The effect of time delays on transmission dynamics of schistosomiasis

A 6-dimension dynamical schistosomiasis model incorporating five time delays is established in this paper. Two equilibrium points: a disease free equilibrium and an endemic equilibrium, are calculated respectively. The stability behaviors at the disease free equilibrium are analysed. Both analytical and numerical results are presented that prepatent periods in infection can affect the schistosomiasis transmission significantly. Thus, two effective measures on schistosomiasis prevention and control are obtained: lengthening the prepatent period in susceptible snails, and prolonging the incubation periods in miracidia and cercaria by temperature control or drug restraint. And then, numerical simulations are given to illustrate the validity and effectiveness of the model. At last a discussion is provided about our results and further work.     

A schistosomiasis compartment model with incubation and its optimal control

A new mathematical model included an exposed compartment is established in consideration of incubation period of schistosoma in human body. The basic reproduction number is calculated to illustrate the threshold of disease outbreak. The existence of the disease free equilibrium and the endemic equilibrium are proved. Studies about stability behaviors of the model are exploited. Moreover, control measure assessments are investigated in order to seek out effective control interventions for anti‐schistosomiasis. Then, the corresponding optimal control problem according to the model is presented and solved. Theoretical analyses and numerical simulations induce several prevention and control strategies for anti‐schistosomiasis. At last, a discussion is provided about our results and further work. Copyright © 2017 John Wiley & Sons, Ltd.     

Qualitative study of a quarantine/isolation model with multiple disease stages

Recent studies suggest that, for disease transmission models with latent and infectious periods, the use of gamma distribution assumption seems to provide a better fit for the associated epidemiological data in comparison to the use of exponential distribution assumption. The objective of this study is to carry out a rigorous mathematical analysis of a communicable disease transmission model with quarantine (of latent cases) and isolation (of symptomatic cases), in which the waiting periods in the infected classes are assumed to have gamma distributions. Rigorous analysis of the model reveals that it has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. The model has a unique endemic equilibrium when the threshold quantity exceeds unity. The endemic equilibrium is shown to be locally and globally-asymptotically stable for special cases. Numerical simulations, using data related to the 2003 SARS outbreaks, show that the cumulative number of disease-related mortality increases with increasing number of disease stages. Furthermore, the cumulative number of new cases is higher if the asymptomatic period is distributed such that most of the period is spent in the early stages of the asymptomatic compartments in comparison to the cases where the average time period is equally distributed among the associated stages or if most of the time period is spent in the later (final) stages of the asymptomatic compartments. Finally, it is shown that distributing the average sojourn time in the infectious (asymptomatic) classes equally or unequally does not effect the cumulative number of new cases.     

具有常数输入的SEIR模型的稳定性分析

讨论了易感者类和潜伏者类均为常数输入,潜伏期、染病期和恢复期均具有传染力,且传染率为一般传染率的SEIR传染病模型.利用Hurwitz判据证明了地方病平衡点的局部渐近稳定性,进一步利用复合矩阵理论得到了地方病平衡点全局渐近稳定的充分条件.     

A two-strain epidemic model with mutant strain and vaccination

In this paper, it is assumed that the spread of a pathogen can mutate in the host to create a second, cocirculating, mutant strain. Vaccinated individuals perhaps becomes infected after being in contact with individuals infected with mutant strain. A?two-strain epidemic model with vaccination is firstly investigated. The existence and stability properties of equilibria in this model are examined. By analyzing the characteristic equation and constructing Lyapunov functions, the conditions for local and global stability of the infection-free, boundary and endemic equilibria are established. The existence of Hopf bifurcation from the endemic equilibrium is also examined as this equilibrium loses its stability. Our theoretical results are confirmed by numerical simulations.     

Fixed period of temporary immunity after run of anti-malicious software on computer nodes

SIRS epidemic model has been developed with a fixed period of temporary immunity, following temporary recovery from the infection of malicious objects in place of an exponentially distributed period of temporary immunity. When a node is recovered from the infected class, it recovers temporarily, acquiring temporary immunity with probability p (0  p  1) and dies from the attack of malicious object with probability (1 − p). Temporary immunity is observed in the computer network when anti-malicious software is run after a node gets affected by a malicious object(s). The model consists of a set of integro-differential equations. The stability of the results is stated in terms of threshold parameter. It has been observed that the endemic equilibrium for this model may be unstable thus giving an example of a generalization which leads to new possibilities for the behavior of the model. Numerically it has been verified that the endemic equilibrium is not asymptotically stable for all parameter values.     

Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis

A deterministic model for studying the transmission dynamics of bovine tuberculosis in a single cattle herd is presented and qualitatively analyzed. A notable feature of the model is that it allows for the importation of asymptomatically infected cattle (into the herd) because re‐stocking from outside sources. Rigorous analysis of the model shows that the model has a globally‐asymptotically stable disease‐free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. In the absence of importation of asymptomatically infected cattle, the model has a unique endemic equilibrium whenever the reproduction number exceeds unity (this equilibrium is globally asymptotically stable for a special case). It is further shown that, for the case where asymptomatically infected cattle are imported into the herd, the model has a unique endemic equilibrium. This equilibrium is also shown to be globally asymptotically stable for a special case. Copyright © 2011 John Wiley & Sons, Ltd.     

Mathematical Analysis of SIR Epidemic Model with Piecewise Infection Rate and Control Strategies

The limitation of contact between susceptible and infected individuals plays an important role in decreasing the transmission of infectious diseases. Prevention and control strategies contribute to minimizing the transmission rate. In this paper, we propose SIR epidemic model with delayed control strategies, in which delay describes the response and effect time. We study the dynamic properties of the epidemic model from three aspects: steady states, stability and bifurcation. By eliminating the existence of limit cycles, we establish the global stability of the endemic equilibrium, when the delay is ignored. Further, we find that the delayed effect on the infection rate does not affect the stability of the disease-free equilibrium, but it can destabilize the endemic equilibrium and bring Hopf bifurcation. Theoretical results show that the prevention and control strategies can effectively reduce the final number of infected individuals in the population. Numerical results corroborate the theoretical ones.     

Qualitative analysis on a diffusive SIS epidemic system with logistic source and spontaneous infection in a heterogeneous environment

In this paper, we consider an SIS epidemic reaction–diffusion model with spontaneous infection and logistic source in a heterogeneous environment. The uniform bounds of solutions are established, and the global asymptotic stability of the constant endemic equilibrium is discussed in the case of homogeneous environment. This paper aims to analyze the asymptotic profile of endemic equilibria (when it exists) as the diffusion rate of the susceptible or infected population is small or large. Our results on this new model reveal that varying total population and spontaneous infection can enhance persistence of infectious disease, which may provide some implications on disease control and prediction.     

一类具有感染时滞的HIV模型的稳定性分析

在基本病毒动力学模型的基础上,建立了一个具有HollingⅡ型感染率且带有时滞的HIV模型.通过稳定性分析,讨论了模型无病平衡点以及正平衡点的稳定性态.最后借助Matlab对模型进行了数值模拟.     

一类时滞病毒动力学模型的稳定性分析

建立并分析了一类带有两个时滞的病毒动力学模型.通过讨论,获得了有时滞情况下无病平衡点以及正平衡点的稳定性态.     

一类具有常恢复率且总人口变化的SEIS传染病模型的稳定性

This paper considers an SEIS epidemic model with infectious force in the latent period and a general population-size dependent contact rate.A threshold parameter R is identified.If R≤1,the disease-free equilibrium O is globally stable.IfR>1,there is a unique endemic equilibrium and O is unstable.For two important special cases of bilinear and standard incidence,sufficient conditions for the global stability of this endemic equilibrium are given.The same qualitative results are obtained provided the threshold is more than unity for the corresponding SEIS model with no infectious force in the latent period.Some existing results are extended and improved.     

A delayed SIR epidemic model with saturation incidence and a constant infectious period

In this paper, an SIR epidemic model with saturation incidence and a time delay describing a constant infectious period is investigated. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. When the basic reproduction number is greater than unity, it is proved that the disease is uniformly persistent in the population, and explicit formulae are obtained to estimate the eventual lower bound of the fraction of infectious individuals. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are derived for the global attractiveness of the endemic equilibrium. Numerical simulations are carried out to illustrate the main results.     

Global dynamics of a delayed SEIS infectious disease model with logistic growth and saturation incidence

In this paper, a delayed Susceptible‐Exposed‐Infectious‐Susceptible (SEIS) infectious disease model with logistic growth and saturation incidence is investigated, where the time delay describes the latent period of the disease. By analyzing corresponding characteristic equations, the local stability of a disease‐free equilibrium and an endemic equilibrium is discussed. The existence of Hopf bifurcations at the endemic equilibrium is established. By using the persistence theory for infinite dimensional dynamic systems, it is proved that if the basic reproduction number is greater than unity, the system is permanent. By means of suitable Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global stability of the disease‐free equilibrium and the endemic equilibrium, respectively. Numerical simulations are carried out to illustrate the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS＊

In this article,we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlinear incidence rates and distributed...     

Global behavior of an SEIRS epidemic model with time delays

This is a study of dynamic behavior of an SEIRS epidemic model with time delays. It is shown that disease-free equilibrium is globally stable if the reproduction number is not greater than one. When the reproduction number is greater than 1, 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 fraction of infectious individuals can be computed. Local stability of endemic equilibrium is also discussed.     

Global dynamics of a Vector‐Borne disease model with two delays and nonlinear transmission rate

In this paper, we investigate a Vector‐Borne disease model with nonlinear incidence rate and 2 delays: One is the incubation period in the vectors and the other is the incubation period in the host. Under the biologically motivated assumptions, we show that the global dynamics are completely determined by the basic reproduction number R₀. The disease‐free equilibrium is globally asymptotically stable if R₀≤1; when R₀>1, the system is uniformly persistent, and there exists a unique endemic equilibrium that is globally asymptotically. Numerical simulations are conducted to illustrate the theoretical results.     

Two time-delay dynamic model on the transmission of malicious signals in wireless sensor network

Deployed in a hostile environment, motes of a Wireless sensor network (WSN) could be easily compromised by the attackers because of several constraints such as limited processing capabilities, memory space, and limited battery life time etc. While transmitting the data to their neighbour motes within the network, motes are easily compromised due to resource constraints. Here time delay can play an efficient role to reduce the adversary effect on motes. In this paper, we propose an epidemic model SEIR (Susceptible–Exposed–Infectious–Recovered) with two time delays to describe the transmission dynamics of malicious signals in wireless sensor network. The first delay accounts for an exposed (latent) period while the second delay is for the temporary immunity period due to multiple worm outbreaks. The dynamical behaviour of worm-free equilibrium and endemic equilibrium is shown from the point of stability which switches under some threshold condition specified by the basic reproduction number. Our results show that the global properties of equilibria also depends on the threshold condition and that latent and temporary immunity period in a mote does not affect the stability, but they play a positive role to control malicious attack. Moreover, numerical simulations are given to support the theoretical analysis.     

一类具免疫应答和非线性感染函数的时滞HIV-1感染模型的全局稳定性

考虑到HIV-1感染过程中免疫反应和非线性感染函数,建立了一类具有三个分布时滞的HIV-1感染动力学模型.得到了关于病毒感染的基本再生数R0和CTLs免疫反应的基本再生数R1 ＜R0.通过构造Lyapunov泛函证明了系统具有阈值动力学性质,即当R0≤1时,系统存在全局渐近稳定的无感染平衡点;当R1≤1＜R0时,系统出...