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

a unified treatment of integer parameter models abstract it is shown that there is an analog to the score function the derivative of the log likelihood with respect to the parameter here called the difference score which plays an important theoretical role in integer parameter models a unified treatment of integer parameter models can be obtained by recognizing that many commonly used models have a linear difference score lds it is established that many of the exact results available for exponential family models also hold for lds models in particular we find an information lower bound show the uniqueness of the maximum likelihood estimator establish monotone likelihood ratio and establish optimality properties in two sided tests it is also shown that if one has a multiparameter likelihood in which there is a conditional likelihood depending on n alone then these exact results apply there as well it is also established when and why the usual likelihood methodology will apply to a class of nonstandard models in which the natural approxima

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

global stability for two species lotka volterra systems with delay abstract in this paper a two species delayed lotka volterra system without delayed intraspecific competitions is considered it is proved that the system is globally stable for all off diagonal delays 12 21 0 if and only if the interaction matrix of the system satisfies condition wdd

a martingale estimating equation for a capture recapture experiment in discrete time we use martingale theory and a method of moments technique to derive a class of estimators for the size of a closed population in a capture recapture experiment in discrete time where the capture probabilities on the successive capture occasions are allowed to be different the same capture probability is assumed to apply to all animals in the population on the same occasion explicit expressions are given for these estimators and their associated standard errors which involve only simple computation an optimal estimator for the population size is found asymptotic results are readily obtained by an application of a martingale central limit theorem two examples are given to compare the results

global dynamics for lotka volterra systems with infinite delay and patch structure we study some aspects of the global dynamics of an n dimensional lotka volterra system with infinite delay and patch structure such as extinction persistence existence and global asymptotic stability of a positive equilibrium both the cases of an irreducible and reducible linear community matrix are considered and no restriction on the signs of the intra and inter specific delayed terms is imposed although the system is not cooperative our approach often uses comparative results with an auxiliary cooperative system some models in recent literature are generalized and results improved

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

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

on polarizations in invariant theory abstract given a reductive algebraic group g and a finite dimensional algebraic g module v we study how close is the algebra of g invariant polynomials on v n to the subalgebra generated by polarizations of g invariant polynomials on v we address this problem in a more general setting of g actions on arbitrary affine varieties

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

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

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

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

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

linear spaces of nilpotent matrices abstract we consider several questions on spaces of nilpotent matrices we present sufficient conditions for triangularizability and give examples of irreducible spaces we give a necessary and sufficient condition in terms of the trace for all linear combinations of a given set of operators to be nilpotent we also consider the question of the dimension of a space l of nilpotents on f n in particular we give a simple new proof of a theorem due to m gerstenhaber concerning the maximal dimension of such spaces

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

multilist population estimation with incomplete and partial stratification multilist capture recapture methods are commonly used to estimate the size of elusive populations in many situations lists are stratified by distinguishing features such as age or sex stratification has often been used to reduce biases caused by heterogeneity in the probability of list membership among members of the population however it is increasingly common to find lists that are structurally not active in all strata we develop a general method to deal with cases when not all lists are active in all strata using an expectation maximization em algorithm we use a flexible log linear modeling framework that allows for list dependencies and differential probabilities of ascertainment in each list finally we apply our method of estimating population size to two examples

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

stability and bifurcation analysis of a reaction diffusion equation with spatio temporal delay abstract the spatio temporal patterns of a general reaction diffusion equation with non local delay are considered by analyzing the characteristic equations and applying the complex plane theory the stability of the constant steady state and the possible hopf bifurcations are obtained with the mean delay as a bifurcation parameter the absolute stable conditional stable and unstable region can be explicitly divided by the coefficients of the term without delay and the term with non local delay by investigating the effect of any weight function on the dynamics of the system we establish the existence of spatially homogeneous and inhomogeneous bifurcating periodic solutions and the algorithm for determining the direction and stability of hopf bifurcation is derived by computing the normal form on the center manifold the result shows that for the case of strong kernel the average delay may induce the stability switches finally the theoretical results are applied to a diffusive food limited population model numerical simulations show the existence of the orbit connecting the unstable spatially inhomogeneous periodic solutions to stable spatially homogeneous periodic solutions

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

the criteria for globally stable equilibrium in n dimensional lotka volterra systems abstract in this paper a set of sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium point in various submodels of the classic n dimensional lotka volterra system the submodels are the following systems competition cooperative or predator prey chain system and competition cooperative or predator prey model between one and multispecies the criteria in this paper are in explicit forms of the parameters and thus are easily verifiable

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

monte carlo confidence intervals des intervalles de confiance non parametriques obtenus par la methode de monte carlo sont utiles quand les intervalles obtenus par les methodes analytiques sont indisponibles ou ne correspondent pas au niveau nominal on fournit des formules permettant de generer un intervalle de niveau donne et de quantifier la variation due a la methode de monte carlo

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

stability and hopf bifurcation in a model of gene expression with distributed time delays in this paper we consider the effect of distributed time delays on dynamics of a mathematical model of gene expression both the weak and strong delay kernels are discussed sufficient conditions for the local stability of the unique equilibrium are obtained taking the average delay as a bifurcation parameter we investigate the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions by using the method of multiple time scales finally numerical simulation is carried out to illustrate our theoretical results it shows both subcritical and supercritical hopf bifurcations can happen

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

capture recapture when time and behavioral response affect capture probabilities summary we consider a capture recapture model in which capture probabilities vary with time and with behavioral response two inference procedures are developed under the assumption that recapture probabilities bear a constant relationship to initial capture probabilities these two procedures are the maximum likelihood method both unconditional and conditional types are discussed and an approach based on optimal estimating functions the population size estimators derived from the two procedures are shown to be asymptotically equivalent when population size is large enough the performance and relative merits of various population size estimators for finite cases are discussed the bootstrap method is suggested for constructing a variance estimator and confidence interval an example of the deer mouse analyzed in otis et al 1978 wildlife monographs62 93 is given for illustration

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

inference on population size in binomial detectability models many models for biological populations including simple mark recapture models and distance sampling models involve a binomially distributed number n of observations x 1 x n on members of a population of size n two popular estimators of n where is a vector parameter are the maximum likelihood estimator and the conditional maximum likelihood estimator c c based on the conditional distribution of x 1 x n given n we derive the large n asymptotic distributions of and c c and give formulae for the biases of log and log c we show that the difference c is remarkably of order 1 and we give a simple formula for the leading part of this difference simulations indicate that in many cases this formula is very accurate and that confidence intervals based on the asymptotic distribution have excellent coverage an extension to product binomial models is given

efficiency of martingale methods in recapture studies summary the asymptotic variances of various estimators of the population size from a recapture study are calculated the estimation methods compared are maximum likelihood and a class of natural martingale estimators with both time variation and behavioural response it is found that i the martingale methods appropriate for time variation or behavioural response are fully efficient ii there is no loss of efficiency at all in allowing for time variation iii there is large loss of efficiency in allowing for behavioural response iv there is a natural martingale method that allows for both time variation and behavioural response

quantifying precision of mark recapture estimates using the bootstrap and related methods summary bootstrap techniques yield variance estimates under any model for which parameter estimates can be calculated and are useful in cases where analytic variances are not available in closed form or are available only if more restrictive assumptions are made here the application of bootstrap techniques to mark recapture models is discussed the approach also allows generation of robust confidence intervals which extend beyond the permissible parameter range only if the mark recapture model itself allows out of range parameter estimates if an animal population is assumed to be closed i e no death birth or migration two further methods of obtaining confidence limits for population size are suggested the first is based on a robbins monro search for each limit and the second applies the concept of a randomisation or permutation test in the absence of nuisance parameters both methods are exact apart from monte carlo variation and the limitations imposed by a discrete distribution for the second if all possible permutations are enumerated monte carlo variation is eliminated

the necessary and sufficient condition for global stability of a lotka volterra cooperative or competition system with delays abstract this paper presents the necessary and sufficient condition for the global stability of the following lotka volterra cooperative or competition system with time delays x t x t r 1 ax t x t 11 y t 12 y t y t r 2 ay t x t 21 y t 22 it is showed that the positive equilibrium of the system is globally asymptotically stable for all delays ij 0 i j 1 2 if and only if

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

invariant differential operators on a reductive lie algebra and weyl group representations let v be a finite dimensional vector space and let g be a subgroup of gl v set d v equal to the algebra of differential operators on v with polynomial coefficients and d v g equal to the g invariants in d v if g is a reductive lie algebra over c then t c g is a cartan subgroup of g and if g is the adjoint group of g then w is the weyl group of g harish chandra introduced an algebra homomorphism 5 of d g g to d t w h3 a is given by the obvious restriction mapping on the subalgebra of invariant polynomials and on the invariant constant coefficient differential operators and ker is

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

some practical aspects of a conditional likelihood approach to capture experiments the use of conditional likelihood methods in the analysis of capture data allows the modeling of capture probabilities in terms of observable characteristics of the captured individuals and the trapping occasions the resulting models may then be used to estimate the size of the population here the use of conditional likelihood procedures to construct models for capture probabilities is discussed and illustrated by an example

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