Discrimination and bifurcation analysis of tumor immune interaction in fractional form |
Taj, Muhammad
(Department of Mathematics, University of Azad Jammu and Kashmir)
Khadimallah, Mohamed A. (Prince Sattam Bin Abdulaziz University, College of Engineering, Civil Engineering Department) Hussain, Muzamal (Department of Mathematics, Govt. College University Faisalabad) Rashid, Yahya (Prince Sattam Bin Abdulaziz University, College of Engineering, Civil Engineering Department) Ishaque, Waqas (Department of Mathematics, University of Azad Jammu and Kashmir) Mahmoud, S.R. (GRC Department, Faculty of Applied studies, King Abdulaziz University) Din, Qamar (Department of Mathematics University of Poonch Rawalakot 12350) Alwabli, Afaf S. (Department of Biological Sciences, Rabigh-Faculty of Science & Arts, King Abdulaziz University) Tounsi, Abdelouahed (YFL (Yonsei Frontier Lab), Yonsei University) |
1 | Abbas, S., Banerjee, M. and Momani, S. (2011), "Dynamical analysis of fractional-order modified logistic model", Comput. Mathe. Applicat., 62(3), 1098-1104. https://doi.org/10.1016/j.camwa.2011.03.072 DOI |
2 | Agarwal, R. and Karahanna, E. (2000), "Time flies when you're having fun: Cognitive absorption and beliefs about information technology usage", MIS Quarterly, 665-694. https://doi.org/10.2307/3250951 DOI |
3 | Ahmed, E., El-Sayed, A.M.A. and El-Saka, H.A. (2007), "Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models", J. Mathe. Anal. Applicat., 325(1), 542-553. https://doi.org/10.1016/j.jmaa.2006.01.087 DOI |
4 | Ahmed, E., Hashish, A. and Rihan, F.A. (2012), "On fractional order cancer model", J. Fract. Calculus Appl. Anal., 3(2), 1-6. https://doi.org/10.1155/2013/816803 DOI |
5 | Kuznetsov, V.A., Makalkin, I.A., Taylor, M.A. and Perelson, A.S. (1994), "Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis", Bull. Mathe. Biol., 56(2), 295-321. https://doi.org/10.1016/S0092-8240(05)80260-5 DOI |
6 | Bray, F., Jemal, A., Grey, N., Ferlay, J. and Forman, D. (2012), "Global cancer transitions according to the Human Development Index (2008-2030): a population-based study", lancet oncol., 13(8), 790-801. https://doi.org/10.1016/S1470-2045(12)70211-5 DOI |
7 | Chabner, B.A. and Roberts, T.G. (2005), "Chemotherapy and the war on cancer", Nature Rev. Cancer, 5(1), 65-72. https://doi.org/10.1038/nrc1529 DOI |
8 | Chen, W.-C. (2008), "Nonlinear dynamics and chaos in a fractional-order financial system", Chaos Solitons Fractals, 36(5), 1305-1314. https://doi.org/10.1016/j.chaos.2006.07.051 DOI |
9 | Duan, W.L., Fang, H. and Zeng, C. (2019), "The stability analysis of tumor-immune responses to chemotherapy system with gaussian white noises", Chaos Solitons Fractals, 127, 96-102. https://doi.org/10.1016/j.chaos.2019.06.030 DOI |
10 | Tenreiro Machado, J.A., Silva, M.F., Barbosa, R.S., Jesus, I.S., Reis, C.M., Marcos, M.G. and Galhano, A.F. (2010), "Some applications of fractional calculus in engineering", Mathe. Problems Eng. https://doi.org/10.1155/2010/639801 DOI |
11 | Banerjee, S. (2008), "Immunotherapy with interleukin-2: a study based on mathematical modeling", Int. J. Appl. Mathe. Comput. Sci., 18(3), 389-398. https://doi.org/10.2478/v10006-008-0035-6 DOI |
12 | Rowlands, M. and Gunnell, D. (2009), "HArrIS r, VATTEN LJ, HOLLy JM, MAr-TIN rM. Circulating insulin-like growth factor peptides and prostate cancer risk: A systematic review and meta-analysis", Int. J. Cancer, 124, 2416-2429. https://doi.org/10.1002/ijc.24202 DOI |
13 | Oldham, K.B. (2010), "Fractional differential equations in electrochemistry", Adv. Eng. Software, 41(1), 9-12. https//doi.org/10.1016/j.advengsoft.2008.12.012 DOI |
14 | Panda, S.K. and Katariya, P.V. (2015), "Stability and free vibration behaviour of laminated composite panels under thermomechanical loading", Int. J. Appl. Computat. Mathe., 1(3), 475-490. DOI |
15 | Arafa, A.A.M., Rida, S.Z. and Khalil, M. (2013), "The effect of anti-viral drug treatment of human immunodeficiency virus type 1 (HIV-1) described by a fractional order model", Appl. Mathe. Modell., 37(4), 2189-2196. https://doi.org/10.1016/j.apm.2012.05.002 DOI |
16 | Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., Int. J., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603 DOI |
17 | Banerjee, S. and Sarkar, R.R. (2008), "Delay-induced model for tumor-immune interaction and control of malignant tumor growth", Bio Syst., 91(1), 268-288. https://doi.org/:10.1016/j.biosystems.2007.10.002 DOI |
18 | Podlubny, I. (1999), "Fractional differential equations", Vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, CA, USA. https//doi.org/10.12691/ajma-1-1-3 |
19 | Rihan, F.A., Hashish, A., Al-Maskari, F., Hussein, M.S., Ahmed, E., Riaz, M.B. and Yafia, R. (2016), "Dynamics of tumor-immune system with fractional-order", J. Tumor Res., 2(1), 109-115. |
20 | Safaei, B., Khoda, F.H. and Fattahi, A.M. (2019), "Non-classical plate model for single-layered graphene sheet for axial buckling", Adv. Nano Res., Int. J., 7(4), 265-275. https://doi.org/10.12989/anr.2019.7.4.265 DOI |
21 | Simsek, M. (2011), "Forced vibration of an embedded single-walled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory", Steel Compos. Struct., Int. J., 11(1), 59-76. https://doi.org/10.12989/scs.2011.11.1.059 DOI |
22 | Tarasov, V.E. (2008), "Fractional vector calculus and fractional Maxwell's equations", Annals Phys., 323(11), 2756-2778. https://doi.org/10.1016/j.aop.2008.04.005 DOI |
23 | Thamareerat, N., Luadsong, A. and Aschariyaphotha, N. (2017), "Stability results of a fractional model for unsteady-state fluid flow problem", Adv. Differ. Equ., 74. https://doi.org/10.1186/s13662-017-1116-3 DOI |
24 | Din, Q. (2017), "Complexity and chaos control in a discrete-time prey-predator model", Commun. Nonlinear Sci. Numer. Simul., 49, 113-134. https://doi.org/10.1016/j.cnsns.2017.01.025 DOI |
25 | De Pillis, L.G. and Radunskaya, A. (2003), "The dynamics of an optimally controlled tumor model: A case study", Mathe. Comput. Modell., 37(11), 1221-1244. https://doi.org/10.1016/S0895-7177(03)00133 DOI |
26 | De Pillis, L.G., Gu, W. and Radunskaya, A.E. (2006), "Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations", J. Theor. Biol., 238(4), 841-862. https://doi:10.1016/j.jtbi.2005.06.037 DOI |
27 | DeLisi, C. and Rescigno, A. (1977), "Immune surveillance and neoplasia-1 a minimal mathematical model", Bull. Mathe. Biol., 39(2), 201-221. https://doi.org/10.1016/S0092-8240(77)80008-6 DOI |
28 | Marincola, F.M., White, D.E., Wise, A.P. and Rosenberg, S.A. (1995), "Combination therapy with interferon alfa-2a and interleukin-2 for the treatment of metastatic cancer", J. Clinical Oncology, 13(5), 1110-1122. https://doi.org/10.1385/1-59745-011 DOI |
29 | Din, Q. and Ishaque, W. (2020), "Bifurcation analysis and chaos control in discrete-time eco-epidemiological models of pelicans at risk in the Salton Sea", Int. J. Dyn. Control, 8(1), 132-148. https://doi.org/10.1007/s40435-019-00508-x DOI |
30 | Katariya, P.V. and Panda, S.K. (2016), "Thermal buckling and vibration analysis of laminated composite curved shell panel", Aircr. Eng. Aerosp. Technol., 88(1), 97-107. https://doi.org/10.1108/AEAT-11-2013-0202 DOI |
31 | Zhang, J., Nan, J., Du, W., Chu, Y. and Luo, H. (2016), "Dynamic analysis for a fractional-order autonomous chaotic system", Discrete Dyn. Nature Soc., 2016. https://doi.org/10.1155/2016/8712496. DOI |
32 | De Boer, R.J., Hogeweg, P., Dullens, H.F., De Weger, R.A. and Den Otter, W. (1985), "Macrophage T lymphocyte interactions in the anti-tumor immune response a mathematical model", J. Immunol., 134(4), 2748-2758. https://doi.org/10.1016/S0022-5193(05)80142-0 DOI |
33 | Wen, G., Chen, S. and Jin, Q. (2008), "A new criterion of period-doubling bifurcation in maps and its application to an inertial impact shaker", J. Sound Vib., 311(1-2), 212-223. https://doi.org/10.1016/j.jsv.2007.09.003 DOI |
34 | Thariat, J., Hannoun-Levi, J.M., Myint, A.S., Vuong, T. and Gerard, J.P. (2013), "Past, present, and future of radiotherapy for the benefit of patients", Nature Rev. Clinical Oncol., 10(1), 52. https://doi.org/10.1038/nrclinonc.2012.203 DOI |
35 | Ucar, E., Ozdemir, N. and Altun, E. (2019), "Fractional order model of immune cells influenced by cancer cells", Mathe. Modell. Natural Phenomena, 14(3), 308. https://doi.org/10.1051/mmnp/2019002 DOI |
36 | Villasana, M. and Radunskaya, A. (2003), "A delay differential equation model for tumor growth", J. Mathe. Biol., 47(3), 270-294. https://doi.org/10.1007/s00285-003-0211-0 DOI |
37 | Duan, W.L. (2020), "The stability analysis of tumor-immune responses to chemotherapy system driven by Gaussian colored noises", Chaos Solitons Fractals, 141, 110303. https://doi.org/10.1016/j.chaos.2020.110303 DOI |
38 | Wen, G. (2005), "Criterion to identify Hopf bifurcations in maps of arbitrary dimension", Phys. Rev. E, 72(2), 026201. https://doi.org/10.1103/PhysRevE.72.026201 DOI |
39 | Wyld, L., Audisio, R.A. and Poston, G.J. (2015), "The evolution of cancer surgery and future perspectives", Nature Rev Clinical Oncol., 12(2), 115. https://doi.org/10.1038/nrclinonc.2014.191 DOI |
40 | Yuste, S.B., Acedo, L. and Lindenberg, K. (2004), "Subdiffusionlimited A+ B→ C reaction-subdiffusion process", Phys. Rev. E, 69(3), 36-126. https://doi.org/10.1103/PhysRevE.69.036126 DOI |
41 | Kar, V.R., Mahapatra, T.R. and Panda, S.K. (2017), "Effect of different temperature load on thermal postbuckling behaviour of functionally graded shallow curved shell panels", Compos. Struct., 160, 1236-1247. https://doi.org/10.1016/j.compstruct.2016.10.125 DOI |
42 | Itik, M., Salamci, M.U. and Banks, S.P. (2009), "Optimal control of drug therapy in cancer treatment", Nonlinear Anal.: Theory, Methods Applicat., 71(12), e1473-e1486. https://doi.org/10.3906/elk-1001-411 DOI |
43 | Kar, V.R. and Panda, S.K. (2016), "Post-buckling behaviour of shear deformable functionally graded curved shell panel under edge compression", Int. J. Mech. Sci., 115, 318-324. https://doi.org/10.1016/j.ijmecsci.2016.07.014 DOI |
44 | Kar, V.R., Panda, S.K. and Mahapatra, T.R. (2016), "Thermal buckling behaviour of shear deformable functionally graded single/doubly curved shell panel with TD and TID properties", Adv. Mater. Res., Int. J., 5(4), 205-221. |
45 | Karami, B., Janghorban, M. and Tounsi, A. (2017), "Effects of triaxial magnetic field on the anisotropic nanoplates", Steel Compos. Struct, Int. J., 25(3), 361-374. https://doi.org/10.12989/scs.2017.25.3.361 DOI |
46 | Gonzalez-Parra, G., Arenas, A.J. and Chen-Charpentier, B.M (2014), "A fractional order epidemic model for the simulation of outbreaks of influenza A (H1N1)", Mathe. Methods Appl. Sci., 37(15), 2218-2226. https://doi.org/10.1002/mma.2968 DOI |
47 | Edelstein-Keshet, L. (2005), "Mathematical models in biology", SIAM. https://doi.org/10.1137/1.9780898719147 |
48 | Eladdadi, A., Pillis, L.D. and Kim, P. (2018), "Modelling tumour-immune dynamics, disease progression and treatment", Taylor and Francis. https://doi.org/10.1080/23737867.2018.1483003 |
49 | El-Sayed, A.M.A., Elsonbaty, A., Elsadany, A.A. and Matouk, A.E. (2016), "Dynamical analysis and circuit simulation of a new fractional-order hyperchaotic system and its discretization", Int. J. Bifurcat. Chaos, 26(13), 1650222. https://doi.org/10.1142/S0218127416502229 DOI |
50 | Galach, M. (2003), "Dynamics of the Tumor-Immune System Competition the Effect of Time Delay", Int. J. Appl. Mathe. Comput. Sci, 13, 395-406. https://doi.org/10.1155/DDNS/2006/58463 DOI |
51 | Gorenflo, R. and Mainardi, F. (1997), "Fractional calculus", Fractals and fractional calculus in continuum mechanics, Springer, 223-276. https://doi.org/10.1007/978-3-7091-2664-6 |
52 | Hilfer, R. (2000), "Applications of fractional calculus in physics", World scientific, Singapore. https://doi.org/10.1142/3779 |
53 | Huang, C., Cao, J. and Xiao, M. (2016), "Hybrid control on bifurcation for a delayed fractional gene regulatory network." Chaos Solitons Fractals, 87, 19-29. https://doi.org/10.1007/s11431-018-9376-2 DOI |
54 | Arenas, A.J., Gonzalez-Parra, G. and Chen-Charpentier, B.M. (2016), "Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order", Mathe. Comput. Simul., 121, 48-63. https://doi.org/10.1155/2017/8273430 DOI |
55 | Rutter, E.M. and Kuang, Y. (2017), "Global dynamics of a model of joint hormone treatment with dendritic cell vaccine for prostate cancer", Discrete Continuous Dyn. Syst.-B, 22(3), 1001. http://dx.doi.org/10.3934/dcdsb.2017050 DOI |
56 | Jun, D., Guang-jun, Z., Yong, X., Hong, Y. and Jue, W. (2014), "Dynamic behavior analysis of fractional-order Hindmarsh-Rose neuronal model", Cognitiveneurodynamics, 8(2), 167-175. https://dx.doi.org/10.1007%2Fs11571-013-9273-x DOI |
57 | Elsadany, A. and Matouk, A. (2015), "Dynamical behaviors of fractional-order Lotka-Volterra predator-prey model and its discretization", J. Appl. Mathe. Comput., 49(1-2), 269-283. http://dx.doi.org/10.1007%2Fs12190-014-0838-6 DOI |
58 | Karami, B., Janghorban, M. and Tounsi, A. (2018), "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., Int. J., 27(2), 201-216. https://doi.org/10.12989/scs.2018.27.2.201 DOI |
59 | Katariya, P.V. and Panda, S.K. (2020), "Numerical analysis of thermal post-buckling strength of laminated skew sandwich composite shell panel structure including stretching effect", Steel Compos. Struct., Int. J., 34(2), 279-288. https://doi.org/10.12989/sss.2017.20.5.595 DOI |
60 | Katariya, P.V., Panda, S.K., Hirwani, C.K., Mehar, K. and Thakare, O. (2017), "Enhancement of thermal buckling strength of laminated sandwich composite panel structure embedded with shape memory alloy fibre", Smart Struct. Syst., Int. J., 20(5), 595-605. https://doi.org/10.12989/sss.2017.20.5.595 DOI |
61 | Matouk, A. and Elsadany, A. (2016), "Dynamical analysis, stabilization and discretization of a chaotic fractional-order GLV model", Nonlinear Dyn., 85(3), 1597-1612. https//doi.org/10.1007/s11071-016-2781-6 DOI |
62 | Curti, B.D., Ochoa, A.C., Urba, W.J., Alvord, W.G., Kopp, W.C., Powers, G., Hawk, C., Creekmore, S.P., Gause, B.L., Janik, J.E. and Holmlund, J.T. (1996), "Influence of interleukin-2 regimens on circulating populations of lymphocytes after adoptive transfer of anti-CD3-stimulated T cells: results from a phase I trial in cancer patients", Journal of Immunotherapy with Emphasis on Tumor Immunology: Official Journal of the Society for Biological Therapy, 19(4), 296-308. https://doi.org/ 10.1097/00002371-199607000-00005. DOI |
63 | Kirschner, D. and Panetta, J.C. (1998), "Modeling immunotherapy of the tumor-immune interaction", J. Mathe. Biol., 37(3), 235-252. https://doi.org/10.1007/s002850050127 DOI |
64 | Laskin, N. (2002), "Fractional schrodinger equation", Phys. Rev. E, 66(5), 056108. https://doi.org/10.1103/PhysRevE.66.056108 DOI |
65 | Liao, W., Lin, J.X. and Leonard, W.J. (2011), "IL-2 family cytokines: new insights into the complex roles of IL-2 as a broad regulator of T helper cell differentiation", Current Opinion Immunol., 23(5), 598-604. https//doi.org/10.1016/j.coi.2011.08.003 DOI |
66 | Madani, H., Hosseini, H. and Shokravi, M. (2016), "Differential cubature method for vibration analysis of embedded FG-CNT-reinforced piezoelectric cylindrical shells subjected to uniform and non-uniform temperature distributions", Steel Compos. Struct., Int. J., 22(4), 889-913. https://doi.org/10.12989/scs.2016.22.4.889 DOI |
67 | Matouk, A.E., Elsadany, A.A., Ahmed, E. and Agiza, H.N. (2015), "Dynamical behavior of fractional-order Hastings-Powell food chain model and its discretization", Commun. Nonlinear Sci. Numer. Simul., 27(1-3), 153-167. http://dx.doi.org/http://dx.doi.org/10.10 16/j.cnsns.2015.03.004 DOI |
68 | Mehar, K., Panda, S.K., Devarajan, Y. and Choubey, G. (2019), "Numerical buckling analysis of graded CNT-reinforced composite sandwich shell structure under thermal loading", Compos. Struct., 216, 406-414. https://doi.org/10.1016/j.compstruct.2019.03.002 DOI |
69 | Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D. and Feliu-Batlle, V. (2010), "Fractional-order systems and controls: fundamentals and applications", Springer Science and Business Media. https//doi.org/:10.1007/978-1-84996-335-0 |
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