Multiplicative noise is found to divide the growth law of tumors into two parts in a logistic model, which is driven by additive and multiplicative noises simul- taneously. The Fokker-Planck equation was also derived to explain the fact that the influence of the intensity of multi- plicative noise on the growth of tumor cells has a stochastic resonance-like characteristic. An appropriate intensity of multiplicative noise is benefit to the growth of the tumor cells. The correlation between two sorts of noises weakens the sto- chastic resonance-like characteristic. Homologous noises promote the growth of the tumor cells.
Considering the growth of tumor cells modeled by an enzyme dynamic process under an immune surveillance, we studied in anti-tumor immunotherapy the single-variable growth dynamics of tumor cells subject to a multiplicative noise and an external therapy intervention simultaneously. The law of tumor growth of the above anti-tumor immunotherapy model was revealed through numerical simula- tions to the relevant stochastic dynamic differential equation. Two simulative parameters of therapy, i.e., therapy intensity and therapy duty-cycle, were introduced to characterize a treatment process similar to a tumor clinic therapy. There exists a critical therapy boundary which, in an expo- nent-decaying form, divides the parameter region of therapy into an invalid and a valid treatment zone, respectively. A greater critical therapy duty-cycle is necessary to achieve a valid treatment for a lower therapy intensity while the critical therapy intensity decreases accordingly with an enhancing immunity. A primary clinic observation of the patients with the typical non-hodgekin’s lymphoma was carried out, and there appears a basic agreement between clinic observations and dynamic simulations.
We report the nonequilibrium dynamical phase transition (NDPT) appearing in a kinetic Ising spin system (ISS) subject to the joint application of a deterministic ex-ternal field and the stochastic mutually correlated noises simultaneously. A time-dependent Ginzburg-Landau sto-chastic differential equation, including an oscillating modu-lation and the correlated multiplicative and additive white noises, was addressed and the numerical solution to the rele-vant Fokker-Planck equation was presented on the basis of an average-period approach of driven field. The correlated white noises and the deterministic modulation induce a kind of dynamic symmetry-breaking order, analogous to the sto-chastic resonance in trend, in the kinetic ISS, and the reen-trant transition has been observed between the dynamic disorder and order phases when the intensities of multiplicative and additive noises were changing. The dependencies of a dynamic order parameter Q upon the intensities of additive noise A and multiplicative noise M, the correlation λ between two noises, and the amplitude of applied external field h were investigated quantitatively and visualized vividly. Here a brief discussion is given to outline the underlying mechanism of the NDPT in a kinetic ISS driven by an external force and correlated noises.
