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We consider a susceptible-exposed-infected-removed (SEIR) epidemic model with discontinuous treatment strategies. The treatment rate has at most a finite number of jump discontinuities in every compact interval. By using Lyapunov theory for discontinuous differential equations and other techniques on non-smooth analysis, the basic reproductive number $\mathcal{R}_{0}$ is proved to be a sharp threshold value which completely determines the dynamics of the model. If $\mathcal{R}_{0}\leq 1$, then there only exists a disease-free equilibrium which is globally stable. If $\mathcal {R}_{0}>1$, the disease-free equilibrium becomes unstable and there exists a unique endemic equilibrium which is globally stable. The numerical simulations indicate that strengthening treatment measures after infective individuals reach some level is beneficial to disease control. Furthermore, we discuss that the disease will die out in a finite time, which is impossible for the corresponding SEIR model with continuous treatment. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study a heroin epidemic model with distributed time delays. The basic reproduction number for the model is identified and the threshold property of is established. It is shown that drug-free equilibrium is globally asymptotically stable if . When , there is a disease endemic equilibrium which is locally asymptotically stable, it is proved that the disease is uniformly persistent in the population, and explicit formulae are obtained by which the eventual lower bound of the drug user individuals can be computed. [Copyright &y& Elsevier]

Abstract: A deterministic model is developed to describe the effects of vaccination and treatment on the spread of tuberculosis. It is shown that the basic reproduction number characterizes the disease transmission dynamics: if , there exists only the disease-free equilibrium which is globally asymptotically stable, and if then there is a disease endemic equilibrium and the disease persists. Global stability of the endemic equilibrium is also discussed. Analysis of the dependence of on the vaccination rate, vaccine efficacy and treatment rate shows that the spread of tuberculosis can be controlled if the vaccine rate or the efficacy or the treatment rate reaches a certain threshold. [Copyright &y& Elsevier]

Abstract: A nonautonomous eco-epidemic model with disease in the prey is formulated and studied. Some sufficient and necessary conditions on the permanence and extinction of the infective prey are established by introducing the new research method. Some sufficient conditions on the global attractivity of the model are presented by constructing a Lyapunov function. Finally, an example is given to show that the periodic model is global attractivity if the infective prey is permanent. [ABSTRACT FROM AUTHOR]

Abstract: A nonautonomous SIRS epidemic model with distributed delay is investigated. Two new threshold values, and are derived. The model is permanent as and implies the extinction of the disease. Using the Liapunov functional method, global behavior of the model is studied. [Copyright &y& Elsevier]

Abstract: In this paper, a nonautonomous SIRS epidemic model with time delay is studied. We introduce some new threshold values and and further obtain the disease will be permanent when and the disease extinct when . Using the method of Liapunov functional, some sufficient conditions are derived for the global attractivity of the system. The known results are extended. [Copyright &y& Elsevier]

Abstract: In this paper an autonomous SIRS epidemic model with time delay is studied. The basic reproductive number is obtained which determines whether the disease is extinct or not. When the basic reproductive number is greater than 1, it is proved that the disease is permanent in the population, and explicit formula are obtained by which the eventual lower bound of the fraction of infectious individuals can be computed. Throughout the total paper, we mainly use the technique of Lyapunov functional to establish the global stability of the infection-free equilibrium and the local stability of the endemic equilibrium but need another sufficient condition. [Copyright &y& Elsevier]

In this paper, using the forward Euler and backward Euler methods, we present four discrete epidemic models with the nonlinear incidence rate. We discuss the effect of two discretizations on the stability of the endemic equilibrium for these models. Numerical simulations are performed to illustrate our analytic results. [ABSTRACT FROM AUTHOR]