Traveling waves in a nonlocal dispersal SIR epidemic model

This paper is concerned with traveling wave solutions of a nonlocal dispersal SIR epidemic model. The existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the minimal wave speed. This threshold dynamics are proved by Schauder’s fixed point theorem and the Laplace transform. The main difficulties are that the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity.     

具有染病者输入的离散SIR传染病模型分析

引入相应的概率建立了具有染病者输入的离散SIR传染病模型,确定了决定其动力学性态的阂值.在阈值之下模型仅存在无病平衡点,且无病平衡点是全局渐近稳定的;在阈值之上模型是一致持续的,有唯一的地方病平衡点存在,且地方病平衡点是局部渐近稳定的.     

Damped oscillating traveling waves of a diffusive SIR epidemic model

This paper is concerned with the damped oscillating traveling waves of a diffusive SIR model with vital dynamics. Our result implies that the vital dynamics can induce damped spatio-temporal oscillations. Biologically, this shows that the vital dynamics can cause multiple outbreaks of infection with decreasing maximum outbreak sizes.     

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.     

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.     

Existence of two periodic solutions for a non-autonomous SIR epidemic model

A non-autonomous SIR model with periodic transmission rate and a constant removal rate is formulated. By using the continuation theorem of coincidence degree theory, sufficient conditions for the existence of at least two positive periodic solutions are obtained. The stability of the periodic solution for small seasonality is investigated. Numerical simulations are done to show our theoretical results.     

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.     

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.     

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.     

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.     

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.     

The differential susceptibility SIR epidemic model with stage structure and pulse vaccination

The differential susceptibility SIR epidemic model with stage structure and pulse vaccination is introduced. By the comparison theorem, some sufficient conditions for the globally attractivity of an infection-free periodic solution and the permanence of this system are presented. Two numerical simulations are also given to illustrate our main results.     

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.     

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.     

Dynamics of a discretized SIR epidemic model with pulse vaccination and time delay

We derive a discretized SIR epidemic model with pulse vaccination and time delay from the original continuous model. The sufficient conditions for global attractivity of an infection-free periodic solution and permanence of our model are obtained. Improving discretization, our results are corresponding to those in the original continuous model.     

Dynamics of a discretized SIR epidemic model with pulse vaccination and time delay

We derive a discretized SIR epidemic model with pulse vaccination and time delay from the original continuous model. The sufficient conditions for global attractivity of an infection-free periodic solution and permanence of our model are obtained. Improving discretization, our results are corresponding to those in the original continuous model.