A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence.

The main objective of this research is to investigate a new fractional mathematical model involving a nonsingular derivative operator to discuss the clinical implications of diabetes and tuberculosis coexistence. The new model involves two distinct populations, diabetics and nondiabetics, while each of them consists of seven tuberculosis states: susceptible, fast and slow latent, actively tuberculosis infection, recovered, fast latent after reinfection, and drug-resistant. The fractional operator is also considered a recently introduced one with Mittag-Leffler nonsingular kernel. The basic properties of the new model including non-negative and bounded solution, invariant region, and equilibrium points are discussed thoroughly. To solve and simulate the proposed model, a new and efficient numerical method is established based on the product-integration rule. Numerical simulations are presented, and some discussions are given from the mathematical and biological viewpoints. Next, an optimal control problem is defined for the new model by introducing four control variables reducing the number of infected individuals. For the control problem, the necessary and sufficient conditions are derived and numerical simulations are given to verify the theoretical analysis.

[1]  Roberto Garrappa,et al.  On linear stability of predictor–corrector algorithms for fractional differential equations , 2010, Int. J. Comput. Math..

[2]  D. Baleanu,et al.  New aspects of poor nutrition in the life cycle within the fractional calculus , 2018, Advances in Differential Equations.

[3]  N. Ahmed,et al.  Molecular epidemiology of tuberculosis in India: moving forward with a systems biology approach. , 2011, Tuberculosis.

[4]  Devendra Kumar,et al.  Numerical solution of predator-prey model with Beddington-DeAngelis functional response and fractional derivatives with Mittag-Leffler kernel. , 2019, Chaos.

[5]  D. Baleanu,et al.  New features of the fractional Euler-Lagrange equations for a physical system within non-singular derivative operator , 2019, The European Physical Journal Plus.

[6]  Ernst Hairer,et al.  FAST NUMERICAL SOLUTION OF NONLINEAR VOLTERRA CONVOLUTION EQUATIONS , 1985 .

[7]  Wolfgang Hackbusch,et al.  A numerical method for solving parabolic equations with opposite orientations , 1978, Computing.

[8]  G. Ippolito,et al.  The global dynamics of diabetes and tuberculosis: the impact of migration and policy implications. , 2017, International journal of infectious diseases : IJID : official publication of the International Society for Infectious Diseases.

[9]  Roberto Garrappa,et al.  Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial , 2018 .

[10]  Dumitru Baleanu,et al.  Numerical solutions of the fractional Fisher's type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods. , 2019, Chaos.

[11]  Dumitru Baleanu,et al.  On an accurate discretization of a variable-order fractional reaction-diffusion equation , 2019, Commun. Nonlinear Sci. Numer. Simul..

[12]  Samuel Bowong,et al.  Analysis of The Impact of Diabetes on The Dynamical Transmission of Tuberculosis , 2012 .

[13]  Abdon Atangana,et al.  On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems. , 2019, Chaos.

[14]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[15]  Alan D. Freed,et al.  Detailed Error Analysis for a Fractional Adams Method , 2004, Numerical Algorithms.

[16]  Carla M. A. Pinto,et al.  Diabetes mellitus and TB co-existence: Clinical implications from a fractional order modelling , 2019, Applied Mathematical Modelling.

[17]  K. Dooley,et al.  Tuberculosis and diabetes mellitus: convergence of two epidemics. , 2009, The Lancet. Infectious diseases.

[18]  I. Khan.  New idea of Atangana and Baleanu fractional derivatives to human blood flow in nanofluids. , 2019, Chaos.

[19]  Dumitru Baleanu,et al.  Two-strain epidemic model involving fractional derivative with Mittag-Leffler kernel. , 2018, Chaos.

[20]  J. Watmough,et al.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. , 2002, Mathematical biosciences.

[21]  A. Young,et al.  Approximate product-integration , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[22]  Kai Diethelm,et al.  A fractional calculus based model for the simulation of an outbreak of dengue fever , 2013 .

[23]  D. Baleanu,et al.  On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel , 2018 .

[24]  Dumitru Baleanu,et al.  A New Formulation of the Fractional Optimal Control Problems Involving Mittag–Leffler Nonsingular Kernel , 2017, J. Optim. Theory Appl..

[25]  I. Su,et al.  Effect of type 2 diabetes mellitus on the clinical severity and treatment outcome in patients with pulmonary tuberculosis: a potential role in the emergence of multidrug-resistance. , 2011, Journal of the Formosan Medical Association = Taiwan yi zhi.

[26]  Marina Popolizio,et al.  Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions , 2018 .

[27]  M. Ezzati,et al.  Effect of diabetes on tuberculosis control in 13 countries with high tuberculosis: a modelling study. , 2015, The lancet. Diabetes & endocrinology.

[28]  Dumitru Baleanu,et al.  Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel , 2016, 1607.00262.

[29]  Dumitru Baleanu,et al.  Fractional modeling of blood ethanol concentration system with real data application. , 2019, Chaos.