A Study Of Extinction And Persistence Of Tumor Cells Based On Stochastic Differential Equation

The treatment for tumor is always a challenge for the medicine,so the research on tumor has attracted more and more attention.There are great important theoretical values and clinical significance for the prevention and treatment of tumor to research the inherent law of tumor evolution and the sufficient conditions for extinction and persistence of tumor cells.Especially,the application of the theory of stochastic differential equation in tumor research gives new ideas and methods to understand the underlying mechanism of tumor evolution and the tumor treatment.The effect of the stochastic fluctuation on the tumor extinction and persistence is investigated,with based on the stochastic differential equation and tumor immune model.The main work and conclusion are as follows:Firstly,threshold problem for extinction and persistence of tumor cells with the abduction of environmental noise is investigated.The extinction and persistence of tumor cells are defined by the concepts of extinction,weak persistence,strong persistence in the mean and stochastic persistence.In addition,making use of the methods of Ito's formula and Lyapunov function,the threshold for extinction,weak persistence and stochastic persistence of tumor cells is carried out by the strict mathematical proofs,instead of the approaches of Fokker-Planck equation and effective potential function.It is found that the strength of immunization plays key role in determining the survival and extinction of cancer cells.With the increase of immunological intensity,the tumor will experience the process from stochastic persistence to weak persistence until extinction.Secondly,we research that the environmental noise induces extinction of tumor cells under immune surveillance.We apply the pure mathematical methods of Ito's integral and Lyapunov function to derive the sufficient conditions for extinction,strong persistence in the mean and stochastic persistence of tumor cells.By numerical simulations,it is indicated that extinction and persistence of tumor cells not only rely on the strength of immunization but also the intensity of noise.When the tumor cells is under the immune surveillance,immune fluctuation could induces and accelerates the extinction of tumor cells and is beneficial in the process of tumor treatment.When the tumor cells are not enough controlled by the immune surveillance,immune fluctuation will enhance the randomness of tumor cells and is not beneficial in tumor therapy.Thirdly,the extinction and persistence of tumor cells influenced by periodic treatment are studied.The threshold conditions of intensity of cyclic treatment that accelerates the extinction and persistence of tumor cells is carried out by the strict mathematical proofs.In addition,it is indicated that extinction and survival of tumor cells not only rely on the strength of periodic treatment but also the intensity of noise.With the increasing intensity of periodic treatment,the tumor cells will experience the process from weak persistence to extinction.When periodic treatment is not enough to eliminate the tumor cells,the external noise could induce and accelerate the extinction of tumor cells and it shows that noise is beneficial in the process of tumor treatment.