食饵具有流行病的阶段结构捕食模型

本文考虑了一类食饵具有流行病和阶段结构的脉冲时滞捕食模型．利用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当脉冲周期在一定范围内时，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E（EIL）之下．我们的结论为现实的害虫管理提供了可靠的策略依据．     

不动点指数及其对传染病传播模型的非线性积分方程的应用*

本文研究了一类具边界控制条件的正算子的不动点指数问题,得到了几个有较好应用价值的不动点指数为1或0的条件,以及一个更为广泛的锥拉伸与锥压缩不动点定理,推广和改进了[3,8,9]中的相应结果。作为应用,文中也讨论了传染病传播模型的非线性积分方程具正连续周期解的条件。     

New modelling approach concerning integrated disease control and cost-effectivity

Two new models for controlling diseases, incorporating the best features of different control measures, are proposed and analyzed. These models would draw from poultry, livestock and government expertise to quickly, cooperatively and cost-effectively stop disease outbreaks. The combination strategy of pulse vaccination and treatment (or isolation) is implemented in both models if the number of infectives reaches the risk level (RL). Firstly, for one time impulsive effect we compare three different control strategies for both models in terms of cost. The theoretical and numerical results show that there is an optimal vaccination and treatment proportion such that integrated pulse vaccination and treatment (or isolation) reaches its minimum in terms of cost. Moreover, this minimum cost of integrated strategy is less than any cost of single pulse vaccination or single treatment. Secondly, a more realistic case for the second model is investigated based on periodic impulsive control strategies. The existence and stability of periodic solution with the maximum value of the infectives no larger than RL is obtained. Further, the period T of the periodic solution is calculated, which can be used to estimate how long the infectious population will take to return back to its pre-control level (RL) once integrated control tactics cease. This implies that we can control the disease if we implement the integrated disease control tactics every period T. For periodic control strategy, if we aim to control the disease such that the maximum number of infectives is relatively small, our results show that the periodic pulse vaccination is optimal in terms of cost.     

Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases

In this paper, a non-linear mathematical model for the effects of awareness programs on the spread of infectious diseases such as flu has been proposed and analyzed. In the modeling process it is assumed that disease spreads due to the contact between susceptibles and infectives only. The growth rate of awareness programs impacting the population is assumed to be proportional to the number of infective individuals. It is further assumed that due to the effect of media, susceptible individuals form a separate class and avoid contact with the infectives. The model is analyzed by using stability theory of differential equations. The model analysis shows that the spread of an infectious disease can be controlled by using awareness programs but the disease remains endemic due to immigration. The simulation analysis of the model confirms the analytical results.     

Dynamics of SIS Epidemic Model with Varying Total Population and Multivaccination Control Strategies

In this paper, two susceptible‐infected‐susceptible epidemic models with varying total population size, continuous vaccination, and state‐dependent pulse vaccination are formulated to describe the transmission of infectious diseases, such as diphtheria, measles, rubella, pertussis, and so on. The first model incorporates the proportion of infected individuals in population as monitoring threshold value; we analytically show the existence and orbital asymptotical stability of positive order‐1 periodic solution for this control model. The other model determines control strategy by monitoring the proportion of susceptible individuals in population; we also investigate the existence and global orbital asymptotical stability of the disease‐free periodic solution. Theoretical results imply that the disease dies out in the second case. Finally, using realistic parameter values, we carry out some numerical simulations to illustrate the main theoretical results and the feasibility of state‐dependent pulse control strategy.     

A delayed epidemic model with stage-structure and pulses for pest management strategy

From a biological pest management standpoint, epidemic diseases models have become important tools in control of pest populations. This paper deals with an impulsive delay epidemic disease model with stage-structure and a general form of the incidence rate concerning pest control strategy, in which the pest population is subdivided into three subgroups: pest eggs, susceptible pests, infectious pests that do not attack crops. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic susceptible pest-eradication solution of the system and observe that the susceptible pest-eradication periodic solution is globally attractive, provided that the amount of infective pests released periodically is larger than some critical value. When the amount of infective pests released is less than another critical value, the system is shown to be permanent, which implies that the trivial susceptible pest-eradication solution loses its attractivity. Our results indicate that besides the release amount of infective pests, the incidence rate, time delay and impulsive period can have great effects on the dynamics of our system.     

Vaccination strategies based on feedback control techniques for a general SEIR-epidemic model

This paper presents several simple linear vaccination-based control strategies for a SEIR (susceptible plus infected plus infectious plus removed populations) propagation disease model. The model takes into account the total population amounts as a refrain for the illness transmission since its increase makes more difficult contacts among susceptible and infected. The vaccination control objective is the asymptotically tracking of the removed-by-immunity population to the total population while achieving simultaneously that the remaining populations (i.e. susceptible plus infected plus infectious) tend asymptotically to zero.     

On vaccination controls for the SEIR epidemic model

This paper presents a simple continuous-time linear vaccination-based control strategy for a SEIR (susceptible plus infected plus infectious plus removed populations) disease propagation model. The model takes into account the total population amounts as a refrain for the illness transmission since its increase makes more difficult contacts among susceptible and infected. The control objective is the asymptotically tracking of the removed-by-immunity population to the total population while achieving simultaneously the remaining population (i.e. susceptible plus infected plus infectious) to asymptotically tend to zero.     

Parametric control of solutions to an evolution problem in a neighborhood of an unstable stationary regime

We study a controlled system of ordinary differential equations in a neighborhood of an unstable stationary regime. We seek for a control under which the solution remains in the neighborhood however long. We find conditions under which such control is possible and prove an existence theorem. The results are of a constructive character and can be applied to controlling actual processes.     

K-means聚类算法在SIR传染病模型中的应用研究

基于SIR传染病模型,建立了具有K-means聚类算法的SIR元胞自动机模拟模型.通过对分别服从高斯分布和随机均匀分布的两类初始感染源的分析与模拟,给出了疾病感染半径与隔离半径对疾病传播的影响.结果显示:在两种不同类型的初试分布下,感染者的最大值分别与疾病感染传播半径和隔离半径呈正相关与负相关关系,感染者数量随时间的变化率亦呈现相同的变化规律.初始数据的不同分布类型只影响这种正负相关关系的增速.研究结果可为控制和消除传染病提供有效合理的隔离措施,为卫生部门提供防控传染病的理论支持.     

Low‐density barriers for controlling plant crowd diseases: How far and fast can pathogens spread?

We explore numerically the possibility of controlling the spread of plant diseases characterized by relatively low dispersal (crowd diseases) through the introduction of a spatial barrier with low density of susceptible hosts. We use the diffusion approximation to Kendall's spatially extended version of the Kermack–McKendrick epidemic model and illustrate our findings within the context of a representative viral disease that affects cocoa trees. RECOMMENDATIONS FOR MANAGERS:

• Our numerical results suggest that using low‐density barriers of hosts in crowd plant diseases might be an effective way of halting the spatial dispersal of pathogens. The introduction of these barriers may reduce the economic impact when compared with other methods of controlling the disease spread.
• Before using the model to approximate suitable sizes of barriers, it is necessary to execute an exhaustive assessment of the model appropriateness for any particular disease under consideration.
• Our results suggest that to improve the efficiency of low‐density barriers it is important to explore their use in combination of current alternative control methods.

     

Generalized Markov Models of Infectious Disease Spread: A Novel Framework for Developing Dynamic Health Policies

We propose a class of mathematical models for the transmission of infectious diseases in large populations. This class of models, which generalizes the existing discrete-time Markov chain models of infectious diseases, is compatible with efficient dynamic optimization techniques to assist real-time selection and modification of public health interventions in response to evolving epidemiological situations and changing availability of information and medical resources. While retaining the strength of existing classes of mathematical models in their ability to represent the within-host natural history of disease and between-host transmission dynamics, the proposed models possess two advantages over previous models: (1) these models can be used to generate optimal dynamic health policies for controlling spreads of infectious diseases, and (2) these models are able to approximate the spread of the disease in relatively large populations with a limited state space size and computation time.     

Effects of population density on the spread of disease

The effect of population density on the epidemic outbreak of measles or measles-like infectious diseases was evaluated. Using average-number contacts with susceptible individuals per infectious individual as a measure of population density, an analytical model for the distribution of the nonstationary stochastic process of susceptible contact is presented. A 5-dimensional lattice simulation model of disease spread was used to evaluate the effects of four different population densities. A zero-inflated Poisson probability model was used to quantify the nonstationarity of the contact rate in the stochastic epidemic process. Analysis of the simulation results identified a decrease in a susceptible contact rate from four to three, resulted in a dramatic effect on the distribution of contacts over time, the magnitude of the outbreak, and, ultimately, the spread of disease. © 2001 John Wiley & Sons, Inc.     

Controlling absence seizures by deep brain stimulus applied on substantia nigra pars reticulata and cortex

Epilepsy is a typical neural disease in nervous system, and the control of seizures is very important for treating the epilepsy. It is well known that the drug treatment is the main strategy for controlling the epilepsy. However, there are about 10–15 percent of patients, whose seizures cannot be effectively controlled by means of the drug. Alternatively, the deep brain stimulus (DBS) technology is a feasible method to control the serious seizures. However, theoretical explorations of DBS are still absent, and need to be further made. Presently, we will explore to control the absence seizures by introducing the DBS to a basal ganglia thalamocortical network model. In particular, we apply DBS onto substantia nigra pars reticulata (SNr) and the cortex to explore its effects on controlling absence seizures, respectively. We can find that the absence seizure can be well controlled within suitable parameter ranges by tuning the period and duration of current stimulation as DBS is implemented in the SNr. And also, as the DBS is applied onto the cortex, it is shown that for the ranges of present parameters, only adjusting the duration of current stimulation is an effective control method for the absence seizures. The obtained results can have better understanding for the mechanism of DBS in the medical treatment.     

A study on time-limited control of single-pest with stage-structure

Two kinds of time-limited pest control models of single-pest with stage-structure, which can be described by the boundary value problem of ordinary differential equation and impulsive differential equation, are presented according to the ways of artificial control (continuous control and impulsive control). The conditions under which the corresponding model has a solution are given. If the model has a solution, the corresponding aim of pest control can be achieved. The theoretical results show that both the mature and the immature pest should be controlled synchronously, otherwise the aims of pest control can not be achieved in a finite time. Finally, some discussions and numerical simulations show that the impulsive control is more practical than the continuous control.     

考虑污染和种内关系的种群扩散最优控制模型

针对污染和种内关系均影响细菌种群扩散这一管理生态学问题,本文建立了基于非线性拟抛物方程的最优控制模型,将外界环境向细菌种群输入的毒素率作为控制变量,运用控制理论和方法探讨污染和种内关系双重影响下种群扩散系统的最优控制问题。利用Schauder不动点定理证明了该种群扩散系统的适定性;同时,通过建立新的Carleman型估计,给出了容许控制和最优控制的存在性。最后,通过数值算例分析了理论推导的结果,在算例中都找到一对时间最优控制,验证了种群扩散系统最优控制模型的有效性。该研究结果对现代传染病预防具有借鉴意义,也为有效控制瘟疫的爆发和流行提供理论参考。     

Global properties of a two-scale network stochastic delayed human epidemic dynamic model

Complex population structure and the large-scale inter-patch connection human transportation underlie the recent rapid spread of infectious diseases of humans. Furthermore, the fluctuations in the endemicity of the diseases within patch dwelling populations are closely related with the hereditary features of the infectious agent. We present an SIR delayed stochastic dynamic epidemic process in a two-scale dynamic structured population. The disease confers temporary natural or infection-acquired immunity to recovered individuals. The time delay accounts for the time-lag during which naturally immune individuals become susceptible. We investigate the stochastic asymptotic stability of the disease free equilibrium of the scale structured mobile population, under environmental fluctuations and the impact on the emergence, propagation and resurgence of the disease. The presented results are demonstrated by numerical simulation results.     

Hopf bifurcation in a control system for the Washout filter-based delayed neural equation

Washout filter is a simple filter that can be designed easily. In this paper, a system for controlling a neural equation with discrete time delay based on Washout filter is presented. The transcendental equation of the corresponding linearized system is analyzed. In this control system, it is found that Hopf bifurcation occurs when the control parameters are chosen properly and that a chaotic orbit can be controlled to a stable periodic solution. The stability condition for bifurcating periodic solutions and the direction of Hopf bifurcation are studied by applying the normal form theory and the center manifold theorem. Some numerical results are also presented to illustrate the correctness of our results.     

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.     

A Plant Virus Disease Model with Periodic Environment and Pulse Roguing

Regular roguing is an effective method to control plant virus diseases. In this paper, a compartmental mathematical model is established to represent the dynamics of plant disease in a periodic environment, including impulsive roguing control strategy. The basic reproductive number and its relation to the persistence of the disease is discussed via using next infection operation. Numerical simulations are performed to demonstrate the theoretical findings, and to illustrate the effect of control measures. Our results show that, (i) when the infection rate is high, it may be impossible to eradicate the disease by simply roguing the infectious plant, so how to effectively identify the latent plant is a key issue in disease control, (ii) increasing the replanting rate is bad for disease control, (iii) the published autonomous research model with continuous roguing may overestimate the infectious risk inherent to impulsive control.