Fractional Order In Vitro Fertilization Model Real Data Analysis with Novel Application of Inequalities via Stability and Computational Techniques

Manal Ghannam¹, Bilgen Kaymakamzade^1,2, Muhammad Farman^1,3,4, Kottakkaran Sooppy Nisar^5,*, Mohammed Altaf Ahmed⁶
1 Department of Mathematics, Mathematical Research Center, Near East University, Mersin, Turkey
2 Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
3 Faculty of Medicine, Department of Biostatistics and Medical Informatics, Karadeniz Technical University, Trabzon, Turkey
4 International Center for Interdisciplinary Research in Sciences, The University of Lahore, Lahore, Pakistan
5 Department of Mathematics, College of Science and Humanities, Prince Sattam bin Abdulaziz University, Al-Kharj, Saudi Arabia
6 Department of Computer Engineering, College of Computer Engineering & Sciences, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia
* Corresponding Author: Kottakkaran Sooppy Nisar. Email: n.sooppy@psau.edu.sa
(This article belongs to the Special Issue: Mathematical Aspects of Computational Biology and Bioinformatics-III)

Computer Modeling in Engineering & Sciences https://doi.org/10.32604/cmes.2026.081075

Received 23 February 2026; Accepted 12 May 2026; Published online 10 June 2026

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Abstract

In Vitro Fertilization (IVF) has been a major medical advancement in the field of fertility treatment. It has helped millions of individuals and couples overcome infertility by providing a workable option. It involves removing eggs from the ovaries of a female, fertilizing those eggs with male sperm in a monitored lab condition. In this work, we developed a new model to show the success of In Vitro Fertilization rates in women through a fractional- order compartmental modeling framework by using real data. The developed model is analyzed statistically, and the biological feasibility of the model. The Lipschitz condition, expressed by the Lipschitz inequality, suggests that the change in a function’s output is limited by a constant known as the Lipschitz constant in relation to the input changes between any two points in its domain. This characteristic ensures that solution routes remain consistent, resulting in well-behaved and singular solutions. The linear growth inequality describes the condition of linear growth by asserting that one variable can be limited by another that rises linearly. This inequality is crucial because it prevents differential equation solutions from rising too rapidly, ensuring that they remain within established boundaries. We used Volterra integral inequality with a Volterra-type Lyapunov function to analyze the fractional derivatives of Lyapunov functions in a fractional-order system, which is necessary to establish global asymptotic stability. This extends the standard Lyapunov stability theory to fractional-order systems, particularly in the analysis of complex models like the infectious disease model, by providing a method to constrain the behavior of the system without solving the differential equations explicitly. Additionally, a sensitivity analysis of several aspects is derived through mathematical simulations. Additionally, we use numerical simulations to validate our theoretical findings at different fractional orders.