dynamics of the deterministic and stochastic siqs epidemic model with non linear incidence the deterministic and stochastic siqs models with non linear incidence are introduced for deterministic model the basic reproductive rate r 0 is derived moreover if r 0 1 there exists a unique endemic equilibrium which is globally asymptotically stable for stochastic model sufficient condition for extinction of the disease that is regardless of the value of r 0 is presented in addition if the intensities of the white noises are sufficiently small and r 0 1 then there exists a unique stationary distribution to stochastic model numerical simulations are also carried out to confirm the analytical results

applications of fixed point methods to discrete variational and quasi variational inequalities in this paper discrete analogues of variational inequalities v i and quasi variational inequalities q v i encountered in stochastic control and mathematical physics are discussed it is shown that those discrete v i s and q v i s can be written in the fixed point formx tx such that eithert or some power oft is a contraction this leads to globally convergent iterative methods for the solution of discrete v i s and q v i s which are very suitable for implementation on parallel computers with single instruction multiple data architecture particularly on massively parallel processors m p p s

analysis of a hybrid switching svir epidemic model with vaccination and levy noise abstract this paper is concerned with the dynamical behavior of a hybrid switching svir epidemic model with vaccination and levy noise besides a standard geometric brownian motion another two driving processes are considered a stationary poisson point process and a continuous time finite state markov chain to begin with we obtain sufficient conditions for persistence in the mean of the disease then we establish sufficient conditions for extinction of the disease furthermore in the case of persistence we also obtain sufficient conditions for the existence of positive recurrence of the solutions by constructing a suitable stochastic lyapunov function with regime switching

global behavior of a multi group sis epidemic model with age structure abstract we study global dynamics of a system of partial differential equations the system is motivated by modelling the transmission dynamics of infectious diseases in a population with multiple groups and age dependent transition rates existence and uniqueness of a positive endemic equilibrium are established under the quasi irreducibility assumption which is weaker than irreducibility on the function representing the force of infection we give a classification of initial values from which corresponding solutions converge to either the disease free or the endemic equilibrium the stability of each equilibrium is linked to the dominant eigenvalue s a where a is the infinitesimal generator of a quasi irreducible semigroup generated by the model equations in particular we show that if s a 0 then the disease free equilibrium is globally stable if s a 0 then the unique endemic equilibrium is globally stable

periodic solution for the stochastic chemostat with general response function abstract this paper addresses a stochastic chemostat model with periodic dilution rate and general class of response functions the general functional response is assumed to satisfy two classifications of conditions and these assumptions on the functional response are relative weak that are valid for many forms of growth response for the chemostat with periodic dilution rate we derive the sufficient criteria for the existence of the stochastic nontrivial positive periodic solution by utilizing khasminskii s theory on periodic markov process

symplectic quantization of self dual master lagrangian we consider the master lagrangian of deser and jackiw interpolating between the self dual and the maxwell chern simons lagrangian and quantize it following the symplectic approach as well as the traditional dirac scheme we demonstrate the equivalence of these procedures in the subspace of the second class constraints we then proceed to embed this mixed first and second class system into an extended first class system within the framework of both approaches and construct the corresponding generator for this extended gauge symmetry in both formulations

global threshold dynamics of a stochastic differential equation sis model abstract in this paper we further investigate the global dynamics of a stochastic differential equation sis susceptible infected susceptible epidemic model recently proposed in gray et al 2011 8 we present a stochastic threshold theorem in term of a stochastic basic reproduction number r 0 s the disease dies out with probability one if r 0 s 1 and the disease is recurrent if r 0 s 1 we prove the existence and global asymptotic stability of a unique invariant density for the fokker planck equation associated with the sde sis model when r 0 s 1 in term of the profile of the invariant density we define a persistence basic reproduction number r 0 p and give a persistence threshold theorem the disease dies out with large probability if r 0 p 1 while persists with large probability if r 0 p 1 comparing the stochastic disease prevalence with the deterministic disease prevalence we discover that the stochastic prevalence is bigger than the deterministic prevalence if the deterministic basic reproduction number r 0 d 2 this shows that noise may increase severity of disease finally we study the asymptotic dynamics of the stochastic sis model as the noise vanishes and establish a sharp connection with the threshold dynamics of the deterministic sis model in term of a limit stochastic threshold theorem

a stochastic differential equation sis epidemic model with two independent brownian motions 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 in sis model based on their previous work we consider another perturbation on the parameter 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

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

dynamic programming and maximum principle for discrete goursat systems abstract we obtain the dynamic programming equations for discrete goursat systems we prove that they are necessary and sufficient for optimality and we prove formulas for the exact or approximate solution of these equations using dynamic programming arguments we obtain semi dynamic programming equations and then we prove a maximum principle of pontryagin s type

dynamic programming approach to the optimal control of systems governed by goursat darboux equations the method of dynamic programming is extended to a class of two dimensional time problems the goursat darboux systems hyperbolic systems with initial boundary data on characteristics a formal derivation of a maximum principle of pontryagin s type is obtained by using the semi dynamic programming equations controlled goursat darboux systems have applications to the probiems of gas absorption and drying processes

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 in a stochastic heroin model with seasonal variation abstract in this paper a stochastic non autonomous heroin model is formulated we obtain sufficient conditions for extinction and permanence in mean with probability one of the positive solutions in addition we prove that there exists at least one positive periodic solution under some conditions finally numerical simulations are performed to illustrate our theoretical results

a stochastic model of aids and condom use in this paper we introduce stochasticity into a model of aids and condom use via the technique of parameter perturbation which is standard in stochastic population modelling we show that the model established in this paper possesses non negative solutions as desired in any population dynamics we also carry out a detailed analysis on asymptotic stability both in probability one and in pth moment our results reveal that a certain type of stochastic perturbation may help to stabilise the underlying system

threshold behavior of a stochastic sis model with levy jumps in this paper the dynamics of a stochastic sis model with levy jumps are investigated we first prove that this model has a unique global positive solution starting from the positive initial value then taking the accumulated jump size into account we find a threshold of the model denoted by r 0 which completely determines the extinction and prevalence of the disease if r 0 1 the disease dies out exponentially with probability one if r 0 1 the solution of the model tends to a point in time average which leads to the stochastical persistence of the disease from the view of epidemiology the existence of threshold is useful in determining treatment strategies and forecasting epidemic dynamics moreover we find that levy noise can suppress disease outbreak finally we introduce some numerical simulations to support the main results obtained

extinction and persistence of a stochastic nonlinear sis epidemic model with jumps abstract in this paper brownian motion and l e vy jumps are introduced to a sis type epidemic model with nonlinear incidence rate the dynamical behavior of the considered model is investigated in order to reveal the extinction and permanence of the disease two threshold values r 0 r 0 are showed we find that if r 0 1 the disease may die out and when r 0 1 the disease may be persistent finally the numerical simulations are presented to illustrate our mathematical results

on solvability of functional equations and system of functional equations arising in dynamic programming the purpose of this paper is to study solvability of two classes of functional equations and a class of system of functional equations arising in dynamic programming of multistage decision processes by using fixed point theorems a few existence and uniqueness theorems of solutions and iterative approximation for solving these classes of functional equations are established under certain conditions some existence theorems of coincidence solutions for the class of system of functional equations are shown some examples are given to demonstrate the advantage of our results than existing ones in the literature

optimal control of goursat darboux systems with discontinuous co state abstract we study an optimal control problem for goursat darboux systems over domains that are more general than rectangles with costs that include integral terms supported in the two dimensional domain as well as terms with singular one dimensional and zero dimensional support we prove that the co state satisfies a system of hamiltonian equations and a set of side conditions that involves jump discontinuities across a network of characteristic curves we present a method of solution of the hamiltonian equations and in the process of doing so we establish the unique solvability of these equations

solutions to two functional equations arising in dynamic programming this paper deals with the existence uniqueness and iterative approximation of solutions for two functional equations arising in dynamic programming of multistage decision processes the results presented in this paper extend improve and unify the results due to bellman bhakta and choudhury bhakta and mitra liu liu and ume and others

dynamics of a stochastic siqr epidemic model with standard incidence abstract in this paper a stochastic siqr epidemic model with quarantine and standard incidence is investigated we first consider the existence and uniqueness of the global positive solution to the stochastic system further we obtain conclusions that the large noise can make the disease die out exponentially and the stationary distribution will exist under certain conditions

stationary distribution and extinction of sir model with nonlinear incident rate under markovian switching abstract taking into account of both white and colored noises a stochastic epidemic model with nonlinear incident rate under regime switching is formulated based on this model we investigate the dynamic behaviors such as ergodicity and extinction of the sir model with beddington deangelis incidence rate and markov switching first we study the existence of the unique positive solution of system 1 3 secondly by using lyapunov functions we prove that the system has a ergodic stationary distribution under certain sufficient conditions then we obtain the conditions for extinction finally numerical simulations are employed to illustrate our theoretical analysis

optimal control and the value of information for a stochastic epidemiological sis model abstract we analyze a continuous time stochastic sis epidemiological model describing the transmission dynamics of infectious diseases where infected individuals can become susceptible again after recovery the infectivity period of infected individuals can be influenced dynamically by a decision maker with the aim to minimize the expected aggregated economic costs due to illness and drug treatment this stochastic control problem is investigated under two alternative assumptions about the information available if a complete and exact measurement of the size of the infected population is available at any time then the optimal control is sought in a state feedback form and the related hamilton jacobi bellman hjb equation can be used if no state measurement is available then the optimal control is sought in an open loop form and the problem can be reformulated as an optimal control problem for the kolmogorov forward equation in both cases deriving optimality conditions requires non standard arguments due to the degeneracy of the involved hjb and kolmogorov equations based on the obtained theoretical results the role of the information pattern is numerically investigated

stuckelberg field shifting quantization of free particle on d dimensional sphere in this paper we quantize a free particle on a d dimensional sphere in an unambiguous way by converting the second class constraint using the stuckelberg field shifting formalism furthermore we argue that this formalism is equivalent to the bfft batalin fradkin fradkina tyutin constraint conversion method and show that the energy spectrum is identical to the pure laplace beltrami operator without additional terms arising from the curvature of the sphere we work out the gauge symmetry generators with results consistent with those obtained through the nonlinear implementation of the gauge symmetry

optimal control of volterra integral equations in two independent variables we prove an extremality condition for optimal control problems for systems of integral equations of volterra type with two independent variables these systems generalize both the hyperbolic control problems in two independent variables and the optimal control of ordinary i e with one independent variable volterra integral equations

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

threshold behavior in a stochastic siqr epidemic model with standard incidence and regime switching abstract this paper is concerned with the dynamical behavior of a stochastic siqr epidemic model with standard incidence which is disturbed by both white and telegraph noises firstly we obtain sufficient conditions for persistence in the mean of the disease then we establish sufficient conditions for extinction of the disease in addition in the case of persistence we get sufficient conditions for the existence of positive recurrence of the solutions by constructing a suitable stochastic lyapunov function with regime switching meanwhile the threshold between persistence in the mean and extinction of the stochastic system is also obtained finally some numerical simulations are introduced to demonstrate the analytical results

stochastic analysis of a novel nonautonomous periodic siri epidemic system with random disturbances abstract in this paper a new stochastic nonautonomous siri epidemic model is formulated given that the incidence rates of diseases may change with the environment we propose a novel type of transmission function the main aim of this paper is to obtain the thresholds of the stochastic siri epidemic model to this end we investigate the dynamics of the stochastic system and establish the conditions for extinction and persistence in mean of the disease by constructing some suitable lyapunov functions and using stochastic analysis technique furthermore we show that the stochastic system has at least one nontrivial positive periodic solution finally numerical simulations are introduced to illustrate our results

the optimal control problem in the processes described by the goursat problem for a hyperbolic equation in variable exponent sobolev spaces with dominating mixed derivatives in this paper a necessary and sufficient condition such as the pontryagin s maximum principle for an optimal control problem with distributed parameters is given by a hyperbolic equation of the second order with l p x coefficients the results can be used in the theory of optimal processes for distribution pontryagin maximum principle for various controlled processes described by hyperbolic equations of second order with discontinuous coefficients in variable exponent sobolev spaces with dominant mixed derivatives

asymptotic behavior and stability of a stochastic model for aids transmission a stochastic model for aids transmission is presented in this paper the existence and uniqueness of its solution is proved and the solution belongs to an interval satisfying the actual condition furthermore the disease equilibrium can be demonstrated to be existent and both its almost surely exponential stability and its pth moment exponential stability can be concluded based on parameter analysis the first quadrant can be integrated into several parts according to the comparison between almost surely exponential stability and pth moment exponential stability and the potential meaning of the stability is further demonstrated finally based on the above theoretical analysis some conditions for aids to decrease and then to die out are obtained by computer simulation

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

threshold of a regime switching sirs epidemic model with a ratio dependent incidence rate abstract in this paper we study the dynamical behavior of a regime switching sirs epidemic model with a ratio dependent incidence rate we propose a stochastic reproduction number r 0 s which can be regarded as a threshold to use in identifying the stochastic extinction and persistence if r 0 s 1 the disease dies out exponentially with probability one while if r 0 s 1 there exists a unique ergodic stationary distribution of the positive solutions to the system which implies the stochastic persistence of the infectious disease

stability of an seir epidemic model with independent stochastic perturbations abstract for an epidemic model of the type mentioned we prove a theorem on almost sure exponential stability of the disease free equilibrium for small values of the diffusion parameter we describe the stability of the disease free equilibrium point in terms of an appropriate analogue r of the basic reproduction number r 0 of the deterministic special case whenever 0 then r r 0 for small values of the stability theorem guarantees almost sure exponential stability whenever r 1 we also discuss the effect of increasing

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

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

permanence and extinction for a single species system with jump diffusion abstract in this study we consider the effect of jump diffusion random environmental perturbations on the permanence and extinction of a single species dispersal periodic system in poor patchy environments with the possibility of species loss during their dispersion among patches first we prove that there is a unique global positive solution to the system with any initial positive value with probability 1 and we obtain the sufficient conditions that stochastically ensure the ultimate boundedness as well as the asymptotic polynomial growth of the population system next we establish the sufficient conditions for the almost sure permanence in the mean stochastic permanence and extinction of the system in particular by constructing an appropriate integrating factor we obtain the sufficient conditions for the almost sure weak permanence of the patch the conditions obtained for permanence generalize the sufficient conditions established previously on the system without random environmental perturbations finally we discuss the biological implications of the main results

pattern dynamics of a spatial epidemic model with time delay we presented an epidemic model with delay emergence of spatial pattern is obtained rich dynamics is shown by analysis and numerical results observed data confirms our theoretical results the nonlinear incidence rate can explain the complicated infectious process of disease and time delay describing the latent period widely exists in the process of disease contagion in this paper a spatiotemporal epidemic model with nonlinear incidence rate is investigated in particular we considered that the time delay is relatively small in this case the characteristic equation are derived we obtain two mechanisms of instability of the positive constant stationary state that is one is the diffusion induced instability and the other one is delay induced instability moreover the results of numerical simulation validate our theoretical analyses the obtained results may well catch some major features for epidemic models

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

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

dynamics of stochastic seis epidemic model with varying population size abstract we introduce the stochasticity into a deterministic model which has state variables susceptible exposed infected with varying population size in this paper the infected individuals could return into susceptible compartment after recovering we show that the stochastic model possesses a unique global solution under building up a suitable lyapunov function and using generalized ito s formula the densities of the exposed and infected tend to extinction when some conditions are being valid moreover the conditions of persistence to a global solution are derived when the parameters are subject to some simple criteria the stochastic model admits a stationary distribution around the endemic equilibrium which means that the disease will prevail to check the validity of the main results numerical simulations are demonstrated as end of this contribution

persistence and distribution of a stochastic susceptible infected removed epidemic model with varying population size abstract in this paper the dynamics of a stochastic susceptible infected removed model in a population with varying size is investigated we firstly show that the stochastic epidemic model has a unique global positive solution with any positive initial value then we verify that random perturbations lead to extinction when some conditions are being valid moreover we prove that the solution of the stochastic epidemic model is persistent in the mean by building up a suitable lyapunov function and using generalized ito s formula further the stochastic epidemic model admits a stationary distribution around the endemic equilibrium when parameters satisfy some sufficient conditions to end this contribution and to check the validity of the main results numerical simulations are separately carried out to illustrate these results

long time behaviors of a stochastic cooperative lotka volterra system with distributed delay abstract this article focuses on a stochastic cooperative lotka volterra system with distributed delay we first transfer the stochastic system with weak kernel into a degenerate stochastic system made up of four equations for the deterministic system global stability of the positive equilibrium is investigated for the stochastic system with distributed delay sharp sufficient conditions for the persistence of two species are established what is more we obtain the existence and uniqueness of the stationary distribution by constructing suitable lyapunov function and proving the global attraction of the positive solution the results show that the weaker white noises can ensure the existence of a unique stationary distribution and the stronger white noises can result in the extinction of one or two species though the positive equilibrium is globally stable without white noises

a stochastic model for the evolution of bone metastasis persistence and recovery abstract bone metastasis is a complex disease that modifies the natural dynamics of the bone remodeling process which is vital for maintaining a healthy bone structure nowadays there are deterministic differential models that describe the bone metastasis evolution by considering interaction mechanisms between cancer cells and basic bone cells such as osteoclasts and osteoblasts based on these deterministic models a stochastic differential model is proposed to describe the bone metastasis evolution in order to analyze the effect of environmental fluctuations inclusions conditions for the existence and uniqueness of a nonnegative solution of the proposed stochastic model are given to characterize whether the invasion of cancer persists or is eradicated finally we illustrate different dynamical behaviors of the stochastic solution by means of numerical simulations

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

the stationary distribution and ergodicity of a stochastic phytoplankton zooplankton model with toxin producing phytoplankton under regime switching abstract this paper addresses a stochastic phytoplankton zooplankton model with toxin producing phytoplankton under regime switching the stochastic model in this paper includes white and colored noises for the model we first prove the existence of the global positive solution of the model and then by using khasminskii s method and lyapunov function we derive sufficient conditions for the existence of a unique ergodic stationary distribution finally numerical simulations are carried out to support our results

connection between the spectrum and the moments of the ground state density in n dimensional space we show that a series of recurrent inequalities derived in n 3 have the same formal expressions in any dimension n 2 they are derived from the multipole sum rules and provide us with upper bounds for the moments of the ground state density depending only on the lowest multipole excitation energy these bounds are transformed into approximate recurrent relations by means of an empirical correction factor the 1 r potential and the harmonic oscillator play a key role in establishing this factor which is exact for these two potentials by construction for a large class of potentials we show that this factor tends to 1 as n in such cases at the large n limit the lowest state for each multipole excitation exhausts the sum rule it thus acquires the characteristics of the one phonon excitation typical of the harmonic oscillator

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

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

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

dynamical behavior of a stochastic sei epidemic model with saturation incidence and logistic growth abstract in this paper we investigate the dynamical behavior of a stochastic sei epidemic model with saturation incidence and logistic growth firstly by constructing a suitable stochastic lyapunov function we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the solutions to the model then we obtain sufficient conditions for extinction of the disease in two cases that is the first case is that the susceptible population survival and infected and exposed populations extinction the second case is that all the populations extinction

stationary distribution and extinction of the ds i a model disease with periodic parameter function and markovian switching this paper introduces the ds i a model with periodic parameter function and markovian switching first we will prove that the solution of the system is positive and global furthermore we draw a conclusion that there exists nontrivial positive periodic solution for the stochastic system and we establish sufficient conditions for extinction of system moreover we construct stochastic lyapunov functions with regime switching to obtain the existence of ergodic stationary distribution of the solution to ds i a model perturbed by white and telephone noises and we also establish sufficient conditions for extinction of system with regime switching finally we test our theory conclusion by simulations

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

stationary distribution and extinction of a stochastic sirs epidemic model with standard incidence abstract in this paper we consider a stochastic sirs epidemic model with standard incidence by constructing suitable stochastic lyapunov function we establish sufficient conditions for the existence of ergodic stationary distribution of the model moreover we also establish sufficient conditions for extinction of the disease

dynamics of a stochastic sis model with double epidemic diseases driven by levy jumps abstract this paper is to investigate the dynamics of a stochastic sis epidemic model with saturated incidence rate and double epidemic diseases which make the research more complex the environment variability in this study is characterized by white noise and jump noise sufficient conditions for the extinction and persistence in the mean of two epidemic diseases are obtained it is shown that the two diseases can coexist under appropriate conditions finally numerical simulations are introduced to illustrate the results developed

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 sis epidemic model with markovian switching abstract population systems are often subject to environmental noise motivated by takeuchi et al 7 we will discuss in this paper the effect of telegraph noise on the well known sis epidemic model we establish the explicit solution of the stochastic sis epidemic model which is useful in performing computer simulations we also establish the conditions for extinction and persistence for the stochastic sis epidemic model and compare these with the corresponding conditions for the deterministic sis epidemic model we first prove these results for a two state markov chain and then generalise them to a finite state space markov chain computer simulations based on the explicit solution and the euler maruyama scheme are performed to illustrate our theory we include a more realistic example using appropriate parameter values for the spread of s t r e p t o c o c c u s p n e u m o n i a e in children