Abstract
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.