The Stability Analysis For Stochastic Dynamics System Of Tumor-immune Responses To Chemotherapy

The tumor-immune system is of great significance and application to human and biology and has been widely studied.When chemotherapy is used to treat tumor,the system evolves into the tumor-immune responses to chemotherapy system,and whether the tumor is treated depends on its stability.Therefore,it is of great significance and application value to study the stability of tumor-immune responses to chemotherapy system.In this paper,we firstly study the second-order algorithm for stochastic simulation white noise and the method of the unified colored noise approximation of multidimensional stochastic dynamic system,and then use them to study the stability of the stochastic dynamic system of tumor-immune responses to chemotherapy.The main research contents and results are as follows:1.The second-order algorithm for stochastic simulation white noise and the method of the unified colored noise approximation of multidimensional stochastic dynamic system are established.For multidimensional stochastic dynamic system driven by Gaussian white noises,which come from different sources,the second-order algorithm for stochastic simulation white noise is established.For multidimensional stochastic dynamic system driven by correlated Gaussian colored noises,the formula of the unified colored noise approximation is presented to use easily,it includes the classical unified colored noise approximation for one-dimensional stochastic dynamic system.The unified colored noise approximation of multidimensional stochastic dynamic system with Gaussian colored and white noises is extended to the case for the correlated noises and the different self-correlation time of colored noises.The specifically analytical formulas for the common one-dimensional,two-dimensional,and three-dimensional stochastic dynamic systems with correlated Gaussian colored noises are also presented respectively.2.The stability of the tumor-immune responses to chemotherapy system driven by Gaussian white noises is researched.The second-order algorithm for stochastic simulation white noise of multidimensional stochastic dynamic system is used to simulate the maximum Lyapunov exponent of system,then it is found that one of the two equilibrium states is asymptotically stable at weak noises and another equilibrium state is always unstable regardless of the noises intensity.The orbit of system is simulated by the secondorder algorithm for stochastic simulation white noise of multidimensional stochastic dynamic system to confirm the conclusions obtained by the maximum Lyapunov exponent.3.The stability of the tumor-immune responses to chemotherapy system driven by Gaussian colored noises is researched.By means of the unified colored noise approximation for multidimensional stochastic dynamic system,the system is simplified into a system with Gaussian white noises.Then the analytical formula of the maximum Lyapunov exponent of system is derived by Arnold method.The maximum Lyapunov exponent is put forward to study and determine the enhanced stability phenomenon of dynamic system.The results show that,both colored noises have different effects on the stability of system,which may transform between asymptotically stable and unstable as the intensity or self-correlation time of each colored noise varies,among the system will occur the noise-enhanced stability phenomenon,which is determined by the maximum Lyapunov exponent.4.The stability of the delayed tumor-immune responses to chemotherapy system driven by Gaussian colored noises is researched.By means of small delay expansion and the extended unified colored noise approximation for multidimensional stochastic dynamic system,the system is simplified into a system driven by Gaussian white noises.Then the analytical formula of the maximum Lyapunov exponent of system is derived by Arnold method.It is found that delay and Gaussian colored noises jointly regulate the complex stability of system: anyone variation in delay as well as intensity and self-correlation time of each colored noise may change to make the stability transform between asymptotically stable and unstable,thereinto,the system will occur the phenomena of noise-enhanced stability and delay-enhanced stability,which is determined by the maximum Lyapunov exponent.The second-order algorithm for stochastic simulation white noise and the method of the unified colored noise approximation for multidimensional stochastic dynamic system established by this paper provides theoretical support to simulate and simplify multidimensional stochastic dynamic system.The maximum Lyapunov exponent is put forward to determine the enhanced stability phenomenon of dynamic system,including the phenomena of noise-enhanced stability and delay-enhanced stability.The research results on the stability of the tumor-immune responses to chemotherapy system provide some theoretical basis for chemotherapy method to treat tumor.