Analytical Approximation and an Efficient Adaptive Algorithm for a Delay-Induced Tumor–Immune–Normal Cell Model

S. Daniel; A.R. Manikandan

doi:https://doi.org/10.14445/22315373/IJMTT-V72I8P103

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 8 | Year 2026 | Article Id. IJMTT-V72I8P103 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I8P103

Analytical Approximation and an Efficient Adaptive Algorithm for a Delay-Induced Tumor–Immune–Normal Cell Model

S. Daniel, A.R. Manikandan

Received	Revised	Accepted	Published
21 Jun 2026	24 Jul 2026	14 Aug 2026	29 Aug 2026

Citation :

S. Daniel, A.R. Manikandan, "Analytical Approximation and an Efficient Adaptive Algorithm for a Delay-Induced Tumor–Immune–Normal Cell Model," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 8, pp. 11-26, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I8P103

Abstract

The delay-induced tumor–immune–normal cell model of Das et al. (Adv. Contin. Discrete Models 2022:15) established local and global stability results for the tumor-free equilibrium and a Hopf-bifurcation analysis for the coexisting equilibrium, supported by numerical simulation. The present note extends that work along two directions that were left open in the original paper. First, we derive a semi-analytical solution of the full nonlinear delay system using the classical method of steps combined with the Adomian Decomposition Method (ADM); we show that because the initial history is constant, the delay system reduces exactly to an ordinary differential system on the first characteristic interval [0,τ], which yields a genuine (verifiable) power-series solution rather than a purely numerical approximation, and we describe the recursive procedure that extends this construction to every subsequent interval [nτ,(n+1)τ]. Second, we identify and correct an order-reduction phenomenon that afflicts the naive fixed-step Runge–Kutta scheme used for this type of model (its empirical order drops from four to two because of low-order interpolation of the delayed argument), and we propose an adaptive continuous embedded Runge–Kutta 5(4) scheme with cubic-Hermite dense output (“ACDP-DDE”) that restores fourth-order accuracy and reduces the number of function evaluations needed for a given accuracy by more than an order of magnitude. Both extensions are verified numerically against the equilibrium values, the critical delay τ₀ ≈ 18.919, and the qualitative dynamics (damped spiral, limit cycle, post-Hopf oscillation) reported in the original article.

Keywords

Adomian Decomposition Method (ADM), Adaptive Runge–Kutta Method, Delay Differential Equations (DDEs), HOPF Bifurcation, Tumor–Immune–Normal Cell Model.

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