具有病毒感染的Holling-Tanner捕食-食饵模型的定性分析

利用稳定性理论,讨论了一类带扩散项食饵染病的捕食-食饵模型正常数解的一致渐近稳定性,得出一定条件下,在常数解的某个邻域内系统不存在非常数正解的结论.同时,利用极值原理和分歧理论,研究了正平衡解的上下界估计和非常数正解的存在性.文中的结果表明,在一定的条件下,受病毒影响的捕食,食饵种群是可以共存的.     

关于日本血吸虫病动力学模型的周期解

日本血吸虫病是在我国广为流传的传染病和寄生虫病,对人体的健康造成了极其严重的危害,关于日本血吸虫病的传播动力学模型引起了广泛的讨论.在吴建宏等建立的双宿主日本血吸虫病的自治动力学模型的基础上,考虑到湖泊型地区钉螺数量的季节性变化因素,本文考察了相应的非自治的传播动力学模型,研究了其周期解的稳定性,并在此基础上进行了数值模拟.研究表明,在一定的参数条件下,无论开始时疾病传播情况如何,疾病终将趋于消亡;否则,在一定的初始条件下,疾病传播形成周期性的地方病.数值模拟发现,在一定的参数条件下,钉螺数量的季节性变化振幅充分大时,可使疾病趋于消亡;此外,同时对患病的人与牛进行治疗,也有利于使疾病消亡.本文中还研究了Barbour双宿主模型的非自治动力学模型,不仅对其周期解的稳定性进行了讨论,还得到该系统周期解的存在性条件.     

一类空间非齐次的SIR传染病模型的稳态解

为研究环境的空间异质性和种群扩散对传染病持续和消除的影响,本文提出了一类空间非齐次的SIR传染病模型.首先,构造模型的基本再生数R0,并分析染病者的扩散对R0的影响.若R01,则无病平衡点全局渐近稳定;若R01,则无病平衡点不稳定.其次,在低危险区域,我们运用分歧理论研究了地方病平衡点的存在性和稳定性.结果表明,减少染病者的扩散并不有利于传染病的消除,但地方病平衡点的不稳定性表明最终传染病可以得到控制.     

具有脉冲效应的Holling—Tanner捕食-被捕食系统的正周期解

为了描述突然性因素对捕食一被捕食系统的影响,本文建立了具有固定时刻脉冲影响的Holling-Tanner捕食-被捕食系统模型,研究了系统的正周期解的存在性.利用基于叠合度理论的连续定理,证明了模型的正T-周期解的存在性,给出了系统存在正周期解的条件,该条件由两个种群的内禀增长率、脉冲周期及一个脉冲周期内的脉冲影响总量给出,这说明脉冲影响可能导致种群系统的平衡状态发生改变.     

一类具时滞和收获的捕食模型的稳定性与Hopf分支

本文研究一类具有常数收获率和时滞的捕食模型,其中时滞描述了捕食种群的妊娠期。通过分析特征方程,得到了正平衡点局部稳定的条件。当时滞τ增加时,正平衡点失去稳定性,当τ跨过临界值时系统将出现Hopf分支。应用中心流形定理和规范型理论,得到了确定Hopf分支方向和分支周期解的稳定性的计算公式。最后对所得理论结果进行了数值模拟。     

具有年龄阶段的离散SCIRS模型在我国流脑中的应用

本文主要研究了具有三个年龄阶段的离散SCIRS模型的动力学性态.首先,利用再生矩阵的方法定义了模型的基本再生数R0,证明了当R01时,模型存在唯一的无病平衡点并且是全局渐近稳定的,当R01时,除了无病平衡点,模型还存在唯一的地方病平衡点.其次,利用法定传染病报告的流脑数据,把模型应用到我国流脑的流行传播中.针对模型中很多参数的不确定性,对基本再生数中的参数进行了敏感性分析.最后,在模型的基础上考虑流脑发病的季节因素对模型加以改进,预测分析了我国流脑的发病情况,数值模拟的结果显示季节因素对疾病进展率的影响程度大于对疾病传染率的影响,为控制流脑在我国的流行传播提供建议.     

具有一般传染率的 SIRS 年龄结构模型的分支研究

考虑到年龄在一些传染病流行过程中的重要影响,建立了一个具有一般传染率的 SIRS 年龄结构仓室模型。通过将模型改写为抽象柯西问题并利用 Hille-Yosida 算子相关定理,分析了模型的动力学性态,讨论了平衡点的稳定性以及平衡点失稳时产生 Hopf 分支的条件。结果表明,当基本再生数小于 1 时,免疫年龄不影响无病平衡点的全局稳定性;当基本再生数大于 1 时,免疫年龄扰动导致地方病平衡点的稳定性改变,从而产生 Hopf 分支。同时,数值模拟验证了理论结果并显示了免疫年龄对模型动力学性态的影响。     

一类具有交叉扩散的捕食–食饵模型正解的存在性分析

带交叉扩散的捕食–食饵模型作为描述生态系统中种群相互作用关系的一种非常重要的生物模型,它的动力学行为在很大程度上依赖于其正解的性质.本文主要运用最大值原理、局部分歧理论、全局分歧理论等,研究了一类具有交叉扩散和MonodHaldane型功能反应函数的捕食模型正解的存在性问题,并给出正解存在的充分条件以及捕食者与食饵在一定条件下可以共存的结果.     

两类具有空间扩散的SIR传染病模型的斑图形成(英文)

空间传染病模型可用于估计空间斑图的形成及传染病的传播速率,因此能更精确地描述传染病的传播过程.在自然界中,考虑到易感者有能力识别感染者群体并有远离他们的趋势,我们研究了两类具有自我扩散和交叉扩散的SIR传染病模型的空间斑图形成.通过在正平衡态处将模型线性化,分析相应的特征方程,得到了系统出现图灵不稳定的充分条件.数值模拟表明模型会出现点状斑图和条状-点状斑图.研究结果丰富了传染病动力学模型中的斑图形成,并可用于预测传染病在空间中的传播过程,为传染病的预防和控制的科学决策提供了理论依据.     

考虑媒体作用的传染病模型的分析与控制?

针对媒体宣传教育对人们行为方式和生活习惯的影响,本文考虑了由于媒体影响而导致易感性不同的一个SEI传染病模型。分析了模型可能出现的后项分支及其平衡点的稳定性和持久性。结果表明,当基本再生数小于1时,模型的无病平衡点全局稳定;当基本再生数大于1时,地方病平衡点一致持久。同时,利用控制理论,本文也研究了媒体的宣传作用对易感者进行影响和教育的最优控制措施,给出了使目标函数值最小的最优控制,并用数值模拟显示了模型解的动力学性态及控制措施对防止疾病蔓延所起的作用。     

一个具有暂时免疫和非线性接触率的SIS流行病模型的分析

本文研究一个具有时滞,一般接触率,常数出生和疾病引起死亡的流行病模型.假设时滞表示暂时免疫期,即恢复者再次变成易感者所需要的时间,同时在模型中考虑了对易感者和恢复者的接种.本文得到了基本再生数R0.分析了模型的无病平衡点和地方病平衡点的存在性.通过Hurwitz准则,研究了无病平衡点和地方病平衡点的局部渐近稳定性.通过Liapunov泛函和Lasalle不变原理,证明了无病平衡点的全局渐近稳定性及在双线性接触率的情况下地方病平衡点的全局渐近稳定性.研究结果表明:R0与对易感者的有效接种率P有关,并且通过增加接种率P可以根除疾病.最后给出了数值模拟.     

Predators indirectly control vector-borne disease: linking predator–prey and host–pathogen models

Pathogens transmitted by arthropod vectors are common in human populations, agricultural systems and natural communities. Transmission of these vector-borne pathogens depends on the population dynamics of the vector species as well as its interactions with other species within the community. In particular, predation may be sufficient to control pathogen prevalence indirectly via the vector. To examine the indirect effect of predators on vectored-pathogen dynamics, we developed a theoretical model that integrates predator–prey and host–pathogen theory. We used this model to determine whether predation can prevent pathogen persistence or alter the stability of host–pathogen dynamics. We found that, in the absence of predation, pathogen prevalence in the host increases with vector fecundity, whereas predation on the vector causes pathogen prevalence to decline, or even become extinct, with increasing vector fecundity. We also found that predation on a vector may drastically slow the initial spread of a pathogen. The predator can increase host abundance indirectly by reducing or eliminating infection in the host population. These results highlight the importance of studying interactions that, within the greater community, may alter our predictions when studying disease dynamics. From an applied perspective, these results also suggest situations where an introduced predator or the natural enemies of a vector may slow the rate of spread of an emerging vector-borne pathogen.     

一类具有食饵恐惧和避难所的反应扩散捕食模型的稳定性与 Hopf 分支

在捕食生态系统中,恐惧因子和食饵避难所都有重要的作用。为此,对一类带恐惧因子和食饵避难所的捕食-食饵反应扩散模型进行了研究。通过分析平衡点特征方程,得到了平衡点的局部渐近稳定性;将不受保护食饵比例作为分支参数,给出了正平衡点 Hopf 分支存在的条件。结果表明：避难所的存在会导致 Hopf 分支,产生空间齐次周期解。扩散的加入会产生新的Hopf分支点,产生空间非齐次周期解。这说明通过设立适当的食饵避难所或者减小捕食者的扩散,有助于物种共存。最后,利用 Matlab 进行数值模拟验证了所得的结论。     

Logistic增长的时滞SIS模型分析

本文研究了一类具有生理阶段结构的Logistic增长的SIS传染病模型,给出了系统边界平衡点和正平衡点全局渐近稳定的条件。即得到了传染病最终消除和成为地方病的阈值。     

The contagious nature of imprisonment: an agent-based model to explain racial disparities in incarceration rates

We build an agent-based model of incarceration based on the susceptible–infected–suspectible (SIS) model of infectious disease propagation. Our central hypothesis is that the observed racial disparities in incarceration rates between Black and White Americans can be explained as the result of differential sentencing between the two demographic groups. We demonstrate that if incarceration can be spread through a social influence network, then even relatively small differences in sentencing can result in large disparities in incarceration rates. Controlling for effects of transmissibility, susceptibility and influence network structure, our model reproduces the observed large disparities in incarceration rates given the differences in sentence lengths for White and Black drug offenders in the USA without extensive parameter tuning. We further establish the suitability of the SIS model as applied to incarceration by demonstrating that the observed structural patterns of recidivism are an emergent property of the model. In fact, our model shows a remarkably close correspondence with California incarceration data. This work advances efforts to combine the theories and methods of epidemiology and criminology.     

基于SEIRS模型的机场飞行区人为风险传染过程研究

为更科学合理地结合风险演化规律和传染过程进行风险管控,构建了基于SEIRS（susceptible-exposed-infected-recovered-susceptible,易感群体-潜伏群体-感染群体-免疫群体-易感群体）传染病模型的机场飞行区人为风险传染过程模型,界定了模型的假设条件,并分析了模型的平衡点及稳定性,模型在平衡点均为渐进稳定的。运用Matlab对模型进行模拟仿真,风险传染初期、中期和末期潜伏群体、感染群体和免疫群体最终均趋于零平衡。其中潜伏群体呈较为平滑的递减趋势,感染群体在急速减少后逐渐趋于零平衡,免疫群体均短期内快速增长后迅速回落。不同阶段各群体的变化率有显著区别。末期潜伏群体在急速减少后呈缓慢上升趋势,最终趋于零平衡。结论为风险管控不同阶段的策略重点提供了理论依据。     

Exogenous re-infection and the dynamics of tuberculosis epidemics: local effects in a network model of transmission

Infection with Mycobacterium tuberculosis leads to tuberculosis (TB) disease by one of the three possible routes: primary progression after a recent infection; re-activation of a latent infection; or exogenous re-infection of a previously infected individual. Recent studies show that optimal TB control strategies may vary depending on the predominant route to disease in a specific population. It is therefore important for public health policy makers to understand the relative frequency of each type of TB within specific epidemiological scenarios. Although molecular epidemiologic tools have been used to estimate the relative contribution of recent transmission and re-activation to the burden of TB disease, it is not possible to use these techniques to distinguish between primary disease and re-infection on a population level. Current estimates of the contribution of re-infection therefore rely on mathematical models which identify the parameters most consistent with epidemiological data; these studies find that exogenous re-infection is important only when TB incidence is high. A basic assumption of these models is that people in a population are all equally likely to come into contact with an infectious case. However, theoretical studies demonstrate that the social and spatial structure can strongly influence the dynamics of infectious disease transmission. Here, we use a network model of TB transmission to evaluate the impact of non-homogeneous mixing on the relative contribution of re-infection over realistic epidemic trajectories. In contrast to the findings of previous models, our results suggest that re-infection may be important in communities where the average disease incidence is moderate or low as the force of infection can be unevenly distributed in the population. These results have important implications for the development of TB control strategies.     

具有一般形式非线性饱和传染率染病年龄结构SIS模型渐近分析

：研究一类具有一般形式非线性饱和传染率染病年龄结构SIS流行病传播数学模型动力学性态,得到疾病绝灭和持续生存的阈值条件——基本再生数。当基本再生数小于或等于1时,仅存在无病平衡点,且在其小于1的情况下,无病平衡点全局渐遗稳定,疾病将逐渐消除;当基本再生数大于1时,存在不稳定的无病平衡点和唯一的局部渐近稳定的地方病平衡点,疾病将持续存在。已有的两类模型可视为本模型的特例,其相关结论可作为本文的推论。     

Impact of scale on the effectiveness of disease control strategies for epidemics with cryptic infection in a dynamical landscape: an example for a crop disease.

We use a spatially explicit, stochastic model to analyse the effectiveness of different scales of local control strategies in containing the long-term, multi-seasonal spread of a crop disease through a dynamically changing population of susceptible crops in which there is cryptic infection. The model distinguishes between susceptible, infested and symptomatic fields. It is motivated by rhizomania disease on sugar beet in the UK as an exemplar of a spatially structured and partially asymptomatic epidemic. Our results show the importance of matching the scales of local control strategies to prevent intensification and regional spread of disease with the inherent temporal and spatial scales of an epidemic. A simple field-scale containment strategy, whereby the susceptible crop is no longer grown on fields showing symptoms, fails for this system with cryptic infection because the locally applied control lags behind the epidemic. A farm-scale strategy, whereby growers respond to the disease status of neighbouring farms by transferring their quota for sugar beet to farmers in regions of reduced risk, succeeds. We conclude that a soil-borne pathogen such as rhizomania could be managed by movement of susceptible crops in the landscape using a strategy that matches the temporal and spatial scales of the epidemic and which take account of risk aversion among growers. We show some parallels and differences in effectiveness between a 'culling' strategy involving crop removal around emerging foci and the local deployment of partially resistant varieties that reduce amplification and transmission of inoculum. Some relationships between the control of plant and livestock diseases are briefly discussed.     

Effect of data quality on estimates of farm infectiousness trends in the UK 2001 foot-and-mouth disease epidemic

Most of the mathematical models that were developed to study the UK 2001 foot-and-mouth disease epidemic assumed that the infectiousness of infected premises was constant over their infectious periods. However, there is some controversy over whether this assumption is appropriate. Uncertainty about which farm infected which in 2001 means that the only method to determine if there were trends in farm infectiousness is the fitting of mechanistic mathematical models to the epidemic data. The parameter values that are estimated using this technique, however, may be influenced by missing and inaccurate data. In particular to the UK 2001 epidemic, this includes unreported infectives, inaccurate farm infection dates and unknown farm latent periods. Here, we show that such data degradation prevents successful determination of trends in farm infectiousness.