The dynamics of Helicobacter pylori-associated stomach cancer under Ψ–Hilfer fractional derivative
Ahmet Yildirim Xani 1 and Mehran Mahdavi 2,
1
Department of Mathematics, Howard University, Washington, DC 20059 USA
2
Department of Mathematics, Bowie State University, 14000 Jericho Park Road, Bowie, MD 20715 USA
Received on December 31, 2025, revised on January 15, 2026
Accepted January 19, 2026
Communicated by Gaston M. N'Guérékata
| Abstract. Helicobacter pylori (HP) is an infection that affects approximately 50% of the world’s population and is one of the main pathogenic risk factors for stomach cancer. Therefore, a rapid, non-invasive diagnostic approach is necessary to avert gastritis, peptic ulcer, and ultimately, stomach cancer associated with HP. HP accounts for the majority of the etiological burden associated with gastric cancer; however, the majority of in vivo studies have failed to adequately induce malignancy through HP infection alone with HP infection alone. This paper creates a new fractional-order differential equations (FODEs) model of HP-induced stomach cancer using the Ψ-Hilfer fractional derivative. FODEs support real-world population dynamics so this model propagates a FODE approach to HP-induced stomach cancer relative to treatment of HP infection as transmission dynamics support direct and indirect transmission with exclusion and inclusion approaches - successful treatment works as a control (removal) factor. This model involves six populations - susceptibles, exposed, infected, treated, stomach cancer patients, and and environmental bacterial concentration. The model’s unique solution is established, and the properties of positivity and boundedness are verified. Local and global stability assessed for equilibrium point where coexistence is observed. The Adams-Bashforth-Moulton method (Predictor-Corrector scheme) is applied for numerical simulations to assess fractional dynamics and population behavior relative to parameters of interest. This study shows that by increasing the treatment rate of HP infection, the incidence rate of HP infection and stomach cancer significantly drop; thus, effective treatment alongside reduction in the contact rate is essential for population-induced elimination of stomach cancer. | |
| Keywords: | Ψ–Hilfer fractional derivative, stomach cancer, Helicobacter pylori, Adams-Bashforth-Moulton method. |
| 2010 Mathematics Subject Classification: 26A33; 34A08, 92D30. | |
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