Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks

In this paper, we study the spreading of infections in complex heterogeneous networks based on an SIRS epidemic model with birth and death rates. We find that the dynamics of the network-based SIRS model is completely determined by a threshold value. If the value is less than or equal to one, then the disease-free equilibrium is globally attractive and the disease dies out. Otherwise, the disease-free equilibrium becomes unstable and in the meantime there exists uniquely an endemic equilibrium which is globally asymptotically stable. A series of numerical experiments are given to illustrate the theoretical results. We also consider the SIRS model in the clustered scale-free networks to examine the effect of network community structure on the epidemic dynamics.     

Dynamical behavior of computer virus on Internet

In this paper, we presented a computer virus model using an SIRS model and the threshold value R₀ determining whether the disease dies out is obtained. If R₀ is less than one, the disease-free equilibrium is globally asymptotically stable. By using the time delay as a bifurcation parameter, the local stability and Hopf bifurcation for the endemic state is investigated. Numerical results demonstrate that the system has periodic solution when time delay is larger than a critical values. The obtained results may provide some new insight to prevent the computer virus.     

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.     

GLOBAL STABILITY OF AN SIRS EPIDEMIC MODEL WITH DELAYS

In this article, an SIRS epidemic model spread by vectors (mosquitoes) which have an incubation time to become infectious is formulated. It is shown that a disease-free equilibrium point is globally stable if no endemic equilibrium point exists. Further, the endemic equilibrium point (if it exists) is globally stable with a respect "weak delay". Some known results are generalized.     

Susceptible-infected-recovered models with natural birth and death on complex networks

This paper proposes two modified susceptible-infected-recovered (SIRS) models on homogenous and heterogeneous networks, respectively. In the study of the homogenous network model, Lyapunov functions are used to study the globally asymptotically stable of the equilibria of the model. It is proved that if the basic reproduction number of the model is less than one, then the disease-free equilibrium is globally asymptotically stable, otherwise, the endemic equilibrium is globally asymptotically stable. In the study of the heterogeneous network model, the existences of the disease-free equilibrium and epidemic equilibrium of the model are discussed. A threshold value is given. It is proved that if the threshold value of the model is less than one, then the disease-free equilibrium is globally asymptotically stable. The simulation examples on the two SIRS models are given. Copyright © 2013 John Wiley & Sons, Ltd.     

Lyapunov functions for a dengue disease transmission model

In this paper, we study a model for the dynamics of dengue fever when only one type of virus is present. For this model, Lyapunov functions are used to show that when the basic reproduction ratio is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is also globally asymptotically stable.     

QUALITATIVE ANALYSIS OF AN EPIDEMIC MODEL WITH VACCINATION

An SIRS epidemic modei with vaccination, temporary immunity and vary-ing total population size is considered. The threshold of existence of endemic equilibrium is found. The disease-free equilibrium is globally asymptotically stable below the threshold, the endemic equilibrium is globally asymptotically stable above the threshold.     

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．     

Global asymptotic stability of a generalized SIRS epidemic model with transfer from infectious to susceptible

In this paper, we propose a generalized SIRS epidemic model with varying total population size caused by the death rate due to the disease and transfer from infectious to susceptible, where the incidence rate employed includs a wide range of monotonic and concave incidence rates. Applying the geometric approach developed by Smith, Li and Muldowey, we prove that the endemic equilibrium is globally asymptotically stable provided that the rate of loss of immuity $\delta$ is in a critical interval $[\eta,\bar\delta)$ when the basic reproduction number $R_0$ is greater than unity.     

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SIRS传染病模型

This paper considers an SIRS epidemic model that incorporates constant immigration rate, a general population-size dependent contact rate and proportional transfer rate from the infective class to susceptible class. A threshold parameter a is identified. If σ≤1, the disease-free equilibrium is globally stable. If σ＞1, a unique endemic equilibrium is locally asymptotically stable. For two important special cases of mass action incidence and standard incidence,global stability of the endemic equilibrium is proved provided the threshold is larger than unity. Some previous results are extended and improved.     

GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS＊

In this article,we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlinear incidence rates and distributed...     

Global stability of nonresident computer virus models

In this paper, applying Lyapunov functional techniques to nonresident computer virus models, we establish global dynamics of the model whose threshold parameter is the basic reproduction number R₀ such that the virus‐free equilibrium is globally asymptotically stable when R₀ ≤ 1, and the infected equilibrium is globally asymptotically stable when R₀ > 1 under the same restricted condition on a parameter, which appeared in the literature on delayed susceptible‐infected‐recovered‐susceptible (SIRS) epidemic models. We use new techniques on permanence and global stability of this model for R₀ > 1. Copyright © 2014 John Wiley & Sons, Ltd.     

一类具有非线性发生率的时滞传染病模型的全局稳定性

充分考虑人口统计效应、疾病的潜伏期与传播规律的复杂性,研究了一类具有非线性发生率的时滞SIRS传染病模型的动力学行为.通过分析对应的线性化近似系统的特征方程,证明了无病平衡点的局部稳定性.利用Lyapunov-LaSalle不变集原理,当基本再生数R₀<1时,证明了无病平衡点是全局渐近稳定的;当R₀>1时,得到了地方病平衡点全局渐近稳定的充分条件.所得结论可为人们有效预防和控制传染病传播提供一定的理论依据.     

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.     

Stability of a delayed SIRS epidemic model with a nonlinear incidence rate

In this paper, an SIRS epidemic model with a nonlinear incidence rate and a time delay is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. By comparison arguments, it is proved that if the basic reproductive number R₀<1, the disease-free equilibrium is globally asymptotically stable. If R₀>1, by means of an iteration technique, sufficient conditions are derived for the global asymptotic stability of the endemic equilibrium.     

两类带有确定潜伏期的SEIS传染病模型的分析

通过研究两类带有确定潜伏期的SEIS传染病模型,发现对种群的常数输入和指数输入会使疾病的传播过程产生本质的差异．对于带有常数输入的情形,找到了地方病平衡点存在及局部渐近稳定的阈值,证明了地方病平衡点存在时一定局部渐近稳定,并且疾病一致持续存在．对于带有指数输入的情形,发现地方病平衡点当潜伏期充分小时是局部渐近稳定的,当潜伏期充分大时是不稳定的．     

Global stability of a virus infection model with two delays and two types of target cells

In this paper, incorporating the delay of viral cytopathicity within target cells, we first presented a basic model of viral infection with delay, and then extended it into a model with two delays and two types of target cells. For the models proposed here, both their basic reproduction numbers are found. By constructing Lyapunov functionals, necessary and sufficient conditions ensuring the global stability of the models with delays are given. The obtained results show that, when the basic reproduction number is not greater than one, the infection-free equilibrium is globally stable in the feasible region, which implies that the virus infection goes extinct eventually; when it is greater than one, the infection equilibrium is globally stable in the feasible region, which implies that the virus infection persists in the body of host.     

Dynamics of an age‐structured host‐vector model for malaria transmission

In this paper, we propose a host‐vector model for malaria transmission by incorporating infection age in the infected host population and nonlinear incidence for transmission from infectious vectors to susceptible hosts. One novelty of the model is that the recovered hosts only have temporary immunity and another is that successfully recovered infected hosts may become susceptible immediately. Firstly, the existence and local stability of equilibria is studied. Secondly, rigorous mathematical analyses on technical materials and necessary arguments, including asymptotic smoothness and uniform persistence of the system, are given. Thirdly, by applying the fluctuation lemma and the approach of Lyapunov functionals, the threshold dynamics of the model for a special case were established. Roughly speaking, the disease‐free equilibrium is globally asymptotically stable when the basic reproduction number is less than one and otherwise the endemic equilibrium is globally asymptotically stable when no reinfection occurs. It is shown that the infection age and nonlinear incidence not only impact on the basic reproduction number but also could affect the values of the endemic steady state. Numerical simulations were performed to support the theoretical 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.