Dynamic behaviour of a discrete-time SIR model with information dependent vaccine uptake

This paper proposes and analyzes a discrete-time deterministic SIR model with information dependent immunization behaviour, where vaccination coverage at birth during any period of time is a general phenomenological function of the risk of infection that is perceived at the beginning of the period. Results on existence of equilibria, their local stability, and system persistence are proved. Then, by considering the noteworthy subcase of a piecewise linear ‘prevalence-dependent’ coverage function, the local stability of the endemic state is proved and conditions for its global asymptotic stability are given. Some insight on both Neimarck-Sacher and period-doubling bifurcations are provided. Overall we show that prevalence-dependent coverage is an essentially stabilising force. However period-doubling bifurcations are possible though under stressed parameter constellations.     

Global behaviour of an SIR epidemic model with time delay

We study the stability of a delay susceptible–infective–recovered epidemic model with time delay. The model is formulated under the assumption that all individuals are susceptible, and we analyse the global stability via two methods—by Lyapunov functionals, and—in terms of the variance of the variables. The main theorem shows that the endemic equilibrium is stable. If the basic reproduction number ?₀ is less than unity, by LaSalle invariance principle, the disease‐free equilibrium E_s is globally stable and the disease always dies out. By applying the integral averaging theory, we also investigate the stability in variance of the model. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

一类具时滞SIR传染病模型的动力行为

分析传染病模型的稳定性,并考虑到已感染者对易感染者的作用的时滞影响.文中首先在R_01时,构造一个Lyapunov泛函,证明了无病平衡点的全局渐近稳定性.当R_01时,证明了正平衡点的局部渐近稳定性和持久性.     

一类具有非线性传染率的时滞SIR模型的分析

分析并建立具有时滞及非线性传染率的SIR传染病模型.通过分析在无病平衡点和正平衡点处的特征方程,可得到在这两个平衡点处的局部渐近稳定性,然后我们得到了系统在两个平衡点处的全局渐近稳定性,最后我们证明了系统的持久性.     

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.     

Stability analysis of a stochastic delayed SIR epidemic model with general incidence rate

In this paper, a stochastic delayed epidemic model with a generalized incidence rate is proposed and discussed. The positivity of solutions is established. A linearized form of the model is given and the stability conditions of the endemic equilibrium are obtained by using the technique of Lyapunov functionals.     

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.     

Permanence of an SIR epidemic model with density dependent birth rate and distributed time delay

In this paper, we investigate the permanence of an SIR epidemic model with a density-dependent birth rate and a distributed time delay. We first consider the attractivity of the disease-free equilibrium and then show that for any time delay, the delayed SIR epidemic model is permanent if and only if an endemic equilibrium exists. Numerical examples are given to illustrate the theoretical analysis. The results obtained are also compared with those from the analog system with a discrete time delay.     

Dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate

In this paper, the dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate are analyzed. The existence and uniqueness of the global positive solution is proved by constructing the equivalent system without pulses. The threshold which determines the extinction and persistence of the disease is obtained. The global attraction of disease-free periodic solution is addressed. Sufficient condition for the existence of a positive periodic solution is established. These results are supported by computer simulations.     

Local stability of an SIR epidemic model and effect of time delay

We describe an SIR epidemic model with a discrete time lag, analyse the local stability of its equilibria as well as the effects of delay on the reproduction number and on the dynamical behaviour of the system. The model has two equilibria—a necessary condition for local asymptotic stability is given. The proofs are based on linearization and the application of Lyapunov functional approach. An upper bound of the critical time delay for which the model remains valid is derived. Numerical simulations are carried out to illustrate the effect of time delay which tends to reduce the epidemic threshold. Copyright © 2009 John Wiley & Sons, Ltd.     

基于两斑块和人口流动的SIR传染病模型的稳定性

根据传染病动力学原理,考虑人口在两斑块上流动且具有非线性传染率,建立了一类基于两斑块和人口流动的SIR传染病模型.利用常微分方程定性与稳定性方法,分析了模型永久持续性和非负平衡点的存在性,通过构造适当的Lyapunov函数和极限系统理论,获得无病平衡点和地方病平衡点全局渐近稳定的充分条件.研究结果表明:基本再生数是决定疾病流行与否的阈值,当基本再生数小于等于1时,感染者逐渐消失,病毒趋于灭绝;当基本再生数大于1并满足永久持续条件时,感染者持续存在且病毒持续流行并将成为一种地方病.     

The effect of impulsive vaccination on an SIR epidemic model

In this paper, an SIR epidemic model is constructed and analyzed. We get the result that if the parameters satisfy the condition β>α+γ+b, then the disease will be ultimately permanent. Under this condition, we consider how the impulsive vaccination affects the original system. The sufficient condition for the global asymptotical stability of the disease-eradication solution is obtained. We also get that if the impulsive vaccination rate is less than some value, the disease will be permanent, and the disease cannot be controlled. People can select appropriate vaccination rate according to our theoretical result to control diseases.     

Numerical modelling of an SIR epidemic model with diffusion

A spatial SIR reaction-diffusion model for the transmission disease such as whooping cough is studied. The behaviour of positive solutions to a reaction-diffusion system with homogeneous Neumann boundary conditions are investigated. Sufficient conditions for the local and global asymptotical stability are given by linearization and by using Lyapunov functional. Our result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small. These results are verified numerically by constructing, and then simulating, a robust implicit finite-difference method. Furthermore, the new implicit finite-difference method will be seen to be more competitive (in terms of numerical stability) than the standard finite-difference method.     

The Longtime Behavior of an SIR Epidemic Model with Free Boundaries

In \cite{KLZ2013} , the authors studied an SIR epidemic model with free boundary. They first proved the global existence and uniqueness of solution, and then gave the criteria for spreading and vanishing. They also obtained the longtime behavior for the case of vanishing. However, the longtime behavior when spreading happens remains open. In this short paper, we aim to solve this open problem.