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In this paper, we propose a periodic SIS epidemic model with time delay and transport-related infection in a patchy environment. The basic reproduction number R 0 is derived which determines the global dynamics of the model system: if R 0  < 1, the disease-free periodic state is globally attractive while there exists at least one positive periodic state and the disease persists if R 0  > 1. Numerical simulations are performed to confirm the analytical results and to explore the dependence of R 0 on the transport-related infection parameters and the amplitude of fluctuations., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

A delayed SEIRS epidemic model with pulse vaccination and saturation incidence rate is investigated. Using Krasnoselskii's fixed-point theorem, we obtain the existence of infection-free periodic solution of the impulsive delayed epidemic system. We define some new threshold values R(1), R(2) and R(3). Further, using the comparison theorem, we obtain the explicit formulae of R(1) and R(2). Under the condition R(1) < 1, the infection-free periodic solution is globally attractive, and that R(2) > 1 implies that the disease is permanent. Theoretical results show that the disease will be extinct if the vaccination rate is larger than θ* and the disease is uniformly persistent if the vaccination rate is less than θ(*). Our results indicate that a long latent period of the disease or a large pulse vaccination rate will lead to eradication of the disease. Moreover, we prove that the disease will be permanent as R(3) > 1.

An SVEIR epidemic model with continuous age‐dependent vaccination, latency, infection, and disease relapse is proposed and analyzed in this paper. The dynamical behaviors including the derivation of basic reproduction number R0, the existence and the stability of steady states, and the uniform persistence of the model are investigated. The results indicate that if R0≤1, the disease‐free steady state is globally asymptotically stable, and the disease dies out, whereas if R0>1, the disease is uniformly persistent, and the endemic steady state is also globally asymptotically stable, and the disease remains at the endemic level. The research shows the global dynamics of the model are sharply determined by its basic reproduction number R0. Finally, numerical examples support our main theoretical results. [ABSTRACT FROM AUTHOR]

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]

In this paper, by applying a nonstandard finite difference scheme, we formulate a discretized SIRVS epidemic model which takes into account vaccination. Under quite weak assumptions, the threshold value conditions on permanence and extinction of disease are established. Some new threshold values in product forms R 0 * and R 1 * are obtained. We show that the disease is permanent if R 0 * > 1 , and if R 1 * < 1 , then the disease is extinct. When the model degenerates into a periodic model, a sharp threshold value R 0 is obtained for permanence versus extinction of disease. In order to illustrate our analytic analysis, some numerical simulations are also included in the end. [ABSTRACT FROM AUTHOR]

Hepatitis B is an infectious disease caused by the hepatitis B virus (HBV) which affects livers. In this paper, we formulate a hepatitis B model to study the transmission dynamics of hepatitis B in Xinjiang, China. The epidemic model involves an exponential birth rate and vertical transmission. For a better understanding of HBV transmission dynamics, we analyze the dynamic behavior of the model. The modified reproductive number σ is obtained. When σ < 1, the disease-free equilibrium is locally asymptotically stable, when σ > 1, the disease-free equilibrium is unstable and the disease is uniformly persistent. In the simulation, parameters are chosen to fit public data in Xinjiang. The simulation indicates that the cumulated HBV infection number in Xinjiang will attain about 600,000 cases unless there are stronger or more effective control measures by the end of 2017. Sensitive analysis results show that enhancing the vaccination rate for newborns in Xinjiang is very effective to stop the transmission of HBV. Hence, we recommend that all infants in Xinjiang receive the hepatitis B vaccine as soon as possible after birth. [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 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: In this paper, a nonautonomous SIRS epidemic model with density dependent birth rate is proposed and studied. Threshold conditions for the permanence and extinction of the disease are established. Some new threshold values of integral form are obtained. We prove that the disease is permanent if , and extinct if or . For the periodic and almost periodic cases, these threshold conditions act as sharp threshold values for the permanence and extinction of the disease. Global asymptotic stability of periodic solution for the periodic system is derived. Some examples are given to illustrate the main results of this paper. [Copyright &y& Elsevier]

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: The aim of this paper is to analyze an SIRVS epidemic model in which pulse vaccination strategy (PVS) is included. We are interested in finding the basic reproductive number of the model which determine whether or not the disease dies out. The global attractivity of the disease-free periodic solution (DFPS for short) is obtained when the basic reproductive number is less than unity. The disease is permanent when the basic reproductive number is greater than unity, i.e., the epidemic will turn out to endemic. Our results indicate that the disease will go to extinction when the vaccination rate reaches some critical value. [Copyright &y& Elsevier]

Abstract: We study an SIR model with a seasonal contact rate and a staged treatment strategy, which is an extension of our previous work [Z. Bai, Y. Zhou, Existence of two periodic solutions for a non-autonomous SIR epidemic model, Appl. Math. Model. 35 (2011) 382–391]. It is proved that the persistence and extinction of the disease are determined by the basic reproductive number () and a threshold parameter (). It is obtained that the model exhibits two different bistable behaviors under certain conditions: the stable disease-free state coexists with a stable endemic periodic solution, and three endemic periodic solutions coexist with two of them being stable. Numerical simulations are presented to illustrate theoretical results. [Copyright &y& Elsevier]