A Tumor-Immune System with Switching Threshold Depending on Tumor Cells and Its Change Rate

Tumor prevention and control, taking into account the load of tumor cells or immune cells, are often described by Filippov systems, usually with a smooth function on the right-hand side describing the tumor immunity. However, these systems only consider the load of tumor cells or immune cells as a threshold function, which suggest that patients will only be treated if the load of tumor cells or immune cells exceeds a certain threshold. As a result, there may be an excessive growth of tumor cells or a rapid decrease in immune cells, which can lead to undesirable results. Considering this problem, we propose a weighted switching system using the load of tumor cells and its change rate as the threshold. Theoretical analysis and numerical simulation of the system are performed by changing the threshold value. We find that in the proposed nonlinear weighted threshold control method for the system, on the one hand, the system can produce rich dynamical behaviors, including many possible cases of bistability and sliding segments of the system, etc., and on the other hand, it is also very important for the prevention and control of tumor.