Nonlinear fractional mathematical model of tuberculosis (TB) disease with incomplete treatment under Atangana-Baleanu derivative

Abstract The current work investigates the mathematical model of Tuberculosis disease with incomplete treatment under the Atangana-Baleanu-Caputo (ABC) derivative with fractional order. Upon exploiting fixed point approach and nonlinear analysis, we derive some theoretical results about solution existence and its stability. The famous fractional Adam Bashforth technique is applied to compute numerical solution to the considered model. The aforesaid tool is based on fundamental theorem of fractional calculus and Lagrange interpolation polynomials. Additionally, various numerical plots are given corresponding to different fractional order in ( 0 , 1 ] .

[1]  Haci Mehmet Baskonus,et al.  New Numerical Results for the Time-Fractional Phi-Four Equation Using a Novel Analytical Approach , 2020, Symmetry.

[2]  Tailei Zhang,et al.  Global stability for a tuberculosis model , 2011, Math. Comput. Model..

[3]  J. Rappaz,et al.  On a two-dimensional magnetohydrodynamic problem. I. Modelling and analysis , 1992 .

[4]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[5]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[6]  D. G. Prakasha,et al.  A new study of unreported cases of 2019-nCOV epidemic outbreaks , 2020, Chaos, Solitons & Fractals.

[7]  K. Shah,et al.  A NEW NUMERICAL TREATMENT FOR FRACTIONAL DIFFERENTIAL EQUATIONS BASED ON NON-DISCRETIZATION OF DATA USING LAGUERRE POLYNOMIALS , 2020 .

[8]  R. Wallis.  Mathematical Models of Tuberculosis Reactivation and Relapse , 2016, Front. Microbiol..

[9]  Ghaus ur Rahman,et al.  Numerical solution of fractional order smoking model via laplace Adomian decomposition method , 2017, Alexandria Engineering Journal.

[10]  A. O. Egonmwan,et al.  Analysis of a mathematical model for tuberculosis with diagnosis , 2018, Journal of Applied Mathematics and Computing.

[11]  A. Atangana,et al.  On solutions of fractal fractional differential equations , 2021, Discrete & Continuous Dynamical Systems - S.

[12]  M. Khan,et al.  A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative , 2020, Discrete & Continuous Dynamical Systems - S.

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

[14]  Abdon Atangana,et al.  Analysis of fractal fractional differential equations , 2020 .

[15]  Fatmawati,et al.  A mathematical model of tuberculosis (TB) transmission with children and adults groups: A fractional model , 2020 .

[16]  M. Shitikova,et al.  Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids , 1997 .

[17]  A. Akgül.  A novel method for a fractional derivative with non-local and non-singular kernel , 2018, Chaos, Solitons & Fractals.

[18]  U. Diwekar,et al.  A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions , 2014 .

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

[20]  A. Geser,et al.  The use of mathematical models in the study of the epidemiology of tuberculosis. , 1962, American journal of public health and the nation's health.

[21]  F. Haq,et al.  Numerical analysis of fractional order model of HIV-1 infection of CD4+ T-cells , 2017 .

[22]  R. Agarwal,et al.  A study of fractional Lotka‐Volterra population model using Haar wavelet and Adams‐Bashforth‐Moulton methods , 2020, Mathematical Methods in the Applied Sciences.

[23]  Kottakkaran Sooppy Nisar,et al.  Transmission dynamics of fractional order Typhoid fever model using Caputo–Fabrizio operator , 2019, Chaos, Solitons & Fractals.

[24]  B. Samet,et al.  A study on fractional host–parasitoid population dynamical model to describe insect species , 2020, Numerical Methods for Partial Differential Equations.

[25]  Ahmet Bekir,et al.  A novel analytical method for time-fractional differential equations , 2016 .

[26]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[27]  O. Kuipers,et al.  Identification of Differentially Expressed Genes during Bacillus subtilis Spore Outgrowth in High-Salinity Environments Using RNA Sequencing , 2016, Front. Microbiol..

[28]  Manar A. Alqudah,et al.  Semi-analytical study of Pine Wilt Disease model with convex rate under Caputo–Febrizio fractional order derivative , 2020 .

[29]  Abdon Atangana,et al.  Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties , 2018, Physica A: Statistical Mechanics and its Applications.

[30]  D. Baleanu,et al.  A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative , 2020 .

[31]  KAMAL SHAH,et al.  EXISTENCE AND UNIQUENESS OF SOLUTIONS TO FRACTIONAL ORDER MULTI-POINT BOUNDARY VALUE PROBLEMS , 2015 .

[32]  Mati ur Rahman,et al.  Investigating a nonlinear dynamical model of COVID-19 disease under fuzzy caputo, random and ABC fractional order derivative , 2020, Chaos, Solitons & Fractals.

[33]  S. Geerlings,et al.  Immune dysfunction in patients with diabetes mellitus (DM). , 1999, FEMS immunology and medical microbiology.

[34]  Z. Khan,et al.  Stability analysis of a dynamical model of tuberculosis with incomplete treatment , 2020, Advances in Difference Equations.

[35]  D. Okuonghae,et al.  A mathematical model of tuberculosis transmission with heterogeneity in disease susceptibility and progression under a treatment regime for infectious cases , 2013 .

[36]  M. Goyal,et al.  An efficient technique for a time fractional model of lassa hemorrhagic fever spreading in pregnant women , 2019, The European Physical Journal Plus.

[37]  Martin Pitt,et al.  An analysis of the academic literature on simulation and modelling in health care , 2009, J. Simulation.

[38]  K. Nisar,et al.  Mathematical model for spreading of COVID‐19 virus with the Mittag–Leffler kernel , 2020, Numerical methods for partial differential equations.

[39]  Boping Zhou,et al.  MERS, SARS, and Ebola: The Role of Super-Spreaders in Infectious Disease , 2015, Cell Host & Microbe.

[40]  Zhien Ma,et al.  Global stability of two models with incomplete treatment for tuberculosis , 2010 .

[41]  Karam Allali,et al.  Mathematical analysis of a fractional differential model of HBV infection with antibody immune response , 2020 .

[42]  Dumitru Baleanu,et al.  A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying , 2019, Advances in Difference Equations.

[43]  Hamid Reza Mohammadi Daniali,et al.  Variational iteration method for solving the epidemic model and the prey and predator problem , 2007, Appl. Math. Comput..

[44]  Mati ur Rahman,et al.  Investigation of fractional order tuberculosis (TB) model via Caputo derivative , 2020 .

[45]  Abdon Atangana,et al.  Modelling and analysis of fractal-fractional partial differential equations: Application to reaction-diffusion model , 2020 .

[46]  B. Samet,et al.  A wavelet based numerical scheme for fractional order SEIR epidemic of measles by using Genocchi polynomials , 2020, Numerical Methods for Partial Differential Equations.

[47]  B. Samet,et al.  A new Rabotnov fractional‐exponential function‐based fractional derivative for diffusion equation under external force , 2020, Mathematical Methods in the Applied Sciences.

[48]  J. F. Gómez‐Aguilar,et al.  A chaos study of tumor and effector cells in fractional tumor-immune model for cancer treatment , 2020 .

[49]  Yongjin Li,et al.  Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel , 2019, Mathematical Methods in the Applied Sciences.

[50]  D. G. Prakasha,et al.  New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques , 2020, Numerical Methods for Partial Differential Equations.

[51]  A. Ullah,et al.  On analysis of the fractional mathematical model of rotavirus epidemic with the effects of breastfeeding and vaccination under Atangana-Baleanu (AB) derivative , 2020 .

[52]  Abdon Atangana,et al.  Can transfer function and Bode diagram be obtained from Sumudu transform , 2020 .

[53]  S. Momani,et al.  Numerical investigations on COVID‐19 model through singular and non‐singular fractional operators , 2020, Numerical Methods for Partial Differential Equations.

[54]  Aziz Khan,et al.  Analytical solutions of time-fractional wave equation by double Laplace transform method , 2019, The European Physical Journal Plus.

[55]  M. Jleli,et al.  A fractional model for population dynamics of two interacting species by using spectral and Hermite wavelets methods , 2020, Numerical Methods for Partial Differential Equations.

[56]  Zhencai Zhu,et al.  Stochastic resonance in an overdamped system with a fractional power nonlinearity: Analytical and re-scaled analysis , 2019, The European Physical Journal Plus.

[57]  B. Samet,et al.  Chaotic behaviour of fractional predator-prey dynamical system , 2020 .

[58]  Tugba Akman Yildiz,et al.  New aspects of time fractional optimal control problems within operators with nonsingular kernel , 2020, Discrete & Continuous Dynamical Systems - S.

[59]  Yong Li,et al.  Mathematical modeling of tuberculosis data of China. , 2015, Journal of theoretical biology.

[60]  Devendra Kumar,et al.  An efficient numerical scheme for fractional model of HIV-1 infection of CD4+ T-cells with the effect of antiviral drug therapy , 2020 .

[61]  H. M. Baskonus,et al.  Regarding new positive, bounded and convergent numerical solution of nonlinear time fractional HIV/AIDS transmission model , 2020 .

[62]  A mathematical model of COVID-19 using fractional derivative: outbreak in India with dynamics of transmission and control , 2020, Advances in difference equations.

[63]  J. Manafian,et al.  APPLICATION OF THE LAPLACE ADOMIAN DECOMPOSITION AND IMPLICIT METHODS FOR SOLVING BURGERS ’ EQUATION , 2015 .

[64]  A. Arafa,et al.  Approximate analytical solution of the fractional epidemic model , 2012 .

[65]  H. M. Baskonus,et al.  New approach for the model describing the deathly disease in pregnant women using Mittag-Leffler function , 2020 .