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In this paper, a novel spreading dynamical model for *Echinococcosis* with distributed time delays is proposed. For the model, we firstly give the basic reproduction number $\mathcal{R}_0$ and the existence of a unique endemic equilibrium when $\mathcal{R}_0>1$. Furthermore, we analyze the dynamical behaviors of the model. The results show that the dynamical properties of the model is completely determined by $\mathcal{R}_0$. That is, if $\mathcal{R}_0<1$, the disease-free equilibrium is globally asymptotically stable, and if $\mathcal{R}_0>1$, the model is permanent and the endemic equilibrium is globally asymptotically stable. According to human *Echinococcosis* cases from January 2004 to December 2011 in Xinjiang, China, we estimate the parameters of the model and study the transmission trend of the disease in Xinjiang, China. The model provides an approximate estimate of the basic reproduction number $\mathcal{R}_0=1.23$ in Xinjiang, China. From theoretic results, we further find that *Echinococcosis* is endemic in Xinjiang, China. Finally, we perform some sensitivity analysis of several model parameters and give some useful measures on controlling the transmission of *Echinococcosis*.

We study the optimal nonlinearity control problem for the nonlinear Schrödinger equation $iu_{t} = -\triangle u+V(x)u+h(t)|u|^α u$, which is originated from the Fechbach resonance management in Bose-Einstein condensates and the nonlinearity management in nonlinear optics. Based on the global well-posedness of the equation for $0<α<\frac{4}{N}$, we show the existence of the optimal control. The Fréchet differentiability of the objective functional is proved, and the first order optimality system for $N≤ 3$ is presented.

In this paper, we study the dynamics of deterministic and stochastic models for a predator-prey, where the predator species is subject to an SIS form of parasitic infection. The deterministic model is a system of ordinary differential equations for a predator-prey model with disease in the predator only. The existence and local stability of the boundary equilibria and the uniform persistence for the ODE model are investigated. Based on these results, some threshold values for successful invasion of disease or prey species are obtained. A new stochastic model is derived in the form of continuous-time Markov chains. Branching process theory is applied to the continuous-time Markov chain models to estimate the probabilities for disease outbreak or prey species invasion. The deterministic and stochastic threshold theories are compared and some relationships between the deterministic and stochastic thresholds are derived. Finally, some numerical simulations are introduced to illustrate the main results and to highlight some of the differences between the deterministic and stochastic models.

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