New numerical simulations for some real world problems with Atangana–Baleanu fractional derivative

Abstract In this work, we introduce ABC-Caputo operator with ML kernel and its main characteristics are discussed. Viral diseases models for AIDS and Zika are considered, and finally, as third model, the macroeconomic model involving ABC fractional derivatives is investigated, respectively. It is presented that the AB Caputo derivatives satisfy the Lipschitz condition along with superposition property. The numerical methods for solving the fractional models are presented by means of ABC fractional derivative in a detailed manner. Finally the simulation results obtained in this paper according to the suitable values of parameters are also manifested.

[1]  Y. A. Terefe,et al.  Mathematics of a model for Zika transmission dynamics , 2018, Theory in Biosciences.

[2]  R. Lanciotti,et al.  Genetic and Serologic Properties of Zika Virus Associated with an Epidemic, Yap State, Micronesia, 2007 , 2008, Emerging infectious diseases.

[3]  Haci Mehmet Baskonus,et al.  Active Control of a Chaotic Fractional Order Economic System , 2015, Entropy.

[4]  Kashif Ali Abro,et al.  A comparison of heat and mass transfer on a Walter’s-B fluid via Caputo-Fabrizio versus Atangana-Baleanu fractional derivatives using the Fox-H function , 2019, The European Physical Journal Plus.

[5]  Zakia Hammouch,et al.  Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative , 2019, Chaos, Solitons & Fractals.

[6]  I. K. Youssef,et al.  Solving Poisson’s Equations with fractional order using Haarwavelet , 2017 .

[7]  F. Nyabadza,et al.  Modelling HIV/AIDS in the presence of an HIV testing and screening campaign. , 2011, Journal of theoretical biology.

[8]  J. F. Gómez‐Aguilar,et al.  A dynamical model of asymptomatic carrier zika virus with optimal control strategies , 2019 .

[9]  Kaifa Wang,et al.  Propagation of HBV with spatial dependence. , 2007, Mathematical biosciences.

[10]  Ebenezer Bonyah,et al.  A Theoretical Model for the Transmission Dynamics of the Buruli Ulcer with Saturated Treatment , 2014, Comput. Math. Methods Medicine.

[11]  Hasan Bulut,et al.  Novel wave surfaces to the fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation , 2017 .

[12]  Sergei S. Pilyugin,et al.  Stability analysis of a deterministic model of Zika/Dengue co-circulation , 2019 .

[13]  Delfim F. M. Torres,et al.  A TB-HIV/AIDS coinfection model and optimal control treatment , 2015, 1501.03322.

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

[15]  A. Prakash,et al.  Analysis and numerical simulation of fractional order Cahn–Allen model with Atangana–Baleanu derivative , 2019, Chaos, Solitons & Fractals.

[16]  Abdon Atangana,et al.  Mathematical analysis and computational experiments for an epidemic system with nonlocal and nonsingular derivative , 2019, Chaos, Solitons & Fractals.

[17]  Hasan Bulut,et al.  Chaos in the fractional order logistic delay system: Circuit realization and synchronization , 2016 .

[18]  H. Kheiri,et al.  Optimal control of a fractional-order model for the HIV/AIDS epidemic , 2018, International Journal of Biomathematics.

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

[20]  Marin P. Allen.  Chronicling the Risk and Risk Communication by Governmental Officials During the Zika Threat , 2018, Risk analysis : an official publication of the Society for Risk Analysis.

[21]  K. M. Owolabi.  Behavioural study of symbiosis dynamics via the Caputo and Atangana–Baleanu fractional derivatives , 2019, Chaos, Solitons & Fractals.

[22]  Hang Xu,et al.  Analytical approximations for a population growth model with fractional order , 2009 .

[23]  Hasan Bulut,et al.  Regarding on the prototype solutions for the nonlinear fractional-order biological population model , 2016 .

[24]  Zhenqing Li,et al.  Complex dynamics of a reaction–diffusion epidemic model , 2012 .

[25]  R. Bagley,et al.  A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity , 1983 .

[26]  A. Atangana,et al.  Analysis of 3D IS-LM macroeconomic system model within the scope of fractional calculus , 2019, Chaos, Solitons & Fractals: X.

[27]  Agraj Tripathi,et al.  Modelling the spread of AIDS epidemic with vertical transmission , 2006, Appl. Math. Comput..

[28]  D. Brzezinski.  Comparison of Fractional Order Derivatives Computational Accuracy - Right Hand vs Left Hand Definition , 2017 .

[29]  Junhai Ma,et al.  Complexity analysis research of financial and economic system under the condition of three parameters’ change circumstances , 2012 .

[30]  Ilyas Khan,et al.  Dual thermal analysis of magnetohydrodynamic flow of nanofluids via modern approaches of Caputo–Fabrizio and Atangana–Baleanu fractional derivatives embedded in porous medium , 2019, Journal of Thermal Analysis and Calorimetry.

[32]  Lang Wu,et al.  A mechanistic nonlinear model for censored and mismeasured covariates in longitudinal models, with application in AIDS studies. , 2018, Statistics in medicine.

[33]  Rui Xu,et al.  An HBV model with diffusion and time delay. , 2009, Journal of theoretical biology.

[34]  Vicente Feliu-Batlle,et al.  Fractional robust control of main irrigation canals with variable dynamic parameters , 2007 .

[35]  W. O. Kermack,et al.  Contributions to the Mathematical Theory of Epidemics. II. The Problem of Endemicity , 1932 .

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

[37]  Zakia Hammouch,et al.  Spatiotemporal patterns in the Belousov–Zhabotinskii reaction systems with Atangana–Baleanu fractional order derivative , 2019, Physica A: Statistical Mechanics and its Applications.

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

[39]  Ilyas Khan,et al.  Convection in ethylene glycol-based molybdenum disulfide nanofluid , 2018, Journal of Thermal Analysis and Calorimetry.

[40]  C. Ferreira,et al.  A MATHEMATICAL ANALYSIS OF ZIKA VIRUS EPIDEMIC IN RIO DE JANEIRO AS A VECTOR-BORNE AND SEXUALLY TRANSMITTED DISEASE , 2019, Journal of Biological Systems.

[41]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[42]  B. Mukhopadhyay,et al.  Existence of epidemic waves in a disease transmission model with two-habitat population , 2007, Int. J. Syst. Sci..

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

[44]  Hiroshi Nishiura,et al.  Test-and-treat approach to HIV/AIDS: a primer for mathematical modeling , 2017, Theoretical Biology and Medical Modelling.

[45]  Asıf Yokuş,et al.  Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation , 2019, Applied Mathematics and Nonlinear Sciences.

[46]  Hasan Bulut,et al.  On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method , 2015 .

[47]  Joanna Rencławowicz,et al.  Traveling Waves and Spread Rates for a West Nile Virus Model , 2006, Bulletin of mathematical biology.

[48]  D. Brzezinski.  Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus , 2018, Applied Mathematics and Nonlinear Sciences.

[49]  O. M. Otunuga.  Global stability for a 2n+1 dimensional HIV/AIDS epidemic model with treatments. , 2018, Mathematical biosciences.

[50]  Peixuan Weng,et al.  Spreading speed and traveling waves for a multi-type SIS epidemic model , 2006 .

[51]  Bilal Ilyas,et al.  TRAVELING WAVES FOR A SIMPLE DIFFUSIVE EPIDEMIC MODEL , 1995 .

[52]  Kaifa Wang,et al.  Dynamics of an HBV model with diffusion and delay. , 2008, Journal of theoretical biology.

[53]  H. M. Baskonus,et al.  Analysis of the dynamics of hepatitis E virus using the Atangana-Baleanu fractional derivative , 2019, The European Physical Journal Plus.

[54]  Victor Fabian Morales-Delgado,et al.  Application of the Caputo‐Fabrizio and Atangana‐Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect , 2019, Mathematical Methods in the Applied Sciences.