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Solving age-dependent infectious diseases and tumor growth models using the contraction approach
Syed Khayyam Shah1, Muhammad Sarwar2,3, Kamal Shah4,3
1Department of Sustainable Environment and Energy Systems (SEES), Middle East Technical University, Northern Cyprus Campus, 99738 Kalkanli, Guzelyurt, Mersin 10, Turkey.
Methodsx
|August 14, 2025
Summary
This study uses fixed-point theory and novel contraction techniques to prove the existence and uniqueness of solutions for models of infectious diseases and tumor growth. This enhances mathematical rigor and computational validation in biological modeling.
Area of Science:
- Mathematical Biology
- Differential Equations
- Fixed-Point Theory
Background:
- Biological models for disease spread and tumor growth often involve complex nonlinearities.
- Ensuring model stability and solution consistency is crucial for accurate predictions.
- Existing mathematical frameworks may struggle with the intricacies of these biological systems.
Purpose of the Study:
- To establish existence and uniqueness theorems for solutions in biological models.
- To investigate criteria for unique solutions in tumor growth and infectious disease models.
- To enhance the mathematical proficiency of epidemiological and oncological modeling.
Main Methods:
- Application of fixed-point theory, specifically contraction mapping principles.
- Development and utilization of generalized contraction techniques (e.g., F-contraction, α-F-contraction, rational type (ψ, φ)-contraction, Geraghty-type contraction).
- Analysis of differential equations governing biological systems.
Main Results:
- Demonstrated existence and uniqueness of solutions for models of age-dependent disease infectiousness, infectious disease transmission, and tumor growth dynamics.
- Established novel results ensuring model stability through rigorous mathematical analysis.
- Provided computational techniques for validating biological models.
Conclusions:
- Fixed-point methodologies offer robust mathematical foundations for biological modeling.
- Generalized contraction techniques effectively manage nonlinearities in biological systems.
- This work advances the investigation of disease dynamics and treatment effectiveness.
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