Global stability of a DS–DI epidemic model with age of infection

A model with differential susceptibility, differential infectivity (DS–DI), and age of infection is formulated in this paper. The susceptibles are divided into n groups according to their susceptibilities. The infectives are divided into m groups according to their infectivities. The total population size is assumed constant. Formula for the reproductive number is derived so that if the reproduction number is less than one, the infection-free equilibrium is locally stable, and unstable otherwise. Furthermore, if the reproductive number is less than one, the infection-free equilibrium is globally asymptotically stable. If the reproductive number is greater than one, it is shown that there exists a unique endemic equilibrium which is globally asymptotically stable. This result is obtained through a Lyapunov function.     

Generalized Ambrosetti–Rabinowitz condition for minimal period solutions of autonomous Hamiltonian systems

We show, under an iterative condition, generalizing that of Ambrosetti–Rabinowitz and using a variational method, the existence of a T-periodic solution for the autonomous superquadratic second order Hamiltonian system with even potential

$$\begin{aligned} \ddot{z} +V'(z)=0, \qquad z\in \mathbb R^N, \qquad N\in \mathbb N^* \end{aligned}$$

for any prescribed period \(T>0\). Moreover, under a certain symmetry condition, such a solution possesses T or T / 3 as its minimal period.

     

Equivalence of $n$-point Gauss–Chebyshev rule and $4n$-point midpoint rule in computing the period of a Lotka–Volterra system

Two integrals (3.6), (4.7) for the period of a periodic solution of the Lotka–Volterra system are presented in terms of two inverse functions of restricted on , , respectively. In computing this period numerically, the integral (3.6), which possesses a weak singularity of the square root type at each endpoint of the integration, is an excellent example of using the Gauss–Chebyshev integration rule of the first kind; while the integral (4.7), which is an integral of a smooth periodic function over its period , is an excellent example of using the midpoint rule, but not the trapezoidal rule, suggested by Waldvogel [39, 40], due to a removable singularity of the integrand at , , , , and , respectively. This paper shows, in computing the period of a periodic solution of the Lotka–Volterra system, the -point Gauss–Chebyshev integration rule of the first kind applied to the integral (3.6) becomes the -point midpoint rule to the integral (4.7). Dedicated to R. Bruce Kellogg on the occasion of his 75th birthday.     

Stability and bifurcation analysis of a prey–predator model with age based predation

It is observed that in large animals only adult predators take part in direct predation while suckling feed on milk of adult predators and juveniles are dependent on the dead prey stock killed by the adult predators. Some parts of the dead prey population is consumed by adult predators and remaining parts are consumed by juveniles and the remaining portion decays naturally. In light of this, a mathematical model is proposed to study the stability and bifurcation behaviour of a prey–predator system with age based predation. All the feasible equilibria of the system are obtained and the conditions for the existence of the interior equilibrium are determined. The local stability analysis of all the feasible equilibria is carried out and the possibility of Hopf-bifurcation of the interior equilibrium is studied. Finally, numerical simulation is conducted to support the analytical results.     

A time-periodic reaction–diffusion epidemic model with infection period

In this paper, we propose a time-periodic and diffusive SIR epidemic model with constant infection period. By introducing the basic reproduction number \({\mathcal{R}_0}\) via a next generation operator for this model, we show that the disease goes extinction if \({\mathcal{R}_0 < 1}\) ; while the disease is uniformly persistent if \({\mathcal{R}_0 > 1}\).     

Mathematical analysis of a population model with an age–weight structured two-stage life history: asymptotic behavior of solutions

In this paper, we propose a model for the dynamics of a physiologically structured population of individuals whose life cycle is divided into two stages: the first stage is structured by the weight, while the second one is structured by the age, the exit from the first stage occurring when a threshold weight is attained. The model originates in a complex one dealing with a fish population and covers a large class of situations encompassing two-stage life histories with a different structuring variable for each state, one of its key features being that the maturation process is determined in terms of a weight threshold to be reached by individuals in the first stage. Mathematically, the model is based on the classical Lotka–MacKendrick linear model, which is reduced to a delayed renewal equation including a constant delay that can be viewed as the time spent by individuals in the first stage to reach the weight threshold. The influence of the growth rate and the maturation threshold on the long-term behavior of solutions is analyzed using Laplace transform methods.     

Higher levels of ℤ/plℤ and period equations

We first study the d ^th levels of the finite prime fields ? _p . These are the numbers

On the points of order 3 of the period parallelogram of the Weierstrass elliptic function

We examine the significance of the values of the Weierstrass ℘ function at the points of order three in the period parallelogram. Based upon the well-known duplication formulae and the differential equation of the Weierstrass ℘ function, we derive a set of identities involving the values of ℘ at these points.      

Track–Monitoring of Wheel–Rail–Systems

Inspection and maintenance of railway networks is a complex and expensive task. Special measurement vehicles are used to record the geometrical properties of railway lines within required time intervals. Due to the extent of measurement data the quality of railway track is evaluated considering only a few parameters. Although safety and comfort of wheel–rail–systems depend on the dynamical behavior, current inspection vehicles are not equipped to measure dynamic properties. In this paper, we will discuss a novel approach to evaluate the quality of railway tracks based on wheel–rail dynamics: Wheelset dynamics of subway trains are analyzed by Karhunen–Loève Transformation to extract the principal dynamics from the collected measurement data. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Realisations of Kummer–Chern–Eisenstein-motives

Inspired by work of G. Harder we construct via the motive of a Hilbert modular surface an extension of a Tate motive by a Dirichlet motive. We compute the realisation classes and indicate how this is linked to the Hodge-1-motive of the given Hilbert modular surface.     

Construction of Runge–Kutta methods of Crouch–Grossman type of high order

An approach is described to the numerical solution of order conditions for Runge–Kutta methods whose solutions evolve on a given manifold. This approach is based on least squares minimization using the Levenberg–Marquardt algorithm. Methods of order four and five are constructed and numerical experiments are presented which confirm that the derived methods have the expected order of accuracy.     

Cardinality of the Set of Delta–Closed Classes of Functions of Multi–Valued Logic

A closure under composition operation and weak version of inversion operation is considered on the set of functions of k–valued logic. The cardinality of the set of all such closed classes is calculated.     

On theorems of Delsarte–McEliece and Chevalley–Warning–Ax–Katz

We present a theorem that generalizes the result of Delsarte and McEliece on the p-divisibilities of weights in abelian codes. Our result generalizes the Delsarte–McEliece theorem in the same sense that the theorem of N. M. Katz generalizes the theorem of Ax on the p-divisibilities of cardinalities of affine algebraic sets over finite fields. As the Delsarte–McEliece theorem implies the theorem of Ax, so our generalization implies that of N. M. Katz. The generalized theorem gives the p-divisibility of the t-wise Hamming weights of t-tuples of codewords (c ⁽¹⁾, . . . ,c ^(t)) as these words range over a product of abelian codes, where the t-wise Hamming weight is defined as the number of positions i in which the codewords do not simultaneously vanish, i.e., for which ${(c^{(1)}_i,\ldots,c^{(t)}_i)\not=(0,\ldots,0)}$ . We also present a version of the theorem that, for any list of t symbols s ₁, . . . ,s _t , gives p-adic estimates of the number of positions i such that ${(c^{(1)}_i,\ldots,c^{(t)}_i)=(s_1,\ldots,s_t)}$ as these words range over a product of abelian codes.     

Regularity properties of viscosity solutions of integro-partial differential equations of Hamilton–Jacobi–Bellman type

We study the regularity properties of integro-partial differential equations of Hamilton–Jacobi–Bellman type with the terminal condition, which can be interpreted through a stochastic control system, composed of a forward and a backward stochastic differential equation, both driven by a Brownian motion and a compensated Poisson random measure. More precisely, we prove that, under appropriate assumptions, the viscosity solution of such equations is jointly Lipschitz and jointly semiconcave in (t,x)∈Δ×R^d$(t,x)\in \Delta \times {\mathbb{R}}^d$, for all compact time intervals Δ$\Delta$ excluding the terminal time. Our approach is based on the time change for the Brownian motion and on Kulik’s transformation for the Poisson random measure.     

Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words

We construct irreducible representations of affine Khovanov–Lauda–Rouquier algebras of arbitrary finite type. The irreducible representations arise as simple heads of appropriate induced modules, and thus our construction is similar to that of Bernstein and Zelevinsky for affine Hecke algebras of type A. The highest weights of irreducible modules are given by the so-called good words, and the highest weights of the ‘cuspidal modules’ are given by the good Lyndon words. In a sense, this has been predicted by Leclerc.     

Blow-up of Compressible Navier–Stokes–Korteweg Equations

Based on the results of Xin (Commun. Pure Appl. Math. 51(3):229–240, 1998), Zhang and Tan (Acta Math. Sin. Engl. Ser. 28(3):645–652, 2012), we show the blow-up phenomena of smooth solutions to the non-isothermal compressible Navier–Stokes–Korteweg equations in arbitrary dimensions, under the assumption that the initial density has compact support. Here the coefficients are generalized to a more general case which depends on density and temperature. Our work extends the previous corresponding results.     

Minimal period estimates on P-symmetric periodic solutions of first-order mild superquadratic Hamiltonian systems

With the aid of P-index iteration theory,we consider the minimal period estimates on P-symmetric periodic solutions of nonlinear P-symmetric Hamiltonian systems with mild superquadratic growth.     

Inversion of the Cauchy–Bunyakovskii–Schwarz inequality

In the present paper, the inequality inverse to the Cauchy–Bunyakovskii–Schwarz inequality and generalizing other well-known inversions of this inequality is proved.