Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate

In this paper, an SEIR epidemic disease model with time delay and nonlinear incidence rate is studied, and the dynamical behavior of the model under pulse vaccination is analyzed. Using the discrete dynamical system determined by the stroboscopic map, we show that there exists an infection-free periodic solution. Further, we show that the infection-free periodic solution is globally attractive when the period of impulsive effect is less than some critical value. Using a new modelling method, we obtain a sufficient condition for the permanence of the epidemic model with pulse vaccination. We show that time delay, pulse vaccination can bring different effects on the dynamic behavior of the model by numerical analysis. Our results also show the time delay is “profitless”. The main feature of this paper is to introduce time delay and impulse into the SEIR epidemic model and to give pulse vaccination strategies.     

Dynamics of fisheries with prey reserve and harvesting

A model for two fish species and one predator in a patchy environment is formulated using a deterministic model to study the dynamics of fishery in two homogeneous patches, a free fishing zone and a refuge for prey reserve in which fishing is prohibited. The system is analysed around steady states; the criteria for local and global stabilities are established. The existence of bionomic equilibrium of the system is determined and the conditions for their existence are derived. The optimal harvesting policy is studied by using Pontryagin's maximal principle. Sensitivity analysis is carried out and it is observed that the populations are more sensitive to growth, dispersal and predation rates, least sensitive to the catchability coefficient. Statistical analysis is employed to estimate the parameters and to assess both the uncertainty in the model parameters and in the model-based predictions. Graphical representations of the model are provided.     

Continuous and impulsive harvesting strategies in a stage-structured predator–prey model with time delay

In this paper, the continuous and impulsive harvesting policies are considered in a predator–prey system with stage-structure. In the case in which a continuous harvesting is used, it is shown that the mature predator becomes extinct under appropriate conditions. In the case in which an impulsive harvesting is used, using the discrete dynamical system determined by the stroboscopic map, we obtain the mature predator-eradication periodic solution of the system which is globally attractive. The conditions of permanence are established by the method of comparison involving multiple Liapunov functions and auxiliary function. These results indicate that: a short period of pulse harvest is a sufficient condition for the eradication of the population; the impulsive harvest policy is more effective than the continuous one from eradicating predator point of view. Our results offer a more economical and safe strategy in controlling pest in contrast with biological control and chemical control. Furthermore, we give a summary of the dynamic behavior when the impulsive period takes values in different intervals. Finally, numerical results show that the impulsive system we considered has more complex dynamics including quasi-periodic oscillation and chaos.     

Algorithmic solution of Abel’s equation

The solution scheme for Abel’s equation proposed in this article avoids to a large extent thead hoc methods that have been discovered in the last two centuries since Abel introduced the equation named after him. On the one hand, it describes an algorithmic method for obtaining almost all closed form solutions known in the literature. It is based on Lie’s symmetry analysis. Secondly, for equations without a symmetry, a new method is proposed that allows to generate solutions of all equations within an equivalence class if a single representative has been solved before. It is based on functional decomposition of the absolute invariant of the equation at hand for which computer algebra algorithms have become available recently.     

Solving Third Order Differential Equations with Maximal Symmetry Group

The largest group of Lie symmetries that a third-order ordinary differential equation (ode) may allow has seven parameters. Equations sharing this property belong to a single equivalence class with a canonical representative v ^′′′(u)=0. Due to this simple canonical form, any equation belonging to this equivalence class may be identified in terms of certain constraints for its coefficients. Furthermore a set of equations for the transformation functions to canonical form may be set up for which large classes of solutions may be determined algorithmically. Based on these steps a solution algorithm is described for any equation with this symmetry type which resembles a similar scheme for second order equations with projective symmetry group. Received March 9, 2000; revised June 8, 2000     

Equivalence Classes,Symmetries and Solutions of Linear Third-order Differential Equations

 The subject of this article are third-order differential equations that may be linearized by a variable change. To this end, at first the equivalence classes of linear equations are completely described. Thereafter it is shown how they combine into symmetry classes that are determined by the various symmetry types. An algorithm is presented allowing it to transform linearizable equations by hyperexponential transformations into linear form from which solutions may be obtained more easily. Several examples are worked out in detail. Received February 18, 2002; revised May 10, 2002 Published online: October 24, 2002     

Numerical simulation of multi-species biofilms in porous media for different kinetics

There are bacteria that can form strong biofilms in porous media. These biofilms can be used as biobarriers to restrict the flow of pollutants. For certain contaminants, a second species of bacteria that can actually react with the contaminants can be added to the biobarrier to actually degrade the pollutants. We propose some mathematical models for the formation of these reacting biobarriers under different hypotheses, and numerically solve the resulting equations for the flow, transport and reactions. Qualitative comparisons with some experimental results are also given.     

Quasi-Interpolants with Tension Properties from and in CAGD

We present C² quasi-interpolating schemes with tension properties. The B-splines like functions used in the quasi-interpolanting schemes are parametric cubic curves and their shape can be easily controlled via tension parameters which have an immediate geometric interpretation. Applications to the problem of approximation of curves with shape-constraints are discussed.     

On the Efficient Evaluation of Coalescence Integrals in Population Balance Models

The solution of population balance equations is a function f(t,r,x) describing a population density of particles of the property x at time t and space r. For instance, the additional independent variable x may denote the particle size. The describing partial differential equation contains additional sink and source terms involving integral operators. Since the coordinate x adds at least one further dimension to the spatial directions and time coordinate, an efficient numerical treatment of the integral terms is crucial. One of the more involved integral terms appearing in population balance models is the coalescence integral, which is of the form ∫ ₀ ^x κ(x–y, y) f(y) f(x–y)dy. In this paper, we describe an evaluation method of this integral which needs only operations, where n is the number of degrees of freedom with respect to the variable x. This cost can also be obtained in the case of a grid geometrically refined towards x=0.     

On the ideal convergence of sequences of fuzzy numbers

In this paper we introduce and study the concepts of I-convergence, I^∗-convergence and I-Cauchy sequence for sequences of fuzzy numbers where I denotes the ideal of subsets of N, the set of positive integers.     

On the Quadratic Convergence of the Levenberg-Marquardt Method without Nonsingularity Assumption

Recently, Yamashita and Fukushima [11] established an interesting quadratic convergence result for the Levenberg-Marquardt method without the nonsingularity assumption. This paper extends the result of Yamashita and Fukushima by using _k=||F(x_k)||, where [1,2], instead of _k=||F(x_k)||² as the Levenberg-Marquardt parameter. If ||F(x)|| provides a local error bound for the system of nonlinear equations F(x)=0, it is shown that the sequence {x_k} generated by the new method converges to a solution quadratically, which is stronger than dist(x_k,X^*)0 given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.     

Application of the Polynomial Chaos Expansion to the simulation of chemical reactors with uncertainties

In this paper we consider the simulation of probabilistic chemical reactions in isothermal and adiabatic conditions. Models for reactions under isothermal conditions result in advection equations, adiabatic conditions yield the reactive Euler equations. In order to treat with scattering data, the equations are projected onto the polynomial chaos space. Scattering data can largely affect the estimation of quantities in the system, including variable optimization. This is demonstrated on a selective non-catalytic reduction of nitric oxide.     

具有时滞和饱和接触率的SIRS模型的Hopf分支

应用微分方程分支理论,研究了具有时滞和饱和接触率的SIRS模型,以时滞[τ]为分支参数,运用Hopf分支理论,得到当时滞[τ]充分小时正平衡点是局部渐近稳定的,当[τ]经过一系列临界值时模型出现Hopf分支。用Matlab软件进行数值仿真验证了结论的正确性。     

A new fundamental solution for differential Riccati equations arising in control

The matrix differential Riccati equation (DRE) is ubiquitous in control and systems theory. The presence of the quadratic term implies that a simple linear-systems fundamental solution does not exist. Of course it is well-known that the Bernoulli substitution may be applied to obtain a linear system of doubled size. Here, however, tools from max-plus analysis and semiconvex duality are brought to bear on the DRE. We consider the DRE as a finite-dimensional solution to a deterministic linear/quadratic control problem. Taking the semiconvex dual of the associated semigroup, one obtains the solution operator as a max-plus integral operator with quadratic kernel. The kernel is equivalently represented as a matrix. Using the semigroup property of the dual operator, one obtains a matrix operation whereby the kernel matrix propagates as a semigroup. The propagation forward is through some simple matrix operations. This time-indexed family of matrices forms a new fundamental solution for the DRE. Solution for any initial condition is obtained by a few matrix operations on the fundamental solution and the initial condition. In analogy with standard-algebra linear systems, the fundamental solution can be viewed as an exponential form over a certain idempotent semiring. This fundamental solution has a particularly nice control interpretation, and might lead to improved DRE solution speeds.     

Experimental design for parameter estimation of two time-scale model of photosynthesis and photoinhibition in microalgae

A method for parameter estimation of two time-scale model of photosynthesis and photoinhibition is presented.     

On the Crossing Number of Complete Graphs

Let (G) denote the rectilinear crossing number of a graph G. We determine (K₁₁)=102 and (K₁₂)=153. Despite the remarkable hunt for crossing numbers of the complete graph K_n – initiated by R. Guy in the 1960s – these quantities have been unknown forn>10 to date. Our solution mainly relies on a tailor-made method for enumerating all inequivalent sets of points (order types) of size 11. Based on these findings, we establish a new upper bound on (K_n) for general n. The bound stems from a novel construction of drawings of K_n with few crossings.