Global stability of an SIS epidemic model with delays

In this paper, we consider the global dynamics of the S(E)IS model with delays denoting an incubation time. By constructing a Lyapunov functional, we prove stability of a disease‐free equilibrium E₀ under a condition different from that in the recent paper. Then we claim that R₀≤1 is a necessary and sufficient condition under which E₀ is globally asymptotically stable. We also propose a discrete model preserving positivity and global stability of the same equilibria as the continuous model with distributed delays, by means of discrete analogs of the Lyapunov functional.     

具一般非线性隔离函数和接触率的染病年龄SIRS模型平衡点的存在性及稳定性

考虑一类具有一般非线性隔离函数和接触率的染病年龄结构SIRS传染病模型,研究平衡点的存在性及渐近稳定性,得到正平衡点指数渐近稳定的一般性条件.     

Asymptotic behavior of an SEI epidemic model with diffusion

An SEI epidemic model with constant recruitment and infectious force in the latent period is investigated. This model describes the transmission of diseases such as SARS. The behavior 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 the method of upper and lower solutions and its associated monotone iterations. Our result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small.     

具有常数输入的SEIS模型的全局渐近稳定性

讨论一类具有常数输入且传染率为非线性的SEIS流行病传播数学模型,给出了决定疾病灭绝和持续生存的基本再生数R0.当R0<1时,无病平衡点全局渐近稳定;当R0>1时,利用第二加性复合矩阵证明了惟一地方病平衡点全局渐近稳定.     

Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate

In this paper, we investigate a disease transmission model of SIRS type with latent period τ?0 and the specific nonmonotone incidence rate, namely, . For the basic reproduction number R₀>1, applying monotone iterative techniques, we establish sufficient conditions for the global asymptotic stability of endemic equilibrium of system which become partial answers to the open problem in [Hai-Feng Huo, Zhan-Ping Ma, Dynamics of a delayed epidemic model with non-monotonic incidence rate, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 459-468]. Moreover, combining both monotone iterative techniques and the Lyapunov functional techniques to an SIR model by perturbation, we derive another type of sufficient conditions for the global asymptotic stability of the endemic equilibrium.     

Global analysis of an epidemic model with a constant removal rate

In this paper we consider an epidemic model, which is a reduced SIRS model with a constant removal rate of the infective individuals, for its qualitative properties which were not revealed in [W. Wang, S. Ruan, Bifurcations in an epidemic model with constant removal rate of the refectives, J. Math. Anal. Appl. 291 (2004) 775–793]. We first prove the uniqueness of closed orbits if they exist for this epidemic model. Then we discuss the qualitative properties of equilibria at infinity for global tendencies. Therefore, global dynamical behaviors of this system are obtained in the end.     

Globalexponential stability of an epidemic model with saturated and periodic incidence rate

This paper is concerned with a SIR model with saturated and periodic incidence rate and saturated treatment function. By using differential inequality technique, we employ a novel argument to show that the disease‐free equilibrium is globally exponentially stable. The obtained results improve and supplement existing ones. We also use numerical simulations to demonstrate our theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Global dynamics of an SEI model with acute and chronic stages

A model with acute and chronic stages in a population with exponentially varying size is proposed. An equivalent system is obtained, which has two equilibriums: a disease-free equilibrium and an endemic equilibrium. The stability of these two equilibriums is controlled by the basic reproduction number _R0$R_0$. When _R0<1$R_0<1$, the disease-free equilibrium is globally stable. When _R0>1$R_0>1$, the disease-free equilibrium is unstable and the unique endemic equilibrium is locally stable. When _R0>1$R_0>1$ and γ=0,α=0$\gamma =0,\alpha =0$, the endemic equilibrium is globally stable in Γ⁰${\Gamma }^0$.     

Global stability of a generalized epidemic model

The competitive exclusion principle is one of the most interesting and important phenomena in both theoretical epidemiology and biology. We show that the equilibrium in which only the strain with the maximum basic reproductive number exists is globally asymptotically stable by using an average Lyapunov function theorem and some dynamical system theory. This result is anticipated by H.J. Bremermann and H.R. Thieme (1989) [6] where they showed that the equilibrium is locally stable — the global result has not been established previously.     

Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate

This paper deals with the global analysis of a dynamical model for the spread of tuberculosis with a general contact rate. The model exhibits the traditional threshold behavior. We prove that when the basic reproduction ratio is less than unity, then the disease-free equilibrium is globally asymptotically stable and when the basic reproduction ratio is great than unity, a unique endemic equilibrium exists and is globally asymptotically stable under certain conditions. The stability of equilibria is derived through the use of Lyapunov stability theory and LaSalle’s invariant set theorem. Numerical simulations are provided to illustrate the theoretical results.     

Traveling waves in an SEIR epidemic model with a general nonlinear incidence rate

ABSTRACT

We study the traveling waves of reaction-diffusion equations for a diffusive SEIR model with a general nonlinear incidence. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed. Its proof is showed by introducing an auxiliary system, applying Schauder fixed point theorem and then a limiting argument. The non-existence proof is obtained by two-sided Laplace transform when the speed is less than the critical velocity. Finally, we present some examples to support our theoretical results.     

Global stability of an SAIRS epidemic model with vaccinations,transient immunity and treatment

We consider an SAIRS epidemic model with vaccinations and treatment, where asymptomatic and symptomatic infectious individuals are considered in the transmission of the disease. We found the basic reproduction number, ${ℛ}_0$ and using ${ℛ}_0$, we conducted global stability analysis. We proved when ${ℛ}_0<1$, the disease-free equilibrium is globally stable. If ${ℛ}_0>1$, the disease-free equilibrium in unstable and a unique endemic equilibrium exists. We explored the global stability of the endemic equilibrium and noticed it is globally stable under certain conditions. Moreover, we then considered a special case of the SAIRS model, the SAIR model. We proved the disease-free equilibrium is globally stability when ${ℛ}_0<1$ and the endemic equilibrium is globally stable when ${ℛ}_0>1$. Next, we numerically simulated our analytical results and plotted these for various cases. Finally, we performed sensitivity analysis to tell us how each parameter in the system affects disease transmission.