the clifford algebra of an infinite dimensional space the clifford algebra of a real infinite dimensional vector space is constructed this algebra contains all finite and infinite dimensional clifford algebras over the real and complex numbers some of the mathematical properties of these algebras are discussed the automorphisms and involutions are specifically addressed the special case with metric g 1 2 follows as a corollary to the more general results

stability and bifurcation for an seis epidemic model with the impact of media abstract a novel seis epidemic model with the impact of media is introduced by analyzing the characteristic equation of equilibrium the basic reproduction number is obtained and the stability of the steady states is proved the occurrence of a forward backward and hopf bifurcation is derived numerical simulations and sensitivity analysis are performed our results manifest that media can regard as a good indicator in controlling the emergence and spread of the epidemic disease

analysis of a predator prey model with modified leslie gower and holling type ii schemes with stochastic perturbation abstract in this paper we consider a predator prey model with modified leslie gower and holling type ii schemes with stochastic perturbation we show there is a unique positive solution to the system with positive initial value and mainly investigate the long time behavior of the system condition for the system to be extinct is given and persistent condition is established at last numerical simulations are carried out to support our results

mean square dissipative methods for stochastic age dependent capital system with fractional brownian motion and jumps abstract in this paper we analyze mean square dissipativity of numerical methods applied to a class of stochastic age dependent vintage capital system with fractional brownian motion fbm and poisson jumps some sufficient conditions are obtained for ensuring the underlying systems are mean square dissipative it is shown that the mean square dissipativity is preserved by the compensated split step backward euler method and compensated backward euler method without any restriction on stepsize while the split step backward euler method and backward euler method could reproduce mean square dissipativity under a stepsize constraint those results indicate that compensated numerical methods achieve superiority over non compensated numerical methods in terms of mean square dissipativity a numerical example is provided to illustrate the theoretical results

threshold behavior in two types of stochastic three strains influenza virus models abstract in this paper we study two types of stochastic models consisting of three strains of influenza for each type of the stochastic model we establish sufficient conditions for extinction and persistence in the mean of these diseases by using the stochastic lyapunov analysis method meanwhile we obtain the threshold between persistence in the mean and extinction of the diseases

exclusion and persistence in deterministic and stochastic chemostat models abstract we first introduce and analyze a variant of the deterministic single substrate chemostat model in this model microbe removal and growth rates depend on biomass concentration with removal terms increasing faster than growth terms using a comparison principle we show that persistence of all species is possible in this scenario then we turn to modelling the influence of random fluctuations by setting up and analyzing a stochastic differential equation in particular we show that random effects may lead to extinction in scenarios where the deterministic model predicts persistence on the other hand we also establish some stochastic persistence results

a stochastic model for internal hiv dynamics in this paper we analyse a stochastic model representing hiv internal virus dynamics the stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling we show that the model established in this paper possesses non negative solutions as this is essential in any population dynamics model we also carry out analysis on the asymptotic behaviour of the model we approximate one of the variables by a mean reverting process and find out the mean and variance of this process numerical simulations conclude the paper

the threshold of a stochastic siqs epidemic model in this paper we discuss the dynamics of a stochastic siqs epidemic model 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

decroissance exponentielle du noyau de la chaleur sur la diagonale i we give examples based upon large deviation s theory where the heat kernel of a degenerate diffusion has an exponential decay over the diagonal using malliavin calculus we give conditions for a more generalized heat kernel to have an exponential decay over the diagonal we give lower bound in some particular case by using the bismut s condition

stationary distribution of a stochastic cholera model between communities linked by migration abstract in this paper we develop and study a stochastic cholera model between communities linked by migration firstly we show that there exists a unique global positive solution of the considered system with any positive initial value then by using the stochastic lyapunov analysis method we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model our result shows that the migration and the small white noise are not beneficial for eliminating cholera finally some examples together with numerical simulations are introduced to support the theoretical result

algebraic spinor solution to the kahler equation the kahler equation for an inhomogeneous differential form is analyzed in some detail algebraic spinorial solution to the kahler equation lying in a minimal left ideal characterized by a certain idempotent projector has been examined

dynamics of an imprecise sirs model with levy jumps abstract sudden environmental perturbations may affect population dynamics parameters of mathematical models can also be imprecise due to uncertainties and unknown data how sudden environmental noise and parameter imprecision influence the dynamics of epidemic systems remains unclear this paper studies a stochastic susceptible infected recovered susceptible sirs model that includes levy jumps and interval parameters we prove that the model has a unique global positive solution sufficient conditions for persistence and extinction of the disease are obtained large noise intensity is able to suppress the emergence of disease outbreaks numerical simulations are carried out to show the influence of stochastic noise on disease dynamics these results suggest that levy jumps and imprecise parameters can greatly affect the long term behavior of the epidemic system

dynamics of a multigroup sis epidemic model with standard incidence rates and markovian switching abstract in this paper we investigate a multigroup sis epidemic model with standard incidence rates and markovian switching first of all we establish sufficient conditions for extinction of the diseases then we obtain sufficient conditions for persistence in the mean of the diseases in addition in the case of persistence we obtain sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic lyapunov function with regime switching

convergence and asymptotic stability of the explicit steklov method for stochastic differential equations in this paper we develop a new numerical method with asymptotic stability properties for solving stochastic differential equations sdes the foundations for the new solver are the steklov mean and an exact discretization for the deterministic version of the sdes strong consistency and convergence properties are demonstrated for the proposed method moreover a rigorous linear and nonlinear asymptotic stability analysis is carried out for the multiplicative case in a mean square sense and for the additive case in a path wise sense using the pullback limit in order to emphasize the characteristics of the steklov discretization we use as benchmarks the stochastic logistic equation and the langevin equation with a nonlinear potential of the brownian dynamics we show that the steklov method has mild stability requirements and allows long time simulations in several applications

on competitive lotka volterra model in random environments focusing on competitive lotka volterra model in random environments this paper uses regime switching diffusions to model the dynamics of the population sizes of n different species in an ecosystem subject to the random changes of the external environment it is demonstrated that the growth rates of the population sizes of the species are bounded above moreover certain long run average limits of the solution are examined from several angles a partial stochastic principle of competitive exclusion is also derived finally simple examples are used to demonstrate our findings

threshold dynamics and ergodicity of an sirs epidemic model with markovian switching abstract this work studies the threshold dynamics and ergodicity of a stochastic sirs epidemic model with the disease transmission rate driven by a semi markov process the semi markov process used in this paper for describing a randomly changing environment is a very large extension of the most common markov regime switching process we define a basic reproduction number for the semi markov regime switching environment and show that its position with respect to 1 determines the extinction or persistence of the disease in the case of disease persistence we give mild sufficient conditions for ensuring the existence and absolute continuity of the invariant probability measure under the same conditions we also prove the global attractivity of the limit set of the system and the convergence in total variation norm of the transition probability to the invariant measure compared with the existing results in the markov regime switching environment the results generalized require almost no additional conditions

a weighted networked sirs epidemic model abstract population mobility in an sirs epidemic model is considered via graph laplacian diffusion we show the existence and uniqueness of solutions to the sirs model defined on weighed graph by the approach of upper and lower solutions we show that the disease free equilibrium is asymptotically stable if the basic reproduction number is lower than 1 by constructing lyapunov function we show that the endemic equilibrium is globally asymptotically stable for the model with the same diffusion rates if the basic reproduction number is greater than 1 with numerical simulations we apply our generalized weighed graph to watts strogatz network where the degree of node is illustrated to determine the peak number of infectious population it also indicates that the network has an impact on small time behavior of epidemic transmission

a stochastic sis epidemic model with vaccination abstract 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 r 0 s which can be seen as a threshold parameter to utilize in identifying the stochastic extinction and persistence if r 0 s 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 r 0 s 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

a stochastic epidemic model with nonmonotone incidence rate sufficient and necessary conditions for near optimality abstract in this paper we studied the near optimal control of a stochastic susceptible infected recovered susceptible sirs model that includes a nonmonotone incidence rate the near optimal control problem was formulated by reducing the infected population while keeping the treatment cost to a minimum we obtained the sufficient and necessary conditions for the near optimality we showed that any near optimal control can be solved by using the hamiltonian function to approximate the cost function according to the adjoint equations we derived the estimate for the error bound of the near optimality numerical simulations were performed to illustrate the results and confirm the effect of treatment control on the disease dynamics

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

long time dynamics of an sirs reaction diffusion epidemic model abstract in this paper we first deal with a diffusive sirs epidemic system in spatially heterogeneous and time periodic environment with no flux boundary condition we establish the uniform persistence if the basic reproduction number r 0 1 when the diffusion rates are equal and the system parameters are constant we obtain the global stability of the unique endemic equilibrium if r 0 1 and that of the diseases free equilibrium if r 0 1 we also study the sirs epidemic system with spontaneous infection in addition to disease transmission our results show that spontaneous infection enhances the persistence of disease the uniform bounds of solutions which are derived by a feedback argument play a fundamental role in our analysis

analysis of stochastic two prey one predator model with levy jumps abstract taking white noises and levy noises into account a two prey one predator model in random environments is proposed and investigated under some simple assumptions the critical value between persistence in the mean and extinction for each population is obtained then sufficient conditions for stability in distribution of the model are established finally some numerical examples are introduced to validate the analytical findings

stochastic persistence and stationary distribution in a holling tanner type prey predator model abstract in this paper we study a stochastic predator prey model with beddington deangelis type functional response and logistic growth for predators the deterministic model is already well studied and we recall some important results here we construct the stochastic model from the deterministic model by introducing multiplicative noise terms into the growth equations of prey and predator populations for the stochastic model we show that the system admits unique positive global solution starting from the positive initial value then we prove that the system is strongly persistent in mean when the intensity of environmental forcing is less than some threshold magnitudes finally we show that the system has a stationary distribution under certain parametric restrictions numerical simulations are carried out to substantiate the analytical results

dynamics of a modified hindmarsh rose neural model with random perturbations moment analysis and firing activities abstract in this paper an attempt has been made to understand the activity of mean membrane voltage and subsidiary system variables with moment equations i e mean variance and covariance s under noisy environment we consider a biophysically plausible modified hindmarsh rose h r neural system injected by an applied current exhibiting spiking bursting phenomenon the effects of predominant parameters on the dynamical behavior of a modified h r system are investigated numerically it exhibits period doubling period halving bifurcation and chaos phenomena further a nonlinear system has been analyzed for the first and second order moments with additive stochastic perturbations it has been solved using fourth order runge kutta method and noisy systems by euler s scheme it has been demonstrated that the firing properties of neurons to evoke an action potential in a certain parameter space of the large exact systems can be estimated using an approximated model strong stimulation can cause a change in increase or decrease of the firing patterns corresponding to a fixed set of parameter values the firing behavior and dynamical differences of the collective variables of a large exact and approximated systems are investigated

measures induced by analytic functions and a problem of walter rudin let h denote the set of bounded analytic functions in the unit disk u zec izi l and let ho denote those p e h for which p 0 0 every p e h has a nontangential limit at almost every point of u and the resulting function on du is measurable for this and other basic facts about h and related function spaces discussed in this paper see e g d g we denote the boundary function of p also by p and define the induced measure of p by the formula

complete global analysis of a two scale network sirs epidemic dynamic model with distributed delay and random perturbations the model validation results are established the basic reproduction numbers at three human vector contact levels are computed the stochastic stability of equilibria is proven interplay between stability and human mobility are exhibited in real life scenarios a stochastic sirs epidemic dynamic model with distributed time delay for vector borne diseases in two scale network structured populations is presented the distributed delay accounts for the varying incubation period of the disease furthermore the disease dynamics is influenced by random environmental perturbations in the disease transmission process as well as the two scale human mobility process the basic reproduction numbers at three human vector contact levels in the two scale population are computed and the results for the stochastic asymptotic stability of the equilibria are presented moreover the asymptotic stability results are exhibited in several real life scenarios and the significance of the results are presented numerical simulation results are presented

permanence and extinction for the stochastic sir epidemic model abstract 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 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 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

unique stationary distribution and ergodicity of a stochastic logistic model with distributed delay abstract dynamics of a stochastic logistic model with distributed delay are considered we first transfer a scalar stochastic logistic model with strong kernel or weak kernel into an equivalent stochastic system through the linear chain technique then we obtain the sufficient and necessary conditions for extinction and persistence of the species with probability one moreover in the case of persistence we prove that there exists a unique stationary distribution by the markov semigroups theory the results show that the stronger white noise results in the extinction of the species and the weaker white noise guarantees the existence of a unique stationary distribution though for the deterministic model with strong kernel or weak kernel the average delay may induce the existence of a group of small amplitude periodic solutions

stationary distribution of a stochastic predator prey model with distributed delay and general functional response abstract in this paper we study a stochastic predator prey model with distributed delay and general functional response we transform the model with weak kernel case into an equivalent system through the linear chain technique since the diffusion matrix is degenerate the uniform ellipticity condition does not hold the markov semigroup theory is used to derive the existence of a unique stable stationary distribution we verify the densities of the distributions of the positive solutions can converge in l 1 to an invariant density

nontrivial periodic solution for a stochastic brucellosis model with application to xinjiang china abstract brucellosis is a kind of zoonotic disease caused by gram negative bacteria of the genus brucella in this paper we propose a stochastic periodic brucellosis model by introducing the effect of environmental white noise on transmission dynamics of brucellosis by has minskii theory of periodic solution and constructing a novel combination of lyapunov functions we establish the existence of nontrivial positive periodic solution if the condition r 0 s 1 holds based on the reported data of newly acute human brucellosis cases for each season from 2010 to 2014 in xinjiang numerical simulations have been performed to support our result and indicate that brucellosis in xinjiang takes on the feature of long term prevalence and cyclical fluctuation

reducibility and unitary equivalence of analytic multipliers on sobolev disk algebra abstract it is proved that m with 2 blaschke product is reducible on sobolev disk algebra r d if and only if z 2 1 z 2 for some d and unimodular constant also an analytic multiplier m is unitary equivalent to m z n on r d if and only if z n for some unimodular constant

dynamical behavior of an epidemic model with a nonlinear incidence rate abstract in this paper we study the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action by carrying out global qualitative and bifurcation analyses it is shown that either the number of infective individuals tends to zero as time evolves or there is a region such that the disease will be persistent if the initial position lies in the region and the disease will disappear if the initial position lies outside this region when such a region exists it is shown that the model undergoes a bogdanov takens bifurcation i e it exhibits a saddle node bifurcation hopf bifurcations and a homoclinic bifurcation existence of none one or two limit cycles is also discussed

the dynamics of a stochastic vaccinated tuberculosis model with treatment abstract in this paper we study a stochastic model with vaccination and treatment to describe the transmission dynamics of tuberculosis firstly by using the stochastic lyapunov analysis method we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model then we obtain sufficient conditions for extinction of the diseases our work greatly improves the results of feng and qiu 2018 finally some examples and numerical simulations are introduced to illustrate our theoretical results

environmental variability in a stochastic epidemic model abstract in this paper we investigate the stochastic dynamics of a simple epidemic model incorporating the mean reverting ornstein uhlenbeck process analytically and numerically we define two threshold parameters the stochastic demographic reproduction number r d s and the stochastic basic reproduction number r 0 s to utilize in identifying the stochastic extinction and persistence of the disease we find that the stochastic disease dynamics can be determined by the environment fluctuations which measured by the intensity of volatility and the speed of reversion the larger intensity of volatility or the smaller speed of reversion can suppress the outbreak of the disease the smaller intensity of volatility or the the higher speed of reversion can enhance the outbreak of the disease furthermore via numerical simulations we find that the stochastic model has an endemic stationary distribution which leads to the stochastic persistence of the disease our results show that mean reverting process is a well established way of introducing stochastic environmental noise into biologically realistic population dynamic models

on invariant measures and the asymptotic behavior of a stochastic delayed sirs epidemic model abstract in this paper we consider a stochastic epidemic model with time delay and general incidence rate we first prove the existence and uniqueness of the global positive solution by using the krylov bogoliubov method we obtain the existence of invariant measures furthermore we study a special case where the incidence rate is bilinear with distributed time delay when the basic reproduction number r 0 1 the analysis of the asymptotic behavior around the disease free equilibrium e 0 is provided while when r 0 1 we prove that the invariant measure is unique and ergodic the numerical simulations also validate our analytical results

geometrical properties of the algebraic spinors for r3 1 the different geometrical properties of majorana even dirac and chevalley spinors in the clifford algebra approach are investigated

the threshold of a periodic stochastic sivs epidemic model with nonlinear incidence abstract in this paper a stochastic periodic sivs epidemic model with nonlinear incidence and vaccination is investigated the threshold conditions on the existence of stochastic positive periodic solutions and the extinction of disease with probability one are established by constructing the new stochastic lyapunov functions and using the new technique to deal with the nonlinear incidence and vaccination for the stochastic epidemic model the numerical simulations are given to illustrate the main theoretical results and present some new interesting conjectures

ergodic stationary distribution and extinction of a stochastic sirs epidemic model with logistic growth and nonlinear incidence abstract a stochastic sirs model with logistic growth and nonlinear incidence rate is probed in this paper we exemplify that the proposed stochastic sirs model reveals a global and positive solution by applying suitable lyapunov functions the sufficient conditions for the existence of ergodic stationary distribution of the solution to the stochastic sirs model are derived furthermore we acquire the sufficient conditions for extinction of the infectious disease

stationary distribution of a stochastic staged progression hiv model with imperfect vaccination abstract in this paper we study a stochastic staged progression hiv model with imperfect vaccination we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by using the stochastic lyapunov analysis method

stochastic asymptotic analysis of a multi host model with vector transmission abstract we present a stochastic epidemic model with vectorial transmission and multi host structure to include environmental noise we stochastically perturb biting rates with general state functional intensities so we derive a stochastic differential equation sde which describes a vector disease with two types of hosts humans and animals and give conditions to assure disease extinction and persistence finally via numerical experiments we extend and illustrate our results using literature parameters for chagas

rudin orthogonality problem on the bergman space in this paper we study the rudin orthogonality problem on the bergman space which is to characterize those functions bounded analytic on the unit disk whose powers form an orthogonal set in the bergman space of the unit disk we completely solve the problem if those functions are univalent in the unit disk or analytic in a neighborhood of the closed unit disk as a consequence it is shown that an analytic multiplication operator on the bergman space is unitarily equivalent to a weighted unilateral shift of finite multiplicity n if and only if its symbol is a constant multiple of the n th power of a mobius transform which was obtained via the hardy space theory of the bidisk in sun et al 2008 10

a regime switching model for species subject to environmental noises and additive allee effect abstract taking into account the environmental noises and allee effect we develop a regime switching model for a single species and then we consider dynamics of species population such as persistence ergodicity and global attractivity more precisely firstly we prove the global existence of the positive solution and then study the extinction of species non persistence in mean persistence in mean and stochastic permanence it is followed by the study of positive recurrence ergodicity and the global attractivity finally we conclude our study by a numerical example and discussions

asymptotic behaviour of the stochastic lotka volterra model this paper examines the asymptotic behaviour of the stochastic extension of a fundamentally important population process namely the lotka volterra model the stochastic version of this process appears to have far more intriguing properties than its deterministic counterpart indeed the fact that a potential deterministic population explosion can be prevented by the presence of even a tiny amount of environmental noise shows the high level of difference which exists between these two representations

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

long time behavior of a stochastic epidemic model with varying population size abstract in this paper we investigate the persistence and extinction of a stochastic epidemic model with a varying population environment in the long term behavior our model consists of two stochastic differential equations one for the susceptible individuals in which the transmission rate is disturbed by white noise one for the exposed individuals in which the same perturbation occurs and one ordinary differential equation in which describes the infective individuals in a varying population environment we derive sufficient conditions for the extinction and persistence of the epidemic model depending on the constant contact rate moreover we carry out several numerical simulations to illustrate the main results of this contribution

sde sis epidemic model with demographic stochasticity and varying population size an sde sis epidemic model with demographic stochasticity is developed existence of a unique non negative solution is shown an upper bound is found for the infectives in terms of the population size the infectives and susceptibles go extinct in finite time a s results are supported numerically some with real examples in this paper we look at the two dimensional stochastic differential equation sde susceptible infected susceptible sis epidemic model with demographic stochasticity where births and deaths are regarded as stochastic processes with per capita disease contact rate depending on the population size first we look at the sde model for the total population size and show that there exists a unique non negative solution then we look at the two dimensional sde sis model and show that there exists a unique non negative solution which is bounded above given the total population size furthermore we show that the number of infecteds and the number of susceptibles become extinct in finite time almost surely lastly we support our analytical results with numerical simulations using theoretical and realistic disease parameter values

a clifford algebra quantization of dirac s electron positron field the quantum field theory of free dirac particles four component massive spin 1 2 particles is derived by a segal quantization procedure first details are given on how the spinor space of dirac is actually a minimal left ideal of the clifford algebra associated with a lorentz inner product space and how the homogeneous group actions break the natural two component quaternion structure to give familiar four component complex spinors second wigner s procedure for constructing unitary representations of the poincare group is used to construct the appropriately induced infinite dimensional representation of the inhomogeneous group starting from the above four dimensional nonunitary representation third and finally segal s procedure for quantizing classical fermion fields is adapted to this infinite dimensional hilbert space to obtain the clifford algebra of annihilation creation operators for spin 1 2 particles the familiar fock space appears as a minimal left ideal in this second clifford algebra

extinction and stationary distribution of an epidemic model with partial vaccination and nonlinear incidence rate abstract a stochastic susceptible infected vaccinated model with a general incidence rate and varying population size is considered in this paper by means of constructing a suitable lyapunov function the sufficient conditions for the extinction are derived further some conditions that guarantee the existence of a unique stationary distribution is obtained the main results are illustrated by computer simulations

stochastic spatial models in the models we will consider space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites our main aim here is to explain an idea of durrett and levin 1994 the behavior of these models can be predicted from the properties of the mean field ode i e the equations for the densities of the various types that result from pretending that all sites are always independent we will illustrate this picture through a discussion of eight families of examples from statistical mechanics genetics population biology epidemiology and ecology some of our findings are only conjectures based on simulation but in a number of cases we are able to prove results for systems with fast stirring by exploiting connections between the spatial model and an associated reaction diffusion equation

the stationary distribution in a stochastic sis epidemic model with general nonlinear incidence abstract in this paper a stochastic sis type epidemic model with general nonlinear incidence is investigated the diffusion process in the model is indicated to be degenerate a new technique is proposed to investigate the existence of stationary distribution by using the theory of integral markov semigroup the threshold criterion is established to ensure the existence of unique stationary distribution for the model the numerical examples are carried out to illustrate our theoretical results

a stochastic sirs epidemic model with logistic growth and general nonlinear incidence rate abstract in this paper we consider a stochastic sirs epidemic model with logistic growth and general nonlinear incidence rate we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic lyapunov function which provides us a good description of persistence we find that the hypothetical conditions on the nonlinear function are relative weak and valid for many forms of incidence rate

stability of markov semigroups and applications to parabolic systems abstract a new theorem for asymptotic stability of markov semigroups is proved this result is applied to semigroups generated by parabolic systems describing the evolution of densities of two state diffusion processes

stationary distribution of a stochastic cholera model with imperfect vaccination abstract in this paper a stochastic cholera model with imperfect vaccination is proposed and investigated we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic lyapunov function our result may provide some new insights for elimination of cholera some numerical simulation is provided to demonstrate our main result

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

critical result on the threshold of a stochastic sis model with saturated incidence rate abstract in this paper we discuss the threshold r 0 of a stochastic sis epidemic model with saturated incidence based on 10 if r 0 1 the disease will die out for any intensity noise 2 0 which is not proved in 10 then it is obtained that the disease will also go to extinction in probability if r 0 1 which has been left as an open problem in 10 and many other literatures and when r 0 1 we get the stochastic persistence of the disease which is different from the persistence in mean in previous literatures by using chebyshev s inequality and it shows the smaller noise will strengthen the stability of systems besides the existence of the stationary distribution is gotten finally numerical simulations are given to illustrate the results

periodic solution of a stochastic hbv infection model with logistic hepatocyte growth a stochastic hbv model within logistic hepatocyte growth is developed existence of at least one periodic solution induced by white noise is shown existence of a unique stationary distribution is displayed results are supported numerically in this paper we present a stochastic hepatitis b virus hbv infection model with logistic hepatocyte growth we show that the stochastic differential equation sde hbv model has at least one periodic solution induced by the white noise and a unique stationary distribution which is stable one of the most interesting findings is that random perturbations may be beneficial to format the periodic solution to the sde hbv model

the effect of media coverage on threshold dynamics for a stochastic sis epidemic model abstract media coverage is one of the important measures for controlling infectious diseases but the effect of media coverage on diseases spreading in a stochastic environment still needs to be further investigated here we present a stochastic susceptible infected susceptible sis epidemic model incorporating media coverage and environmental fluctuations by using feller s test and stochastic comparison principle we establish the stochastic basic reproduction number r 0 s which completely determines whether the disease is persistent or not in the population if r 0 s 1 the disease will go to extinction if r 0 s 1 the disease will also go to extinction in probability which has not been reported in the known literatures and if r 0 s 1 the disease will be stochastically persistent in addition the existence of the stationary distribution of the model and its ergodicity are obtained numerical simulations based on real examples support the theoretical results the interesting findings are that i the environmental fluctuation may significantly affect the threshold dynamical behavior of the disease and the fluctuations in different size scale population and ii the media coverage plays an important role in affecting the stationary distribution of disease under a low intensity noise environment

stationary distribution and ergodicity of a stochastic food chain model with levy jumps abstract in this paper a three species stochastic food chain model with levy jumps is proposed and analyzed sharp sufficient criteria for the existence and uniqueness of an ergodic stationary distribution are established the effects of levy jumps on the existence of the stationary distribution are revealed in some cases the levy jumps could make the stationary distribution appear while in some cases the levy jumps could make the stationary distribution disappear some numerical simulations are introduced to illustrate the theoretical results

a stochastic differential equation sis epidemic model incorporating ornstein uhlenbeck process abstract in this paper based on the results of gray et al 2011 we propose a new sde sis model incorporating mean reverting ornstein uhlenbeck process and prove that the stochastic basic reproduction number r 0 s can be used to identify the stochastic extinction and persistence for the sde mode if r 0 s 1 under mild extra conditions the disease will be extinct a s while if r 0 s 1 the disease will persist a s epidemiologically we find that smaller speed of reversion or bigger intensity of volatility can suppress the disease outbreak thus in order to control the spread of the disease we must decrease the speed of reversion or increase the intensity of volatility

optimal harvesting policy of logistic population model in a randomly fluctuating environment abstract in this paper we investigate the optimal harvest policy of a stochastic logistic population model the value of this study lies in two aspects mathematically we establish a stochastic threshold theorem to govern whether the population persists or not in the case of population persistence we prove the existence uniqueness and global asymptotic stability of the invariant density of the fokker planck equation associated with the sde model and we further show the relation between these two models in addition we give the unique optimal harvesting effort and the corresponding maximum of expectation of sustainable yield which gives us the profile of the optimal harvesting policy of the sde model ecologically we find that big harvesting effort or big intensity of noise will lead the population to extinct risk almost surely in addition we find that under a fixed randomness strategy and proper harvesting the maximum sustainable yield increases systematically as the harvesting effort increases but overexploitation will reduce the level of maximize sustainable yield and eventually make the whole population extinct with probability one hence in order to obtain the optimal harvesting policy we must decrease the harvesting effort and the intensity of noise furthermore we find that our parameter perturbation method in this paper is more beneficial to the exploitation of renewable resources than the classical one given by beddington and may 1977 the results show that different perturbation method can exhibit different stochastic population dynamics

dynamics of a stochastic sir epidemic model with saturated incidence in this paper the dynamics of a stochastic sir epidemic model with saturated incidence is investigated firstly we prove that the system has a unique global positive solution with any positive initial value then we verify that random effect may lead the disease to extinction under a simple condition thirdly we establish a sufficient condition for persistence in the mean of the disease moreover we show that there is a stationary distribution to the stochastic system under certain parametric restrictions finally some numerical simulations are carried out to confirm the analytical results

a stochastic sis epidemic model with saturating contact rate abstract in this paper we formulate and investigate the dynamics of a sis epidemic model with saturating contact rate and its corresponding stochastic differential equation version for the deterministic epidemic model we show that the disease free equilibrium is global asymptotically stable if the basic reproduction number is less than unity and if the basic reproduction number is greater than unity model 1 2 admits a unique endemic equilibrium which is locally asymptotically stable by analyzing the corresponding characteristic equations for the stochastic epidemic model the existence and uniqueness of the positive solution are proved by employing the lyapunov analysis method the basic reproduction number r 0 s is proved to be a sharp threshold which determines whether there is an endemic outbreak or not if r 0 s 1 and under mild extra conditions the disease can be eradicated almost surely whereas if r 0 s 1 it has a stationary distribution which leads to the stochastic persistence of the disease finally numerical simulations are presented to illustrate our theoretical results

coexistence of limit cycles and homoclinic loops in a sirs model with a nonlinear incidence rate recently ruan and wang j differential equations 188 2003 pp 135 163 studied the global dynamics of a sirs epidemic model with vital dynamics and a nonlinear saturated incidence rate under certain conditions they showed that the model undergoes a bogdanov takens bifurcation i e it exhibits saddle node hopf and homoclinic bifurcations they also considered the existence of none one or two limit cycles in this paper we investigate the coexistence of a limit cycle and a homoclinic loop in this model one of the difficulties is to determine the multiplicity of the weak focus we first prove that the maximal multiplicity of the weak focus is 2 then feasible conditions are given for the uniqueness of limit cycles the coexistence of a limit cycle and a homoclinic loop is obtained by reducing the model to a universal unfolding for a cusp of codimension 3 and studying degenerate hopf bifurcations and degenerate bogdanov takens bifurcations of limit cycles and homoclinic loops of order 2

dynamics for a class of stochastic sis epidemic models with nonlinear incidence and periodic coefficients abstract in this paper we investigate the dynamics of a class of periodic stochastic sis epidemic models with general nonlinear incidence f s i some sufficient conditions on the permanence in the mean and extinction of positive solutions with probability one are established by using the khasminskii s boundary periodic markov processes the existence of stochastic nontrivial periodic solution for the models is also obtained the numerical simulations are given to illustrate the main theoretical results and some interesting conjectures are presented

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

optimal harvesting strategy for a stochastic mutualism system in a polluted environment with regime switching abstract in this paper we stochastically perturb the harvesting mutualism system into stochastic system in a polluted environment by white noise telegraph noise and levy jump we first prove the existence and uniqueness of the global positive solution of the stochastic system and establish the sufficient condition for persistence in mean of the two cooperative species then using aggregation and ergodic methods the explicit forms of optimal harvesting strategy and the maximum of sustainable yield are obtained we also give some examples and numerical simulations to illustrate the theoretical results the impacts of toxicant in organism telegraph noise and levy jump on the persistence in mean and optimal harvesting strategy are revealed

a stochastic sirs epidemic model with infectious force under intervention strategies abstract 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

threshold behavior in a stochastic sir epidemic model with logistic birth abstract in this paper we consider a stochastic sir epidemic model with logistic birth by using the stochastic lyapunov function method we show that the stochastic basic reproduction number r 0 s can be used to determine the threshold dynamics of the stochastic system if r 0 s 1 we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model while if r 0 s 1 under some extra conditions we obtain sufficient conditions for extinction of the disease finally some examples and numerical simulations are provided to illustrate the theoretical results

modelling a stochastic hiv model with logistic target cell growth and nonlinear immune response function abstract a stochastic hiv viral model with both logistic target cell growth and nonlinear immune response function is formulated to investigate the effect of white noise on each population the existence of the global solution is verified by employing a novel combination of lyapunov functions we obtain the existence of the unique stationary distribution for small white noises we also derive the extinction of the virus for large white noises numerical simulations are performed to highlight the effect of white noises on model dynamic behaviour under the realistic parameters it is found that the small intensities of white noises can keep the irregular blips of hiv virus and ctl immune response while the larger ones force the virus infection and immune response to lose efficacy

a stochastic sirs epidemic model with nonlinear incidence rate a stochastic sirs model with ratio dependent incidence rate is developed the global dynamics of the deterministic model is shown the stochastic dynamics of the sde model is given the existence of a unique stationary distribution of the sde model is displayed in this paper we investigate the global dynamics of a general sirs epidemic model with a ratio dependent incidence rate and its corresponding stochastic differential equation version for the deterministic model we show that the basic reproduction number r0 determines whether there is an endemic outbreak or not if r0 1 the disease persists for the stochastic model we show that its related reproduction number r0s can determine whether there is a unique disease free stationary distribution or a unique endemic stationary distribution in addition we provide analytic results regarding the stochastic boundedness and permanence extinction one of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak which can provide us some useful control strategies to regulate disease dynamics

global analysis and numerical simulations of a novel stochastic eco epidemiological model with time delay abstract in this paper a new stochastic eco epidemiological model is proposed and investigated and the high dimensional model includes time delay and a general incidence rate which make the study more complex first the existence and uniqueness of the global positive solution with any positive initial value are verified second using stochastic analysis methods and some differential inequality techniques we explore the long time asymptotic properties of the stochastic delayed system furthermore we prove that the solution of the stochastic delayed system has a unique stationary distribution and it has ergodic property third the sufficient conditions of permanence in mean or extinction for the ecological populations are obtained finally we introduce a series of computer simulations to show the performance of the theoretical 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

positive steady states in an epidemic model with nonlinear incidence rate abstract in this paper we investigate a diffusive epidemic model with nonlinear incidence rate under homogeneous neumann boundary condition the value of this study lies in two aspects mathematically we show the stability of the constant positive steady state solution the existence and nonexistence the local and global structure of nonconstant positive steady states and epidemiologically we find that the system exhibits turing patterns controlled by diffusion

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