基于防治策略的毒品滥用流行病学模型

针对现有毒品滥用流行病学研究未考虑预防措施的不足，通过引入预防机制，提出了基于防治策略的易感-感染-治疗-康复-易感（SITRS）毒品滥用流行病学模型。首先，通过分析毒品滥用相关人群的演化过程，利用常微分方程构建了一个自治的动力系统；其次，证明了系统无毒平衡点的存在性和局部渐进稳定性；然后，分析了地方病平衡点的唯一存在性，并得到了地方病平衡点的全局渐进稳定性的充分条件；最后，计算了出现后向分支现象的必要条件，并比较了综合防治策略和单一治疗策略下的基本再生数。数值模拟验证了存在后向分支的可能性及平衡点的稳定性。研究结果表明相对于单一治疗措施，采用综合防治策略，通过提高宣传覆盖率和教育有效率，能够进一步降低毒品滥用的基本再生数，更有效地防止毒品滥用的滋生。     

基于防治策略的毒品滥用流行病学模型

针对现有毒品滥用流行病学研究未考虑预防措施的不足,通过引入预防机制,提出了基于防治策略的易感-感染-治疗-康复-易感（SITRS）毒品滥用流行病学模型。首先,通过分析毒品滥用相关人群的演化过程,利用常微分方程构建了一个自治的动力系统;其次,证明了系统无毒平衡点的存在性和局部渐进稳定性;然后,分析了地方病平衡点的唯一存在性,并得到了地方病平衡点的全局渐进稳定性的充分条件;最后,计算了出现后向分支现象的必要条件,并比较了综合防治策略和单一治疗策略下的基本再生数。数值模拟验证了存在后向分支的可能性及平衡点的稳定性。研究结果表明相对于单一治疗措施,采用综合防治策略,通过提高宣传覆盖率和教育有效率,能够进一步降低毒品滥用的基本再生数,更有效地防止毒品滥用的滋生。     

移动自组网病毒传播模型及稳定性分析

考虑移动自组网中节点的移动特性,基于平均场理论提出移动自组网中病毒传播模型,并对建立的方程组进行平衡点存在性和稳定性分析,得出病毒传播的阈值及消亡条件,从而研究节点移动速度、通信半径、免疫成功率和免疫失效率对移动自组网中病毒传播行为和传播临界特性的影响。结果表明:当病毒基本再生数R0<1时,网络全局渐近稳定在无病毒平衡点;当R0>1时,网络全局渐近稳定在地方病平衡点。最后通过数值仿真验证了该模型的正确性。     

考虑无症状和变异者的SAIVR传染病模型

考虑无症状者、变异者的存在以及易感者通过其他方式直接变为免疫者等因素，在传统传染病模型的基础上建立了一个新的SAIVR传染病模型。根据SAIVR模型的传播规则，利用微分方程理论给出了该模型的传播动力学方程，分析了该模型的无病平衡点和地方病平衡点的存在性，利用下一代矩阵方法计算出该模型在无病平衡点处的基本再生数，根据Routh-Hurwitz判据得到了该模型在平衡点处的局部渐近稳定性条件，利用Lyapunov理论证明了模型的全局稳定性。仿真实验表明，考虑无症状者和变异者的SAIVR模型准确预测了传染病的爆发时间、爆发规模和消亡时间，有助于减少传染病在人群中的传播率，增加感染者、变异者的免疫率，有效控制SAIVR传染病的传播。     

具免疫时滞的HBV传染病模型的稳定性和Hopf分支

考虑了溶解性和非溶解性机制下的一类具有免疫时滞的HBV感染动力学模型。分析了无感染平衡点及感染无免疫平衡点的全局稳定性,讨论了感染免疫平衡点的局部稳定性和Hopf分支的存在条件。数值模拟结果表明：当易感细胞生成率的取值使得基本再生数满足平衡存在条件且低于某一临界值时,时滞对平衡点的稳定性没有影响;当大于该临界值时,随着时滞增大,平衡点不稳定,出现一系列Hopf分支,最终表现为周期波动模式。     

对等网络中被动型蠕虫传播动力学建模与分析

鉴于被动型蠕虫对网络的安全构成了威胁，考虑网络异质性和一次搜索允许的跳数，提出一个带有补充率和移除率的新动力学模型，推导出决定被动型蠕虫能否传播的阈值--基本再生数，分析了无蠕虫平衡点和正平衡点的稳定性问题。最后通过数值模拟验证了理论的正确性，并对参数敏感性进行了分析。研究结果发现，当基本再生数大于1时，跳数的增加不但加快被动型蠕虫的传播，而且使感染节点的数量也增加；当基本再生数小于1时，跳数的增加使感染节点的数量迅速减少，最终被动型蠕虫灭绝。     

基于SEIPQR模型的工控蠕虫防御策略

随着社会的发展和技术的进步,计算机病毒也发生了进化,变得越来越复杂,越来越隐蔽。其中蠕虫病毒更是最早的计算机病毒发展进化成为可以在工控系统上感染并进行传播的工控蠕虫病毒,极大影响工业生产的安全。单一的网络隔离或者打补丁免疫,已经跟不上蠕虫病毒的传播速度。针对该现状,分析蠕虫病毒在工控系统上的传播方式以及特点,在原有网络隔离和补丁的基础上提出一种针对工控蠕虫的防御策略,以达到有效防御蠕虫病毒的目的。该防御策略基于传染病模型的基本思想提出了一个模拟蠕虫传播趋势的数学模型 SEIPQR。该模型包含易感染（susceptible）状态、暴露（exposed）状态、打补丁（patched）状态、感染（infected）状态、隔离（quarantine）状态以及免疫（recovered）状态 6 种状态,创建模型的 6 种状态转换图,对状态转换图得到微积分方程组,在系统设备数量一定的情况下,对方程组进行变换,通过求解基本再生数R0的方法对方程组进行求解,并分析当暴露主机和感染主机的数量为0时模型的6种方程表达式,根据Routh-Hurwitz准则得出当R₀＜1时,系统是渐进稳定的;当R₀＞1时,系统是不稳定的。通过数值仿真对比在不同打补丁概率、不同隔离率以及不同感染率3种情况下SEIPQR模型的动力学特性,并得到模型的无病平衡点和地方病平衡点。数据仿真结果表明,在整个系统感染蠕虫病毒时,对易感染设备及时地打补丁以及进行网络隔离可以有效抑制工控蠕虫的传播。     

具常数移出率和阶段结构的SIR传染病模型

对一类具常数移出率和阶段结构传染病模型进行了分析,得到了传染病最终消除和成为地方病的阈值,当它小于1时,无病平衡点是全局渐近稳定的,此时疾病消除.当它大于1时,地方病平衡点是局部渐近稳定的,此时传染病成为地方病.     

不同输入率与移出率的蠕虫病毒传播模型

为研究网络中节点输入率和移出率的差异性对蠕虫病毒传播的影响,基于仓室建模思想分析各个仓室之间的转化关系,构建一种具有不同输入率和移出率的蠕虫病毒传播SEIR模型。计算模型的平衡点和基本再生数,给出平衡点稳定性规范,并通过Hurwitz定理、LaSlle不变性原理和Bendixson定理证明平衡点的稳定性。在此基础上,利用数值仿真验证理论分析的结果,分析影响蠕虫病毒传播的关键因素,进而提出抑制蠕虫病毒传播的建议措施。     

一类具饱和传染率和两阶段结构的传染病模型

对一类具饱和传染率和阶段结构传染病模型进行了分析,得到了传染病最终消除和成为地方病的阈值,当它小于1时,无病平衡点是全局渐近稳定的,此时疾病消除;当它大于1时,地方病平衡点是局部渐近稳定的,此时传染病成为地方病。     

Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity

In this paper, a delayed SIRS epidemic model with saturation incidence and temporary immunity is investigated. The immunity gained by experiencing a disease is temporary, whenever infected the diseased individuals will return to the susceptible class after a fixed period. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.     

Dynamics of Infectious Diseases with Media Coverage and Two Time Delay

An epidemic model is formulated and analysed on the prevalence of infectious diseases using awareness campaign driven by media with the aim to investigate the effect of awareness and delay on disease outbreak. Two time-delay factors are considered, one is for the time lag in reporting number of infected individuals and another is for the delay between the awareness campaign and the time of taking measures by susceptible individual. The system exhibits two equilibria: the disease-free equilibrium and endemic equilibrium. The disease-free equilibrium is stable for any delay when the basic reproduction number is less than unity. The endemic equilibrium exhibits Hopf-bifurcation for both the delays. Numerical simulations prove the results of analytical outcomes and the significance of awareness and delay in controlling infectious diseases.     

Global dynamics of a two-strain avian influenza model

A deterministic model for the transmission dynamics of avian influenza in birds (wild and domestic) and humans is developed. The model, which allows for the transmission of an avian strain and its mutant (assumed to be transmissible between humans), as well as the isolation of individuals with symptoms of any of the two strains, has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. Further, the model has a unique endemic equilibrium whenever this threshold quantity exceeds unity. It is shown, using a non-linear Lyapunov function and LaSalle invariance principle, that this endemic equilibrium is globally asymptotically stable for a special case of the avian-only system. Numerical simulations show that, on average, the isolation of individuals with the avian strain is more beneficial than isolating those with the mutant strain. Furthermore, disease burden increases with increasing mutation rate of the avian strain.

     

Mathematical analysis of a disease transmission model with quarantine,isolation and an imperfect vaccine

A new mathematical model for the transmission dynamics of a disease subject to the quarantine (of latent cases) and isolation (of symptomatic cases) and an imperfect vaccine is designed and analyzed. The model undergoes a backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. It is shown that the backward bifurcation phenomenon can be removed if the vaccine is perfect or if mass action incidence, instead of standard incidence, is used in the model formulation. Further, the model has a unique endemic equilibrium when the threshold quantity exceeds unity. A nonlinear Lyapunov function, of the Goh–Volterra type, is used to show that the endemic equilibrium is globally-asymptotically stable for a special case. Numerical simulations of the model show that the singular use of a quarantine/isolation strategy may lead to the effective disease control (or elimination) if its effectiveness level is at least moderately high enough. The combined use of the quarantine/isolation strategy with a vaccination strategy will eliminate the disease, even for the low efficacy level of the universal strategy considered in this study. It is further shown that the imperfect vaccine could induce a positive or negative population-level impact depending on the size (or sign) of a certain associated epidemiological threshold.     

Global dynamics of an HIV-1 infection model with distributed intracellular delays

In this paper, an HIV-1 infection model with distributed intracellular delays is investigated, where the intracellular delays account for the time the target cells are contacted by the virus particles and the time the contacted cells become actively infected meaning that the contacting virions enter cells and the time the virus has penetrated into a cell and the time the new virions are created within the cell and are released from the cell, respectively. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and LaSalle’s invariance principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable; and if the basic reproduction ratio is greater than unity, the chronic-infection equilibrium is globally asymptotically stable.     

Structured parametric epidemic models

A stage-structured model for a theoretical epidemic process that incorporates immature, susceptible and infectious individuals in independent stages is formulated. In this analysis, an input interpreted as a birth function is considered. The structural identifiability is studied using the Markov parameters. Then, the unknown parameters are uniquely determined by the output structure corresponding to an observation of infection. Two different birth functions are considered: the linear case and the Beverton–Holt type to analyse the structured epidemic model. Some conditions on the parameters to obtain non-zero disease-free equilibrium points are given. The identifiability of the parameters allows us to determine uniquely the basic reproduction number ?₀ and the stability of the model in the equilibrium is studied using ?₀ in terms of the model parameters.     

Dynamical behavior of a vector-host epidemic model with demographic structure

In the paper, we propose a model that tracks the dynamics of many diseases spread by vectors, such as malaria, dengue, or West Nile virus (all spread by mosquitoes). Our model incorporates demographic structure with variable population size which is described by nonlinear birth rate and linear death rate. The stability of the system is analyzed for the existence of the disease-free and endemic equilibria points. We find the basic reproduction number R₀ in terms of measurable epidemiological and demographic parameters is the threshold condition that determines the dynamics of disease infection: if R₀<1 the disease fades out, and for R₀>1 the disease remains endemic. The threshold condition provides important guidelines for accessing control of the vector diseases, and implies that it is an efficient way to halt the spread of vector epidemic by reducing the carrying capacity of the environment for the vector and the host. Moreover, sufficient conditions are also obtained for the global stability of the unique endemic equilibrium E^*.     

The combined impact of external computers and network topology on the spread of computer viruses

This paper aims to study the combined impact of external computers and network topology on the spread of computer viruses over the Internet. By assuming that the network underlying a recently proposed model capturing virus spreading behaviour under the influence of external computers follows a power-law degree distribution, a new virus epidemic model is proposed. A comprehensive study of the model shows the global stability of the virus-free equilibrium or the global attractivity of the viral equilibrium, depending on the basic reproduction number R₀. Next, the impacts of different model parameters on R₀ are analysed. In particular, it is found that (a) higher network heterogeneity benefits virus spreading, (b) higher-degree nodes are more susceptible to infections than lower-degree nodes, and (c) a lower rate at which external computers enter the Internet could restrain virus spreading. On this basis, some practical measures of inhibiting virus diffusion are suggested.