A robust study of the dynamics of tumor–immune interaction for public health via fractional framework

It is evident that a tumor is a dangerous lump of tissue developed through the uncontrollable division of cells, replacing healthy tissue with abnormal tissue. It is cancerous and spreads through the lymphatic system or blood within the body of a host individual while the human immune system, consisting of interrelated special cells, tissues, and organs, is employed for the protection of the body from microorganisms, foreign diseases, infections, and toxins. Thus, the conceptualization and understanding of the intersections of tumor–immune cells are valuable. In this article, the natural process of tumor–immune cell interactions is formulated through a mathematical framework. The intricate dynamics of tumor–immune interactions are then represented by means of operators of fractional calculus involving nonlocal and nonsingular kernels. The definitions and basic properties of non-integer derivatives are introduced for the investigation of the proposed system. In addition, a new numerical scheme is introduced for the dynamics, showing the chaos and oscillation of the tumor–immune system. The proposed dynamics of tumor–immune interaction are highlighted with the effect of different input factors. Our findings not only contribute to a thorough comprehension of the complex interactions between input parameters and tumor dynamics, but critical factors that have a major impact on the dynamics are also identified. These outcomes are pivotal for refining and optimizing the proposed system to enhance its predictive accuracy and efficacy in modeling tumor behavior.