Search

Releasing Wolbachia-infected mosquitoes to invade the wild mosquito population is a method of mosquito control. In this paper, a stochastic mosquito population model with Wolbachia invasion perturbed by environmental fluctuation is studied. Firstly, the well-posedness, positivity, and Markov-Feller property of the solution for this model are proved. Then a group of sharp threshold-type conditions is provided to characterize the long-term behavior of the model, which pinpoints the almost necessary and sufficient conditions for the persistence and extinction of Wolbachia-infected and uninfected mosquito populations. Our results indicate that even for a low initial Wolbachia infection frequency, a successful Wolbachia invasion into the wild mosquito population can be driven by stochastic environmental fluctuations. Finally, some numerical experiments are carried out to support our theoretical results. [ABSTRACT FROM AUTHOR]

This paper examines the stationary distribution of the stochastic Lotka-Volterra model with infinite delay. Since the solutions of stochastic functional or delay differential equations depend on their history, they are non-Markov, which implies that traditional techniques based on the Markov property, cannot be applicable. This paper uses the variable substituting technique to obtain a higher-dimensional stochastic differential equation without delay. In this paper it is shown that the new equation satisfies the strong Feller property and strong Markov property, by which the invariant measure follows from the Krylov-Bogoliubov Theorem. Since distribution of the original model is actually the marginal distribution of the new equation, these results also show that there exists the stationary distribution for the original equation. Moreover, when the noise intensity is strongly dependent on the population size, by Hörmander's theorem, this paper also shows that the stationary distribution of this stochastic Lotka-Volterra system with infinite delay holds a C ∞ -smooth density. [ABSTRACT FROM AUTHOR]

In this work, we study the exponential stability of the stationary distribution of a McKean-Vlasov equation, of nonlinear hyperbolic type which was recently derived in [1,2]. We complement the convergence result proved in [2] using tools from dynamical systems theory. Our proof relies on two principal arguments in addition to a Picard-like iteration method. First, the linearized semigroup is positive which allows to precisely pinpoint the spectrum of the infinitesimal generator. Second, we use a time rescaling argument to transform the original quasilinear equation into another one for which the nonlinear flow is differentiable. Interestingly, this convergence result can be interpreted as the existence of a locally exponentially attracting center manifold for a hyperbolic equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the stochastic SIR equation with general incidence functional responses and in which both natural death rates and the incidence rate are perturbed by white noises. We derive a sufficient and almost necessary condition for the extinction and permanence for SIR epidemic system with multi noises { d S (t) = [ a 1 − b 1 S (t) − I (t) f (S (t) , I (t)) ] d t + σ 1 S (t) d B 1 (t) − I (t) g (S (t) , I (t)) d B 3 (t) , d I (t) = [ − b 2 I (t) + I (t) f (S (t) , I (t)) ] d t + σ 2 I (t) d B 2 (t) + I (t) g (S (t) , I (t)) d B 3 (t). Moreover, the rate of all convergences of the solution are also established. A number of numerical examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

The linkage disequilibrium coefficient r 2 is a measure of statistical dependence of the alleles possessed by an individual at different genetic loci. It is widely used in association studies to search for the locations of disease-causing genes on chromosomes. Most studies to date treat r 2 as a fixed property of two loci in a finite population, and investigate the sampling distribution of estimators due to the statistical sampling of individuals from the population. Here, we instead consider the distribution of r 2 itself under a process of genetic sampling through the generations. Using a classical two-locus model for genetic drift, mutation, and recombination, we investigate the probability density function of r 2 at stationarity. This density function provides a tool for inference on evolutionary parameters such as mutation and recombination rates. We reconstruct the approximate stationary density of r 2 by calculating a finite sequence of the distribution's moments and applying the maximum entropy principle. Our approach is based on the diffusion approximation, under which we demonstrate that for certain models in population genetics, moments of the stationary distribution can be obtained without knowing the probability distribution itself. To illustrate our approach, we show how the stationary probability density of r 2 can be used in a maximum likelihood framework to estimate mutation and recombination rates from sample data of r 2. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce two perturbations in the classical deterministic susceptible–infected–susceptible epidemic model. Greenhalgh and Gray [4] in 2011 use a perturbation on β in SIS model. Based on their previous work, we consider another perturbation on the parameter μ + γ and formulate the original model as a stochastic differential equation (SDE) with two independent Brownian Motions for the number of infected population. We then prove that our Model has a unique and bounded global solution I (t). Also we establish conditions for extinction and persistence of the infected population I (t). Under the conditions of persistence, we show that there is a unique stationary distribution and derive its mean and variance. Computer simulations illustrate our results and provide evidence to back up our theory. [ABSTRACT FROM AUTHOR]

By using biclique partitions of digraphs, this paper gives reduction formulas for the number of oriented spanning trees, stationary distribution vector and Kemeny's constant of digraphs. As applications, we give a method for enumerating spanning trees of undirected graphs by vertex degrees and biclique partitions. The biclique partition formula also extends the results of Knuth and Levine from line digraphs to general digraphs. [ABSTRACT FROM AUTHOR]

Signatures of recent historical admixture are ubiquitous in human populations. We present a mechanistic model of admixture with two source populations, encompassing recurrent admixture periods and study the distribution of admixture fractions for finite but arbitrary genome size. We provide simulation-based methods to estimate the introgression parameters and discuss the implications of reaching stationarity on estimability of parameters when there are recurrent admixture events with different rates. [ABSTRACT FROM AUTHOR]

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 R 0 is established: the disease can be eradicated almost surely if R 0 < 1 , while the disease persists almost surely if R 0 > 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 R 0 > 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. [ABSTRACT FROM AUTHOR]

We study a two dimensional version of Neuhauser's long range sexual reproduction model and prove results that give bounds on the critical values λ f for the process to survive from a finite set and λ e for the existence of a nontrivial stationary distribution. Our first result comes from a standard block construction, while the second involves a comparison with the “generic population model” of Bramson and Gray (1991) [3] . An interesting new feature of our work is the suggestion that, as in the one dimensional contact process, edge speeds characterize critical values. We are able to prove the following for our quadratic contact process when the range is large but suspect they are true for two dimensional finite range attractive particle systems that are symmetric with respect to reflection in each axis. There is a speed c ( θ ) for the expansion of the process in each direction. If c ( θ ) > 0 in all directions, then λ > λ f , while if at least one speed is positive, then λ > λ e . It is a challenging open problem to show that if some speed is negative, then the system dies out from any finite set. [ABSTRACT FROM AUTHOR]

Recently, Burden and Tang (2016) provided an analytical expression for the stationary distribution of the multivariate neutral Wright–Fisher model with low mutation rates. In this paper we present a simple, alternative derivation that illustrates the approximation. Our proof is based on the discrete multivariate boundary mutation model which has three key ingredients. First, the decoupled Moran model is used to describe genetic drift. Second, low mutation rates are assumed by limiting mutations to monomorphic states. Third, the mutation rate matrix is separated into a time-reversible part and a flux part, as suggested by Burden and Tang (2016). An application of our result to data from several great apes reveals that the assumption of stationarity may be inadequate or that other evolutionary forces like selection or biased gene conversion are acting. Furthermore we find that the model with a reversible mutation rate matrix provides a reasonably good fit to the data compared to the one with a non-reversible mutation rate matrix. [ABSTRACT FROM AUTHOR]

This work derives sufficient conditions for the coexistence and exclusion of a stochastic competitive Lotka–Volterra model. The conditions obtained are close to necessary. In addition, convergence in distribution of positive solutions of the model is also established. A number of numerical examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

Focusing on stochastic dynamics involving continuous states as well as discrete events, this paper investigates stochastic logistic model with regime switching and obtains sufficient conditions for stochastic permanence, which are much weaker than the existing results in the literature. Furthermore, under the conditions of stochastic permanence, the existence and uniqueness of stationary distribution is proved. An interesting fact is: The regime switching can suppress the non-permanence. A couple of examples and numerical simulations are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, the strictly critical parameter λ ˜ is identified as the break-even concentration for the stochastic chemostat model. λ ˜ > S 0 suffices for extinction of the microorganism for any noise intensity σ 2 > 0 is proved firstly, which completely solves the open problem proposed in [10] . Then, by using the Feller test, we show that the microorganism will go extinct in probability if λ ˜ = S 0 , which has not been studied in the known literature. When λ ˜ < S 0 , in addition to the previously discussed persistence in mean, the stochastic persistence of the microorganism is studied by use of Chebyshev's inequality. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some properties of the doubly skewed CIR process which is an extension of skew Brownian motion. The weak existence and pathwise uniqueness of this process is considered in the beginning. Then we present the explicit formulae for the stationary density function and show that the process is positive Harris recurrent and geometrically ergodic. We also provide the explicit Laplace transforms of the first hitting times by using the basic theory of linear diffusions. As a natural corollary, the expectations of the first hitting times are given in the end. [ABSTRACT FROM AUTHOR]

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 R 0 S can be used to govern the stochastic dynamics of SDE model. If R 0 S < 1 , under mild extra conditions, the SDE system has a disease-free absorbing set which means the extinction of disease with probability one. If R 0 S > 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. [ABSTRACT FROM AUTHOR]

In this paper, we propose a graph based algorithm that efficiently segments common objects from multiple images. We first generate a number of object proposals from each image. Then, an undirected graph is constructed based on proposal similarities and co-saliency maps. Two different methods are followed to extract the proposals containing common objects. They are: (1) degree centrality of nodes obtained after graph thresholding and (2) site entropy rate of nodes calculated on the stationary distribution of Markov chain constructed on the graph. Finally, we obtain the co-segmented image region by selecting the more salient of the object proposals obtained by the two methods, for each image. Multiple instances of the common object are also segmented efficiently. The proposed method has been compared with many existing co-segmentation methods on three standard co-segmentation datasets. Experimental results show its effectiveness in co-segmentation, with larger IoU values as compared to other co-segmentation methods. [ABSTRACT FROM AUTHOR]

The rotor-router model is a popular deterministic analogue of random walk. In this paper we prove that all orbits of the rotor-router operation have the same size on a strongly connected directed graph (digraph) and give a formula for the size. By using this formula we address the following open question about orbits of the rotor-router operation: Is there an infinite family of non-Eulerian strongly connected digraphs such that the rotor-router operation on each digraph has a single orbit? It turns out that on a strongly connected digraph the stationary distribution of the simple random walk coincides with the frequency of vertices in a rotor walk. In this common aspect a rotor walk simulates a random walk. This gives one similarity between two models on (finite) digraphs. [ABSTRACT FROM AUTHOR]

We consider the following random process on the complete graph: repeatedly draw edges (with replacement) and with probability p assign the vertices of the edge blue and with probability 1 − p assign the vertices of the edge red. This is a random walk on a state space of red/blue colorings of the complete graph and so has a stationary distribution. We derive this stationary distribution as well as answer some related questions. [ABSTRACT FROM AUTHOR]

We present a determinantal formula for the steady state probability of each state of the TASEP (Totally Asymmetric Simple Exclusion Process) with open boundaries, a 1D particle model that has been studied extensively and displays rich combinatorial structure. These steady state probabilities are computed by the enumeration of Catalan tableaux, which are certain Young diagrams filled with α 's and β 's that satisfy some conditions on the rows and columns. We construct a bijection from the Catalan tableaux to weighted lattice paths on a Young diagram, and from this we enumerate the paths with a determinantal formula, building upon a formula of Narayana that counts unweighted lattice paths on a Young diagram. Finally, we provide a formula for the enumeration of Catalan tableaux that satisfy a given condition on the rows, which corresponds to the steady state probability that in the TASEP on a lattice with n sites, precisely k of the sites are occupied by particles. This formula is an α / β generalization of the Narayana numbers. [ABSTRACT FROM AUTHOR]

This work focuses on population dynamics of two species described by Kolmogorov systems of competitive type under telegraph noise that is formulated as a continuous-time Markov chain with two states. Our main effort is on establishing the existence of an invariant (or a stationary) probability measure. In addition, the convergence in total variation of the instantaneous measure to the stationary measure is demonstrated under suitable conditions. Moreover, the Ω -limit set of a model in which each species is dominant in a state of the telegraph noise is examined in detail. [ABSTRACT FROM AUTHOR]

Abstract: This work examines almost sure stability of a pure random delay system whose delay time is modeled by a finite state continuous-time Markov chain with two-time scales. The Markov chain contains a fast-varying part and a slowly-changing part. Using the properties of the weighted occupation measure of the Markov chain, it is shown that the overall systemʼs almost-sure-asymptotic stability can be obtained by using the “averaged” delay. This feature implies that even if some longer delay times may destabilize the system individually, the system may still be stable if their impact is balanced. In other words, the Markov chain becomes a stabilizing factor. Numerical results are provided to demonstrate our results. [Copyright &y& Elsevier]

Abstract: This paper discusses a randomized n-species Lotka–Volterra competition system. We show that this system is stable in time average under certain conditions. Furthermore, there is a stationary distribution of this system, if extra conditions are satisfied. Also we give the extinction condition of this system. Finally, numerical simulations are carried out to support our results. [Copyright &y& Elsevier]

Abstract: We compare the deterministic method and the stochastic method for a polymerization network when the number of available subunits is small. For the stochastic method, we prove there is a recursive method to compute the expected molecule numbers of various components in the reaction network, using the stationary probability distribution of molecule numbers which we illustrate to have a multivariate Poisson form. For the deterministic method, ordinary differential equations for the component concentrations are built following the mass action law. The steady state of the system is extracted to estimate the corresponding molecule numbers. Identities involving the propensity function parameters for the stochastic method and the reaction rate constants in the deterministic method are used to connect the two methods. Computations are conducted for a group of combinations of total number of subunits and reaction rate constant ratios, and the results are compared. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss a stochastic density dependent predator–prey system with Beddington–DeAngelis functional response. First, we show that this system has a unique positive solution as this is essential in any population dynamics model. Then, we investigate the asymptotic behavior of this system. When the white noise is small, the stochastic system imitates the corresponding deterministic system. Either there is a stationary distribution, or the predator population will die out. While if the white noise is large, besides the extinction of the predator population, both species in the system may also die out, which does not happen in the deterministic system. Finally, simulations are carried out to conform to our results. [Copyright &y& Elsevier]

Abstract: In this paper, we show there is a stationary distribution of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation and it has ergodic property. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the dynamics of Kolmogorov systems of competitive type under the telegraph noise. The telegraph noise switches at random two Kolmogorov competition-type deterministic models. The aim of this work is to describe the omega-limit set of the system and investigates properties of stationary density. [Copyright &y& Elsevier]

Abstract: The heat shock response (HSR) is a highly evolutionarily conserved defence mechanism allowing the cell to promptly react to elevated temperature conditions and other forms of stress. It has been subject to intense research for at least two main reasons. First, it is considered a promising candidate for deciphering the engineering principles underlying regulatory networks. Second, heat shock proteins (main actors of the HSR) play crucial role in many fundamental cellular processes. Therefore, profound understanding of the heat shock response would have far-reaching ramifications for the cell biology. Recently, a new deterministic model of the eukaryotic heat shock response has been proposed in the literature. It is very attractive since it consists of only the minimum number of components required by any functional regulatory network, while yet being capable of biological validation. However, it admits small molecule populations of some of the considered metabolites. In this paper a stochastic model corresponding to the deterministic one is constructed and the outcomes of these two models are confronted. The aim with this comparison is to show that, in the case of the heat shock response, the approximation of a discrete system with a continuous model is a reasonable approach. This is not always the truth, especially when the numbers of molecules of the considered species are small. By making the effort of performing and analysing 1000 stochastic simulations, we investigate the range of behaviour the stochastic model is likely to exhibit. We demonstrate that the obtained results agree well with the dynamics displayed by the continuous model, which strengthens the trust in the deterministic description. A proof of the existence and uniqueness of the stationary distribution of the Markov chain underlying the stochastic model is given. Moreover, the obtained view of the stochastic dynamics and the performed comparison to the outcome of the continuous formulation provide more insight into the dynamics of the heat shock response mechanism. [Copyright &y& Elsevier]

Although many people infected with Zika virus do not get sick or only have mild symptoms, pregnant women infected with Zika virus may have babies with severe birth defects. To understand the transmission dynamics of Zika virus, we develop deterministic and stochastic models, taking into account mosquito-human transmission, sexual transmission, and the release of Wolbachia -infected male mosquitoes. For the deterministic model, we analyze the stability of the disease-free equilibrium and evaluate the impact of the parameters on the basic reproduction number by sensitivity analysis. We found the upper and lower bounds for the release rate of Wolbachia -infected male mosquitoes to be cost effective. Since the transmission of infectious diseases is affected by many stochastic factors, we incorporate the impact of environmental white noise on death rates of adult mosquitoes, mosquitoes at aquatic stages, and humans to develop a stochastic model for Zika. We proved that the stochastic model has a unique and bounded global solution and derived the sufficient conditions for the extinction of Zika. We also proved the existence of a unique ergodic stationary distribution of the solutions. Numerical simulations show that the stochasticity may drive the disease to extinction when the intensity of the white noise is large enough, and the release of Wolbachia -infected mosquitoes shortens the time it takes for the wild mosquitoes to become extinct. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are given for the existence of limiting distribution of a conservative affine process on the canonical state space R ⩾ 0 m × R n , where m , n ∈ Z ⩾ 0 with m + n > 0. Our main theorem extends and unifies some known results for OU-type processes on R n and one-dimensional CBI processes (with state space R ⩾ 0). To prove our result, we combine analytical and probabilistic techniques; in particular, the stability theory for ODEs plays an important role. [ABSTRACT FROM AUTHOR]