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.
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. . 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.
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.
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
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.
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.
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.
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
d$x=v(t)$d$t+\sigma(x)$d$w$ and prove
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.
A variational problem
Minimize $\quad \int_a^bL(x,u(x),u'(x),u''(x))dx,\quad
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.
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.