Hybrid discrete‐continuum modeling of tumor‐immune interactions: Fractional time and space analysis with immunotherapy

In this paper, we introduce an innovative hybrid discrete‐continuum model that provides a comprehensive framework for understanding the intricate interactions between tumor cells, immune cells, and the effects of immunotherapy. This study distinguishes itself by addressing the limitations of traditional diffusion models, which often fail to capture the irregular and nonlocal movements of cells within the complex tumor microenvironment. To overcome these challenges, we employ fractional time derivatives and the fractional Laplacian operator, offering a more accurate representation of anomalous diffusion processes that are critical in cancer dynamics. Our research begins with a rigorous mathematical analysis, where we establish the global existence of a unique mild solution, laying a solid theoretical foundation for the model. A key innovation of our study is the introduction of the “invasion threshold,” a critical parameter inspired by the next‐generation operator used in epidemiological modeling. This threshold provides a powerful tool for determining the existence and stability of equilibrium points, offering deep insights into the conditions that either promote or inhibit tumor growth under immunotherapeutic interventions. By integrating a hybrid discrete‐continuum approach, we capture the individual behavior of each cell, allowing for a more detailed exploration of the dynamic interplay within the tumor microenvironment. This model not only advances our understanding of tumor‐immune interactions but also holds potential for informing more effective therapeutic strategies, making a significant contribution to the field of mathematical oncology.