dynamics of a stochastic avian human influenza epidemic model with mutation abstract in this paper an avian human influenza epidemic model perturbed by environmental white noise is investigated this model allows the transmission of avian influenza and its mutant our stochastic avian influenza model has no equilibria because of perturbation of environmental noise but similar dynamics as those in the deterministic model can be obtained for the bird sub system critical value between extinction and persistence of the infected bird population is obtained for the full stochastic avian human system different thresholds are obtained persistence and global qualitative behaviors are discussed it is shown using has minskii theory and stochastic lyapunov function that an ergodic stationary distribution exists finally numerical simulations are given to support the theoretical results

generalized thermostatistics based on deformed exponential and logarithmic functions abstract the equipartition theorem states that inverse temperature equals the log derivative of the density of states this relation can be generalized by introducing a proportionality factor involving an increasing positive function x it is shown that this assumption leads to an equilibrium distribution of the boltzmann gibbs form with the exponential function replaced by a deformed exponential function in this way one obtains a formalism of generalized thermostatistics introduced previously by the author it is shown that tsallis thermostatistics with a slight modification is the most obvious example of this formalism and corresponds with the choice x x q

mathematical structures derived from the q multinomial coefficient in tsallis statistics abstract we present q stirling s formula q multinomial coefficient one to one correspondence between q multinominal coefficient and tsallis entropy q pascal s triangle and a conjecture on the q central limit theorem in tsallis statistics for the generalization of the well known fundamental formulas to systems exhibiting power law behaviors the main approach is based on the q product uniquely determined by tsallis entropy which has already been successfully applied to our recent proof of the law of error in tsallis statistics

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

parameter estimation in non extensive thermostatistics abstract equilibrium statistical physics is considered from the point of view of statistical estimation theory this involves the notions of statistical model of estimators and of exponential family a useful property of the latter is the existence of identities obtained by taking derivatives of the logarithm of the partition sum it is shown that these identities still exist for models belonging to generalised exponential families in which case they involve escort probability distributions the percolation model serves as an example a previously known identity is derived it relates the average number of sites belonging to the finite cluster at the origin the average number of perimeter sites and the derivative of the order parameter

periodic solutions and stationary distribution of mutualism models in random environments abstract this paper is concerned with mutualism models in random environments for the periodic mutualism model disturbed by white noise using has minskii theory of periodic solution we show that this model admits a nontrivial positive periodic solution then sufficient conditions for the existence and global attractivity of the boundary periodic solutions are established for the mutualism model disturbed by both white noise and telephone noise sufficient conditions for positive recurrence and the existence of ergodic stationary distribution of the solution are established finally examples are introduced to illustrate the results developed

deformed exponentials and logarithms in generalized thermostatistics abstract criteria are given that deformed logarithmic and exponential functions should satisfy with a pair of such functions one can associate another function called the deduced logarithmic function it is shown that generalized thermostatistics can be formulated in terms of deformed exponential functions together with the associated deduced logarithmic functions

group entropies correlation laws and zeta functions the notion of group entropy is proposed it enables the unification and generaliztion of many different definitions of entropy known in the literature such as those of boltzmann gibbs tsallis abe and kaniadakis other entropic functionals are introduced related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic the associated thermostatistics are discussed the mathematical structure underlying our construction is that of formal group theory which provides the general structure of the correlations among particles and dictates the associated entropic functionals as an example of application the role of group entropies in information theory is illustrated and generalizations of the kullback leibler divergence are proposed a new connection between statistical mechanics and zeta functions is established in particular tsallis entropy is related to the classical riemann zeta function

two parameter entropies s k r and their dualities this paper investigates the dualities of two parameter entropies sk r kaniadakis et al 2005 two kinds of dualities are found namely inverse r r and inverse scaling r rs k ks which recovers the additive q 2 q and multiplicative q 1 q dualities in the tsallis limit

generalized statistics framework for rate distortion theory abstract variational principles for the rate distortion rd theory in lossy compression are formulated within the ambit of the generalized nonextensive statistics of tsallis for values of the nonextensivity parameter satisfying 0 q 1 and q 1 alternating minimization numerical schemes to evaluate the nonextensive rd function are derived numerical simulations demonstrate the efficacy of generalized statistics rd models

multifractal spectrum of phase space related to generalized thermostatistics abstract we consider a self similar phase space with specific fractal dimension d being distributed with spectrum function f d related thermostatistics is shown to be governed by the tsallis formalism of the non extensive statistics where the non additivity parameter equals to q 1 q 1 and the multifractal function q q d q f d q is the specific heat determined with multifractal parameter q 1 at that the equipartition law is shown to take place optimization of the multifractal spectrum function f d arrives at the relation between the statistical weight and the system complexity it is shown that the statistical weight exponent q can be modeled by hyperbolic tangent deformed in accordance with both tsallis and kaniadakis exponential functions to describe arbitrary multifractal phase space explicitly the spectrum function f d is proved to increase monotonically from minimum value f 1 at d 0 to maximum one f 1 at d 1 at the same time the number of monofractals increases with the growth of the phase space volume at small dimensions d and falls down in the limit d 1

law of error in tsallis statistics in order to theoretically explain the ubiquitous existence of power law behavior such as chaos and fractals in nature tsallis entropy has been successfully applied to the generalization of the traditional boltzmann gibbs statistics the fundamental information measure of which is shannon entropy tsallis entropy s sub q is a one parameter generalization of shannon entropy s sub 1 in the sense that lim sub q spl rarr 1 s sub q s sub 1 the generalized statistics using tsallis entropy are referred to as tsallis statistics in order to present the law of error in tsallis statistics as a generalization of gauss law of error and prove it mathematically we apply the new multiplication operation determined by q logarithm and q exponential the fundamental functions in tsallis statistics to the definition of the likelihood function in gauss law of error the present maximum likelihood principle mlp leads us to determine the so called q gaussian distribution which coincides with one of the tsallis distributions derived from the maximum entropy principle for tsallis entropy under the second moment constraint

on q non extensive statistics with non tsallisian entropy we combine an axiomatics of renyi with the qq deformed version of khinchin axioms to obtain a measure of information i e entropy which accounts both for systems with embedded self similarity and non extensivity we show that the entropy thus obtained is uniquely solved in terms of a one parameter family of information measures the ensuing maximal entropy distribution is phrased in terms of a special function known as the lambert w function we analyze the corresponding high and low temperature asymptotics and reveal a non trivial structure of the parameter space salient issues such as concavity and schur concavity of the new entropy are also discussed

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

the extinction and persistence of the stochastic sis epidemic model with vaccination abstract in this paper we discuss the dynamics of a stochastic sis epidemic model with vaccination when the noise is large the infective decays exponentially to zero regardless of the magnitude of r 0 when the noise is small sufficient conditions for extinction exponentially and persistence in the mean are established the results are illustrated by computer simulations

a stochastic differential equation sis epidemic model in this paper we extend the classical susceptible infected susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation sde for the number of infectious individuals i t we then prove that this sde has a unique global positive solution i t and establish conditions for extinction and persistence of i t we discuss perturbation by stochastic noise in the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance the results are illustrated by computer simulations including two examples based on real life diseases

multiplicative duality q triplet and μ ν q relation derived from the one to one correspondence between the μ ν multinomial coefficient and tsallis entropy sq abstract we derive the multiplicative duality q 1 q and other typical mathematical structures as the special cases of the q relation behind tsallis statistics by means of the multinomial coefficient recently the additive duality q 2 q in tsallis statistics is derived in the form of the one to one correspondence between the q multinomial coefficient and tsallis entropy a slight generalization of this correspondence for the multiplicative duality requires the multinomial coefficient as a generalization of the q multinomial coefficient this combinatorial formalism provides us with the one to one correspondence between the multinomial coefficient and tsallis entropy s q which determines a concrete relation among three parameters and q i e 1 1 q which is called q relation in this paper as special cases of the q relation the additive duality and the multiplicative duality are recovered when 1 and q respectively as other special cases when 2 q a set of three parameters q is identified with the q triplet q sen q rel q stat recently conjectured by tsallis moreover when 1 q the relation 1 1 q sen 1 min 1 max in the multifractal singularity spectrum f is recovered by means of the q relation

dynamical behavior of a stochastic svir epidemic model with vaccination abstract in this paper we investigate the dynamical behavior of svir models in random environments firstly we show that if r 0 s 1 the disease of stochastic autonomous svir model will die out exponentially if r 0 s 1 the disease will be prevail moreover this system admits a unique stationary distribution and it is ergodic when r 0 s 1 results show that environmental white noise is helpful for disease control secondly we give sufficient conditions for the existence of nontrivial periodic solutions to stochastic svir model with periodic parameters finally numerical simulations validate the analytical results

the threshold of a stochastic sis epidemic model with vaccination this paper is concerned with the long time behavior of a stochastic sis epidemic model with vaccination sufficient conditions for extinction and persistence in mean are obtained we find a threshold of the stochastic model which determines the outcome of the disease in case the white noises are small at last some numerical simulations are carried out to support our results

the threshold dynamics in a stochastic sis epidemic model with vaccination and nonlinear incidence under regime switching abstract in this paper a stochastic sis epidemic model with vaccination and nonlinear incidence under regime switching is investigated this model concerned about white noise and color noise a new technique to deal with the nonlinear incidence and vaccination for the stochastic epidemic models is introduced the sufficient conditions on the extinction and the permanence in the mean with probability one are established by constructing new lyapunov functions with regime switching

heavy tailed targets and ab normal asymptotics in diffusive motion abstract we show that under suitable confinement conditions the ordinary fokker planck equation may generate non gaussian heavy tailed probability density functions pdfs like for example cauchy or more general levy stable distributions in its long time asymptotics in fact all heavy tailed pdfs known in the literature can be obtained this way for the underlying diffusion type processes our main focus is on their transient regimes and specifically the crossover features when an initially infinite number of pdf moments decreases to a few or none at all the time dependence of the variance if in existence t with 0 2 may in principle be interpreted as a signature of subdiffusive normal diffusive or superdiffusive behavior under confining conditions the exponent is generically well defined in substantial periods of time however there is no indication of any universal time rate hierarchy due to a proper choice of the driver and or external potential

a threshold of a delayed stochastic epidemic model with crowly martin functional response and vaccination abstract in this paper we study a delayed stochastic sir epidemic model with crowly martin functional response and vaccination first we prove the existence and the uniqueness of the positive solution therefore we establish a stochastic threshold r s as a sufficient condition for the extinction and persistence in mean of the stochastic epidemic system finally numerical simulations are presented to support our theoretical results

brief paper dynamics of a multigroup sir epidemic model with stochastic perturbation in this paper we introduce stochasticity into a multigroup sir susceptible infective and recovered model the stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modeling in the deterministic models the basic reproduction number r 0 is a threshold which completely determines the persistence or extinction of the disease we carry out a detailed analysis on the asymptotic behavior of the stochastic model also regarding of the value of r 0 if r email protected 1 the solution of the model is oscillating around a steady state which is the disease free equilibrium of the corresponding deterministic model whereas if r 0 1 there is a stationary distribution which means that the disease will prevail

thermodynamic equilibrium and its stability for microcanonical systems described by the sharma taneja mittal entropy it is generally assumed that the thermodynamic stability of equilibrium states is reflected by the concavity of entropy we inquire in the microcanonical picture about the validity of this statement for systems described by the two parametric entropy s kappa r of sharma taneja and mittal we analyze the composability rule for two statistically independent systems a and b described by the entropy s kappa r with the same set of the deformation parameters it is shown that in spite of the concavity of the entropy the composability rule modifies the thermodynamic stability conditions of the equilibrium state depending on the values assumed by the deformation parameters when the relation s kappa r a union b s kappa r a s kappa r b holds superadditive systems the concavity condition does imply thermodynamics stability otherwise when the relation s kappa r a union b s kappa r a s kappa r b holds subadditive systems the concavity condition does not imply thermodynamical stability of the equilibrium state

persistence and extinction for an age structured stochastic svir epidemic model with generalized nonlinear incidence rate abstract we formulate an epidemic model with age of vaccination and generalized nonlinear incidence rate where the total population consists of the susceptible the vaccinated the infected and the removed we then reach a stochastic svir model when the fluctuation is introduced into the transmission rate by using ito s formula and lyapunov methods we first show that the stochastic epidemic model admits a unique global positive solution with the positive initial value we then obtain the sufficient conditions of the stochastic epidemic model moreover the threshold tells the disease spreads or not is derived if the intensity of the white noise is small enough and r 0 1 then the disease eventually becomes extinct with negative exponential rate if r 0 1 then the disease is weakly permanent the persistence in the mean of the infected is also obtained when the indicator r 0 1 which means the disease will prevail in a long run as a consequence several illustrative examples are separately carried out with numerical simulations to support the main results of this paper

fast algorithms for weighted myriad computation by fixed point search this paper develops fast algorithms to compute the output of the weighted myriad filter myriad filters form a large and important class of nonlinear filters for robust non gaussian signal processing and communications in impulsive noise environments just as the weighted mean and the weighted median are optimized for the gaussian and laplacian distributions respectively the weighted myriad is based on the class of spl alpha stable distributions which can accurately model impulsive processes the weighted myriad is an m estimator that is defined in an implicit manner no closed form expression exists for it and its direct computation is a nontrivial and prohibitively expensive task in this paper the weighted myriad is formulated as one of the fixed points of a certain mapping an iterative algorithm is proposed to compute these fixed points and its convergence is proved rigorously using these fixed point iterations fast algorithms are developed for the weighted myriad numerical simulations demonstrate that these algorithms compute the weighted myriad with a high degree of accuracy at a relatively low computational cost

generalized thermostatistics and mean field theory abstract the present paper studies a large class of temperature dependent probability distributions and shows that entropy and energy can be defined in such a way that these probability distributions are the equilibrium states of a generalized thermostatistics which is obtained from the standard formalism by deformation of exponential and logarithmic functions since this procedure is non unique specific choices are motivated by showing that the resulting theory is well behaved in particular with the choices made in the present paper the equilibrium state of any system with a finite number of degrees of freedom is automatically thermodynamically stable and satisfies the variational principle the equilibrium probability distribution of open systems deviates generically from the boltzmann gibbs distribution if the interaction with the environment is not too strong then one can expect that a slight deformation of the exponential function appearing in the boltzmann gibbs distribution can reproduce the observed distribution with its temperature dependence an example of a system where this statement holds is a single spin of the ising chain however because all systems of the present generalized thermostatistics are automatically stable one must not expect that all open systems can be described in this way indeed systems exhibiting a phase transition in their thermodynamic limit can be unstable even when the interaction with the environment is weak therefore their equilibrium probability distribution cannot be described by a simple deformation of the boltzmann gibbs distribution in order to be able to handle such systems as well the second part of the paper discusses a further extension of the class of probability distributions using mean field techniques connections are discussed that exist between the present formalism tsallis thermostatistics and the superstatistics formalism of beck and cohen in particular the present generalization sheds some light onto the historical development of the tsallis formalism it is pointed out that temperature dependence of the tsallis distribution has hardly been verified experimentally

a note on bounded entropies abstract the aim of the paper is to study the link between non additivity of some entropies and their boundedness we propose an axiomatic construction of the entropy relying on the fact that entropy belongs to a group isomorphic to the usual additive group this allows to show that the entropies that are additive with respect to the addition of the group for independent random variables are nonlinear transforms of the renyi entropies including the particular case of the shannon entropy as a particular example we study as a group a bounded interval in which the addition is a generalization of the addition of velocities in special relativity we show that tsallis havrda charvat entropy is included in the family of entropies we define finally a link is made between the approach developed in the paper and the theory of deformed logarithms

qualitative study of a stochastic sis epidemic model with vertical transmission abstract in this paper we study a stochastic sis epidemic model with vertical infection by constructing suitable stochastic lyapunov function we establish sufficient conditions for the existence of a stationary probability measure of the model in addition we also establish sufficient conditions for extinction and persistence of the disease

non linear kinetics underlying generalized statistics abstract the purpose of the present effort is threefold firstly it is shown that there exists a principle that we call kinetical interaction principle kip underlying the non linear kinetics in particle systems independently on the picture kramers boltzmann used to describe their time evolution secondly the kip imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy thirdly the kip allows on one hand to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and on the other hand suggests how we can introduce naturally new distributions finally as a working example of the approach to the non linear kinetics here presented a new non extensive statistics is constructed and studied starting from a one parameter deformation of the exponential function holding the relation f x f x 1

asymptotic properties of hybrid diffusion systems in response to the increasing needs for control and optimization of hybrid systems this work is concerned with such asymptotic properties as recurrence also known as weak stochastic stability in the literature and ergodicity of regime switching diffusions using liapunov functions necessary and sufficient conditions for positive recurrence are developed then ergodicity of positive recurrent regime switching diffusions is obtained by constructing cycles using the associated discrete time markov chains

the ergodicity and extinction of stochastically perturbed sir and seir epidemic models with saturated incidence abstract in this paper we include stochastic perturbations into sir and seir epidemic models with saturated incidence and investigate their dynamics according to the basic reproduction number r 0 the long time behavior of the two stochastic systems is studied mainly we utilize stochastic lyapunov functions to show under some conditions the solution has the ergodic property as r 0 1 while exponential stability as r 0 1 at last we make simulations to conform our analytical results

stationary distribution of a stochastic siqr epidemic model with saturated incidence and degenerate diffusion abstract the objective of this paper is to study stationary distribution of a stochastic siqr epidemic model with saturated incidence and degenerate diffusion since the diffusion matrix is degenerate the uniform ellipticity condition is not satisfied the markov semigroup theory will be used to obtain the existence of a unique stable stationary distribution a threshold dynamic determined by the basic reproduction number r 0 s is established the disease can be eradicated almost surely if r 0 s 1 whereas if r 0 s 1 it has an endemic stationary distribution which leads to the stochastic persistence of the disease

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

legendre structure of the thermostatistics theory based on the sharma taneja mittal entropy abstract the statistical proprieties of complex systems can differ deeply for those of classical systems governed by boltzmann gibbs entropy in particular the probability distribution function observed in several complex systems shows a power law behavior in the tail which disagrees with the standard exponential behavior showed by gibbs distribution recently a two parameter deformed family of entropies previously introduced by sharma taneja and mittal stm has been reconsidered in the statistical mechanics framework any entropy belonging to this family admits a probability distribution function with an asymptotic power law behavior in the present work we investigate the legendre structure of the thermostatistics theory based on this family of entropies we introduce some generalized thermodynamical potentials study their relationships with the entropy and discuss their main proprieties specialization of the results to some one parameter entropies belonging to the stm family are presented

scaled bregman divergences in a tsallis scenario abstract there exist two different versions of the kullback leibler divergence k ld in tsallis statistics namely the usual generalized k ld and the generalized bregman k ld problems have been encountered in trying to reconcile them a condition for consistency between these two generalized k ld forms is derived by recourse to the additive duality of tsallis statistics it is also shown that the usual generalized k ld subjected to this additive duality known as the dual generalized k ld is a scaled bregman divergence this leads to an interesting conclusion the dual generalized mutual information is a scaled bregman information the utility and implications of these results are discussed

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators