一类具有分布时滞和非线性发生率的媒介传染病模型的全局稳定性

建立了一类具有分布时滞和非线性发生率的SIR媒介传染病模型,分析得到了决定疾病是否一致持续存在的基本再生数．而且当基本再生数不大于1时,疾病最终灭绝;当基本再生数大于1时,模型存在惟一的地方病平衡点,并且疾病一致持续存在于种群之中．通过构造Lyapunov泛函,证明了在一定条件下地方病平衡点只要存在就全局稳定．同时指出了证明地方病平衡点全局稳定时可适用的Lyapunov泛函的不惟一性.     

Global asymptotic stability of a generalized SIRS epidemic model with transfer from infectious to susceptible

In this paper, we propose a generalized SIRS epidemic model with varying total population size caused by the death rate due to the disease and transfer from infectious to susceptible, where the incidence rate employed includs a wide range of monotonic and concave incidence rates. Applying the geometric approach developed by Smith, Li and Muldowey, we prove that the endemic equilibrium is globally asymptotically stable provided that the rate of loss of immuity $\delta$ is in a critical interval $[\eta,\bar\delta)$ when the basic reproduction number $R_0$ is greater than unity.     

Modeling and dynamics of HIV transmission among high-risk groups in Guangzhou city, China

In this paper, a multicompartmental model is formulated to study how HIV is transmitted among different HIV high-risk groups, including MSM (men who have sex with men), FRs (foreigner residents), FSWs (female sex workers), and IDUs (injection drug users). The explicit expression for the basic reproduction number is obtained via the next generation matrix approach. We show that the disease free equilibrium is locally as well as globally asymptotically stable (the disease goes to extinction) when the basic reproduction number is less than unity, and the disease is always present when the basic reproduction number is larger than unity. As an illustration of our theoretical results, we conduct numerical simulations. We also conduct a case study where model parameters are estimated from the demographic and epidemiological data from Guangzhou. Using the parameter estimates, we predict the HIV/AIDS trend for each high-risk group. Furthermore, our study suggests that reducing the transmission routes of the disease and increasing condom use will be useful for control of HIV transmission.     

Analysis of pulse vaccination strategy in SIRVS epidemic model

In this paper, we investigate the dynamics of a SIRVS epidemic model with pulse vaccination strategy and saturation incidence. We show that the disease is eradicated when the basic reproduction number is less than unity, and the disease is permanent when the basic reproduction number is greater than unity. Finally, by means of numerical simulation, we obtain the parameters reach some critical value, and the disease will go to extinction.     

A predator–prey–disease model with immune response in infected prey

In this paper, a predator–prey–disease model with immune response in the infected prey is formulated. The basic reproduction number of the within-host model is defined and it is found that there are three equilibria: extinction equilibrium, infection-free equilibrium and infection-persistent equilibrium. The stabilities of these equilibria are completely determined by the reproduction number of the within-host model. Furthermore, we define a basic reproduction number of the between-host model and two predator invasion numbers: predator invasion number in the absence of disease and predator invasion number in the presence of disease. We have predator and infection-free equilibrium, infection-free equilibrium, predator-free equilibrium and a co-existence equilibrium. We determine the local stabilities of these equilibria with conditions on the reproduction and invasion reproduction numbers. Finally, we show that the predator-free equilibrium is globally stable.     

Threshold of Effective Degree SIR Model

The effective degree SIR model is a precise model for the SIR disease dynamics on a network. The original ODE model is only applicable for a network with finite degree distributions. The new generating function approach rewrites with model as a PDE and allows infinite degree distributions. In this paper, we first prove the existence of a global solution. Then we analyze the linear and nonlinear stability of the disease-free steady state of the PDE effective degree model, and show that the basic reproduction number still determines both the linear and the nonlinear stability. Our method also provides a new tool to study the effective degree SIS model, whose basic reproduction number has been elusive so far.     

Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates

In this paper, we introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that global dynamics are completely determined by the basic reproduction number R₀. It shows that, the basic reproduction number R₀ is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result.     

Stability analysis of delay differential equation models of HIV-1 therapy for fighting a virus with another virus

Considering two kinds of delays accounting, respectively, for (i) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and (ii) a virus production period for new virions to be produced within and released from the infected cells, we develop and analyze a mathematical model for HIV-1 therapy by fighting a virus with another virus. For the different values of the basic reproduction number and the second basic reproduction number, we investigate the stability of the infection-free equilibrium, the single-infection equilibrium and the double-infection equilibrium. We conclude that increasing delays will decrease the values of the basic reproduction number and the second basic reproduction number. Our results have potential applications in HIV-1 therapy. The approach we use here is a combination of analysis of characteristic equations, Fluctuation Lemma and Lyapunov function.     

Basic Reproduction Number in a Growing Population

The basic reproduction number of a fast disease epidemic on a slowly growing network may increase to a maximum then decrease to its equi- librium value while the population increases, which is not displayed by classical homogeneous mixing disease models. In this paper, we show that, by properly keeping track of the dynamics of the per capita contact rate in the population due to population dynamics, classical homogeneous mixing models show simi- lar non-monotonic dynamics in the basic reproduction number. This suggests that modeling the dynamics of the contact rate in classical disease models with population dynamics may be important to study disease dynamics in growing populations.     

Global stability of a delayed epidemic model with latent period and vaccination strategy

In this paper, a mathematical model describing the transmission dynamics of an infectious disease with an exposed (latent) period and waning vaccine-induced immunity is investigated. The basic reproduction number is found by applying the method of the next generation matrix. It is shown that the global dynamics of the model is completely determined by the basic reproduction number. By means of appropriate Lyapunov functionals and LaSalle’s invariance principle, it is proven that if the basic reproduction number is less than or equal to unity, the disease-free equilibrium is globally asymptotically stable and the disease fades out; and if the basic reproduction number is greater than unity, the endemic equilibrium is globally asymptotically stable and therefore the disease becomes endemic.     

Dynamics of a generalized predator‐prey model with delay and impulse via the basic reproduction number

In this paper, we study a generalized predator‐prey model with delay and impulse. The existence of the predator‐free periodic solution is investigated. We employ the approach and techniques coming from epidemiology and calculate the basic reproduction number for the predator. Using the basic reproduction number, we consider the global attraction of the predator‐free periodic solution and permanence of the model. As for application, an example is discussed. Furthermore, some numerical simulations are given to illustrate our results.     

Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza

The basic reproduction number and the point of endemic equilibrium are two very important factors in any deterministic compartmental epidemic model as the basic reproduction number and the point of endemic equilibrium represent the nature of disease transmission and disease prevalence respectively. In this article the sensitivity analysis based on mathematical as well as statistical techniques has been performed to determine the importance of the epidemic model parameters. It is observed that 6 out of the 11 input parameters play a prominent role in determining the magnitude of the basic reproduction number. It is shown that the basic reproduction number is the most sensitive to the transmission rate of disease. It is also shown that control of transmission rate and recovery rate of the clinically ill are crucial to stop the spreading of influenza epidemics.     

具有垂直传染的SEIA传染病模型的稳定性

建立和研究一类具有垂直传染的SEIA传染病模型,得到模型基本再生数R₀的表达式,运用Lyapunov函数和第二加性复合矩阵理论证明了当R₀〈1时无病平衡点全局渐近稳定,当R₀〉1时地方病平衡点全局渐近稳定.     

一类具有非线性发生率的传染病模型的稳定性

建立和研究一类具有非线性发生率的传染病模型,得到该模型基本再生数R_0的表达式,运用Lyapunov函数和第二加性复合矩阵理论证明了当R_0<1时无病平衡点全局渐近稳定,此时疾病消失,当R_0>1时地方病平衡点全局渐近稳定,此时疾病在人群中流行.     

Stability analysis and optimal control of a hand-foot-mouth disease (HFMD) model

In this paper, we propose a system of ordinary differential equations to model the hand-foot-mouth disease (HFMD). We derive the expression of the basic reproduction number . When , the system only has the disease free equilibrium, which is globally asymptotically stable; otherwise, the system is persistent. By sensitivity analysis, we identify the control parameters. Then we formulate an optimal control problem to find the optimal control strategy. These results are applied to the spread of HFMD in Mainland China. The basic reproduction number tells us that it is outbreak in China.     

Global dynamics of multi‐group dengue disease model with latency distributions

In this paper, by incorporating latencies for both human beings and female mosquitoes to the mosquito‐borne diseases model, we investigate a class of multi‐group dengue disease model and study the impacts of heterogeneity and latencies on the spread of infectious disease. Dynamical properties of the multi‐group model with distributed delays are established. The results showthat the global asymptotic stability of the disease‐free equilibrium and the endemic equilibrium depends only on the basic reproduction number. Our proofs for global stability of equilibria use the classical method of Lyapunov functions and the graph‐theoretic approach for large‐scale delay systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence

In this paper, the asymptotic behavior of solutions of an autonomous SEIRS epidemic model with the saturation incidence is studied. Using the method of Liapunov–LaSalle invariance principle, we obtain the disease-free equilibrium is globally stable if the basic reproduction number is not greater than one. Moreover, we show that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions of locally and globally asymptotically stable convergence to an endemic equilibrium are obtained base on the permanence.     

医疗资源影响下具有潜伏期的一类传染病模型建立及性态分析

建立了医疗资源影响下的考虑疾病具有潜伏期的一类传染病模型,并分析了模型的动力学性态.发现疾病流行与否由基本再生数和医院病床数共同决定,并得到了病床数的阈值条件.当基本再生数R_0大于1时,系统只存在惟一正平衡点,且通过构造Dulac函数证明了正平衡点只要存在一定是全局渐近稳定的;当R_01,我们得到系统存在两个正平衡点及无正平衡点的条件,且只有当医院的病床数小于阈值时,系统会经历后向分支.因此,可根据实际情况使医院病床的投入量不低于阈值条件,不仅有利于疾病的控制而且不会出现医疗资源过剩的现象.     

Dynamics of an HIV Infection Model with Two Infection Routes and Evolutionary Competition between Two Viral Strains

The existing combination therapy of HIV antiretroviral drugs can lead to the emergence of drug-resistant viruses, and cannot effectively block direct cell-to-cell infections, these factors results in incomplete virus suppression and increased risk of disease progression. In this paper, we formulate an HIV model with two strains representing a drug-sensitive virus and a drug-resistant virus to study the joint mechanism of drug resistance. We first reduce the infection-age model to a system of integro-differential equations with infinite delays. Then the stability of the equilibria and the dynamics of competition between two viruses are studied to illuminate the joint effects of infection-age and two infection routes on the evolution of both drug-sensitive and drug-resistant strains before and during drug treatment. Applying a persistence theorem for infinite dimensional systems, we obtain that the disease is always present when the basic reproduction number is larger than unity. Numerical simulations confirm that the basic reproduction numbers and mutation coefficient are the key threshold parameters for determining the competition results of the two viral strains and indicate the cell-to-cell transmission increases the likelihood that HIV breaks out within the host. Finally, sensitivity analyses suggest that the available combination therapy should be taken once symptoms of resistance appear during drug treatment, and demonstrate that the presence of cell-to-cell transmission attenuates the efficacy of the existing antiretroviral drug treatments.     

Stability and backward bifurcation on a cholera epidemic model with saturated recovery rate

In this paper, a cholera epidemic model with saturated recovery rate is studied. Backward bifurcation leading to bistability possibly occurs, and global dynamics are shown by compound matrices and geometric approaches. Numerical simulations are presented to illustrate the results. Our results suggest that the basic reproduction number itself is not enough to describe whether cholera will prevail or not when the resources for treatment of infectives are limited and suggest that we should pay more attention to the initial state of cholera. Copyright © 2016 John Wiley & Sons, Ltd.