Avian–human influenza epidemic model with diffusion,nonlocal delay and spatial homogeneous environment

In this paper, an avian–human influenza epidemic model with diffusion, nonlocal delay and spatial homogeneous environment is investigated. This model describes the transmission of avian influenza among poultry, humans and environment. The behavior of positive solutions to a reaction–diffusion system with homogeneous Neumann boundary conditions is investigated. By means of linearization method and spectral analysis the local asymptotical stability is established. The global asymptotical stability for the poultry sub-system is studied by spectral analysis and by using a Lyapunov functional. For the full system, the global stability of the disease-free equilibrium is studied using the comparison Theorem for parabolic equations. Our result shows that the disease-free equilibrium is globally asymptotically stable, whenever the contact rate for the susceptible poultry is small. This suggests that the best policy to prevent the occurrence of an epidemic is not only to exterminate the asymptomatic poultry but also to reduce the contact rate between susceptible humans and the poultry environment. Numerical simulations are presented to illustrate the main results.     

具有饱和治疗的禽流感动力学模型的研究

提出了一类带有治疗的禽流感动力学模型,用来分析禽流感从禽类向人类传播的过程.由于治疗禽流感的药物十分有限,提出一个带有饱和治疗的模型.通过讨论得知当禽流感疫情已经发生时,通过控制染病的禽类就可抑制禽流感在人类的传播.     

The spreading frontiers of avian-human influenza described by the free boundary

In this paper,a reaction-diffusion system is proposed to investigate avian-human influenza.Two free boundaries are introduced to describe the spreading frontiers of the avian influenza.The basic reproduction numbers rF0(t)and RF0(t)are defined for the bird with the avian influenza and for the human with the mutant avian influenza of the free boundary problem,respectively.Properties of these two time-dependent basic reproduction numbers are obtained.Sufficient conditions both for spreading and for vanishing of the avian influenza are given.It is shown that if rF0(0)<1 and the initial number of the infected birds is small,the avian influenza vanishes in the bird world.Furthermore,if rF0(0)<1 and RF0(0)<1,the avian influenza vanishes in the bird and human worlds.In the case that rF0(0)<1 and RF0(0)>1,spreading of the mutant avian influenza in the human world is possible.It is also shown that if rF0(t0)>1 for any t0>0,the avian influenza spreads in the bird world.     

Modelling and analyzing the epidemic of human infections with the avian influenza A(H7N9) virus in 2017 in China

In 2013, in mainland China, a novel avian influenza A(H7N9) virus began to infect humans, followed by the annual outbreaks, and had aroused severe fatality in the infected humans. After introducing the statistical characteristics including the geographical distributions of the outbreaks, a SEV‐SIRS eco‐epidemiological model is established and analyzed. In this model, the factor of virus in environment is incorporated into the model as a class; the vaccine measure in poultry is taken into account in purpose of assessing its control effect in 2017 in China; the nonmonotonic contact function is adopted to characterize the psychosocial effect. The stability of disease‐free equilibrium point (DFE) is obtained by the threshold theory; the stability of the endemic equilibrium point is gotten by the Bendixson criterion based on the geometric approach. Sensitivity analyses of system parameters indicate that the measure of vaccination in poultry can play its role but only when the vaccine rate is more than 98% can the disease control effect be effectively exerted, and the virus in environment is an extremely sensitive factor in the disease transmission and the epidemic control.     

Impacts of TV and radio advertisements on the dynamics of an infectious disease: A modeling study

TV and radio advertisements are widely acknowledged as important interventions in raising issues of public health care and play promising role to control the infection through propagating awareness among the individuals. In this paper, a nonlinear susceptible‐infected‐susceptible (SIS) model is proposed and analyzed to see the impacts of TV and radio advertisements on the spread of influenza epidemic. In the model formulation, it is assumed that the susceptible individuals contract infection through the direct contact with infected individuals. The information regarding the protection against the disease is propagated via TV and radio advertisements, and their growth rates are assumed to be proportional to the fraction of infected individuals. However, the growth rate of TV advertisements decreases with the increase in number of aware individuals. The information broadcasted through TV and radio advertisements induces behavioral changes among the susceptible individuals, and they form an isolated aware class. The epidemiological feasible equilibria, their stability properties, and direction of bifurcation are discussed. The expression for modified basic reproduction number is obtained. The model analysis shows that the dissemination rate of awareness among susceptible individuals due to TV and radio advertisements and baseline number of TV and radio advertisements have potential to reduce the epidemic peak and, thus, control the spread of infection. Further, the analytical findings are well supported through numerical simulation.     

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.     

具有饱和反应率和阶段时滞结构的害虫生物防治模型

考虑了一个害虫和天敌都有阶段结构及具有饱和反应率的阶段时滞脉冲捕食者-食饵模型,利用人工周期定量地投放有病的害虫和天敌去治理害虫.借助脉冲时滞微分方程的相关理论和方法获得易感害虫根除周期解全局吸引的充分条件以及天敌与易感害虫可以共存且易感害虫的密度可以控制在经济危害水平之下的充分条件.我们的结论为现实的害虫管理提供了可靠的策略依据.     

Modeling the spread of bird flu and predicting outbreak diversity

Avian influenza, commonly known as bird flu, is an epidemic caused by H5N1 virus that primarily affects birds like chickens, wild water birds, etc. On rare occasions, these can infect other species including pigs and humans. In the span of less than a year, the lethal strain of bird flu is spreading very fast across the globe mainly in South East Asia, parts of Central Asia, Africa and Europe. In order to study the patterns of spread of epidemic, we made an investigation of outbreaks of the epidemic in one week, that is from February 13–18, 2006, when the deadly virus surfaced in India. We have designed a statistical transmission model of bird flu taking into account the factors that affect the epidemic transmission such as source of infection, social and natural factors and various control measures are suggested. For modeling the general intensity coefficient f(r), we have implemented the recent ideas given in the article Fitting the Bill, Nature [R. Howlett, Fitting the bill, Nature 439 (2006) 402], which describes the geographical spread of epidemics due to transportation of poultry products. Our aim is to study the spread of avian influenza, both in time and space, to gain a better understanding of transmission mechanism. Our model yields satisfactory results as evidenced by the simulations and may be used for the prediction of future situations of epidemic for longer periods. We utilize real data at these various scales and our model allows one to generalize our predictions and make better suggestions for the control of this epidemic.     

Analysis of a viral disease model with saturated contact rate

A viral disease model with saturated contact rate is introduced and investigated. The model consists a host species, which is divided into two classes the susceptible and infected, and a virus, which causes a viral disease in the host and as the host induces, the infected releases more virus into the environment. Taking the virus replication rate as the bifurcating parameter, we prove that there exists a threshold value beyond which the endemic equilibrium bifurcates from the free disease one. Further increasing the value, the endemic equilibrium loses its stability, Hopf bifurcation occurs and a periodic solution arises from it. The orbital stability of the periodic orbits is analyzed by applying Poore’s condition. In the last, numerical simulation of the model is employed to explain the mathematical results of this paper.     

Analysis of a nonautonomous model for migratory birds with saturation incidence rate

In this paper, an eco-epidemiological model is newly proposed to consider the role of migratory birds by incorporating the temporal pattern of the avian migration into the model. In the new model, population of birds varies because they are migratory in nature. Under quite weak assumptions, sufficient conditions for the permanence and extinction of the disease is obtained. Moreover, by constructing a Liapunov function, the global attractivity of the model is discussed.     

Mathematical modelling of acute virus influenza A infections

This article models the immune system and the virus dynamics of acute influenza infection mathematically. We use the model to study the virus dynamics of some well-known and severe and mild types of viruses. Linkages to well-known models in the literature are illustrated. Simulations are compared with experimental results in vivo by comparing with results from infected ferrets where infection closely resembles those in humans. Good agreement is achieved between the model calculations and the experimental values for influenza A viruses. For the Spanish flu virus H1N1 peak virus load is high and virtually all cells are infected in the nostril. In general, the H1N1 viruses show much more prolonged infections than the H3N2 in the nostril. We suggest that the reason is that unspecific immunity attacks H3N2-budded viruses but not H1N1 viruses.     

Pulse and constant control schemes for epidemic models with seasonality

Control schemes for infectious disease models with time-varying contact rate are analyzed. First, time-constant control schemes are introduced and studied. Specifically, a constant treatment scheme for the infected is applied to a SIR model with time-varying contact rate, which is modelled by a switching parameter. Two variations of this model are considered: one with waning immunity and one with progressive immunity. Easily verifiable conditions on the basic reproduction number of the infectious disease are established which ensure disease eradication under these constant control strategies. Pulse control schemes for epidemic models with time-varying contact rates are also studied in detail. Both pulse vaccination and pulse treatment models are applied to a SIR model with time-varying contact rate. Further, a vaccine failure model as well as a model with a reduced infective class are considered with pulse control schemes. Again, easily verifiable conditions on the basic reproduction number are developed which guarantee disease eradication. Some simulations are given to illustrate the threshold theorems developed.     

Modelling the dynamics of avian influenza with nonlinear recovery rate and psychological effect

In this paper, a SI-SEIR type avian influenza epidemic model with psychological effect, nonlinear recovery rate and saturation inhibition effect is formulated to study the transmission and control of avian influenza virus. By setting the basic reproductive number as the threshold parameter and constructing Lyapunov function, Dulac function and using the Li-Muldowney''s geometry approach, we prove the local and global stability of disease-free equilibria and endemic equilibrium. Theoretical analysis are carried out to show the role of the saturation inhibition effect, psychological effect and effective medical resources in this model, and numerical simulations are also given to verify the results.     

Regional control for a spatially structured malaria model

A two‐component reaction‐diffusion system to describe the spread of malaria is considered. The model describes the dynamics of the infected mosquitoes and of the infected humans. The spread of the disease is controlled by three actions (controls) implemented in a subdomain of the habitat: killing mosquitoes, treating the infected humans, and reducing the contact rate mosquitoes‐humans. To start with, the problem of the eradicability of the disease is considered, while the cost of the controls is ignored. We prove that it is possible to decrease exponentially both the human and the vector infective population everywhere in the relevant habitat by acting only in a suitable subdomain. Later, the regional control problem of reducing the total cost of the damages produced by the disease, of the controls, and of the intervention in a certain subdomain is treated for the finite time horizon case. An iterative algorithm to decrease the total cost is proposed; apart from the three controls considered above, the logistic structure of the habitat is taken into account. The level set method is used as a key ingredient for describing the subregion of intervention. Some numerical simulations are given to illustrate the applicability of the theoretical results.     

On global stability of an HIV pathogenesis model with cure rate

In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4⁺ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12 : 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4⁺ T cells. Copyright © 2014 John Wiley & Sons, Ltd.     

自适应网络上的SIV模型动力学

建立了自适应网络上的SIV （susceptible-infected-vaccinated）模型.在该模型中,接种的个体由于疫苗的不完全有效而会被感染.假设只有易感态个体采用重连机制进行自我保护,这一机制使得网络拓扑结构和疾病传播共同演化.通过平均场近似并求解方程,发现了丰富的动力学现象,如双稳态和周期振荡.网络的演化也趋于单峰度分布和负相关性.     

Lyapunov exponent of a stochastic SIRS model

We consider a SIRS (susceptible–infected–removed–susceptible) model influenced by random perturbations. We prove that the solutions are positive for positive initial conditions and are global, that is, there is no finite explosion time. We present necessary and sufficient conditions for the almost sure asymptotic stability of the steady state of the stochastic system.     

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.     

Spreading and vanishing in a West Nile virus model with expanding fronts

We study a simplified version of a West Nile virus(WNv) model discussed by Lewis et al.(2006),which was considered as a first approximation for the spatial spread of WNv. The basic reproduction number R_0 for the non-spatial epidemic model is defined and a threshold parameter R_0~D for the corresponding problem with null Dirichlet boundary condition is introduced. We consider a free boundary problem with a coupled system, which describes the diffusion of birds by a PDE and the movement of mosquitoes by an ODE. The risk index R_0~F(t) associated with the disease in spatial setting is represented. Sufficient conditions for the WNv to eradicate or to spread are given. The asymptotic behavior of the solution to the system when the spreading occurs is considered. It is shown that the initial number of infected populations, the diffusion rate of birds and the length of initial habitat exhibit important impacts on the vanishing or spreading of the virus. Numerical simulations are presented to illustrate the analytical results.     

A fractional order epidemic model and simulation for avian influenza dynamics

We present a nonlinear fractional order epidemic model to investigate the spreading dynamical behavior of the avian influenza. The population of the model contains susceptible individuals, asymptomatic but infective latent individuals, and infective individuals. We first establish the existence, uniqueness, nonnegativity, and positive invariance of the solution, then we study the reproduction number of the model and the stability of the disease‐free equilibrium. We observe that the reproduction number varies with the order of the fractional derivative ν. In terms of epidemics, this suggests that varying ν induces a change in the avian's epidemic status. Furthermore, we derive the sufficient conditions for the existence and the stability of the endemic equilibrium. Finally, we carry out some numerical simulations to validate the analytical results. We find from simulations that the solution of the fractional order model tends to a stationary state over a longer period of time with decreasing the value of the fractional derivative, and the size of epidemic decreases with decreasing ν.