Some new mathematical models of the fractional-order system of human immune against IAV infection

H. M. Srivastava; Khaled M. Saad; J. F. Gómez-Aguilar; Abdulrhman A. Almadiy; H. M. Srivastava; Khaled M. Saad; J. F. Gómez-Aguilar; Abdulrhman A. Almadiy

doi:10.3934/mbe.2020268

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 4942-4969. doi: 10.3934/mbe.2020268

Previous Article Next Article

Research article Special Issues

Some new mathematical models of the fractional-order system of human immune against IAV infection

1.
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
2.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, China
3.
Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan
4.
Department of Mathematics, College of Arts and Sciences, Najran University, Najran, Kingdom of Saudi Arabia
5.
Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen
6.
CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C. P. 62490, Cuernavaca Morelos, México
7.
Department of Biology, College of Arts and Sciences, Najran University, Najran, Kingdom of Saudi Arabia

Received: 26 May 2020 Accepted: 07 July 2020 Published: 16 July 2020

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.
Citation: H. M. Srivastava, Khaled M. Saad, J. F. Gómez-Aguilar, Abdulrhman A. Almadiy. Some new mathematical models of the fractional-order system of human immune against IAV infection[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 4942-4969. doi: 10.3934/mbe.2020268

Related Papers:

Abstract

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.

References

[1]	D. S. Jones, M. J. Plank, B. D. Sleeman, Differential Equations and Mathematical Biology, Chapman & Hall (CRC Press), Baton Roca, Florida, 2010.
[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.
[3]	H. M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J., 60 (2020), 73-116.
[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.
[7]	J. M. Amigó, M. Small, Mathematical methods in medicine: Neuroscience, cardiology and pathology, Philos. Trans. A Math. Phys. Eng. Sci., 375 (2017), 20170016.
[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.
[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.
[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.
[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 effects: A fractional calculus approach, Phys. A, 523 (2019), 48-65.
[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.
[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.
[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.
[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.
[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.
[17]	J. Hristov, Derivation of the fractional Dodson equation and beyond: Transient diffusion with a non-singular memory and exponentially fading-out diffusivity, Progr. Fract. Differ. Appl., 3 (2017), 1-16.
[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.
[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.
[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.
[25]	A. K. Alomari, Homotopy-Sumudu transforms for solving system of fractional partial differential equations, Adv. Differ. Equations, 2020 (2020), 222.
[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.
[27]	J. Singh, D. Kumar, D. Baleanu, On the analysis of fractional diabetes model with exponential law, Adv. Differ. Equations, 2018 (2018), 231.
[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.
[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.
[31]	K. Diethelm, A. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52.
[32]	L. Galeone, R. Garrappa, Fractional Adams-Moulton methods, Math. Comput. Simul., 79 (2008), 1358-1367.
[33]	C. Li, C. Tao, On the fractional Adams method, Comput. Math. Appl., 58 (2009), 1573-1588.
[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.
[35]	V. Daftardar-Gejji, Y. Sukale, S. Bhalekar, A new predictor-corrector method for fractional differential equations, Appl. Math. Comput., 244 (2014), 158-182.
[36]	A. Atangana, K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), 3.
[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.
[38]	Y. Zhang, Z. Xu, Y. Cao, Host-Virus Interaction: How Host Cells Defend Against Influenza A Virus Infection, Viruses, 12 (2020), 376.
[39]	E. De Vries, W. Du, H. Guo, C. A. De Haan, Influenza A Virus Hemagglutinin-NeuraminidaseReceptor Balance: Preserving Virus Motility, Trends Microbiol., 28 (2020), 57-67.
[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 dynamics: The power and limitations of mathematical analysis, Proc. Biol. Sci., 270 (2003), 1651-1657.
[43]	I. Podlubny, 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, Academic Press, San Diego, London and Toronto, 1999, 198, 1-340.
[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.
[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.
[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.
[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.
[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.
[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.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)