×

Some new mathematical models of the fractional-order system of human immune against IAV infection. (English) Zbl 1470.92088

Summary: Fractional derivative operators of non-integer order can be utilized as a powerful tool to model nonlinear fractional differential equations. In this paper, we propose numerical solutions for simulating fractional-order derivative operators with the power-law and exponential-law kernels. We construct the numerical schemes with the help the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. These schemes are applied to simulate the dynamical fractional-order model of the immune response (FMIR) to the uncomplicated influenza A virus (IAV) infection, which focuses on the control of the infection by the innate and adaptive immunity. Numerical results are then presented to show the applicability and efficiency on the FMIR.

MSC:

92C32 Pathology, pathophysiology
26A33 Fractional derivatives and integrals
34C60 Qualitative investigation and simulation of ordinary differential equation models
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] D. S. Jones, M. J. Plank, B. D. Sleeman, Differential Equations and Mathematical Biology, Chapman & Hall (CRC Press), Baton Roca, Florida, 2010. · Zbl 1298.92003
[2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006. · Zbl 1092.45003
[3] H. M. Srivastava, Fractional-order derivatives and
[4] S. Kumar, A. Ahmadian, R. Kumar, D. Kumar, J. Singh, D. Baleanu, et al., An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets, Mathematics, 8 (2020), 558.
[5] J. Singh, A. Kilicman, D. Kumar, R. Swroop, F. M. Ali, Numerical study for fractional model of nonlinear predator-prey biological population dynamical system, Therm. Sci., 23 (2019), S2017- S2025.
[6] S. Aljhani, M. S. Md Noorani, A. K. Alomari, Numerical solution of fractional-order HIV model using homotopy method, Discrete Dyn. Nat. Soc., 2020 (2020), 2149037. · Zbl 1459.92111
[7] J. M. Amigó, M. Small, Mathematical methods in
[8] M. A. Khan, S. W. Shah, S. Ullah, J. F. Gómez-Aguilar, A dynamical model of asymptomatic carrier zika virus with optimal control strategies, Nonlinear Anal. Real World Appl., 50 (2019), 144-170. · Zbl 1430.92045
[9] M. A. Taneco-Hernández, V. F. Morales-Delgado, J. F. Gómez-Aguilar, Fractional Kuramoto-Sivashinsky equation with power law and stretched Mittag-Leffler kernel, Phys. A, 527 (2019), 121085. · Zbl 07568251
[10] S. Ullah, M. A. Khan, J. F. Gómez-Aguilar, Mathematical formulation of hepatitis B virus with optimal control analysis, Optim. Control Appl. Methods, 40 (2019), 529-544. · Zbl 1425.92133
[11] V. F. Morales-Delgado, J. F. Gómez-Aguilar, K. M. Saad, M. A. Khane, P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force
[12] V. F. Morales-Delgado, J. F. Gómez-Aguilar, K. M. Saad, R. F. E. Jiméenez, Application of the Caputo-Fabrizio and Atangana-Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect, Math. Methods Appl. Sci., 42 (2019), 1167-1193. · Zbl 1419.92009
[13] E. Bonyah, M. A. Khan, K. O. Okosun, J. F. Gómez-Aguilar, Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea with optimal control, Math. Biosci., 309 (2019), 1-11. · Zbl 1409.92228
[14] V. F. Morales-Delgado, J. F. Gómez-Aguilar, M. A. Taneco-Hernándeza, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel, J. Nonlinear Sci. Appl., 11 (2018), 994- 1014. · Zbl 1438.92034
[15] E. Uçar, S. Uçar, N. Özdemir, Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative. Chaos Solitons Fractals, 118 (2019), 300- 306. · Zbl 1442.92074
[16] M. A. Khan, Z. Hammouch, D. Baleanu, Modeling the dynamics of hepatitis E via the CaputoFabrizio derivative, Math. Model. Natur. Phenom., 14 (2019), 311. · Zbl 1420.92106
[17] J. Hristov, Derivation of the fractional Dodson equation and
[18] K. M. Owolabi, A. Atangana, Numerical simulation of noninteger order system in subdiffusive, diffusive, and superdiffusive scenarios, J. Comput. Nonlinear Dyn., 12 (2017), 031010.
[19] K. M. Saad, K. M. Khader, J. F. Gómez-Aguilar, D. Baleanu, Numerical solutions of the fractional Fisher’s type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods, Chaos, 29 (2019), 023116. · Zbl 1409.35225
[20] M. M. Khader, K. M. Saad, Numerical treatment for studying the blood ethanol concentration systems with different forms of fractional derivatives, Int. J. Modern Phys. C., 31 (2020), 1-13.
[21] K. M. Saad, J. F. Gómez-Aguilar, Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense, Rev. Mex. Fís., 64 (2018), 539-547.
[22] K. M. Saad, J. F. Gómez-Aguilar, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Phys. A, 509 (2018), 703-716. · Zbl 1514.35469
[23] K. M. Saad, New fractional derivative with non-singular kernel for deriving Legendre spectral collocation method, Alexandria Eng. J., 2019, forthcoming.
[24] K. M. Saad, H. M. Srivastava, J. F. Gómez-Aguilar, A fractional quadratic autocatalysis associated with chemical clock reactions involving linear inhibition, Chaos Solitons Fractals, 132 (2020), 109557. · Zbl 1444.92145
[25] A. K. Alomari, Homotopy-Sumudu transforms for solving system of fractional partial differential equations, Adv. Differ. Equations, 2020 (2020), 222. · Zbl 1482.35241
[26] A. Goswami, J. Singh, D. Kumar, Sushila, An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Phys. A, 524 (2019), 563-575. · Zbl 07563873
[27] J. Singh, D. Kumar, D. Baleanu, On the analysis of fractional diabetes model with exponential law, Adv. Differ. Equations, 2018 (2018), 231. · Zbl 1446.34018
[28] K. M. Saad, E. H. F. Al-Shareef, A. K. Alomari, D. Baleanu, J. F. Gómez-Aguilar, On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burgers’ equations using homotopy analysis transform method, Chin. J. Phys., 63 (2020), 149-162.
[29] M. Masjed-Jamei, Z. Moalemi, H. M. Srivastava, I. Area, Some modified Adams-Bashforth methods based upon the weighted Hermite quadrature rules, Math. Methods Appl. Sci., 43 (2020), 1380-1398. · Zbl 1452.65125
[30] K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2020), 3-22. · Zbl 1009.65049
[31] K. Diethelm, A. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52. · Zbl 1055.65098
[32] L. Galeone, R. Garrappa, Fractional Adams-Moulton methods, Math. Comput. Simul., 79 (2008), 1358-1367. · Zbl 1163.65099
[33] C. Li, C. Tao, On the fractional Adams method, Comput. Math. Appl., 58 (2009), 1573-1588. · Zbl 1189.65142
[34] V. Daftardar-Gejji, H. Jafari, Analysis of a system of non autonomous fractional differential equations involving Caputo derivatives, J. Math. Anal. Appl., 328 (2007), 1026-1033. · Zbl 1115.34006
[35] V. Daftardar-Gejji, Y. Sukale, S. Bhalekar, A new predictor-corrector method for fractional differential equations, Appl. Math. Comput., 244 (2014), 158-182. · Zbl 1337.65071
[36] A. Atangana, K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), 3. · Zbl 1406.65045
[37] B. Hancioglua, D. Swigona, G. Clermont, A dynamical model of human immune response to influenza A virus infection, J. Theoret. Biol., 246 (2007), 70-86. · Zbl 1451.92094
[38] Y. Zhang, Z. Xu, Y. Cao, Host-Virus
[39] E. De Vries, W. Du, H. Guo, C. A. De Haan, Influenza A Virus Hemagglutinin-NeuraminidaseReceptor
[40] B. Li, S. M. Clohisey, B. S. Chia, B. Wang, A. Cui, T. Eisenhaure, et al., Genome-wide CRISPR screen identifies host dependency factors for influenza A virus infection, Nat. Commun., 11 (2020), 164.
[41] R. Jia, S. Liu, J. Xu, X. Liang, IL16 deficiency enhances Th1 and cytotoxic T lymphocyte response against influenza A virus infection, Biosci. Trends, 13 (2020), 516-522.
[42] B. Asquith, C. R. M. Bangham, An introduction to lymphocyte and viral
[43] I. Podlubny, Fractional Differential
[44] K. S. Miller, B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley and Sons, New York, Chichester, Brisbane, Toronto and Singapore, 1993. · Zbl 0789.26002
[45] W. Faridi, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.
[46] H. Singh, H. M. Srivastava, Numerical simulation for fractional-order Bloch equation arising in nuclear magnetic resonance by using the Jacobi polynomials, Appl. Sci., 10 (2020), 2850.
[47] H. M. Srivastava, H. I. Abdel-Gawad, K. M. Saad, Stability of traveling waves based upon the Evans function and Legendre polynomials, Appl. Sci., 10 (2020), 846.
[48] H. M. Srivastava, H. Günerhan, B. Ghanbari, Exact traveling wave solutions for resonance nonlinear Schrödinger equation with intermodal dispersions and the Kerr law nonlinearity, Math. Methods Appl. Sci., 42 (2019), 7210-7221. · Zbl 1430.74070
[49] H. M. Srivastava, F. A. Shah, R. Abass, An application of the Gegenbauer wavelet method for the numerical solution of the fractional Bagley-Torvik equation, Russian J. Math. Phys., 26 (2019), 77-93. · Zbl 1415.65180
[50] H. M. Srivastava, R. S. Dubey, M. Jain, A study of the fractional-order mathematical model of diabetes and its resulting complications, Math. Methods Appl. Sci., 42 (2019), 4570-4583. · Zbl 1425.92118
[51] H. M. Srivastava, H. Günerhan, Analytical and approximate solutions of fractional-order susceptible-infected-recovered epidemic model of childhood disease, Math. Methods Appl. Sci., 42 (2019), 935-941. · Zbl 1410.34140
[52] B. Ahmad, M. Alghanmi, A. Alsaedi, H. M. Srivastava, S. K. Ntouyas, The Langevin equation in terms of generalized Liouville-Caputo derivatives with nonlocal boundary conditions involving a generalized fractional integral, Mathematics, 7 (2019), 533.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.