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A new modified semi-analytical technique for a fractional-order Ebola virus disease model

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript


Ebola virus disease is a fatal hemorrhagic fever of humans and primates caused by viruses. There are many mathematical models to investigate this viral disease. In this paper, the classical form of the Ebola virus disease model has been modified by using new fractional derivatives. The resulting fractional forms of the Ebola virus disease model have then been examined by applying a newly-developed semi-analytical method. The optimal perturbation iteration method has been implemented to obtain new approximate solutions to the system of differential equations which better model the Ebola virus disease. New algorithms are constructed by using three types of operators of fractional derivatives. A real-world problem is also solved in order to prove the efficiency of the proposed algorithms. A good agreement has been found with the real values of the parameters. Finally, several graphical illustrations are presented for different values of the involved biological parameters to show the effects of the new approximate solutions. Obtained results prove that the new method is highly accurate in solving these types of fractional models.

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  1. Leroy, E.M., Kumulungui, B., Pourrut, X., Rouquet, P., Hassanin, A., Yaba, P., D’elicat, A., Paweska, F.T., Gonzalez, J.-P., Swanepoel, R.: Fruit bats as reservoirs of Ebola virus. Nature 438(7068), 575–576 (2005)

  2. Akgül, A., Khoshnaw, S.H.A., Mohammed, W.H.: Mathematical model for the Ebola virus disease. J. Adv. Phys. 7, 190–198 (2018).

    Article  Google Scholar 

  3. Dokuyucu, M.A., Dutta, H.: A fractional order model for Ebola virus with the new Caputo fractional derivative without singular kernel. Chaos Solitons Fract. 134, 109717 (2020).

    Article  MathSciNet  Google Scholar 

  4. Koca, I.: Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators. European Phys. J. Plus 133(3), 1–11 (2018).

    Article  Google Scholar 

  5. Atangana, A., Goufo E.F.D.: On the mathematical analysis of Ebola hemorrhagic fever: deathly infection disease in West African countries. BioMed Res. Int. 2014, 261383 (2014).

    Google Scholar 

  6. Qureshi, S., Atangana, A.: Mathematical analysis of Dengue fever outbreak by novel fractional operators with field data. Phys. A Stat. Mech. Appl. 526, 121–127 (2019).

    Article  MathSciNet  Google Scholar 

  7. Area, I., Batarfi, H., Losada, J., Nieto, J.J., Shammakh, W., Torres, A.: On a fractional order Ebola epidemic model. Adv. Differ. Equations 2015, 278 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  8. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego, London, Toronto (1999).

    MATH  Google Scholar 

  9. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, vol. 204. Elsevier (North-Holland) Science Publishers, Amsterdam (2006).

    Google Scholar 

  10. Singh, H.: Analysis for fractional dynamics of Ebola virus model. Chaos Solitons Fract. 138, 1–8 (2020) (Article ID 109992)

  11. Singh, H., Srivastava, H.M., Hammouch, Z., Nisar, K.S.: Numerical simulation and stability analysis for the fractional-order dynamics of COVID-19. Results Phys. 20, 1–8 (2021) (Article ID 103722)

  12. Srivastava, H.M., Area, I., Nieto, J.J.: Power-series solution of compartmental epidemiological models. Math. Biosci. Eng. 18, 3274–3290 (2021).

    Article  MathSciNet  Google Scholar 

  13. Srivastava, H.M., Deniz, S., Saad, K.M.: An efficient semi-analytical method for solving the generalized regularized long wave equations with a new fractional derivative operator. Journal of King Saud University - Science 33(2), 101345 (2021).

    Article  Google Scholar 

  14. Srivastava, H.M., Irfan, M., Shah, F.A.: A Fibonacci wavelet method for solving dual-phase-lag heat transfer model in multi-layer skin tissue during hyperthermia treatment. Energies 14, 1–20 (2021) (Article ID 2254)

  15. Baleanu, D., Diethelm, K., Scalas, E.: Fractional Calculus: Models and Numerical Methods, vol. 3. World Scientific Publishing Company, Singapore, New Jersey, London, Hong Kong (2012).

    Book  MATH  Google Scholar 

  16. Baleanu, D., Dumitru, J.A.T., Machado, Luo, A.C.-J. (eds.): Fractional Dynamics and Control. Springer, Cham (2011)

  17. Baleanu, D., Güvenç, Z.B., Machado, J.A.T. (eds.): New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Berlin, Heidelberg, New York (2010).

    Google Scholar 

  18. Srivastava, H.M., Tomovski, Ž.: Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 211, 198–210 (2009).

    MathSciNet  MATH  Google Scholar 

  19. Srivastava, H.M., Owa, S. (eds.): Univalent Functions, Fractional Calculus, and their Applications. Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Brisbane, Toronto, Chichester (1989)

  20. Cushing, J.M., Saleem, M., Srivastava, H.M., Khan, M.A. and Merajuddin, M. (eds.): Applied Analysis in Biological and Physical Sciences. Springer Proceedings in Mathematics and Statistics, vol. 186. Springer Nature [Springer (India) Private Limited], New Delhi (2016)

  21. Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Therm. Sci. 20, 763–769 (2016).

    Article  Google Scholar 

  22. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl 1(2), 1–13 (2015).

    Google Scholar 

  23. Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl 1(2), 87–92 (2015).

    Google Scholar 

  24. Atangana, A., and Baleanu, D.: Caputo–Fabrizio derivative applied to groundwater flow within confined aquifer. J. Eng. Mech 143(5) (2017) (Article ID D4016005)

  25. Saad, K.M., Al-Shareef, E.H.F., Alomari, A.K., Baleanu, D., Gómez-Aguilar, J.F.: On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burgers’ equations using homotopy analysis transform method. Chin. J. Phys. 63, 149–162 (2020).

    Article  MathSciNet  Google Scholar 

  26. Khader, M.M., Saad, K.M.: Numerical studies of the fractional Korteweg-de Vries, Korteweg-de Vries-Burgers’ and Burger’ equations. Proc. Nat. Acad. Sci. India Sect. A Phys. Sci. (2020).

  27. Saad, K.M., Srivastava, H.M., Gómez-Aguilar, J.F.: A fractional quadratic autocatalysis associated with chemical clock reactions involving linear inhibition. Chaos Solitons Fract. 132 (2020) (Article ID 109557)

  28. Yang, X.-J., Baleanu, D.: Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci. 17, 625–628 (2013).

    Article  Google Scholar 

  29. Qureshi, S., Yusuf, A., Shaikh, A.A., Inc, M.: Balean, D.: Fractional modeling of blood ethanol concentration system with real data application. Chaos 29(1) (2019) Article ID 013143

  30. Odibat, Z., Baleanu, D.: Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives. Appl. Numer. Math. 156, 94–105 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  31. Al-Khedhairi, A., Matouk, A.E., Khan, I.: Chaotic dynamics and chaos control for the fractional-order geomagnetic field model. Chaos Solitons Fract. 128, 390–401 (2019).

    Article  MathSciNet  Google Scholar 

  32. Dokuyucu, M.A.: A fractional order alcoholism model via Caputo-Fabrizio derivative. AIMS Math. 5, 781–797 (2020).

    Article  MathSciNet  Google Scholar 

  33. Kumar, D., Singh, J., Baleanu, D.: A new analysis for fractional model of regularized long wave equation arising in ion acoustic plasma waves. Math. Methods Appl. Sci. 40, 5642–5653 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  34. Bildik, N., Deniz, S.: A new fractional analysis on the polluted lakes system. Chaos Solitons Fract. 122, 17–24 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  35. Yavuz, M., Ozdemir, N., Baskonus, H.M.: Solutions of partial differential equations using the fractional operator involving Mittag–Leffler kernel. Eur. Phys. J. Plus 133 (2018) (Article ID 215)

  36. Bildik, N., Deniz, S., Saad, K.M.: A comparative study on solving fractional cubic isothermal auto-catalytic chemical system via new efficient technique. Chaos Solitons Fract. 132 (2020) (Article ID 109555)

  37. Saad, K.M.: Comparing the Caputo, Caputo-Fabrizio and Atangana–Baleanu derivative with fractional order: fractional cubic isothermal auto-catalytic chemical system. Eur. Phys. J. Plus 133 (2018) (Article ID 94)

  38. Singh, H., Srivastava, H.M., Kumar, D.: A reliable numerical algorithm for the fractional vibration equation. Chaos Solitons Fract. 103, 131–138 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  39. Srivastava, H.M., Günerhan, H., Ghanbari, B.: Exact traveling wave solutions for resonance nonlinear Schrödinger equation with intermodal dispersions and the Kerr law nonlinearity. Math. Methods Appl. Sci. 42, 7210–7221 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  40. Singh, H., Srivastava, H.M.: Numerical simulation for fractional-order Bloch equation arising in nuclear magnetic resonance by using the Jacobi polynomials. Appl. Sci. 10 (2020) (Article ID 2850)

  41. Srivastava, H.M., Baleanu, D., Machado, J.A.T., Osman, M.S., Rezazadeh, H., Arshed, S., Günerhan, H.: Traveling wave solutions to nonlinear directional couplers by modified Kudryashov method. Phys. Scripta 95 (2020) (Article ID 75217)

  42. Srivastava, H.M., Shah, F.A., Abass, R.: An application of the Gegenbauer wavelet method for the numerical solution of the fractional Bagley-Torvik equation. Russian J. Math. Phys. 26, 77–93 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  43. Ahmad, B., Alghanmi, M., Alsaedi, A., Srivastava, H.M., Ntouyas, S.K.: The Langevin equation in terms of generalized Liouville–Caputo derivatives with nonlocal boundary conditions involving a generalized fractional integral. Mathematics 7 (2019) (Article ID 533)

  44. Srivastava, H.M.: Fractional-order derivatives and integrals: Introductory overview and recent developments. Kyungpook Math. J. 60, 73–116 (2020).

    MathSciNet  MATH  Google Scholar 

  45. Srivastava, H.M., Saad, K.M., Gómez-Aguilar, J.F., Almadiy, A.A.: Some new mathematical models of the fractional-order system of human immune against IAV infection. Math. Biosci. Eng. 17, 4942–4969 (2020).

    Article  MathSciNet  Google Scholar 

  46. Srivastava, H.M.: Diabetes and its resulting complications: mathematical modeling via fractional calculus. Public Health Open Access 4(3), 1–5 (2020) (Article ID 2)

  47. Srivastava, H. M., Saad, K.M.: Numerical simulation of the fractal-fractional Ebola virus. Fractal Fract. 4, 1–13 (2020) (Article ID 49)

  48. Srivastava, H.M., Saad, K.M., Khader, M.M.: An efficient spectral collocation method for the dynamic simulation of the fractional epidemiological model of the Ebola virus. Chaos Solitons Fract. 140, 1–7 (2020) (Article ID 110174)

  49. Deniz, S.: Optimal perturbation iteration method for solving nonlinear heat transfer equations. J. Heat Transfer 139 (2017) (Article ID 074503)

  50. Bildik, N., Deniz, S.: A new efficient method for solving delay differential equations and a comparison with other methods. Eur. Phys. J. Plus 132(1), 1–11 (2017).

    Article  Google Scholar 

  51. Bildik, N., Deniz, S.: New analytic approximate solutions to the generalized regularized long wave equations. Bull. Korean Math. Soc. 55, 749–762 (2018).

    MathSciNet  MATH  Google Scholar 

  52. Deniz, S., Bildik, N.: A new analytical technique for solving Lane-Emden type equations arising in astrophysics, Bull. Belgian Math. Soc.-Simon Stevin 24 , 305–320,(2017)

  53. Bildik, N., Deniz, S.: New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques. Discrete Contin. Dyn. Syst. S 13, 503–518 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  54. Bas, E., Özarslan, R.: Real world applications of fractional models by Atangana-Baleanu fractional derivative. Chaos Solitons Fract. 116, 121–125 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  55. Uçar, S., Uçar, E., Özdemir, N., Hammouch, Z.: Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative. Chaos Solitons Fract. 118, 300–306 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  56. Deniz, S.: Semi-analytical investigation of modified Boussinesq-Burger equations. J. BAUN Inst. Sci. Technol. 22(1), 327–333 (2020).

    Google Scholar 

  57. Deniz, S.: Modification of coupled Drinfel’d-Sokolov-Wilson Equation and approximate solutions by optimal perturbation iteration method. Afyon Kocatepe Univ. J. Sci. Eng. 20(1), 35–40 (2020).

    Article  Google Scholar 

  58. Deniz, S.: Semi-analytical approach for solving a model for HIV infection of CD4\(^{+}\) T-cells. TWMS J. Appl. Eng. Math. 11(1), 273–281 (2021).

    Google Scholar 

  59. Deniz, S.: Optimal perturbation iteration method for solving fractional FitzHugh-Nagumo equation. Chaos Solitons Fract. 142, 110417 (2021).

    Article  MathSciNet  Google Scholar 

  60. Deniz, S.: Semi-analytical analysis of Allen-Cahn model with a new fractional derivative. Math. Methods Appl. Sci. 44(3), 2355–2363 (2021).

    Article  MathSciNet  Google Scholar 

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Srivastava, H.M., Deniz, S. A new modified semi-analytical technique for a fractional-order Ebola virus disease model. RACSAM 115, 137 (2021).

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