The effect of constant and pulse vaccination on an SIR epidemic model with infectious period

In this paper, we investigate two delayed SIR models with vaccination and a generalized nonlinear incidence and obtain sufficient conditions for eradication and permanence of the disease, respectively. Our results indicate that a larger vaccination rate will lead to the eradication of a disease. Furthermore, theoretical results show that constant vaccination strategy can lead to disease eradication at relatively low values of vaccination than pulse vaccination strategy, which is different from the results in [1]. In addition, numerical simulations indicate that pulse vaccination strategy or a longer infectious period will make a larger fraction of population infected by disease.     

Pulse vaccination delayed SEIRS epidemic model with saturation incidence

A delayed SEIRS epidemic model with pulse vaccination and saturation incidence rate is investigated. Using the discrete dynamical system determined by the stroboscopic map, we obtain the existence of the disease-free periodic solution and its exact expression. Further, using the comparison theorem, we establish the sufficient conditions of global attractivity of the disease-free periodic solution and the permanence of disease. Our results indicate that a long latent period of the disease or a proper pulse vaccination rate will lead to eradication of the disease.     

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.     

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.     

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.     

SIR epidemic models with general infectious period distribution

We show how epidemics in which individuals’ infectious periods are not necessarily exponentially distributed may be naturally modelled as piecewise deterministic Markov processes. For the standard susceptible–infective–removed (SIR) model, we exhibit a family of martingales which may be used to derive the joint distribution of the number of survivors of the epidemic and the area under the trajectory of infectives. We also show how these results may be extended to a model in which the rate at which an infective generates infectious contacts may be an arbitrary function of the number of susceptible individuals present.     

Analysis of a delayed HIV/AIDS epidemic model with saturation incidence

In this paper, a delayed HIV/AIDS epidemic model with saturation incidence is proposed and analyzed. The equilibria and their stability are investigated. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. It is found that if the threshold R ₀<1, then the disease-free equilibrium is globally asymptotically stable, and if the threshold R ₀>1, the system is permanent and the endemic equilibrium is asymptotically stable under certain conditions.     

Global dynamics of a delayed SEIS infectious disease model with logistic growth and saturation incidence

In this paper, a delayed Susceptible‐Exposed‐Infectious‐Susceptible (SEIS) infectious disease model with logistic growth and saturation incidence is investigated, where the time delay describes the latent period of the disease. By analyzing corresponding characteristic equations, the local stability of a disease‐free equilibrium and an endemic equilibrium is discussed. The existence of Hopf bifurcations at the endemic equilibrium is established. By using the persistence theory for infinite dimensional dynamic systems, it is proved that if the basic reproduction number is greater than unity, the system is permanent. By means of suitable Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global stability of the disease‐free equilibrium and the endemic equilibrium, respectively. Numerical simulations are carried out to illustrate the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Global attractivity and permanence of a delayed SVEIR epidemic model with pulse vaccination and saturation incidence

In this study, we formulate and analyze a new SVEIR epidemic disease model with time delay and saturation incidence, and analyze the dynamic behavior of the model under pulse vaccination. Using the discrete dynamical system determined by the stroboscopic map, we obtain an ‘infection-free’ periodic solution, further, show that the ‘infection-free’ periodic solution is globally attractive for some parameters of the model under appropriate conditions. The permanence of the model is investigated analytically. By computer simulation it is concluded that a large vaccination rate or a short pulse of vaccination or a long latent period are each a sufficient condition for the extinction of the disease.     

Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies

In this paper, we study the traveling wave solutions of a delayed diffusive SIR epidemic model with nonlinear incidence rate and constant external supplies. We find that the existence of traveling wave solutions is determined by the basic reproduction number of the corresponding spatial‐homogenous delay differential system and the minimal wave speed. The existence is proved by applying Schauder's fixed point theorem and Lyapunov functional method. The non‐existence of traveling waves is obtained by two‐sided Laplace transform. Copyright © 2016 John Wiley & Sons, Ltd.     

Global dynamics and bifurcation in delayed SIR epidemic model

An SIR epidemic model with time delay, information variable and saturated incidence rate, where the susceptibles are assumed to satisfy the logistic equation and the incidence term, is of saturated form with the susceptibles. This model exhibits two bifurcations, one is transcritical bifurcation and the other is Hopf bifurcation. The local and global stability of endemic equilibrium is also discussed. Finally, numerical simulations are carried out to explain the mathematical conclusions.     

Dynamics of a delayed epidemic model with non-monotonic incidence rate

A delayed epidemic model with non-monotonic incidence rate which describes the psychological effect of certain serious on the community when the number of infectives is getting larger is studied. The disease-free equilibrium is globally asymptotically stable when R₀<1 and is globally attractive when R₀=1 are derived. On the other hand, The disease is permanent when R₀>1 is also obtained. Numerical simulation results are given to support the theoretical predictions.     

具有常数输入的非自治SIR流行病模型周期解的存在性

利用MAWHIN重合度理论中的延拓定理研究了一类具有常数输入的非自治SIR流行病模型的非平凡周期解的存在性.并用MatLab对其进行了数值模拟,作出了模型的相图和解曲线图形.     

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.     

具有常数迁入率和非线性传染率βI~pS~q的SI模型分析

对一种具有种群动力和非线性传染率的传染病模型进行了研究,建立了具有常数迁入率和非线性传染率βI~pS~q的SI模型.与以往的具有非线性传染率的传染病模型相比,这种模型引入了种群动力,也就是种群的总数不再为常数,因此,该类模型更精确地描述了传染病传播的规律.还讨论了模型的正不变集,运用微分方程稳定性理论分析了模型平衡点的存在性及稳定性,得出了疾病消除平衡点和地方病平衡点的全局渐进稳定的充分条件.进一步的,得出了在某些参数范围内会出现Hopf分支现象,并对上述模型进行了生物学讨论.     

Stability of a stochastic SEIS model with saturation incidence and latent period

In this paper, a stochastic SEIS epidemic model with a saturation incidence rate and a time delay describing the latent period of the disease is investigated. The model inherits the endemic steady state from its corresponding deterministic counterpart. We first show the existence and uniqueness of the global positive solution of the model. Then, by constructing Lyapunov functionals, we derive sufficient conditions ensuring the stochastic stability of the endemic steady state. Numerical simulations are carried out to confirm our analytical results. Furthermore, our simulation results shows that the existence of noise and delay may cause the endemic steady state to be unstable.     

Dynamic behavior of a delay cholera model with constant infectious period

In this paper, a delay cholera model with constant infectious period is investigated. By analyzing the characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium of the model is established. It is proved that if the basic reproductive number $\mathcal{R}_0>1$, the system is permanent. If $\mathcal{R}_0<1$, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the disease-free equilibrium. If $\mathcal{R}_0>1$, also by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.