Backward bifurcation and global stability in an epidemic model with treatment and vaccination
Xiaomei Feng Zhidong Teng Kai Wang Fengqin Zhang
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
A new parallel splitting descent method for structured variational inequalities
Kai Wang Lingling Xu Deren Han
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.
Fluctuation and extinction dynamics in host-microparasite systems
Kaifa Wang Yang Kuang
Although experimental and observational studies have shown that microparasites can induce the deterministic reduction, fluctuation and extinction scenarios for its host population, most existing host-parasite interaction models fail to produce such rich dynamical behaviors simultaneously. We explore the effects of explicit dynamics of parasites under logistic host growth and different infection rate function. Our results show that the explicit dynamics of parasites and standard incidence function can induce the host density fluctuation and extinction scenario in the case of logistic host growth.
keywords: extinction. reduction stability Host-microparasite model fluctuation
Channel coordination mechanism with retailers having fairness preference ---An improved quantity discount mechanism
Chuan Ding Kaihong Wang Shaoyong Lai
Channel coordination is an optimal state with operation of channel. For achieving channel coordination, we present a quantity discount mechanism based on a fairness preference theory. Game models of the channel discount mechanism are constructed based on the entirely rationality and self-interest. The study shows that as long as the degree of attention (parameters) of retailer to manufacturer's profit and the fairness preference coefficients (parameters) of retailers satisfy certain conditions, channel coordination can be achieved by setting a simple wholesale price and fixed costs. We also discuss the allocation method of channel coordination profit, the allocation method ensure that retailer's profit is equal to the profit of independent decision-making, and manufacturer's profit is raised.
keywords: quantity discounts fairness preference coordination. Channel
The effect of immune responses in viral infections: A mathematical model view
Kaifa Wang Yu Jin Aijun Fan
To study the effect of immune response in viral infections, a new mathematical model is proposed and analyzed. It describes the interactions between susceptible host cells, infected host cells, free virus, lytic and nonlytic immune response. Using the LaSalle's invariance principle, we establish conditions for the global stability of equilibria. Uniform persistence is obtained when there is a unique endemic equilibrium. Mathematical analysis and numerical simulations indicate that the basic reproduction number of the virus and immune response reproductive number are sharp threshold parameters to determine outcomes of infection. Lytic and nonlytic antiviral activities play a significant role in the amount of susceptible host cells and immune cells in the endemic steady state. We also present potential applications of the model in clinical practice by introducing antiviral effects of antiviral drugs.
keywords: Viral infection reproductive number stability immune responses uniform persistence.
Uniform persistence and periodic solution of chemostat-type model with antibiotic
Kaifa Wang Aijun Fan
A system of functional differential equations is used to model the single microorganism in the chemostat environment with a periodic nutrient and antibiotic input. Based on the technique of Razumikhin, we obtain the sufficient condition for uniform persistence of the microbial population. For general periodic functional differential equations, we obtain a sufficient condition for the existence of periodic solution, therefore, the existence of positive periodic solution to the chemostat-type model is verified.
keywords: periodic solution. Antibiotic uniform persistence
Action minimizing stochastic invariant measures for a class of Lagrangian systems
Kaizhi Wang
In this paper we discuss a variational method of constructing an action minimizing stochastic invariant measure for positive definite Lagrangian systems. Then we study some main properties of the stochastic minimal measures. Finally we give the definitions of stochastic Mather's functions with respect to the stochastic differential equation d$x=v(t)$d$t+\sigma(x)$d$w$ and prove their differentiability.
keywords: Lagrangian systems Action minimizing stochastic invariant measures variational method. stochastic Mather's functions
Global dynamics of a delay virus model with recruitment and saturation effects of immune responses
Cuicui Jiang Kaifa Wang Lijuan Song

In this paper, we formulate a virus dynamics model with the recruitment of immune responses, saturation effects and an intracellular time delay. With the help of uniform persistence theory and Lyapunov method, we show that the global stability of the model is totally determined by the basic reproductive number $R_0$. Furthermore, we analyze the effects of the recruitment of immune responses on virus infection by numerical simulation. The results show ignoring the recruitment of immune responses will result in overestimation of the basic reproductive number and the severity of viral infection.

keywords: Global stability Lyapunov functional viral dynamics immune response time delay
Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian
Kaizhi Wang Yong Li
A variational problem

Minimize $\quad \int_a^bL(x,u(x),u'(x),u''(x))dx,\quad u\in\Omega$

with a second-order Lagrangian is considered. In absence of any smoothness or convexity condition on $L$, we present an existence theorem by means of the integro-extremal technique. We also discuss the monotonicity property of the minimizers. An application to the extended Fisher-Kolmogorov model is included.

keywords: Existence result Second-order Lagrangians Nonconvex variational problems Monotonicity property.
Novel dynamics of a simple Daphnia-microparasite model with dose-dependent infection
Kaifa Wang Yang Kuang
Many experiments reveal that Daphnia and its microparasite populations vary strongly in density and typically go through pronounced cycles. To better understand such dynamics, we formulate a simple two dimensional autonomous ordinary differential equation model for Daphnia magna-microparasite infection with dose-dependent infection. This model has a basic parasite production number $R_0=0$, yet its dynamics is much richer than that of the classical mathematical models for host-parasite interactions. In particular, Hopf bifurcation, stable limit cycle, homoclinic and heteroclinic orbit can be produced with suitable parameter values. The model indicates that intermediate levels of parasite virulence or host growth rate generate more complex infection dynamics.
keywords: Hopf bifurcation Daphnia magna-microparasite model limit cycle heteroclinic orbit dose-dependent infection homoclinic orbit

Year of publication

Related Authors

Related Keywords

[Back to Top]