In this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system’s parameters. Some numerical simulations are presented to verify the obtained mathematical results.

Cancer is the second leading cause of death worldwide and occurs when cells uncontrollably divide, grow, and invade surrounding tissue. Several treatment strategies have been introduced to increase survival rates; however, each approach has potential side effects. Consequently, demand for alternative treatment approaches, such as immune system enhancement with vitamins, is growing. Here, we introduce a mathematical framework based on ordinary differential equations to examine the interactions among nutrients, immune cells, and cancer cells. Unlike many existing tumor–immune models, the proposed framework reveals that variations in nutrient levels can induce qualitative changes in system dynamics, including bistability and regime transitions between tumor-dominant and immune-controlled states. Long-term behavior, stability, and bifurcation techniques are employed. Numerical simulations are performed using the odeint solver in SciPy, which is based on the LSODA algorithm and is capable of handling both stiff and non-stiff systems. Our results suggest that optimizing nutrient levels can enhance the immune response, potentially leading to safer cancer treatment strategies.

In the present study, the effects of randomness on the model of the immune and cancer cells is discussed. First, we uncover the existence of a bistable response in this interaction where portion of population are cancerous cells. We then discuss how the individual variation within these cells can affect this bistable response. The dominance duration of the tumour and immune cells are highlighted. We calculated the mean of the elimination time of the tumour cells and how can be affected by some key parameters. The analysis reveals that the switching from a stable state to another has a memoryless property.

In this article, we numerically investigate the effects of noise and heterogeneity on a model of immune–tumor cell interactions. We focus on stochastic dynamics and simulation-based analysis of the time required for tumor elimination. We identify the existence of a bistable response, which is disrupted by the introduction of intrinsic noise into the system. In particular, we characterize noise-induced transitions using first-passage time statistics and waiting-time distributions. We discuss various scenarios of tumor elimination, including the impact of vitamin intake and chemotherapy on tumor cell count, mean elimination time, and the duration of tumor dominance. Our results show that increasing chemotherapy reduces the maximum tumor count and decreases the average tumor elimination time, while intrinsic noise promotes memoryless switching toward the tumor-free state. This behavior is explained by the emergence of a quasi-stationary distribution governing the metastable tumor-present regime, leading to exponentially distributed extinction times. Furthermore, this framework enables the decay rate $\lambda$ to be estimated from simulation data and related to treatment parameters $({\beta }_1,\gamma )$. These findings provide a theoretical and statistical justification for memoryless tumor elimination dynamics and offer quantitative insights into stochastic treatment outcomes.

Turing’s reaction–diffusion framework provides a mathematical basis for diffusion-driven pattern formation in interacting systems. An early biological application of the Turing principles is the Gierer and Meinhardt (GM) model, which was developed to describe hydra head regeneration involving two species: an activator and inhibitor. However, biological networks are far beyond simplicity because they contain numerous interconnected components. In this work, we incorporate a coherent feed-forward loop (FFL) into the GM model by assuming that the activation of the inhibitor is mediated through this motif to better capture the complexity of biological systems. Our results indicate that the FFL reshapes the underlying Turing region in two distinct ways. This not only reduces the Turing instability region but also induces spatial patterns, even when the GM model is stable. This reveals that the network topology, rather than the parameters alone, can reshape the onset and structure of the diffusion-driven instability. Our findings suggest that considering the network topology is crucial for unraveling the behavior of real systems.