Research article

Study of mathematical model of Hepatitis B under Caputo-Fabrizo derivative

  • Received: 05 July 2020 Accepted: 14 September 2020 Published: 09 October 2020
  • MSC : 26A33, 34A08, 93A30

  • The current work is devoted to bring out a detail analysis including qualitative and semi-analytical study of Hepatitis B model under the Caputo- Fabrizio fractional derivative (CFFD). For the required results, fixed point theory is used to establish the conditions for existence and uniqueness of solution to the considered model. On the other hand, for semi analytical solutions, we use decomposition method of Adomian coupled with integral transform of Laplace. Moreover, the concerned solutions are presented via graphs to analyze the dynamics of different compartments of the model.

    Citation: Sajjad Ali Khan, Kamal Shah, Poom Kumam, Aly Seadawy, Gul Zaman, Zahir Shah. Study of mathematical model of Hepatitis B under Caputo-Fabrizo derivative[J]. AIMS Mathematics, 2021, 6(1): 195-209. doi: 10.3934/math.2021013

    Related Papers:

  • The current work is devoted to bring out a detail analysis including qualitative and semi-analytical study of Hepatitis B model under the Caputo- Fabrizio fractional derivative (CFFD). For the required results, fixed point theory is used to establish the conditions for existence and uniqueness of solution to the considered model. On the other hand, for semi analytical solutions, we use decomposition method of Adomian coupled with integral transform of Laplace. Moreover, the concerned solutions are presented via graphs to analyze the dynamics of different compartments of the model.


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    [1] WHO, Hepatitis B Fact Sheet No. 204, The World Health Organisation, Geneva, Switzerland, 2013. Available from: http://www.who.int/mediacentre/factsheets/fs204/en/.
    [2] W. M. Lee, Hepatitis B virus infection, New Engl. J. Med., 337 (1997), 1733-1745.
    [3] R. M. Anderson, R. M. May, Infectious disease of humans: dynamics and control, Oxford: Oxford University Press, 1991.
    [4] G. F. Medley, N. A. Lindop, W. J. Edmunds, D. J. Nokes, Hepatitis-B virus endemicity: heterogeneity, catastrophic dynamics and control, Nat. Med., 7 (2001), 619-624.
    [5] J. Mann, M. Roberts, Modelling the epidemiology of hepatitis B in New Zealand, J. Theor. Biol., 269 (2011), 266-272. doi: 10.1016/j.jtbi.2010.10.028
    [6] S. Thornley, C. Bullen, M. Roberts, Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control policy, J. Theor. Biol., 254 (2008), 599-603. doi: 10.1016/j.jtbi.2008.06.022
    [7] S. J. Zhao, Z. Y. Xu, Y. Lu, A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China, Int. J. Epidemiol., 29 (2000), 744-752. doi: 10.1093/ije/29.4.744
    [8] K. Wang, W. Wang, S. Song, Dynamics of an HBV model with diffusion and delay, J. Theor. Biol., 253 (2008), 36-44. doi: 10.1016/j.jtbi.2007.11.007
    [9] A. V. Kamyad, R. Akbari, A. A. Heydari, A. Heydari, Mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis B virus, Comput. Math. Methods Med., 2014 (2014), 1-15.
    [10] H. A. A. El-Saka, The fractional-order SIS epidemic model with variable population size, J. Egypt. Math. Soc., 22 (2014), 50-54.
    [11] R. Toledo-Hernandez, V. Rico-Ramirez, G. A. Iglesias-Silva, U. M. Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions, Chem. Eng. Sci., 117 (2014), 217-228. doi: 10.1016/j.ces.2014.06.034
    [12] J. Pang, J. A. Cui, X. Zhou, Dynamical behavior of a hepatitis B virus transmission model with vaccination, J. Theor. Biol., 265 (2010), 572-578. doi: 10.1016/j.jtbi.2010.05.038
    [13] E. Jung, S. Lenhart, Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Cont. Dyn. B, 2 (2002), 473-482.
    [14] I. Podlubny, Fractional differential equations, mathematics in science and engineering, New York: Academic Press, 1999.
    [15] A. A. Kilbas, H. Srivastava, J. Trujillo, Theory and application of fractional differential equations, Amsterdam: Elseveir, 2006.
    [16] S. A. Khan, K. Shah, G. Zaman, F. Jarad, Existence theory and numerical solutions to smoking model under Caputo-Fabrizio fractional derivative, Chaos, 29 (2019), 013128. doi: 10.1063/1.5079644
    [17] F. Haq, K. Shah, G. Rahman, M. Shahzad, Numerical solution of fractional order smoking model via Laplace Adomian decomposition method, Alex. Eng. J., 57 (2018), 1061-1069. doi: 10.1016/j.aej.2017.02.015
    [18] A. Ali, K. Shah, R. A. Khan, Numerical treatment for traveling wave solutions of fractional Whitham-Broer-Kaup equations, Alex. Eng. J., 57 (2018), 1991-1998. doi: 10.1016/j.aej.2017.04.012
    [19] M. Caputo, M. Fabrizio, A new definition of fractional derivative with out singular kernel, Progr. Fract. Diff. Appl., 1 (2015), 73-85.
    [20] T. Abdeljawad, D. Baleanu, On fractional derivatives with exponential kernel and their 28 discrete versions, Rep. Math. Phys., 80 (2017), 11-27. doi: 10.1016/S0034-4877(17)30059-9
    [21] M. Al-Refai, T. Abdeljawad, Analysis of the fractional diffusion equations with fractional 19 derivative of non-singular kernel, Adv. Differ. Equ., 2017 (2017), 315. doi: 10.1186/s13662-017-1356-2
    [22] T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Differ. Equ., 2017 (2017), 313. doi: 10.1186/s13662-017-1285-0
    [23] S. J. Liao, Beyond perturbation: Introduction to the homotopy analysis method, Boca Raton: Chapman Hall/CRC Press, 2003.
    [24] M. Rafei, D. D. Ganji, H. Daniali, Solution of the epidemic model by homotopy perturbation method, Appl. Math. Comput., 187 (2007), 1056-1062.
    [25] M. Rafei, H. Daniali, D. D. Ganji, Variational iteration method for solving the epidemic model and the prey and predator problem, Appl. Math. Comput., 186 (2007), 1701-1709.
    [26] F. Awawdeh, A. Adawi, Z. Mustafa, Solutions of the SIR models of epidemics using HAM, Chaos Solit. Frac., 42 (2009), 3047-3052. doi: 10.1016/j.chaos.2009.04.012
    [27] O. A. Arqub, A. El-Ajou, Solution of the fractional epidemic model by homotopy analysis method, J. King Saud Univ. Sci., 25 (2013), 73-81.
    [28] S. Z. Rida, A. A. M. Arafa, Y. A. Gaber, Solution of the fractional epidimic model by LADM, Frac. Calc. Appl., 7 (2016), 189-195.
    [29] O. Kiymaz, An algorithm for solving initial value problems using Laplace Adomian decomposition method, Appl. Math. Sci., 3 (2009), 1453-1459.
    [30] A. S. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, J. Appl. Math., 1 (2001), 141-155. doi: 10.1155/S1110757X01000183
    [31] A. Shaikh, A. Tassaddiq, K. S. Nisar, D. Baleanu, Analysis of differential equations involving Caputo-Fabrizio fractional operator and its applications to reaction-diffusion equations, Adv. Differ. Equ., 2019 (2019), 178. doi: 10.1186/s13662-019-2115-3
    [32] A. Atangana, B. S. Talkahtani, Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel, Adv. Mech. Eng., 7 (2015), 1-6.
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