Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains

N H Sweilam; S M Al-Mekhlafi; D Baleanu

doi:10.1016/j.jare.2019.01.007

Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains

J Adv Res. 2019 Jan 19:17:125-137. doi: 10.1016/j.jare.2019.01.007. eCollection 2019 May.

Authors

N H Sweilam¹, S M Al-Mekhlafi², D Baleanu^{3

4}

Affiliations

¹ Cairo University, Faculty of Science, Mathematics Department, 12613 Giza, Egypt.
² Sana'a University, Faculty of Education, Mathematics Department, Sana'a, Yemen.
³ Cankaya University, Department of Mathematics, 06530, Ankara, Turkey.
⁴ Institute of Space Sciences, P.O. Box MG 23, Magurele, 077125 Bucharest, Romania.

Abstract

The objective of this paper is to study the optimal control problem for the fractional tuberculosis (TB) infection model including the impact of diabetes and resistant strains. The governed model consists of 14 fractional-order (FO) equations. Four control variables are presented to minimize the cost of interventions. The fractional derivative is defined in the Atangana-Baleanu-Caputo (ABC) sense. New numerical schemes for simulating a FO optimal system with Mittag-Leffler kernels are presented. These schemes are based on the fundamental theorem of fractional calculus and Lagrange polynomial interpolation. We introduce a simple modification of the step size in the two-step Lagrange polynomial interpolation to obtain stability in a larger region. Moreover, necessary and sufficient conditions for the control problem are considered. Some numerical simulations are given to validate the theoretical results.

Keywords: Atangana-Baleanu fractional derivative; Diabetes and resistant strains; Lagrange polynomial interpolation; Nonstandard two-step Lagrange interpolation method; Tuberculosis model.