Analyzing the impact of threshold strategies on cancer treatment using non-smooth models

Surgical immunotherapy combined treatment is considered a promising cancer treatment approach, but determining the optimal timing and intensity of treatment remains a significant challenge. To address this issue, this paper proposes and investigates two non-smooth tumor-immune models with dynamic threshold strategies. We first establish a state-dependent impulsive model to describe a single immunotherapy strategy. By analyzing the properties of the Poincaré map, we studied the existence and stability of order-k periodic solutions, proved the existence of chaotic behavior using the maximum Lyapunov exponent, and examined the effects of treatment intervals and the number of treatments under different initial conditions on treatment outcomes. Secondly, we establish and analyze a Filippov tumor-immune model with dual dynamic thresholds. We examine the existence of sliding modes and pseudo-equilibria, prove the existence and stability of three order-k periodic solutions, and further explore the emergence of chaotic phenomena through numerical simulations. Finally, based on the real data of the lung cancer patient, parameter estimation was performed using the least squares method, and the treatment effects of the patient under four different treatment strategies were predicted. The results show that compared to a single-threshold strategy, the dual-threshold strategy not only more effectively controls tumor cell density but also ensures that effector cells remain at a safe level.