Baleanu derivative. (November 2018) - Altaf Khan, Muhammad; Ullah, Saif; Farooq, Muhammad
Highlights: A new fractional order mathematical model for TB dynamics with relapse using Atangana–Baleanu derivative is formulated. A better fit to the real data is provided by the proposed model using least square technique. Parameters are fitted and estimated form the TB reported cases in Khyber Pakhtunkhwa from 2002 to 2017. Existence and uniqueness of the model solution are proved using fixed point theorem. A numerical scheme is applied to investigate the solution of the model numerically for different value of the fractional order parameter. Abstract: In this work, a new fractional order epidemic model for the tuberculosis (TB) disease with relapse using Atangana–Baleanu derivative is formulated. The basic reproduction number of the model is investigated using next generation technique. The fixed point theorem is applied to show the existence and uniqueness of solution for the model. A newly proposed numerical scheme in literature is implemented for the iterative solution of the proposed fractional model. The total new and relapse notified TB cases in Khyber Pakhtunkhwa Pakistan from 2002 to 2017 are used to parameterized the model parameters and provided a good fit to the real data. Finally, numerical results are obtained for different values of the fractional order τ and the model parameters, in order to validate the importance of the arbitrary order derivative. It is noticed that the non-integer order derivative provides more realistic and deeper information about the complexity of the dynamics of TB model with relapse.