Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence

In this paper, the asymptotic behavior of solutions of an autonomous SEIRS epidemic model with the saturation incidence is studied. Using the method of Liapunov–LaSalle invariance principle, we obtain the disease-free equilibrium is globally stable if the basic reproduction number is not greater than one. Moreover, we show that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions of locally and globally asymptotically stable convergence to an endemic equilibrium are obtained base on the permanence.     

Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response

In this paper, we study the global dynamics of a viral infection model with a latent period. The model has a nonlinear function which denotes the incidence rate of the virus infection in vivo. The basic reproduction number of the virus is identified and it is shown that the uninfected equilibrium is globally asymptotically stable if the basic reproduction number is equal to or less than unity. Moreover, the virus and infected cells eventually persist and there exists a unique infected equilibrium which is globally asymptotically stable if the basic reproduction number is greater than unity. The basic reproduction number determines the equilibrium that is globally asymptotically stable, even if there is a time delay in the infection.     

Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates

In this paper, we introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that global dynamics are completely determined by the basic reproduction number R₀. It shows that, the basic reproduction number R₀ is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result.     

Threshold dynamics for a cholera epidemic model with periodic transmission rate

A cholera epidemic model with periodic transmission rate is presented. The basic reproduction number is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the cholera eventually disappears if the basic reproduction number is less than one. And if the basic reproduction number is greater than one, there exists a positive periodic solution which is globally asymptotically stable. Numerical simulations are provided to illustrate analytical results.     

Global properties of an improved hepatitis B virus model

Hepatitis B virus (HBV) infection is an important health problem worldwide. In this paper, we introduce an improved HBV model with standard incidence function and cytokine-mediated ‘cure’ based on empirical evidences. By carrying out a global analysis of the modified model and studying the stability of the equilibria, we show that infection-free equilibrium is globally asymptotically stable if the basic reproduction number of virus is less than one and, conversely, the infection equilibrium is globally asymptotically stable if the basic reproduction number of virus is greater than one. The study and information derived from this model and other related models may have an important impact on preventing mortality due to hepatitis B virus in the future.     

具有标准发生率和因病死亡率的离散SIS传染病模型的全局稳定性分析

主要研究了具有标准发生率和因病死亡率的离散SIS传染病模型的动力学性质,利用构造Lyapunov函数,得到模型无病平衡点和地方性平衡点的全局稳定性,即无病平衡点是全局渐近稳定的当且仅当基本再生数R_0≤1,地方病平衡点是全局渐近稳定的当且仅当R_0>1.     

一类带有非线性传染率的SEIS传染病模型的定性分析

借助极限理论和Fonda定理,研究了一类既有常数输入率又有因病死亡率的SEIS传染病模型．所考虑模型的传染率是非线性的,并且得到了该模型的基本再生数,当基本再生数小于1时,该模型仅存在唯一的无病平衡点,它是全局渐近稳定的,且疾病最终灭绝．当基本再生数大于1时,该模型除存在不稳定的无病平衡点外,还存在唯一的局部渐近稳定的地方病平衡点,并且疾病一致持续存在．     

Global stability of a delayed epidemic model with latent period and vaccination strategy

In this paper, a mathematical model describing the transmission dynamics of an infectious disease with an exposed (latent) period and waning vaccine-induced immunity is investigated. The basic reproduction number is found by applying the method of the next generation matrix. It is shown that the global dynamics of the model is completely determined by the basic reproduction number. By means of appropriate Lyapunov functionals and LaSalle’s invariance principle, it is proven that if the basic reproduction number is less than or equal to unity, the disease-free equilibrium is globally asymptotically stable and the disease fades out; and if the basic reproduction number is greater than unity, the endemic equilibrium is globally asymptotically stable and therefore the disease becomes endemic.     

Stability and Hopf Bifurcation of a Virus Infection Model with a Delayed CTL Immune Response

In this paper, a virus infection model with standard incidence rate and delayed CTL immune response is investigated. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the CTL-activated infection equilibrium are established, respectively. By means of comparison arguments, it is verified that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio is less than unity. By using suitable Lyapunov functional and LaSalle＇s invariance principle, it is shown that the CTL-inactivated infection equilibrium of the system is globally asymptotically stable if tile immune response reproduction ratio is less than unity and the basic reproduction ratio is greater than unity. Numerical simulations are carried out to illustrate the theoretical result.     

带有非线性传染率的传染病模型

对一类带有非线性传染率的SEIS传染病模型,找到了其基本再生数.借助动力系统极限理论,得到当基本再生数小于1时,无病平衡点是全局渐近稳定的,且疾病最终灭绝.当基本再生数大于1时,无病平衡点是不稳定的,而唯一的地方病平衡点是局部渐近稳定的.应用Fonda定理,得到当基本再生数大于1时疾病一致持续存在.     

Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination

This paper deals with global dynamics of an SIRS epidemic model for infections with non permanent acquired immunity. The SIRS model studied here incorporates a preventive vaccination and generalized non-linear incidence rate as well as the disease-related death. Lyapunov functions are used to show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one, and that there is an endemic equilibrium state which is globally asymptotically stable when it is greater than one.     

A delayed SIR epidemic model with saturation incidence and a constant infectious period

In this paper, an SIR epidemic model with saturation incidence and a time delay describing a constant infectious period is investigated. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. When the basic reproduction number is greater than unity, it is proved that the disease is uniformly persistent in the population, and explicit formulae are obtained to estimate the eventual lower bound of the fraction of infectious individuals. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are derived for the global attractiveness of the endemic equilibrium. Numerical simulations are carried out to illustrate the main results.     

Stability and Hopf bifurcation of a delayed epidemic model with stage structure and nonlinear incidence rate

In this paper, a stage‐structured SI epidemic model with time delay and nonlinear incidence rate is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease‐free equilibrium, and the existence of Hopf bifurcations are established. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease‐free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Stability Analysis of an SEIS Epidemic Model with Nonlinear Incidence and Time Delay

In this paper, an SEIS epidemic model with nonlinear incidence and time delay is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the model is established. By using suitable Lyapunov functional and LaSalle's invariance principle, it is shown that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity. If the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are derived for the global stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the theoretical results.     

GLOBAL STABILITY OF AN SVIR EPIDEMIC MODEL WITH VACCINATION

In this paper, an SIR epidemic model with vaccination for both the newborns and susceptibles is investigated, where it is assumed that the vaccinated individuals have the temporary immunity. The basic reproduction number determining the extinction or persistence of the infection is found. By constructing a Lyapunov function, it is proved that the disease free equilibrium is globally stable when the basic reproduction number is less than or equal to one, and that the endemic equilibrium is globally stable wh...     

一类SEIV模型的研究(英文)

本文主要研究一类带有饱和感染率且潜伏期也具有传染性的SEIV模型.运用微分方程中的极限理论和Busenberg-Driessche定理,建立了该模型的全局动力学性质;并且证明了当基本再生数R0≤1时,无病平衡点Q0是全局稳定的,当基本再生数R01时,疾病持续.     

Mathematical model for assessing the impact of vaccination and treatment on measles transmission dynamics

A deterministic model for the transmission dynamics of measles in a population with fraction of vaccinated individuals is designed and rigorously analyzed. The model with standard incidence exhibits the phenomenon of backward bifurcation, where a stable disease‐free equilibrium coexists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. This phenomenon can be removed if either measles vaccine is assumed to be perfect or disease related mortality rates are negligible. In the latter case, the disease‐free equilibrium is shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a nonlinear Lyapunov function of Goh‐Volterra type, to be globally asymptotically stable for a special case.     

Global stability of a stage-structured epidemic model with a nonlinear incidence

In this paper, a stage-structured epidemic model with a nonlinear incidence with a factor S^p is investigated. By using limit theory of differential equations and Theorem of Busenberg and van den Driessche, global dynamics of the model is rigorously established. We prove that if the basic reproduction number R₀ is less than one, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Numerical simulations support our analytical results and illustrate the effect of p on the dynamic behavior of the model.     

Analysis of a viral infection model with immune impairment,intracellular delay and general non-linear incidence rate

In this article we study the dynamical behaviour of a intracellular delayed viral infection with immune impairment model and general non-linear incidence rate. Several techniques, including a non-linear stability analysis by means of the Lyapunov theory and sensitivity analysis, have been used to reveal features of the model dynamics. The classical threshold for the basic reproductive number is obtained: if the basic reproductive number of the virus is less than one, the infection-free equilibrium is globally asymptotically stable and the infected equilibrium is globally asymptotically stable if the basic reproductive number is higher than one.