Time-delay induced oscillations in tumor-immune dynamics in physics laboratory: Theory and electronic experiment

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Abstract
We present a comprehensive study of a tumor-immune interaction model with delayed immune activation, combining analytical, numerical, and experimental approaches. A central feature of our formulation is the use of a generalized Hill function for immune activation, where the exponent m introduces tunable cooperativity. This generalization extends beyond conventional Michaelis–Menten or fixed-saturation forms and captures a wider range of nonlinear immune responses. On the analytical side, we derive explicit conditions for transcritical and Hopf bifurcations, clarifying the roles of key biological parameters and the immune response delay in shaping tumor dynamics. Numerically, we investigate the time-series, phase-plane plots, and both single- and two-parameter bifurcation scenarios with respect to the delay and other system parameters that confirmed the observed transitions are in excellent agreement with the analytical predictions. Most importantly, we implement the delayed tumor-immune model in an analog electronic circuit by reformulating the Hill activation function in terms of hyperbolic tangents, enabling direct laboratory exploration of the system. Experimental investigations reveal steady-state and oscillatory behaviors, dependent on time delay, that closely match the numerical simulations, despite the unavoidable real-world effects such as parameter mismatch, noise, and fluctuations. To the best of our knowledge, this is the first realization of a time-delayed tumor-immune model in hardware in the physics laboratory. This provides a novel bridge between mathematical theory, numerical analysis, and experimental validation and opens new directions for probing tumor dormancy, immune oscillations, and relapse under controlled physical conditions.

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