Delayed immune responses can push a tumor-immune model into oscillations

A new mathematical study examines what happens when the immune system’s response to a tumor is treated as delayed, using a fractional-order model of tumor-immune interactions. The work, published in Advances in Continuous and Discrete Models, analyzes a system involving three components: immune effector cells, tumor cells, and interleukin-2, a signaling molecule involved in immune communication.

The model’s key innovation is a time delay that represents the delayed activation and proliferation of effector cells induced by immune stimulation. In biological terms, this reflects the fact that immune cells do not respond instantaneously to signals; their activation and expansion take time. The authors, led by Mengyan Liu, with co-authors including Xiaozhou Feng, Changtong Li, Fei Xie and Xiaofang Guo, published the peer-reviewed article on 9 October 2026.

The study focuses on stability and Hopf bifurcation analysis. For the commensurate-order case of the fractional-order system, the researchers first establish the existence of equilibrium points, the mathematical states at which the populations of tumor cells, effector cells, and interleukin-2 would remain constant over time.

Next, the team analyzes the local stability of the positive equilibrium, the equilibrium in which all three populations are present. This is done by deriving the corresponding characteristic equation, a standard mathematical tool for determining whether small perturbations around an equilibrium decay back to it or grow away from it.

The central analytical move is to treat the time delay as a bifurcation parameter, a quantity whose variation can qualitatively change the system’s behavior. The authors derive sufficient conditions for stability switching, meaning values of the delay at which the positive equilibrium loses its stability, and for the occurrence of Hopf bifurcation at a critical delay value. A Hopf bifurcation marks the point at which a stable steady state gives way to sustained oscillations.

The theoretical results are accompanied by numerical simulations, which serve two purposes: verifying the analytical findings and illustrating how the fractional order and the time delay affect the dynamic behavior of the system.

The simulations support the paper’s main qualitative conclusion: increasing the delay can destabilize the positive equilibrium and lead to sustained oscillations in the modeled tumor-immune dynamics. In other words, the longer the modeled lag in effector cell activation and proliferation, the more likely the system is to shift from a steady state into persistent cycles. The fractional order, meanwhile, affects the critical bifurcation threshold, meaning it changes the delay value at which this stability switch occurs.

The authors frame these findings as theoretical insight into the dynamic mechanisms of tumor-immune interactions, offering a qualitative reference for further mathematical modeling of immunotherapy-related processes. It is important to note the scope of the work: this is a mathematical analysis of a model, not an experimental or clinical study. The results concern the behavior of the equations themselves, and no patient data, experimental measurements, or clinical outcomes are reported in the abstract. Any connection to actual immunotherapy would require further modeling and empirical validation.

Within that scope, the study adds to a growing body of work on delayed tumor-immune models, including prior analyses of fractional-order tumor-immune systems with time delays and of cancer models with delays in tumor-immune interaction and stimulation processes. The contribution here lies in combining the fractional-order framework with a delay-driven stability-switching analysis for a three-component system, and in explicitly quantifying how both the delay and the fractional order shape the onset of oscillations.

Subject of Research: Stability and Hopf bifurcation analysis of a delayed fractional-order tumor-immune interaction model

Article Title: Stability and bifurcation analysis of a delayed fractional-order tumor-immune model

Article References: Liu, M., Feng, X., Li, C., Xie, F., & Guo, X. (2026). Stability and bifurcation analysis of a delayed fractional-order tumor-immune model. Advances in Continuous and Discrete Models. https://doi.org/10.1186/s13662-026-04129-5

Image Credits: AI Generated

DOI: 10.1186/s13662-026-04129-5

Keywords: fractional-order model, tumor-immune dynamics, time delay, Hopf bifurcation, stability analysis, interleukin-2, effector cells, dynamical systems, mathematical modeling, immunotherapy modeling, Stability, bifurcation