Open Access

ARTICLE

Mathematical Modeling of Leukemia within Stochastic Fractional Delay Differential Equations

Ali Raza^1,2,*, Feliz Minhós^2,3,*, Umar Shafique⁴, Muhammad Mohsin⁵

1 Department of Physical Sciences, The University of Chenab, Gujrat, 50700, Pakistan
2 Center for Research in Mathematics and Applications (CIMA), Institute for Advanced Studies and Research (IIFA), University of Évora, Rua Romão Ramalho, 59, Évora, 7000-671, Portugal
3 Department of Mathematics, School of Science and Technology, University of Évora, Rua Romão Ramalho, 59, Évora, 7000-671, Portugal
4 Department of Mathematics, National College of Business Administration and Economics, Lahore, 54660, Pakistan
5 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro, 3810-193, Portugal

* Corresponding Authors: Ali Raza. Email: or ; Feliz Minhós. Email:

(This article belongs to the Special Issue: Analytical and Numerical Solution of the Fractional Differential Equation)

Computer Modeling in Engineering & Sciences 2025, 143(3), 3411-3431. https://doi.org/10.32604/cmes.2025.060855

Received 11 November 2024; Accepted 08 February 2025; Issue published 30 June 2025

Abstract

In 2022, Leukemia is the 13th most common diagnosis of cancer globally as per the source of the International Agency for Research on Cancer (IARC). Leukemia is still a threat and challenge for all regions because of 46.6% infection in Asia, and 22.1% and 14.7% infection rates in Europe and North America, respectively. To study the dynamics of Leukemia, the population of cells has been divided into three subpopulations of cells susceptible cells, infected cells, and immune cells. To investigate the memory effects and uncertainty in disease progression, leukemia modeling is developed using stochastic fractional delay differential equations (SFDDEs). The feasible properties of positivity, boundedness, and equilibria (i.e., Leukemia Free Equilibrium (LFE) and Leukemia Present Equilibrium (LPE)) of the model were studied rigorously. The local and global stabilities and sensitivity of the parameters around the equilibria under the assumption of reproduction numbers were investigated. To support the theoretical analysis of the model, the Grunwald Letnikov Nonstandard Finite Difference (GL-NSFD) method was used to simulate the results of each subpopulation with memory effect. Also, the positivity and boundedness of the proposed method were studied. Our results show how different methods can help control the cell population and give useful advice to decision-makers on ways to lower leukemia rates in communities.

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