Backward bifurcation and global stability in an epidemic model with treatment and vaccination
Xiaomei Feng Zhidong Teng Kai Wang Fengqin Zhang
Discrete & Continuous Dynamical Systems - B 2014, 19(4): 999-1025 doi: 10.3934/dcdsb.2014.19.999
In this paper, we consider a class of epidemic models described by five nonlinear ordinary differential equations. The population is divided into susceptible, vaccinated, exposed, infectious, and recovered subclasses. One main feature of this kind of models is that treatment and vaccination are introduced to control and prevent infectious diseases. The existence and local stability of the endemic equilibria are studied. The occurrence of backward bifurcation is established by using center manifold theory. Moveover, global dynamics are studied by applying the geometric approach. We would like to mention that in the case of bistability, global results are difficult to obtain since there is no compact absorbing set. It is the first time that higher (greater than or equal to four) dimensional systems are discussed. We give sufficient conditions in terms of the system parameters by extending the method in Arino et al. [2]. Numerical simulations are also provided to support our theoretical results. By carrying out sensitivity analysis of the basic reproduction number in terms of some parameters, some effective measures to control infectious diseases are analyzed.
keywords: compound matrices. global stability SVEIR epidemic model backward bifurcation
Dynamical behaviors of an Echinococcosis epidemic model with distributed delays
Kai Wang Zhidong Teng Xueliang Zhang
Mathematical Biosciences & Engineering 2017, 14(5&6): 1425-1445 doi: 10.3934/mbe.2017074

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.

keywords: Echinococcosis distributed delays transmission dynamics basic reproduction number global stability
A new parallel splitting descent method for structured variational inequalities
Kai Wang Lingling Xu Deren Han
Journal of Industrial & Management Optimization 2014, 10(2): 461-476 doi: 10.3934/jimo.2014.10.461
In this paper, we propose a new parallel splitting descent method for solving a class of variational inequalities with separable structure. The new method can be applied to solve convex optimization problems in which the objective function is separable with three operators and the constraint is linear. In the framework of the new algorithm, we adopt a new descent strategy by combining two descent directions and resolve the descent direction which is different from the methods in He (Comput. Optim. Appl., 2009, 42: 195-212.) and Wang et al. (submitted to J. Optimiz. Theory App.). Theoretically, we establish the global convergence of the new method under mild assumptions. In addition, we apply the new method to solve problems in management science and traffic equilibrium problem. Numerical results indicate that the new method is efficient and reliable.
keywords: variational inequalities Alternating direction method parallel computing augmented Lagrangian method separable structure.
Optimal nonlinearity control of Schrödinger equation
Kai Wang Dun Zhao Binhua Feng
Evolution Equations & Control Theory 2018, 7(2): 317-334 doi: 10.3934/eect.2018016

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.

keywords: Nonlinear Schrödinger equation nonlinearity control objective functional Fréchet-differentiability optimal condition
A comparison of deterministic and stochastic predator-prey models with disease in the predator
Hongxiao Hu Liguang Xu Kai Wang
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-27 doi: 10.3934/dcdsb.2018289

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.

keywords: Predator-prey species uniform persistence species invasion continuous-time Markov chain Kolmogorov-Smirnov test Shapiro-Wilk normality test

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