Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation

In this paper, we discuss the multi-group SEIR and SIR model with random perturbation, allowing random fluctuation around the endemic equilibrium of deterministic SEIR and SIR models. We prove that the endemic equilibriums of the two models are both stochastic asymptotically stable. In addition, the stability condition is obtained by the construction of the Lyapunov function and graph theory. Finally, numerical simulations are presented to illustrate our mathematical findings.     

Global dynamics of a multistage SIR model with distributed delays and nonlinear incidence rate

In this paper, a multistage susceptible‐infectious‐recovered model with distributed delays and nonlinear incidence rate is investigated, which extends the model considered by Guo et al. [H. Guo, M. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J. Appl. Math., 72 (2012), 261–279]. Under some appropriate and realistic conditions, the global dynamics is completely determined by the basic reproduction number R₀. If R₀≤1, then the infection‐free equilibrium is globally asymptotically stable and the disease dies out in all stages. If R₀>1, then a unique endemic equilibrium exists, and it is globally asymptotically stable, and hence the disease persists in all stages. The results are proved by utilizing the theory of non‐negative matrices, Lyapunov functionals, and the graph‐theoretical approach. Copyright © 2016 John Wiley & Sons, Ltd.     

一类带有一般接触率和常数输入的流行病模型的全局分析

借助极限系统理论和构造适当的Liapunov函数,对带有一般接触率和常数输入的SIR型和SIRS型传染病模型进行讨论．当无染病者输入时,地方病平衡点存在的阈值被找到A·D2对相应的SIR模型,关于无病平衡点和地方病平衡点的全局渐近稳定性均得到充要条件;对相应的SIRS模型,得到无病平衡点和地方病平衡点全局渐近稳定的充分条件．当有染病者输入时,模型不存在无病平衡点．对相应的SIR模型,地方病平衡点是全局渐近稳定的;对相应的SIRS模型,得到地方病平衡点全局渐近稳定的充分条件．     

Globally stable endemicity for infectious diseases with information-related changes in contact patterns

SIR and SIS epidemic models with information—related changes in contact patterns are introduced. The global stability analysis of the endemic equilibrium is performed by means of the Li–Muldowney geometric approach. Biological implications of the stability conditions are given.     

The long time behavior of DI SIR epidemic model with stochastic perturbation

In this paper, we present a DI SIR epidemic model with two categories stochastic perturbations. The long time behavior of the two stochastic systems is studied. Mainly, we show how the solution goes around the infection-free equilibrium and the endemic equilibrium of deterministic system under different conditions.     

一类传染病模型的稳定性分析

考虑了一个带有时滞的SIR模型.通过分析特征根的分布讨论了无疾病解和地方病解的稳定性,以及在地方病解附近Hopf分支的存在性.最后,利用数值模拟验证我们的理论结果.     

Stochastically perturbed vector-borne disease models with direct transmission

In this paper we study a stochastic epidemic model of vector-borne diseases with direct mode of transmission and its delay modification. More precisely, we extend the deterministic epidemic models by introducing random perturbations around the endemic equilibrium state. By using suitable Lyapunov functions and functionals, we obtain stability conditions for the considered models and study the effect of the delay on the stability of the endemic equilibrium. Finally, numerical simulations for the stochastic model of malaria disease transmission are presented to illustrate our mathematical findings.     

Two-group SIR epidemic model with stochastic perturbation

A stochastic two-group SIR model is presented in this paper.The existence and uniqueness of its nonnegative solution is obtained,and the solution belongs to a positively invariant set.Furthermore,the globally asymptotical stability of the disease-free equilibrium is deduced by the stochastic Lyapunov functional method if R0 ≤ 1,which means the disease will die out.While if R0 1,we show that the solution is fluctuating around a point which is the endemic equilibrium of the deterministic model in time average.In addition,the intensity of the fluctuation is proportional to the intensity of the white noise.When the white noise is small,we consider the disease will prevail.At last,we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.     

具有接种的确定性和随机SIS流行病模型的全局稳定性

We study the stability of endemic equilibriums of the deterministic and stochastic SIS epidemic models with vaccination. The deterministic SIS epidemic model with vaccination was proposed by Li and Ma(2004), for which some sufficient conditions for the global stability of the endemic equilibrium were given in some earlier works. In this paper, we first prove by Lyapunov function method that the endemic equilibrium of the deterministic model is globally asymptotically stable whenever the basic reproduction number is larger than one. For the stochastic version, we obtain some sufficient conditions for the global stability of the endemic equilibrium by constructing a class of different Lyapunov functions.     

A class of Lyapunov functions and the global stability of some epidemic models with nonlinear incidence

In this paper, by investigating an SIR epidemic model with nonlinear incidence, we present a new technique for proving the global stability of the endemic equilibrium, which consists of introducing a variable transformation and constructing a more general Lyapunov function. For the model we obtain the following results. The disease-free equilibrium is globally stable in the feasible region as the basic reproduction number is less than or equal to unity, and the endemic equilibrium is globally stable in the feasible region as the basic reproduction number is greater than unity.The generality of the technique is illustrated by considering certain nonlinear incidences and SIS and SIRS epidemic models.     

一类具有分布时滞和非线性发生率的媒介传染病模型的全局稳定性

建立了一类具有分布时滞和非线性发生率的SIR媒介传染病模型,分析得到了决定疾病是否一致持续存在的基本再生数．而且当基本再生数不大于1时,疾病最终灭绝;当基本再生数大于1时,模型存在惟一的地方病平衡点,并且疾病一致持续存在于种群之中．通过构造Lyapunov泛函,证明了在一定条件下地方病平衡点只要存在就全局稳定．同时指出了证明地方病平衡点全局稳定时可适用的Lyapunov泛函的不惟一性.     

具有免疫接种且总人口规模变化的SIR传染病模型的稳定性

讨论一类具有预防免疫接种且有效接触率依赖于总人口的SIR传染病模型,给出了决定疾病灭绝和持续生存的基本再生数σ的表达式,在一定条件下证明了疾病消除平衡点的全局稳定性,得到了唯一地方病平衡点的存在性和局部渐近稳定性条件.最后研究了具有双线性传染率和标准传染率的两个具体模型,并证明了当σ>1时该模型地方病平衡点的全局渐近稳定性.     

Stability analysis for differential infectivity epidemic models

We present several differential infectivity (DI) epidemic models under different assumptions. As the number of contacts is assumed to be constant or a linear function of the total population size, either standard or bilinear incidence of infection is resulted. We establish global stability of the infection-free equilibrium and the endemic equilibrium for DI models of SIR (susceptible/infected/removed) type with bilinear incidence and standard incidence but no disease-induced death, respectively. We also obtain global stability of the two equilibria for a DI SIS (susceptible/infected/susceptible) model with population-density-dependent birth and death functions. For completeness, we extend the stability of the infection-free equilibrium for the standard DI SIR model previously proposed.     

Dynamics of a Diffusive SIR Epidemic Model with Time Delay

This paper is devoted to a reaction-diffusion system for a SIR epidemic model with time delay and incidence rate. Firstly, the nonnegativity and boundedness of solutions determined by nonnegative initial values are obtained. Secondly, the existence and local stability of the disease-free equilibrium as well as the endemic equilibrium are investigated by analyzing the characteristic equations. Finally, the global asymptotical stability are obtained via Lyapunov functionals.     

Noise-induced Transitions for an SIV Epidemic Model with Medical-resource Constraints

We consider a stochastically forced epidemic model with medical-resource constraints. In the deterministic case, the model can exhibit two type bistability phenomena, i.e., bistability between an endemic equilibrium or an interior limit cycle and the disease-free equilibrium, which means that whether the disease can persist in the population is sensitive to the initial values of the model. In the stochastic case, the phenomena of noise-induced state transitions between two stochastic attractors occur. Namely, under the random disturbances, the stochastic trajectory near the endemic equilibrium or the interior limit cycle will approach to the disease-free equilibrium. Besides, based on the stochastic sensitivity function method, we analyze the dispersion of random states in stochastic attractors and construct the confidence domains (confidence ellipse or confidence band) to estimate the threshold value of the intensity for noise caused transition from the endemic to disease eradication.     

一类带时滞的SIR传染病模型的稳定性

研究了一类带时滞的SIR传染病模型,利用多项式判别系统研究了无病平衡点的全时滞稳定性,利用超越函数零点判别法研究了正平衡点的局部渐近稳定性.     

Global stability for a multi-group SIRS epidemic model with varying population sizes

In this paper, by extending well-known Lyapunov function techniques to SIRS epidemic models, we establish sufficient conditions for the global stability of an endemic equilibrium of a multi-group SIRS epidemic model with varying population sizes which has cross patch infection between different groups. Our proof no longer needs such a grouping technique by graph theory commonly used to analyze the multi-group SIR models.     

Mathematical Analysis of SIR Epidemic Model with Piecewise Infection Rate and Control Strategies

The limitation of contact between susceptible and infected individuals plays an important role in decreasing the transmission of infectious diseases. Prevention and control strategies contribute to minimizing the transmission rate. In this paper, we propose SIR epidemic model with delayed control strategies, in which delay describes the response and effect time. We study the dynamic properties of the epidemic model from three aspects: steady states, stability and bifurcation. By eliminating the existence of limit cycles, we establish the global stability of the endemic equilibrium, when the delay is ignored. Further, we find that the delayed effect on the infection rate does not affect the stability of the disease-free equilibrium, but it can destabilize the endemic equilibrium and bring Hopf bifurcation. Theoretical results show that the prevention and control strategies can effectively reduce the final number of infected individuals in the population. Numerical results corroborate the theoretical ones.     

Permanence of a delayed SIR epidemic model with general nonlinear incidence rate

In this paper, a SIR model with two delays and general nonlinear incidence rate is considered. The local and global asymptotical stabilities of the disease‐free equilibrium are given. The local asymptotical stability and the existence of Hopf bifurcations at the endemic equilibrium are also established by analyzing the distribution of the characteristic values. Furthermore, the sufficient conditions for the permanence of the system are given. Some numerical simulations to support the analytical conclusions are carried out. At last, some conclusions are given. Copyright © 2014 John Wiley & Sons, Ltd.     

Lyapunov functions for SIR and SIRS epidemic models

In this paper, we construct a new Lyapunov function for a variety of SIR and SIRS models in epidemiology. Global stability of the endemic equilibrium states of these systems is established.