Stability analysis of fractional nabla difference COVID-19 model

Microorganisms lives with us in our environment, touching infectious material on the surfaces by hand-mouth which causes infectious diseases and some of these diseases are rapidly spreading from person to person. These days the world facing COVID-19 pandemic disease. This article concerned with existence of results and stability analysis for a nabla discrete ABC-fractional order COVID-19. The nabla discrete ABC-fractional operator as more general and applicable in modeling of dynamical problems due to its non-singular kernel. For the existence and uniqueness theorems and Hyers-Ulam stability, we need to suppose some conditions which will play important role in the proof of our main results. At the end, an expressive example is given to provide an application for the nabla discrete ABC-fractional order COVID-19 model.

[1]  T. Abdeljawad,et al.  On a New Class of Fractional Difference-Sum Operators with Discrete Mittag-Leffler Kernels , 2019, Mathematics.

[2]  K. Shah,et al.  Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative , 2020, Discrete & Continuous Dynamical Systems - S.

[3]  Yonghong Wu,et al.  The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium , 2014, Appl. Math. Lett..

[4]  J. F. Gómez‐Aguilar,et al.  A fractional order HIV‐TB coinfection model with nonsingular Mittag‐Leffler Law , 2020, Mathematical Methods in the Applied Sciences.

[5]  Delfim F. M. Torres,et al.  Mathematical Modelling, Simulation, and Optimal Control of the 2014 Ebola Outbreak in West Africa , 2015, 1503.07396.

[7]  D. Baleanu,et al.  Lattice fractional diffusion equation in terms of a Riesz–Caputo difference , 2015 .

[8]  Thabet Abdeljawad,et al.  On Riemann and Caputo fractional differences , 2011, Comput. Math. Appl..

[9]  Abdon Atangana,et al.  Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative , 2020, Alexandria Engineering Journal.

[10]  S. Rozelle,et al.  Epidemiology, causes, clinical manifestation and diagnosis, prevention and control of coronavirus disease (COVID-19) during the early outbreak period: a scoping review , 2020, Infectious Diseases of Poverty.

[11]  T. Abdeljawad,et al.  A Krasnoselskii Existence Result For Nonlinear Delay Caputo q — Fractional Difference Equations With Applications to Lotka — Volterra Competition Model , 2018 .

[12]  T. Abdeljawad.  Dual identities in fractional difference calculus within Riemann , 2011, 1112.5795.

[13]  Nien Fan Zhang,et al.  On a new definition of the fractional difference , 1988 .

[14]  J. Hyman,et al.  Estimation of the reproduction number of dengue fever from spatial epidemic data. , 2007, Mathematical biosciences.

[15]  D. Baleanu,et al.  On the Stability of Some Discrete Fractional Nonautonomous Systems , 2012 .

[16]  A. Panfilov,et al.  The drift of a vortex in an inhomogeneous system of two coupled fibers , 1991 .

[17]  K. Chatterjee,et al.  Healthcare impact of COVID-19 epidemic in India: A stochastic mathematical model , 2020, Medical Journal Armed Forces India.

[18]  A. Peterson,et al.  Discrete Fractional Calculus , 2016 .

[19]  Michael T. Holm,et al.  The Theory of Discrete Fractional Calculus: Development and Application , 2011 .

[20]  J. Jonnalagadda.  Analysis of a system of nonlinear fractional nabla difference equations , 2015 .

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

[22]  Abdon Atangana,et al.  Numerical approximation of Riemann‐Liouville definition of fractional derivative: From Riemann‐Liouville to Atangana‐Baleanu , 2018 .

[23]  Thabet Abdeljawad,et al.  Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality , 2017, J. Comput. Appl. Math..

[24]  C. Pinto,et al.  The burden of the HIV viral load and of cell-to-cell spread in HIV/HCV coinfection , 2018 .

[25]  K. Tas,et al.  On Sumudu Transform Method in Discrete Fractional Calculus , 2012 .

[26]  George A. Anastassiou,et al.  Nabla discrete fractional calculus and nabla inequalities , 2009, Math. Comput. Model..

[27]  S. Samko.  Hypersingular Integrals and Their Applications , 2001 .

[28]  Zakia Hammouch,et al.  On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative , 2018, Chaos, Solitons & Fractals.

[29]  T. Abdeljawad.  Fractional difference operators with discrete generalized Mittag–Leffler kernels , 2019, Chaos, Solitons & Fractals.

[30]  B. Shulgin,et al.  Pulse vaccination strategy in the SIR epidemic model , 1998, Bulletin of mathematical biology.

[31]  Fatmawati,et al.  The dynamics of COVID-19 with quarantined and isolation , 2020, Advances in Difference Equations.

[32]  Paul W. Eloe,et al.  DISCRETE FRACTIONAL CALCULUS WITH THE NABLA OPERATOR , 2009 .

[33]  Abhishek De,et al.  Coronavirus Disease of 2019 (COVID-19) Facts and Figures: What Every Dermatologist Should Know at this Hour of Need , 2020, Indian journal of dermatology.

[34]  F. Atici,et al.  Modeling with fractional difference equations , 2010 .

[35]  M. R. Ferrández,et al.  Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China , 2020, Communications in Nonlinear Science and Numerical Simulation.

[36]  J. F. Gómez‐Aguilar,et al.  Stability analysis for fractional order advection–reaction diffusion system , 2019, Physica A: Statistical Mechanics and its Applications.

[37]  Preliminary investigation of the association between COVID-19 and suicidal thoughts and behaviors in the U.S. , 2020, Journal of psychiatric research.

[38]  T. Abdeljawad,et al.  On a new class of fractional difference-sum operators based on discrete Atangana-Baleanu sums. , 2019, 1901.08268.

[39]  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.

[40]  J. F. Gómez‐Aguilar,et al.  Stability analysis and numerical solutions of fractional order HIV/AIDS model , 2019, Chaos, Solitons & Fractals.

[41]  P. Eloe,et al.  A transform method in discrete fractional calculus , 2007 .

[42]  T. Abdeljawad.  On Delta and Nabla Caputo Fractional Differences and Dual Identities , 2011, 1102.1625.