A Study On Extinction And Persistence Of Tumor Cells Induced By Environmental Noise

Tumors seriously endanger the health of human beings.Many scientists in medicine,mathematics,biology,chemistry and so on seek for solutions.Among them,it is of important clinical significance and theoretical value to explore the inherent laws of tumor growth by building a model,especially on the random tumor growth model influenced by environmental fluctuations,provides new ideas and new methods for understanding the inherent laws of tumor growth and tumor treatment.Based on a simplified Kuznetsov-Taylor tumor-immune system competition model,this paper studies the effects of environmental noise on the extinction and persistence of tumor cells.The main contents and conclusions are as follows:1.The extinction and persistence of tumor cells under the influence of Gauss white noise are studied in this paper.Firstly,the Kuznetsov-Taylor model and the Galach simplified model describing the competition between the tumor and immune cells are introduced,then the Gaussian white noise is employed to mimic the environmental fluctuations.Secondly,with the method of Ito’s formula,the sufficient condition for extinction,stochastic persistence and strong persistence in the mean of tumor cells are established by the rigorous mathematical proofs.Finally,numerical simulation confirmed our conclusion.Our work reveals some important and interesting biological results.By comparing the conclusions associated with the deterministic model given by Galach,we can see that environmental noise significantly changes the tumor-immune kinetic properties.For example,under the same conditions,the mean extinction time of the stochastic model is less than that of the deterministic model.That is,under immune surveillance,environmental noise can accelerate the extinction of tumor cells,which means that the noise is beneficial for tumor extinction.2.The stochastic responses of a tumor-immune system competition model with environmental noise and periodic treatment are investigated in this paper.Firstly,the environmental noise is taken into account Gaussian white noise and periodic treatment is regarded as Heaviside function.Secondly,sufficient conditions for extinction and strong persistence in the mean of tumor cells are derived by Ito’s formula.According to Theorems and Figures,it is indicated that extinction and survival of tumor cells rely on the strength of periodic treatment.With the increasing intensity of periodic treatment,the tumor cells will experience the process from strongly persistence in the mean to extinction.Accordingly,our theoretical results will be beneficial to design more effective and feasible treatment therapies.3.A tumor-immune system competition model with random switching environment are studied in this paper.The stochastically switching environment is assumed to be telegraph noise expressed as multi-state Markov chain.The threshold for extinction and persistence of tumor cells are derived by proofs.In addition,we confirm our conclusion with numerical simulation.It is found that the random switching environment has great influence on the occurrence and development of tumors.The conclusion of this paper in two state switching environment can be further extended to multi state switching environment.