一类带有非线性传染率的SEIS传染病模型的定性分析

借助极限理论和Fonda定理,研究了一类既有常数输入率又有因病死亡率的SEIS传染病模型．所考虑模型的传染率是非线性的,并且得到了该模型的基本再生数,当基本再生数小于1时,该模型仅存在唯一的无病平衡点,它是全局渐近稳定的,且疾病最终灭绝．当基本再生数大于1时,该模型除存在不稳定的无病平衡点外,还存在唯一的局部渐近稳定的地方病平衡点,并且疾病一致持续存在．     

Large Deviations for Hierarchical Systems of Interacting Jump Processes

We investigate the large deviations principle from the McKean–Vlasov limit for a collection of jump processes obeying a two-level hierarchy interaction. A large deviation upper bound is derived and it is shown that the associated rate function admits a Lagrangian representation as well as a nonvariational one. Moreover, it is proved that the admissible paths for the weak solution of the McKean–Vlasov equation enjoy certain strong differentiability properties.     

A dynamic epidemic control model on uncorrelated complex networks

In this paper, a dynamic epidemic control model on the uncorrelated complex networks is proposed. By means of theoretical analysis, we found that the new model has a similar epidemic threshold as that of the susceptible-infectedrecovered （SIR） model on the above networks, but it can reduce the prevalence of the infected individuals remarkably. This result may help us understand epidemic spreading phenomena on real networks and design appropriate strategies to control infections.     

具有因病死亡且输人为Berverton-Holt函数的离散SIS传染病模型分析

引入相应的概率建立了考虑因病死亡且输入为Berverton-Holt的离散SIS传染病模型,确定了决定其动力性态的阈值,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,且可以猜想地方病平衡点是全局渐近稳定的.     

Global attractivity of a discrete SIRS epidemic model with standard incidence rate

In this paper, a discrete Susceptible‐Infected‐Recovered‐Susceptible (SIRS) epidemic model with standard incidence rate is studied. By means of the iteration technique and the comparison principle of difference equations, the sufficient conditions are obtained for the global attractivity of the endemic equilibrium when the basic reproduction number is greater than unity. Two examples are given to illustrate the main theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on an NSFD scheme for a mathematical model of respiratory virus transmission

We construct a non-standard finite difference (NSFD) scheme for an SIRS mathematical model of respiratory virus transmission. This discretization is in full compliance with the NSFD methodology as formulated by Mickens. By use of an exact conservation law satisfied by the SIRS differential equations, we are able to determine the corresponding denominator function for the discrete first-order time derivatives. Our scheme is dynamically consistent with the SIRS differential equations, since the conservation laws are preserved. Furthermore, the scheme is shown to satisfy a positivity condition for its solutions for all values of the time step size.     

An NSFD scheme for a class of SIR epidemic models with vaccination and treatment

An non-standard finite difference scheme is employed to discuss a class of SIR epidemic model with vaccination and treatment. The dynamical properties of the discretized model are then analysed. The results demonstrate that the discretized epidemic model is dynamically consistent with the continuous model since it maintains essential properties of the corresponding continuous model, such as positivity property and boundness of solutions, equilibrium points and their local stability properties.     

Global stability of a delayed SIR epidemic model with density dependent birth and death rates

An SIR   epidemic model with density dependent birth and death rates is formulated. In our model it is assumed that the total number of the population is governed by logistic equation. The transmission of infection is assumed to be of the standard form, namely proportional to I(t-h)/N(t-h)$I(t-h)/N(t-h)$ where N(t)$N(t)$ is the total (variable) population size, I(t)$I(t)$ is the size of the infective population and a time delay h   is a fixed time during which the infectious agents develop in the vector. We consider transmission dynamics for the model. Stability of an endemic equilibrium is investigated. The stability result is stated in terms of a threshold parameter, that is, a basic reproduction number _R0$R_0$.     

A one-parameter family of stationary solutions in the Susceptible-Infected-Susceptible epidemic model

We study the uncorrelated Susceptible-Infected-Susceptible (SIS) model in epidemiology on top of a one parameter family of networks whose connectivity distribution ranges from scale free (SF) to exponential. For each network, the fraction of the population infected in the long term is a recursively defined hypergeometric function. For highly contagious diseases, with a high infection rate, the fraction of the population infected is lower when the network is SF. For less contagious diseases, the fraction of the population infected is lower when the network is exponential. This result points to an evolutionary advantage for a network being SF—namely an SF network is more resistant to the spread of a deadly disease.