Existence of solution and stability for the fractional order novel coronavirus (nCoV-2019) model

Azhar Hussain, Dumitru Baleanu, Muhammad Adeel, Azhar Hussain, Dumitru Baleanu, Muhammad Adeel

Abstract

The aim of this work is to present a new fractional order model of novel coronavirus (nCoV-2019) under Caputo-Fabrizio derivative. We make use of fixed point theory and Picard-Lindelöf technique to explore the existence and uniqueness of solution for the proposed model. Moreover, we explore the generalized Hyers-Ulam stability of the model using Gronwall's inequality.

Keywords: Fractional Caputo–Fabrizio derivative; Novel coronavirus (nCoV-2019); Picard–Lindelöf technique.

Conflict of interest statement

Competing interestsThe authors declare that they have no conflict of interests.

© The Author(s) 2020.

References

    1. Diethelm K. The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Berlin: Springer; 2010.
    1. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier; 2006.
    1. Miller K.S., Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley; 1993.
    1. Podlubny I. Fractional Differential Equations. San Diego: Academic Press; 1999.
    1. Samko S.G., Kilbas A.A., Marichev O.I. Fractional Integrals and Derivatives. Theory and Applications. Yverdon: Gordon & Breach; 1993.
    1. Mainardi F. Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri A., Mainardi F., editors. Fractals and Fractional Calculus in Continuum Mechanics. Wien: Springer; 1997.
    1. Kumar D., Singh J., Baleanu D. Numerical computation of a fractional model of differential-difference equation. J. Comput. Nonlinear Dyn. 2016;11(6):061004. doi: 10.1115/1.4033899.
    1. Area I., Batarfi H., Losada J., Nieto J.J., Shammakh W., Torres A. On a fractional order Ebola epidemic model. Adv. Differ. Equ. 2015;2015(1):278. doi: 10.1186/s13662-015-0613-5.
    1. Ma M., Baleanu D., Gasimov Y.S., Yang X.J. New results for multidimensional diffusion equations in fractal dimensional space. Rom. J. Phys. 2016;61:784–794.
    1. Ullah S., Khan M.A., Farooq M., Hammouch Z., Baleanu D. A fractional model for the dynamics of tuberculosis infection using Caputo–Fabrizio derivative. Discrete Contin. Dyn. Syst., Ser. S. 2020;13(3):975–993.
    1. Khan M.A., Atangana A. Dynamics of Ebola disease in the framework of different fractional derivatives. Entropy. 2019;21(3):303. doi: 10.3390/e21030303.
    1. Khan M.A., Hammouch Z., Baleanu D. Modeling the dynamics of hepatitis E via the Caputo–Fabrizio derivative. Math. Model. Nat. Phenom. 2019;14(3):311. doi: 10.1051/mmnp/2018074.
    1. Khan M.A., Ullah S., Okosun K.O., Shah K. A fractional order pine wilt disease model with Caputo–Fabrizio derivative. Adv. Differ. Equ. 2018;2108:410. doi: 10.1186/s13662-018-1868-4.
    1. Atangana A., Khan M.A. Validity of fractal derivative to capturing chaotic attractors. Chaos Solitons Fractals. 2019;126:50–59. doi: 10.1016/j.chaos.2019.06.002.
    1. Atangana A., Alkahtani B.S.T. Analysis of the Keller–Segel model with a fractional derivative without singular kernel. Entropy. 2015;17(6):4439–4453. doi: 10.3390/e17064439.
    1. Alsaedi A., Nieto J.J., Venktesh V. Fractional electrical circuits. Adv. Mech. Eng. 2015 doi: 10.1177/1687814015618127.
    1. Caputo M., Fabrizio M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015;1(2):73–85.
    1. Losada J., Nieto J.J. Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015;1(2):87–92.
    1. Tateishi A.A., Ribeiro H.V., Lenzi E.K. The role of fractional time-derivative operators on anomalous diffusion. Front. Phys. 2017;5:52. doi: 10.3389/fphy.2017.00052.
    1. Kumar D., Singh J., Al Qurashi M., Baleanu D. Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel. Adv. Mech. Eng. 2017 doi: 10.1177/1687814017690069.
    1. Owolabi K.M., Atangana A. Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative. Chaos Solitons Fractals. 2017;105:111–119. doi: 10.1016/j.chaos.2017.10.020.
    1. Kumar D., Tchier F., Singh J., Baleanu D. An efficient computational technique for fractal vehicular traffic flow. Entropy. 2018;20:259. doi: 10.3390/e20040259.
    1. Singh J., Kumar D., Baleanu D. New aspects of fractional Biswas–Milovic model with Mittag-Leffler law. Math. Model. Nat. Phenom. 2019;14(3):303. doi: 10.1051/mmnp/2018068.
    1. Chen T., Rui J., Wang Q., Zhao Z., Cui J., Yin L. A mathematical model for simulating the phase-based transmissibility of a novel coronavirus. Infect. Dis. Poverty. 2020;9:24. doi: 10.1186/s40249-020-00640-3.
    1. Khan, M.A., Atangana, A.: Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alex. Eng. J. (2020, in press)
    1. Abdeljawad T., Baleanu D. On fractional derivatives with exponential kernel and their discrete version. Rep. Math. Phys. 2017;80:11–27. doi: 10.1016/S0034-4877(17)30059-9.

Source: PubMed

Sponsorit ja yhteistyökumppanit

Sairaudet

Huumeiden interventiot

3
Tilaa