Khayyam Shah Syed, Sarwar Muhammad, Shah Kamal, Hleili Manel, Abdeljawad Thabet
Department of Sustainable Environment and Energy Systems (SEES), Middle East Technical University, Northern Cyprus Campus, 99738 Kalkanli, Guzelyurt, Mersin 10, Turkey.
Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000, Khyber, Pakhtunkhwa, Pakistan.
MethodsX. 2025 Jul 28;15:103505. doi: 10.1016/j.mex.2025.103505. eCollection 2025 Dec.
This study establishes existence and uniqueness theorems for solution sets in three domains of biological modeling: age-dependent diseases infectiousness, infectious disease transmission, and tumor growth dynamics. We illustrate that fixed-point theory, using contraction mapping concepts, offers solid mathematical foundations for model stability and solution consistency. Our principal contribution is to develop generalized contraction techniques that ensure the existence and uniqueness of solutions for the differential equations describing these biological systems. This mathematical framework improves the mathematical proficiency of epidemiological and oncological modeling and offers computational techniques for model validation. These findings address significant deficiencies in the scientific literature by employing fixed-point methodologies from classical analysis to manage the intricate nonlinearities present in biological systems, thereby paving emerging paths for the investigation of disease dynamics and treatment effectiveness.•Purpose: In this work, we will look for the criteria of existence of unique solutions of the equations in the models like, tumor growth, infectious diseases dependency and spread.•Methodology: Utilizing contraction principle and using different contractions from the literature like, F-contraction α-F-contraction, rational type (ψ, φ)-contraction, and Geraghty-type contraction we come up with the conditions where the mentioned biological models possesses unique solutions.•Findings: Imposing different conditions we established novel results which help us ensure the stability by analyzing the existence and uniqueness of the solution of the problems arising in the aforementioned biological models.
年龄依赖性疾病传染性、传染病传播以及肿瘤生长动力学。我们表明，使用压缩映射概念的不动点理论为模型稳定性和解决方案一致性提供了坚实的数学基础。我们的主要贡献是开发广义压缩技术，以确保描述这些生物系统的微分方程解的存在性和唯一性。这个数学框架提高了流行病学和肿瘤学建模的数学水平，并提供了模型验证的计算技术。这些发现通过采用经典分析中的不动点方法来处理生物系统中存在的复杂非线性，解决了科学文献中的重大缺陷，从而为疾病动态和治疗效果的研究开辟了新途径。
目的：在这项工作中，我们将寻找肿瘤生长、传染病依赖性和传播等模型中方程唯一解的存在性标准。
方法：利用压缩原理，并使用文献中的不同压缩，如F - 压缩、α - F - 压缩、有理型(ψ, φ) - 压缩和格拉赫蒂型压缩，我们得出了上述生物模型具有唯一解的条件。
发现：通过施加不同条件，我们建立了新的结果，通过分析上述生物模型中出现的问题解的存在性和唯一性，帮助我们确保稳定性。
