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Fractional-Order Delay Differential Equations of Hepatitis C Virus

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Delay Differential Equations and Applications to Biology

Part of the book series: Forum for Interdisciplinary Mathematics ((FFIM))

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Abstract

In this chapter, we investigate a fractional-order delay differential model reflecting the dynamics of Hepatitis C virus (HCV) replication in the presence of interferon-\(\alpha \) treatment. We consider a fractional order in the model to represent the intermediate cellular interactions and intracellular delay of the viral life cycle and incorporate a discrete time-delay to justify the short-run memory. The fractional order is also considered with existing model parameters to unify the units of the differential equations. We analyze the steady states and dynamical behavior of the model.

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References

  1. Mainardi, F., Gorenflo, R.: Fact sheet No. 164: hepatitis C. www.who.int.gate2.inist.fr/mediacentre/factsheets/fs164/en/. Accessed 31 Jan 2014

  2. Mainardi, F., Gorenflo, R.: On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118, 283–299 (2000)

    Article  MathSciNet  Google Scholar 

  3. Mukhopadhya, A.: Hepatitis C in India. J. Biosci. 33(4), 465–473 (2008)

    Article  Google Scholar 

  4. Afdhal, H.N.: The natural history of hepatitis C. Semin. Liver Dis. 24(2), 3–8, 5, 6 (2004)

    Google Scholar 

  5. Wasley, A., Alter, M.J.: Epidemiology of hepatitis C: geographic differences and temporal trends. Semin. Liver Dis. 20(1), 1–16 (2000)

    Article  Google Scholar 

  6. Ramirez, I.: Mathematical modeling of immune responses to hepatitis C virus infection. PhD thesis, East Tennessee State University (2014)

    Google Scholar 

  7. Dahari, H., Lo, A., Ribeiro, R.M., Perelson, A.S.: Modeling hepatitis C virus dynamics: liver regeneration and critical drug efficacy. J. Theor. Biol. 47, 371–381 (2007)

    Article  MathSciNet  Google Scholar 

  8. Perelson, A.S.: Modelling viral and immune system dynamics. Nat. Rev. Immunol. 28–36 (2002)

    Google Scholar 

  9. Tanner, J.T.: The stability and intrinsic growth rates of prey and predator populations. Ecology 56, 855–867 (1975)

    Article  Google Scholar 

  10. Zeuzem, S., Herrmann, E.: Dynamics of hepatitis C virus infection. Ann. Hepatol. 1(2), 56–63 (2002)

    Google Scholar 

  11. Rihan, F.A., Abdelrahman, D.H.: Delay differential model for tumor-immune dynamics with HIV infection of CD4\(+\) T-cells. Int. J. Comput. Math. 90(3), 594–614 (2013)

    Article  MathSciNet  Google Scholar 

  12. Rihan, F.A., Lakshmanan, S., Hashish, A.H., Rakkiyappan, R., Ahmed, E.: Fractional order delayed predator-prey systems with Holling type-ii functional response. Nonlinear Dyn. 80(1), 777–789 (2015)

    Article  MathSciNet  Google Scholar 

  13. Rihan, F.A., Al-Mdallal, Q., AlSakaji, H.J., Hashish, A.: A fractional-order epidemic model with time-delay and nonlinear incidence rate. Chaos Solitons Fractals 126, 97–105 (2019)

    Google Scholar 

  14. Chinnathamb, R., Rihan, F.A., Alsakaji, H.J.: A fractional-order model with time delay for tuberculosis with endogenous reactivation and exogenous reinfections. Math. Methods Appl. Sci. 1–20 (2019)

    Google Scholar 

  15. Rakkiyappan, R., Latha, V.P., Rihan, F.A.: A fractional-order model for zika virus infection with multiple delays. Complexity 2019, 1–15 (2019)

    Article  Google Scholar 

  16. Magin, R.L.: Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl. 59, 1586–1593 (2010)

    Article  MathSciNet  Google Scholar 

  17. Ding, Y., Wang, Z., Ye, H.: Optimal control of a fractional-order HIV-immune system with memory. IEEE Trans. Control Syst. Tech. 20(3), 763–769 (2012)

    Article  Google Scholar 

  18. Rihan, F.A., Velmurugan, G.: Dynamics of fractional-order delay differential model for tumor-immune system. Chaos Solitons Fractals 132, 1–14 (2020)

    Google Scholar 

  19. Arshad, S., Baleanu, D., Huang, J., Tang, Y., Al Qurashi, M.M.: Dynamical analysis of fractional order model of immunogenic tumors. Adv. Mech. Eng. 8(7), 1687814016656704 (2016)

    Google Scholar 

  20. Cole, K.S.: Electric conductance of biological systems. In: Cold Spring Harbor Symposia on Quantitative Biology, pp. 107–116 (1993)

    Google Scholar 

  21. Rakkiyappan, R., Preethi Latha, V., Rihan, F.A.: A fractional-order model for zika virus infection with multiple delays. Complexity (2019), 20 p

    Google Scholar 

  22. Ahmed, E., El-Saka, H.A.: On fractional order models for hepatitis C. Nonlinear Biomed. Phys. 4(1), 1–4 (2010)

    Article  Google Scholar 

  23. Pawlotsky, J.M., Bouvier-Alias, M., Hezode, C., Darthuy, F., Remire, J., Dhumeaux, D.: Standardization of hepatitis C virus RNA quantification. Hepatology 32(3), 654–9 (2000)

    Article  Google Scholar 

  24. Lin, W.: Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007)

    Article  MathSciNet  Google Scholar 

  25. **, C., Gu, K., Boussaada, I., Niculescu, S.I.: Stability analysis of a more general class of systems with delay-dependent coefficients. IEEE Trans. Autom. Control 64(5), 1989–1998 (2018)

    Google Scholar 

  26. **, C., Gu, K., Niculescu, S.I., Boussaada, I.: Stability analysis of systems with delay-dependent coefficients: an overview. IEEE Access 6, 27392–27407 (2018)

    Article  Google Scholar 

  27. Li, H.L., Zhang, L., Hu, C., Jiang, Y.L., Teng, Z.: Dynamical analysis of a fractional-order prey-predator model incorporating a prey refuge. J. Appl. Math. Comput

    Google Scholar 

  28. Bard, Y.: Nonlinear Parameter Estimation. Academic, New York (1974)

    MATH  Google Scholar 

  29. Marchuk, G.I., Romanyukha, A.A., Bocharov, G.A.: Mathematical model of antiviral immune response ii: parameter identification for acute viral hepatitis-B. J. Theor. Biol. 151, 41–70 (1991)

    Article  Google Scholar 

  30. Yasui, K., Okanoue, T., Murakami, Y., Itoh, Y., Minami, M., Sakamoto, S., Sakamoto, M., Nishioji, K.: Dynamics of hepatities C verimia following interferon-\(\alpha \) administration. J. Infect. Dis. 177, 1475–1479 (1998)

    Article  Google Scholar 

  31. Dahari, H., Major, M., Zhang, X., Mihalik, K., Rice, C.M., Perelson, A.S., Feinstone, S.M., Neumann, A.U.: Mathematical modeling of primary hepatitis C infection: noncytolytic clearance and early blockage of virion production. Gastroenterology 128, 1056–1066 (2005)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, United Arab Emirates University, Al-Ain, Abu Dhabi, United Arab Emirates

    Fathalla A. Rihan

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  1. Fathalla A. Rihan
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Correspondence to Fathalla A. Rihan .

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Rihan, F.A. (2021). Fractional-Order Delay Differential Equations of Hepatitis C Virus. In: Delay Differential Equations and Applications to Biology. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-16-0626-7_12

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