Research Papers

New Aspects of Immunogenic Tumors Within Different Fractional Operators

[+] Author and Article Information
Malik Zaka Ullah

Department of Mathematics,
Faculty of Science,
King Abdulaziz University,
Jeddah 21589, Saudi Arabia
e-mail: zmalek@kau.edu.sa

Eman S Al-Aidarous

Department of Mathematics,
Faculty of Science,
King Abdulaziz University,
Jeddah 21589, Saudi Arabia
e-mail: ealaidarous@kau.edu.sa

Dumitru Baleanu

Department of Mathematics,
Faculty of Arts and Sciences,
Cankaya University,
Ankara 06530, Turkey;
Institute of Space Sciences,
P.O. Box. MG-23,
Magurele-Bucharest R 76900, Romania
e-mail: dumitru@cankaya.edu.tr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 7, 2018; final manuscript received January 12, 2019; published online February 15, 2019. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 14(4), 041009 (Feb 15, 2019) (8 pages) Paper No: CND-18-1354; doi: 10.1115/1.4042637 History: Received August 07, 2018; Revised January 12, 2019

This paper presents a new mathematical formulation in fractional sense describing the asymptotic behavior of immunogenic tumor growth. The new model is investigated through different fractional operators with and without singular kernel. An efficient numerical technique to solve these equations is also suggested. Comparative results with experimental data verify that the fractional-order growth model covers the real data better than the integer model of tumor growth. Thus, more precise models can be provided by the fractional calculus (FC), which helps us to examine better the complex dynamics. Finally, numerical results confirming the theoretical analysis are provided.

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Grahic Jump Location
Fig. 1

The interaction of immunogenic tumor cells and immune response when the chimeric mice received 5 × 105 viable BCL1 cells: (a) γ = 0.93, (b) γ = 0.94, (c) γ = 0.96, and (d) γ = 0.98

Grahic Jump Location
Fig. 2

The interaction of immunogenic tumor cells and immune response when the chimeric mice received 5 × 107 viable BCL1 cells: (a) γ = 0.9890, (b) γ = 0.9895, (c) γ = 0.99, and (d) γ = 0.9913

Grahic Jump Location
Fig. 3

The optimal tracking control response for the immunogenic tumor growth when the chimeric mice received 5 × 105 viable BCL1 cells: (a) the immune response E(t) and (b) the immunogenic tumor growth T(t)

Grahic Jump Location
Fig. 4

The optimal tracking control response for the immunogenic tumor growth when the chimeric mice received 5 × 107 viable BCL1 cells: (a) the immunogenic tumor growth T(t) and (b) the immune response E(t)



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