Mild Extra Conditions Research Articles

Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

This paper studies the spread dynamics of a stochastic SIRS epidemic model with nonlinear incidence and varying population size, which is formulated as a piecewise deterministic Markov process. A threshold dynamic determined by the basic reproduction number R0 is established: the disease can be eradicated almost surely if R0<1, while the disease persists almost surely if R0>1. The existing method for analyzing ergodic behavior of population systems has been generalized. The modified method weakens the required conditions and has no limitations for both the number of environmental regimes and the dimension of the considered system. When R0>1, the existence of a stationary probability measure is obtained. Furthermore, with the modified method, the global attractivity of the Ω-limit set of the system and the convergence in total variation to the stationary measure are both demonstrated under a mild extra condition.

A stochastic SIS epidemic model with vaccination

In this paper, we investigate the basic features of an SIS type infectious disease model with varying population size and vaccinations in presence of environment noise. By applying the Markov semigroup theory, we propose a stochastic reproduction number R0s which can be seen as a threshold parameter to utilize in identifying the stochastic extinction and persistence: If R0s<1, under some mild extra conditions, there exists a disease-free absorbing set for the stochastic epidemic model, which implies that disease dies out with probability one; while if R0s>1, under some mild extra conditions, the SDE model has an endemic stationary distribution which results in the stochastic persistence of the infectious disease. The most interesting finding is that large environmental noise can suppress the outbreak of the disease.

Aggregation functions with given super-additive and sub-additive transformations

Aggregation functions and their transformations have found numerous applications in various kinds of systems as well as in economics and social science. Every aggregation function is known to be bounded above and below by its super-additive and sub-additive transformations. We are interested in the “inverse” problem of whether or not every pair consisting of a super-additive function dominating a sub-additive function comes from some aggregation function in the above sense. Our main results provide a negative answer under mild extra conditions on the super- and sub-additive pair. We also show that our results are, in a sense, best possible.

On the existence of aggregation functions with given super-additive and sub-additive transformations

In this note we study restrictions on the recently introduced super-additive and sub-additive transformations, A↦A⁎ and A↦A⁎, of an aggregation function A. We prove that if A⁎ has a slightly stronger property of being strictly directionally convex, then A=A⁎ and A⁎ is linear; dually, if A⁎ is strictly directionally concave, then A=A⁎ and A⁎ is linear. This implies, for example, the existence of pairs of functions f≤g sub-additive and super-additive on [0,∞[n, respectively, with zero value at the origin and satisfying relatively mild extra conditions, for which there exists no aggregation function A on [0,∞[n such that A⁎=f and A⁎=g.

Break-even concentration and periodic behavior of a stochastic chemostat model with seasonal fluctuation

This paper formulates a single-species stochastic chemostat model with periodic coefficients due to seasonal fluctuation. When the noise is small, a modified break-even concentration is identified, whose value below or above the averaged concentration of the input nutrient can completely determine whether the microorganism will persist or not, where an accuracy decay rate is given for extinction. In case of persistence, existence of the random positive periodic solution is proved for the considered model. Further, the random periodic solution is shown to be globally attractive under some mild extra condition. The periodic dynamics obtained in this paper are supported by computer simulations.

Stochastic extinction and persistence of a parasite–host epidemiological model

Abstract In this paper, we investigate the stochastic extinction and persistence of a parasite–host epidemiological model. We show that the global dynamics of the stochastic model can be governed by the basic reproduction number R 0 S : if R 0 S 1 , under mild extra conditions, the disease goes to extinction with probability one and the disease-free dynamics occurs; while R 0 S > 1 , under mild extra conditions, the disease persists and endemic dynamics occurs almost surely, the solutions of the stochastic model fluctuate around the steady state of the deterministic model, and a unique stationary distribution can be found. Based on realistic parameters of Daphnia-microparasite system, numerical simulations have been performed to verify/extend our analytical results. Epidemiologically, we find that: (1) Large environment fluctuations can suppress the outbreak of disease; (2) The distributions are governed by R 0 S ; (3) The noise perturbations can be beneficial to control the spread of disease on average.

A stochastic SIRS epidemic model with infectious force under intervention strategies

In this paper, we extend a classical SIRS epidemic model with the infectious forces under intervention strategies from a deterministic framework to a stochastic differential equation (SDE) one through introducing random fluctuations. The value of our study lies in two aspects. Mathematically, by using the Markov semigroups theory, we prove that the reproduction number R0S can be used to govern the stochastic dynamics of SDE model. If R0S<1, under mild extra conditions, the SDE system has a disease-free absorbing set which means the extinction of disease with probability one. If R0S>1, under mild extra conditions, it has an endemic stationary distribution which leads to the stochastical persistence of the disease. Epidemiologically, we find that random fluctuations can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics.

Structure and K-theory of crossed products by proper actions

We study the C*-algebra crossed product C 0 ( X ) ⋊ G of a locally compact group G acting properly on a locally compact Hausdorff space X . Under some mild extra conditions, which are automatic if G is discrete or a Lie group, we describe in detail, and in terms of the action, the primitive ideal space of such crossed products as a topological space, in particular, with respect to its fibring over the quotient space G ∖ X . We also give some results on the K -theory of such C*-algebras. These more or less compute the K -theory in the case of isolated orbits with non-trivial (finite) stabilizers. We also give a purely K -theoretic proof of a result due to Paul Baum and Alain Connes on K -theory with complex coefficients of crossed products by finite groups.

Posets having continuous intervals

In this paper we consider posets in which each order interval [ a, b] is a continuous poset or continuous domain. After developing some basic theory for such posets, we derive our major result: if X is a core compact space and L is a poset equipped with the Scott topology (assumed to satisfy a mild extra condition) for which each interval is a continuous sup-semilattice, then the function space of continuous locally bounded functions from X into L has intervals that are continuous sup-semilattices. This substantially generalizes known results for continuous domains.

Pullback and Pushout Constructions in C*-Algebra Theory

A systematic study of pullback and pushout diagrams is conducted in order to understand restricted direct sums and amalgamated free products of C*-algebras. Particular emphasis is given to the relations with tensor products (both with the minimal and the maximal C*-tensor norm). Thus it is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and stable under crossed products with a fixed group. General tensor products between diagrams are also investigated. The relations between the theory of extensions and pullback and pushout diagrams are explored in some detail. The crowning result is that if three short exact sequences of C*-algebras are given, with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals, algebras and quotients, then the three new C*-algebras will again form a short exact sequence under some mild extra conditions. As a generalization of a theorem of T. A. Loring it is shown that each morphism between a pair of C*-algebras, combined with its extension to the stabilized algebras, gives rise to a pushout diagram. This result has applications to corona extendibility and conditional projectivity. Finally the pullback and pushout constructions are applied to the class of noncommutative CW complexes defined by (S. Eilers, T. A Loring, and G. K. Pedersen J. Reine Angew. Math., 1998, 499, 101–143) to show that this category is stable under tensor products and under restricted direct sums.

A class of “central limit theorems” for convolution products of generalized functions

Introduction and summary. From the naive viewpoint of a nonprobabilist, the remarkable fact about the central limit theorem is that (under suitable conditions) the average of a large number of random variables converges (in a suitable sense) to a limit which is independenit of the things being averaged, but is completely determined by the averaging process itself. In the present paper, this situation is abstracted from its connection with probability. As we shall see, the restriction to positive functionals which characterizes probability is in no way necessary for validity of the limit theorems. Moreover, we are able to dispense completely with measure theory, in both definitions and proofs. In a purely algebraic and function-space context, we find a whole sequence of averaging processes, each with its own limit function; the first two of them generalize the weak law of large numbers and the central limit theorem. We prove convergence in three different senses, under three different sets of assumptions: first convergence of moments, then uniform convergence of characteristic functions, and finally weak convergence of distribution functions. Our first theorem is purely combinatorial; our next, essentially a calculus theorem; in our third we use function-space techniques borrowed from the modern theory of partial differential equations. By doing so, we obtain rather refined and precise results on the strongest norm in which we can assert convergence. In this respect our results may be of interest even in the classical probabilistic cases. In particular (see Corollary 2 below) we show that if a very mild extra regularity condition is imposed on the summands in the classical central limit theorem, then not only do the partial sums converge in law to the Gaussian, but every derivative of the c.d.f. of the partial sum exists and converges to the corresponding derivative of the Gaussian. Theorem 3 possesses an interpretation in terms of electrical engineering, which is given in the Remark following Corollary 2. It also has a surprising application to numerical analysis, which is described in the Postscript. We begin with an abstract algebra A over the real numbers, equipped with a linear functional Y. (If a E A were a random variable, Y(a`) would be its kth