Investigation Of The Tumor-immune System Interactions With A Discrete Time Delay

As is well known,cancer is the important cause of human death,so the pathogenesis and evolution of mechanism have attracted widely attention.In the cancer research,tumor-immune system interactions are an important part.However,the complexity of life phenomenon reflects the dynamical research of biological systems which it is necessary to consider the influence of time delay.This paper mainly studies the tumor and immune system with time delay.The purpose of this study is to reflect the interactions between tumor cells and effector cells,which can provide reference for understand the mechanisms of tumor growth,predict later behaviors of tumor growth and design better treatment strategies.In this paper,we study the interactions and effects of tumor-immune system through qualitative analysis and numerical simulations.The main results of this paper have been shown as below:Firstly,based on the Kuznetsov model,we consider the immune system needs time to identify tumor cells and develop a suitable response.So we introduce a discrete time delay in the interaction between tumor cells and effector cells.The results can reflect the state of tumor dormancy and the regular periodic oscillations behaviors of tumor growth by using the discrete time delay as a bifurcation parameter.The time delay is very significant in the tumor-immune system.Secondly,due to the proliferation of tumor cells need a period of growth time,from the early development of the growth into mature cells.We consider a discrete time delay in the tumor growth to investigate the tumor-immune system.The results reveal that the system changes from equilibrium phase to escape phase by varying the value of time delay.It corresponds to the cancer immunoediting.Thirdly,the helper T cells have two effects in the tumor-immune system.One is the helper role of helper T cells to effector cells,the other is the cytotoxic role of helper T cells to tumor cells.We mainly study the two effects in the tumor-immune system to present a competition model.For the competition model,we analysis the activation rate of effector cells by helper T cells and the activation rate of helper T cells by tumor cells.Numerical simulations of the theoretical model show that the tumor cells can be eliminated in some conditions with increasing the activation rate.At the same time,when the activation rate has a determined value,it can model the relapse behaviors for tumor with increasing the activation rate.The research results will help to deepen clinical understanding of the tumor-immune system,at the same time,it can provide theoretical basis for the understand the cancer immunoediting and provide references for the analysis and treatment of tumor.