A Semi-Analytical Hybrid Methodology for Studying Mathematical Models of Cancer and Immune Dynamics
Keywords:
Akbari-Gangi Method, Atkin’s External Modification, Cancer and Immune Dynamics, Convergence and StabilityAbstract
In this paper, we present a new method for result analytical approximate solutions to systems of cancer and immune dynamics. The new method is developed by combining the Akbari-Gangi method with an external modification of Atkin. The proposed method was tested by applying it to solve nonlinear mathematical models describing cancer and immune dynamics. To demonstrate the effectiveness of the new method in improving results, it was compared with the Akbari-Gangi method and other methods presented in the literature. The obtained results show that this method has high accuracy, good convergence, and acceptable stability, which were illustrated in the form of tables and graphical representations of the solutions and error estimates.
References
S. Kartal, “Mathematical modeling and analysis of tumor-immune system interaction by using Lotka-Volterra predator-prey like model with piecewise constant arguments,” Periodicals of Engineering and Natural Sciences, vol. 2, no. 1, 2014.
V. Volterra, “Fluctuations in the abundance of a species considered mathematically,” Nature, vol. 118, pp. 558–560, 1926.
R. P. Jiménez, L. E. Bergues Cabrales, and J. I. Montijano, “Dynamics of the Lotka-Volterra model for tumor-host systems under constant or periodic perturbation: Implications for cancer therapy,” Mar. 14, 2025.
G. Craciun, F. Nazarov, and C. Pantea, “Persistence and permanence of mass-action and power-law dynamical systems,” SIAM Journal on Applied Mathematics, vol. 73, pp. 305–329, 2013.
L. Xie and Y. Wang, “On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics,” Journal of Mathematical Analysis and Applications, vol. 471, pp. 584–598, 2019.
K. Su, C. Wang, S. Zhang, and S. Liu, “Lotka-Volterra equation-based modeling of the aerobic granulation process in sequencing batch reactors,” International Biodeterioration & Biodegradation, vol. 115, pp. 49–54, 2016.
R. A. Cropp and J. Norbury, “Population interactions in ecology: A rule-based approach to modeling ecosystems in a mass-conserving framework,” SIAM Review, vol. 57, pp. 437–446, 2015.
C. Li and H. Zhu, “Canard cycles for predator-prey systems with Holling types of functional response,” Journal of Differential Equations, vol. 254, pp. 879–910, 2013.
B. M. Holzapfel, F. Wagner, L. Thibaudeau, J.-P. Levesque, and D. W. Hutmacher, “Concise review: Humanized models of tumor immunology in the 21st century: Convergence of cancer research and tissue engineering,” Stem Cells, vol. 33, no. 6, pp. 1696–1704, 2015.
C. Zhu and G. Yin, “On competitive Lotka-Volterra model in random environments,” Journal of Mathematical Analysis and Applications, vol. 357, pp. 154–170, 2009.
V. Badri, M. J. Yazdanpanah, and M. S. Tavazoei, “Global stabilization of Lotka-Volterra systems with interval uncertainty,” IEEE Transactions on Automatic Control, vol. 64, pp. 1209–1213, 2018.
A. Wang, D. Xue, Z. Wang, J. Zhao, and F. Rao, “Dynamics of a stochastic tumor-immune interaction system,” The European Physical Journal Plus, vol. 139, 2024.
J. D. Murray, Mathematical Biology I: An Introduction, 3rd ed. New York, NY, USA: Springer, 2002.
J. Liu, M. Hong, Y. Li, D. Chen, Y. Wu, and Y. Hu, “Programmed cell death tunes tumor immunity,” Frontiers in Immunology, vol. 13, 2022.
J. Biazar and R. Montazeri, “A computational method for solution of the prey and predator problem,” Applied Mathematics and Computation, vol. 163, no. 2, pp. 841–847, 2005.
L. Bougoffa, “Solvability of the predator and prey system with variable coefficients and comparison of the results with modified decomposition,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 383–387, 2006.
M. S. H. Chowdhury, I. Hashim, and S. Mawa, “Solution of the predator-prey problem by a numerical-analytic technique,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1008–1012, 2009.
O. D. Makinde, “Solving a ratio-dependent predator-prey system with constant-effort harvesting using the Adomian decomposition method,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 17–22, 2007.
E. Yusufoglu and E. Baris, “He’s variational iteration method applied to the solution of the prey and predator problem with variable coefficients,” Physics Letters A, vol. 372, no. 21, pp. 3829–3835, 2008.
V. S. Erturk and S. Momani, “Solutions to the predator-prey problem and the epidemic model via the differential transform method,” Kybernetes, vol. 37, no. 8, pp. 1180–1188, 2008.
A. R. Ghotbi, A. Barari, and D. D. Ganji, “Solving a ratio-dependent predator-prey system with constant-effort harvesting using the homotopy perturbation method,” Mathematical Problems in Engineering, vol. 2008, pp. 1–7, 2008.
T. Vijayalakshmi and S. Rathinam, “Application of homotopy perturbation and variational iteration methods for a nonlinear imprecise prey-predator model with stability analysis,” The Journal of Supercomputing, vol. 78, no. 2, pp. 2477–2502, 2022.
I. Poschke, D. Mougiakakos, and R. Kiessling, “Camouflage and sabotage: Tumor escape from the immune system,” Cancer Immunology, Immunotherapy, vol. 60, no. 9, pp. 1221–1229, 2011.
A. Bozyk, K. Wojas-Krawczyk, P. Krawczyk, and J. Milanowski, “Tumor microenvironment—A short review of cellular and interaction diversity,” Biology, vol. 11, no. 6, p. 929, 2022.
S. Jawad, D. Sultan, and M. Winter, “The dynamics of a modified Holling-Tanner prey-predator model with wind effect,” International Journal of Nonlinear Analysis and Applications, vol. 12, Special Issue, pp. 2203–2210, 2021.
M. W. Yasin, N. Ahmed, M. S. Iqbal, A. Raza, M. Rafiq, E. M. T. Eldin, and I. Khan, “Spatiotemporal numerical modeling of a stochastic predator-prey model,” Scientific Reports, vol. 13, no. 1, p. 1990, 2023.
Y. Sun and S. Chen, “Stability and bifurcation in a reaction-diffusion-advection predator-prey model,” Calculus of Variations and Partial Differential Equations, vol. 62, no. 2, p. 61, 2023.
M. S. Arif, K. Abodayeh, M. Shoaib, and A. Ejaz, “On the stability of the diffusive and nondiffusive predator-prey system with consuming resources and disease in prey species,” Mathematical Biosciences and Engineering, vol. 20, no. 3, pp. 5066–5093, 2023.
R. O. Al-Sadi and A. J. Al-Saif, “A new technique to solve predator-prey models using the Shehu transformation-Akbari-Ganji method with Padé approximants,” Central Asian Journal of Mathematical Theory and Computer Sciences, vol. 4, no. 8, 2023.
A. Ahadi, S. M. Mousavi, A. M. Alinia, and H. Khademi, Partial Differential Equations in Applied Mathematics. 2025.
M. H. Hassan and A. J. Al-Saif, “A mathematical model for the velocity of thin film flow of a third-grade fluid down an inclined plane,” Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, vol. 102, no. 1, 2023.
H. Mirgolbabae, S. T. Ledari, and D. D. Ganji, “Semi-analytical investigation on micropolar fluid flow and heat transfer in a permeable channel using AGM,” Journal of the Association of Arab Universities for Basic and Applied Sciences, vol. 24, pp. 213–222, 2017.
H. Mesgarani, N. Aghazadeh, and P. Parmour, “Aitken extrapolation and epsilon algorithm for an accelerated solution of weakly singular nonlinear Volterra integral equations,” Physica Scripta, vol. 81, no. 1, p. 015001, 2010.
A. Shah, L. Yuan, and A. Khan, “Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier–Stokes equations,” Applied Mathematics and Computation, vol. 215, pp. 3201–3213, 2010.
