Dynamical Behaviors Of Stochastic HIV Model And Stochastic Tumor Model With Immunotherapy

In this paper,the dynamical behaviors of a stochastic HIV model and a stochastic tumor model with immunotherapy are investigated.Firstly,we propose a stochastic HIV infection model with immunotherapy.By means of stochastic Lyapunov analysis,we prove the existence and uniqueness of the global positive solution for the stochastic HIV model,and also derive the sufficient conditions for the stochas-tically ultimate boundedness and stochastic permanence of the system,respectively.Moreover,by constructing auxiliary processes,stochastic comparison theorem and strong ergodicity,we find that HIV-infected CD4~+T cells and free HIV will become extinct,while healthy CD4~+T cells and CTL effector cells are persistent when the perturbation of HIV-infected CD4~+T cells and free HIV are strong.Secondly,a stochastic tumor model with the angiogenic switch and adoptive cellular im-munotherapy is established.The model describes the interactions between host cells,effec-tor immune cells,tumor cells and endothelial cells.By constructing appropriate Lyapunov functions,the existence and uniqueness of the global positive solution,stochastically ultimate boundedness and stochastic permanence of the stochastic tumor model are discussed,respec-tively.In addition,applying the stochastic comparison theorem and strong ergodicity,we also derive sufficient conditions for the extinction and persistence of the disease,respectively.Finally,for the above two models,the theoretical results are verified by numerical sim-ulations.Furthermore,we also compare the effect of with immunotherapy and without im-munotherapy on the disease.It is found that immunotherapy is more effective in improving the disease.