The Research On A Filippov Tumor-immune System Describing The Immunotherapy Effects

Researches show that the immune system has the function of recognizing and eliminating tumor cells,based on which,cellular immunotherapy can enhance anti-tumor activity of the immune system and control tumor cells.Although im-munotherapy has made a lot of progress,the most effective immunotherapy strategy with the least interference to the host immune is still under exploration.In this work,Filippov models describing the interaction between tumor and immune cells are formulated based on threshold policy to study the effectiveness of immunother-apy in enhancing the immune system and eliminating tumor cells.Meanwhile,numerical simulations are carried out to verify the theoretical resultsFirstly,we established a Filippov model with one threshold.The Filippov-type model considers that no immunotherapy will be taken if the number of tumor cells is less than a tumor threshold xc;while once the number of tumor cells exceeds xc,immunotherapy will be taken.The global dynamical behaviour of the Filip-pov model,including the existence and stability of the disease-free equilibrium.the endemic equilibrium and the existence of the sliding mode and its dynamic-s,is investigated by defining the basic reproductive number R0 and constructing the appropriate Dulac function.The numerical simulations show that the model solutions ultimately converge to the equilibrium of the subsystem or to the pseudo-equilibrium of the sliding system,as we vary the threshold value xc.Secondly,a Filippov model with two thresholds is established to control the tumor cells and the immune cells:if the number of immune cells exceeds thresh-old yc,no immunotherapy measures will be taken;further,if the number of the immune cells is less than yc and the number of the tumor cells is lower than xc,the immunotherapy is taken to enhance the activity of the immune system and not to interfere too much with the host immune system;meanwhile,if the number of the immune cells is less than yc and the number of the tumor cells exceeds xc,we take a higher level of immunotherapy effectively to enhance the activity of the host immune system and eliminate the tumor cells.It is shown that under differ-ent threshold strategies,solutions of the model ultimately converge to the endemic equilibrium of the subsystems or to the pseudo-equilibrium or pseudo-attractor of the sliding system.The results of the numerical simulation further indicate that proper combinations of the thresholds yc and xc can effectively control tumor cells and make better therapeutic effect.