Research Article
A Caputo Fractional-Order Model for Breast Cancer
Tumor–Immune Dynamics with Optimal Control and Clinical Data Calibration
Taha Hussein El-Ghareeb*
Issue:
Volume 14, Issue 4, August 2026
Pages:
174-185
Received:
1 June 2026
Accepted:
12 June 2026
Published:
29 June 2026
Abstract: Breast cancer remains one of the leading causes of cancer-related mortality worldwide, and its complex interaction with the immune system and therapeutic interventions presents significant challenges for mathematical modeling. Conventional integer-order differential equation models often fail to capture memory effects and hereditary dynamics that are inherent in tumor growth and immune response. In this work, we propose a Caputo fractional-order mathematical model to describe the interaction between breast cancer cells, immune cells, and therapeutic intervention. The model incorporates the fractional-order parameter to account for memory effects in biological tissues and the long-term influence of past states on disease progression. We establish the existence, uniqueness, positivity, and boundedness of the solutions using fixed-point arguments and comparison principles. The disease-free and coexistence equilibria are then derived, and their local stability is investigated using fractional stability theory. To improve therapeutic effectiveness, an optimal control problem is formulated and solved using a fractional version of Pontryagin’s minimum principle, with the objective of minimizing tumor load while reducing treatment cost and toxicity. Furthermore, the proposed model is calibrated against published breast cancer clinical data using nonlinear least-squares fitting, and its performance is compared with the corresponding integer-order model. Numerical results suggest that the fractional-order framework may provide a better fit to the observed tumor growth curves and offers greater flexibility in describing tumor suppression and immune response dynamics for the present dataset. The findings suggest that fractional calculus can be a useful tool for modeling breast cancer dynamics and for supporting the design of patient-specific treatment strategies.
Abstract: Breast cancer remains one of the leading causes of cancer-related mortality worldwide, and its complex interaction with the immune system and therapeutic interventions presents significant challenges for mathematical modeling. Conventional integer-order differential equation models often fail to capture memory effects and hereditary dynamics that a...
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Research Article
A Comparative Analysis Between RSA and ECC Algorithm
Issue:
Volume 14, Issue 4, August 2026
Pages:
186-198
Received:
8 May 2026
Accepted:
25 May 2026
Published:
9 July 2026
DOI:
10.11648/j.ajam.20261404.12
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Abstract: Cryptography is a fundamental technique for ensuring secure and trustworthy communication between a sender and a receiver by transmitting data in encrypted form. Access to the encrypted data is restricted solely to the legitimate recipient who possesses the appropriate decryption key. It serves a vital function in protecting digital communications and ensuring network security. This paper presents a comparative study of two prominent public-key cryptographic algorithms: RSA (Rivest Shamir Adleman) and ECC (Elliptic Curve Cryptography). In the modern digital landscape, RSA has been the dominant method in public-key encryption systems; however, ECC has gained recognition as a powerful alternative. RSA achieves cryptographic security through the intractability of the Integer Factorization Problem, whereas ECC relies on the computational complexity of the Elliptic Curve Discrete Logarithm Problem. Both mechanisms are widely regarded as effective asymmetric encryption schemes and are extensively applied in data security. The objective of this study is to analyze and compare these approaches to identify strategies that can further strengthen security mechanisms and enhance the protection of sensitive information. Therefore, this RSA versus ECC comparison shows that while RSA enjoys broad adoption, ECC provides enhanced efficiency with minimal key lengths, confirming both serve as indispensable asymmetric encryption methods for constructing resilient, next-generation security architectures defending private data against progressing computational power and novel cybersecurity threats across global digital platforms.
Abstract: Cryptography is a fundamental technique for ensuring secure and trustworthy communication between a sender and a receiver by transmitting data in encrypted form. Access to the encrypted data is restricted solely to the legitimate recipient who possesses the appropriate decryption key. It serves a vital function in protecting digital communications ...
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