Numerical behavior of a fractional order dynamical model of RNA silencing

  • Authors

    • A.M.A. El-Sayed Faculty of Engineering, MSA university, Giza, Egypt
    • M. Khalil
    • A.A.M. Arafa
    • Amaal Sayed
    2016-08-29
    https://doi.org/10.14419/ijsw.v4i2.6474
  • Fractional Calculus, RNA Silencing Fractional Order Model, Predictor-Corrector Method.

  • Abstract

    A class of fractional-order differential models of RNA silencing with memory is presented in this paper. We also carry out a detailed analysis on the stability of equilibrium and we show that the model established in this paper possesses non-negative solutions. Numerical solutions are obtained using a predictor-corrector method to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. Numerical simulations are presented to illustrate the results. Also, the numerical simulations show that, modeling the phenomena of RNA silencing by fractional ordinary differential equations (FODE) has more advantages than classical integer-order modeling.

  • References

    1. [1] E. Ahmed and A.S. Elgazzar,†On fractional order differential equations model for nonlocal epidemicsâ€, PHYSICA A, 379 (2007) 607–614. http://dx.doi.org/10.1016/j.physa.2007.01.010.

      [2] E. Ahmed, A.M.A. El-Sayed, H.A.A. El-Saka,†Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies modelsâ€, J. Math. Anal. Appl. 325 (2007) 542–553, http://dx.doi.org/10.1016/j.jmaa.2006.01.087.

      [3] F. Aini Abdullah and Ahmad Izani Md. Ismail, Simulations of the Spread of the Hantavirus Using Fractional Differential Equations, Matematika, 27(2011) 149-158.

      [4] A.A.M. Arafa, S.Z. Rida, M. Khalil,†Fractional modeling dynamics of HIV and CD4+ T-cells during primary infectionâ€, Nonlinear Biomedical Physics 6(2012) 1-7, http://dx.doi.org/10.1186/1753-4631-6-1.

      [5] A.A.M. Arafa, S.Z. Rida, M. Khalil,†The effect of anti-viral drug treatment of human immunodeficiencyâ€, Appl. Math. Model, 37 (2013) 2189–2196,http://dx. doi:10.1016/j.apm.2012.05.002 http://dx.doi.org/10.1016/j.apm.2012.05.002.

      [6] A.A.M. Arafa, S.Z. Rida. M. Khalil,†Solutions of Fractional model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells using HAMâ€, International Journal of Basic and Applied Sciences, 1 (1) (2012) 1-14. http://dx.doi.org/10.14419/ijbas.v1i1.15.

      [7] D. Baulcombe, “RNA silencingâ€, Curr. Biol. 12, (2002) R82–R84, http://dx.doi.org/10.1016/S0960-9822(02)00665-6..

      [8] C.T. Bergstrom, E. McKittrick, R. Antia,†Mathematical models of RNA silencing: Unidirectional amplification limits accidental self-directed reactionsâ€, PNAS, 100 (2003) 11511–11516. http://dx.doi.org/10.1073/pnas.1931639100.

      [9] M. Dalir, M. Bashour,†Applications of Fractional Calculusâ€, Appl. Math. Sci. 4 (2010) 1021–1032.

      [10] E. Demirci, Arzu Unal and Nuri Ozalp,†A fractional order SEIR Model with density dependent death rateâ€, Hacet. J. Math. Stat., 40 (2) (2011) 287 – 295.

      [11] W. Deng,†Smoothness and stability of the solutions for nonlinear fractional differential equationsâ€, Nonlinear Anal-Theor. 72 (2010) 1768-1777, http://dx.doi.org/10.1016/j.na.2009.09.018.

      [12] K. Diethelm,†The Analysis of Fractional Differential Equationsâ€, Springer-Verlag, Berlin, (2010) 181-182.

      [13] Y. Ding, H. Yea,†A fractional-order differential equation model of HIV infection of CD4+T-cellsâ€, Math. Comput. Model. 50 (2009) 386–392, http://dx.doi.org/10.1186/1753-4631-6-1.

      [14] A. Eamens, M.B. Wang, N. A. Smith, P. M. Waterhouse,†RNA Silencing in Plants: Yesterday, Today, and Tomorrowâ€, Plant Physiology,147(2008),456–468, http://dx.doi.org/10.1104/pp.108.117275.

      [15] A.E.M. El-Misiery, E. Ahmed,†On a fractional model for earthquakesâ€, Appl. Comput. Math. 178 (2006) 207–211. http://dx.doi.org/10.1016/j.amc.2005.10.011.

      [16] A.M.A. El-Sayed, I.L. El-Kalla, E.A.A. Ziada,†Analytical and numerical solutions of multi-term nonlinear fractional orders differential equationsâ€, Applied Numerical Mathematics 60 (2010) 788–797, http://dx.doi.org/10.1016/j.apnum.2010.02.007.

      [17] A.M.A. El-Sayed, M.E. Nasr,†Existence of uniformly stable solutions of non-autonomous discontinuous dynamical systemsâ€, J.Egyptian Math. Soc. (2011) 19, 91-94. http://dx.doi.org/10.1016/j.joems.2011.09.006.

      [18] A.M.A. El-Sayed,†On the existence and stability of positive solution for a nonlinear fractional-order differential equation and some applicationsâ€, Alex. J. Math. 1 (2010) 1–10.

      [19] A.M.A. El-Sayed, S.Z. Rida, A.A.M. Arafa,†Exact solutions of some time-fractional biological population models by using generalized differential transform methodâ€, Int. J. Math. Model. Simul. Appl. 3 (2010) 231–239.

      [20] A.M.A. El-Sayed, S.Z. Rida, A.A.M. Arafa,†On the Solutions of Time-fractional Bacterial Chemotaxis in a Diffusion Gradient Chamberâ€, Int. J. Nonlinear Sci. Numer., 7(2009) 485–492.

      [21] A.M.A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka,†Numerical solution for multi-term fractional (arbitrary) orders differential equationsâ€, Comput. Appl. Math., 23(2004)33–54. http://dx.doi.org/10.1590/S0101-82052004000100002.

      [22] M.A.C. Groenenboom, P. Hogeweg,†Modelling the dynamics of viral suppressors of RNA silencingâ€, J. R. Soc. Interface 9(2012), 436–447. http://dx.doi.org/10.1098/rsif.2011.0361.

      [23] C.P. Li and F.R. Zhang,†A survey on the stability of fractional differential equationsâ€, Eur. Phys. J. Special Topics 193 (2011) 27–47. http://dx.doi.org/10.1140/epjst/e2011-01379-1.

      [24] W. Lin,†Global existence theory and chaos control of fractional differential equationsâ€, J. Math. Anal. Appl. 332 (2007) 709–726, http://dx.doi.org/10.1016/j.jmaa.2006.10.040.

      [25] S. Mlotshwa, O. Voinnet, M. Florian Mette, M. Matzke, H. Vaucheret, S. Wei Ding, G. Pruss, V. B. Vance, “RNA Silencing and the Mobile Silencing Signalâ€, The Plant Cell, (2002) 289–301,http://dx. doi: 10.1105/tpc.001677.

      [26] S. Nikolov,†Dynamics and Complexity in a Time Delay Model of RNA Silencing with Periodic Forcingâ€, Bioautomation, 10(2008) 1-12.

      [27] Z. Odibat and N. Shawagfeh,†Generalized Taylor’s formulaâ€, Appl. Math. Comput. 186 (2007) 286–293, http://dx.doi.org/10.1155/2015/507970.

      [28] Z. Odibat, and Shaher Moamni,†An algorithm for the numerical solution of differential equations of fractional orderâ€, J. Appl. Math. & Informatics, 26(2008) 15 – 27.

      [29] L.M. Petrovic, D.T. Spasic, T.M. Atanackovic,†On a mathematical model of a human root dentinâ€, Dental Materials, 21(2005), 125-128. http://dx.doi.org/10.1016/j.dental.2004.01.004.

      [30] S. Pfeffer, G. Meister, M. Landthaler, T. Tuschl,†RNA silencingâ€, B.I.F. Futura, 20 (2005).

      [31] F.A. Rihan,†Numerical Modeling of Fractional-Order Biological Systemsâ€, Abstract and Applied Analysis, 2013, http://dx.doi.org/10.1155/2013/816803.

      [32] V. Suat Ertürk, Zaid M. Odibat, Shaher Momani,†An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+T-cellsâ€, Computers and Mathematics with Applications 62 (2011) 996–1002. http://dx.doi.org/10.1016/j.camwa.2011.03.091.

      [33] What is RNA? : http://www.umassmed.edu/rti/biology/rna-faq/

      [34] Why is RNA important to the cell?: https://www.reference.com/science/rna-important-cell-bcdd937bd3cbc9b3.

  • Downloads

  • How to Cite

    El-Sayed, A., Khalil, M., Arafa, A., & Sayed, A. (2016). Numerical behavior of a fractional order dynamical model of RNA silencing. International Journal of Scientific World, 4(2), 52-56. https://doi.org/10.14419/ijsw.v4i2.6474

    Received date: 2016-07-12

    Accepted date: 2016-08-17

    Published date: 2016-08-29