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In this paper, a competition model between tumor cells and immune system with random switching environment is proposed and investigated. The stochastic switching environment is assumed to be telegraph noise expressed as multi-state Markov chain. The theoretical threshold for extinction and persistence of tumor cells are obtained. Besides, stochastic simulations are performed to substantiate and interpret the conclusions. It is found that stochastic switching environment plays an important part in the process of tumor evolution. The simple model proposed in our paper is conducive to understand the theoretical mechanism between tumor cells and immune system under random environment. [ABSTRACT FROM AUTHOR]

This paper mainly studies the phenomenon of inverse stochastic resonance induced by non-Gaussian noise in a representative Hodgkin–Huxley model. Firstly, spiking manners of neurons under different input currents are investigated and different current thresholds and bifurcation diagram are given. Besides, the limit cycle with different input current are plotted. Then, taking the average firing rate as a measurement, the result shows that a minimum exists in the curve corresponding to the intensity, correlation time and deviation coefficient of non-Gaussian noise, which is named inverse stochastic resonance(ISR). More important, it is found that the mean firing rate will remain unchanged with noise intensity and deviation coefficient when correlation time is sufficiently large and finally trends toward a constant value which is completely determined by input current. Further, the present results will be benefit to investigate the functional roles of randomness in neural spiking. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the phenomenon that stochastic fluctuation induced the competition between tumor extinction and recurrence in the model of tumor growth derived from the catalytic Michaelis–Menten reaction. We analyze the probability transitions between the extinction state and the state of the stable tumor by the Mean First Extinction Time (MFET) and Mean First Return Time (MFRT). It is found that the positional fluctuations hinder the transition, but the environmental fluctuations, to a certain level, facilitate the tumor extinction. The observed behavior could be used as prior information for the treatment of cancer. [Copyright &y& Elsevier]

Abstract: We investigate a stochastic model of tumor growth derived from the catalytic Michaelis–Menten reaction with positional and environmental fluctuations under subthreshold periodic treatment. Firstly, the influences of environmental fluctuations on the treatable stage are analyzed numerically. Applying the standard theory of stochastic resonance derived from the two-state approach, we derive the signal-to-noise ratio (SNR) analytically, which is used to measure the stochastic resonance phenomenon. It is found that the weak environmental fluctuations could induce the extinction of tumor cells in the subthreshold periodic treatment. The positional stability is better in favor of the treatment of the tumor cells. Besides, the appropriate and feasible treatment intensity and the treatment cycle should be highlighted considered in the treatment of tumor cells. [ABSTRACT FROM AUTHOR]

This paper mainly investigates the dynamic behavior of virus infection model with delay effect influenced by external noise. First, a mathematical model describing the evolution of virus under the influences of stochastic noise and delay effect is developed. And necessary conditions for extinction and weak persistence are derived by Ito's formula. A new basic reproduction number R ˜ 0 for virus infection model with delay effect to determine whether extinction or persistence is given. It is found that the delay parameter τ affects the new basic reproduction number R ˜ 0 , and the behavior of extinction and survival of virus. Besides, the threshold for extinction of stochastic model is smaller than the one of deterministic model. In the end, stochastic simulations are applied to illustrate and test the theoretical conclusions. • The effect of delay and stochastic noise on virus infection model is investigated. • Sufficient conditions for extinction and weak persistence of virus are derived. • A new basic reproduction number for stochastic virus infection model is given. • The theoretical results we obtain are verified by stochastic simulation. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, a class of stochastic time-delayed systems driven by correlated Gaussian white noises are considered. Novikov’s theorem is used to derive delay Fokker–Planck equations. In the case of small time delays, approximate stationary solutions are obtained. As a special case, the delay Fokker–Planck equation and the approximate stationary probability density function is obtained for a bistable system. Numerical simulations show that the small-delay approximation is a good approximation in the case of small delay. Furthermore, the effects of the correlated noises and the feedback are investigated. The critical curve separating the unimodal and bimodal regions of the stationary probability distribution is shown to be affected by (the degree of the correlation of the noises) and (the strength of the feedback). Both and can change the curve of the stationary probability density function from a bimodal to a unimodal structure. [Copyright &y& Elsevier]