Two Stochastic Models For Breast Cancer With Pulsed Comprehensive Therapy

In this paper,we study two tumor-immune models for breast cancer with stochastic perturbations and pulsed comprehensive therapy.One is an immunotherapy model for breast cancer which is affected by white noises and pulsed chemotherapy,and the other is a breast cancer model in which chemotherapy drugs are added and which is subjected to white noises and pulsed immunotherapy.The above two models reflect the combination of immunotherapy and chemotherapy and the effects of environmental random disturbances.Firstly,we consider an immunotherapy model for breast cancer with both white noises and pulsed chemotherapy.By using stochastic Lyapunov analysis,we get the existence,uniqueness and the stochastic ultimate boundedness of the global positive solution for the model.And we further obtain the sufficient conditions of the extinction for tumor cells and the persistence of all three kinds of cells through the strong law of large numbers.Next,what we discuss is a model for breast cancer with chemotherapy drugs and it is influenced by both white noises and pulsed immunotherapy.Likewise,by constructing suitable Lyapunov functions,we prove the existence and uniqueness of the global positive solution for the model.And we derive the sufficient conditions for the stochastic ultimate boundedness of the solution and the stochastic permanence of the system.Similarly,we find that when parameters meet some certain criteria,the disease will tend to die out.Finally,according to the numerical simulations and examples,we verify the correctness of our theoretical analysis for the above two breast cancer models.