具有异质性的传染病模型及疫苗接种策略研究

本文以易感人群疫苗接种为背景提出异质多群组的SEIAR模型.利用下一代矩阵方法求出模型的基本再生数和控制再生数.研究了优先混合方式和异质性对再生数的影响.研究结果表明,活性、亚种群规模以及疫苗覆盖率的异质性对再生数都有着重要的影响,而优先混合能够放大这种影响.最后研究了以疫苗接种率为控制变量的最优控制问题.本文工作可以为传染病疫苗接种策略的制定提供借鉴.     

不完全免疫的多阶段绵羊布鲁氏菌病模型的稳定性分析与最优控制

本文研究一个多阶段的不完全免疫的布鲁氏菌病模型,得到模型平衡点的存在唯一性以及基本再生数R₀.通过构造合适的Lyapunov函数,证明了无病平衡点与地方病平衡点的全局渐近稳定性.在考虑控制成本的情况下,利用最优控制理论得到了布鲁氏菌病最优控制策略.最后用数值模拟验证了理论结果.     

一类带有非线性传染率的SEIS传染病模型的定性分析

借助极限理论和Fonda定理,研究了一类既有常数输入率又有因病死亡率的SEIS传染病模型．所考虑模型的传染率是非线性的,并且得到了该模型的基本再生数,当基本再生数小于1时,该模型仅存在唯一的无病平衡点,它是全局渐近稳定的,且疾病最终灭绝．当基本再生数大于1时,该模型除存在不稳定的无病平衡点外,还存在唯一的局部渐近稳定的地方病平衡点,并且疾病一致持续存在．     

具有阶段进程的SLI艾滋病模型的全局稳定性

艾滋病的潜伏期很长,一般会经历几个潜伏阶段才会发病,故将其看作常数是不合理的.建立了一个具有现实分布的数学模型,即将潜伏期划分为n个阶段.对于一个一般的具有双线性发生率的n-阶段的阶段进程模型,研究了其动力学行为.首先给出了模型的基本再生数.进一步得到,当基本再生数小于1时,无病平衡点是全局渐近稳定的且疾病最终会消失;当基本再生数大于1时,唯一的地方病平衡点是全局渐近稳定的且疾病最终会成为一种地方病.     

具有潜伏期和免疫应答的HIV感染模型的全局稳定性

本文考虑具有CTL免疫应答和细胞内部潜伏阶段的HIV感染数学模型,得到其基本再生数,通过构造适用的Lyapunov函数,研究该模型的健康平衡点和感染平衡点的稳定性.当基本再生数不大于1时,健康平衡点在可行域上是全局稳定的,即HIV在个体体内最终灭绝;当基本再生数大于1时,模型存在惟一的感染平衡点在可行域上是全局稳定的,即HIV在个体体内呈现持续存在状态,且其浓度最终趋于一个常数.     

一类同时含有预防接种和治疗的传染病最优控制模型

建立和分析一个同时含有预防接种和治疗的传染病最优控制模型.首先计算基本再生数,并对无病平衡点和地方病平衡点进行稳定性分析.为了同时降低被感染者人数以及治疗成本,一方面使用最优控制理论和Pontryagin原理分析最优控制策略;另一方面从经济角度出发,综合考虑预防接种和治疗的花费,计算成本效益,使得疾病控制过程中的总成本最省.最后通过数值模拟和敏感性分析,验证理论结果以及寻求对传染病流行起决定性作用的参数.     

具有治疗控制的传染病模型分析

针对筛查和药物治疗对染病者传染性产生的影响,本文考虑了具有两种不同传染水平染病者的仓室数学模型.分析了模型平衡点的稳定性态,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,地方病平衡点在一定条件下也是全局稳定的.同时利用控制理论本文也研究了药物治疗的实施对染病者进行干预和影响的最优控制措施,寻找到了使目标函数值最小的治疗控制方法,并用数值模拟显示了模型解的动力学性态及治疗措施对防止疾病蔓延所起的作用.     

具有媒体报道影响的SIRS模型分析

本文研究了一个具有媒体报道影响的感染率的SIRS传染病模型,得到了基本再生数?_0及模型平衡点的存在性.当?_0?_0~*1时,通过构造Lyapunov函数得到了无病平衡点的全局稳定性.进一步,本文研究由媒体报道引起的对易感者通过降低传染率进行管理控制的最优措施,证明了最优控制的存在性且得到最优控制的显式表达式.     

一类具有阶段结构的虫媒传播植物病模型的全局分析

为了研究化学控制和移除病树对虫媒植物病传播控制的影响,本文建立了一类具有阶段结构的虫媒传播植物病时滞模型.首先,利用再生矩阵法计算得到了基本再生数R₀.理论结果表明,在入侵强度不强的情况下,基本再生数是决定疾病是否流行的阈值条件,即当R₀<1时疾病灭绝,而当R₀>1时疾病爆发.进一步,如果不实施移除病树策略,利用振动逼近的方法我们得到了地方病平衡点全局吸引的充分条件.最后通过数值模拟验证了理论结果,并说明喷洒杀虫剂是一种非常有效的控制措施.     

一类具有双线性传染率的HIV/AIDS病毒动力学改进模型

针对HIV/AIDS传播的具有常数移民和指数出生的SI型模型,为了更加符合实际意义,对具有双线性传染率的模型进行局部改进,并对改进后的动力学模型进行了简化.对于改进后的模型,证明了平衡点的存在与局部稳定性,并证明了传染病毒的灭绝与持续性,得到了传染病毒的基本再生数.结果表明:当单位时间内从外界迁入人口中染病者的比例系数c近似等于零时,基本再生数小于1时,传染病毒最终灭绝;当基本再生数大于1时,模型存在唯一的正平衡点,且是局部渐近稳定的,说明传染病毒一致持续存在.     

Modeling the Transmission of West Nile Virus with {\it Wolbachia} in a Heterogeneous Environment

{\it Wolbachia} are maternally transmitted endosymbiotic bacteria. To investigate the effect of {\it Wolbachia} on the spreading and vanishing of West Nile virus, we construct a reaction-diffusion model associated with the {\it Wolbachia} parameter in a heterogeneous environment, which has nonlinear infectious disease parameters. Based on the spectral radius of next infection operator and the related eigenvalue problem, we present a corresponding explicit expression describing the basic reproduction number. Furthermore, utilizing this number, we not only give out the stability of disease-free equilibrium, but also analyze the uniqueness and globally asymptotic behavior of endemic equilibrium. Our theoretical results and numerical simulations indicate that only if {\it Wolbachia} reach a certain magnitude in mosquitoes, it can be effective in the control of West Nile virus.     

一类潜伏期有传染性的传染病模型动力学分析

建立了一类潜伏期具备传染性的传染病传播模型,根据疾病传播规律求解了疾病消失和持续生存的阈值——基本再生数.对系统的稳定性进行了讨论,得到了系统稳定性条件.最后,以COVID-19为例,解释了各种举措在疾病控制中的作用,并对疫情传播扩散做了探讨和预测.     

Impact of farming awareness based roguing,insecticide spraying and optimal control on the dynamics of mosaic disease

Control interventions and farming knowledge are equally important for plant disease control. In this article, a mathematical model has been derived using saturated response functions (nonlinear infection rate) for studying the dynamics of mosaic disease with farming awareness based roguing (removal of infected plants) and insecticide spraying . It is assumed that the use of roguing and spraying depend on the level of awareness about the disease. The model possesses three equilibria namely the trivial, which is always unstable, the disease-free equilibrium which is stable if the basic reproduction number is below unity and the coexisting which may be stable or can exhibit Hopf-bifurcation under certain condition. Finally, we have opted an optimal control problem introducing three control parameters for determining the optimal level of roguing, spraying and cost regarding media awareness for cost-effective control of mosaic disease. Numerical simulations establish the main results suggesting that the awareness campaigns through radio, TV advertisement are important for eradication of the disease. Also, awareness campaign, roguing and spraying should be incorporated with optimal level for cost effective control of mosaic disease.

     

Optimal bed net use for a dengue disease model with mosquito seasonal pattern

Controversial results concerning the effectiveness of bed net in reducing dengue fever transmission make further research necessary in this direction. At this aim, we consider a mathematical model of dengue transmission where the use by individuals of insecticide‐treated bed nets is taken into account, combined or not with insecticide spraying. Furthermore, as climatic factors play a key role in mosquito‐borne diseases, we model the effect of seasonality through a periodic mosquito birth rate. We numerically investigate some specific scenarios according to different rainfall and mean temperature values. We set an optimal control problem to minimize the number of human infections and the cost of efforts placed into bed net adoption and maintenance and insecticide spraying. To assess the most appropriate strategy to eliminate dengue with minimum costs, we perform a comparative cost‐effectiveness analysis, which also shows how the cost‐benefit of intervention efforts is affected by changes in the amplitude of seasonal variation. One general result is that in any case the combination of bed net use and insecticide spraying produces the highest ratio of infections averted, whereas in terms of cost‐benefit only spraying campaigns should be implemented in control programs for regions with no large seasonality.     

Optimal control approach to dengue reduction and prevention in Cali,Colombia

The aim of this paper is to propose optimal strategies for dengue reduction and prevention in Cali, Colombia. For this purpose, we consider two variants of a simple dengue transmission model, epidemic and endemic, each of which is amended with two control variables. These variables express feasible control actions to be taken by an external decision‐maker. First control variable stands for the insecticide spraying and thus targets to suppress the vector population. The second one expresses the protective measures (such as use of repellents, mosquito nets, and insecticide‐treated clothes) that are destined to reduce the number of contacts (bites) between female mosquitoes (principal dengue transmitters) and human individuals. We use the Pontryagin's maximum principle in order to derive the optimal strategies for dengue control and then perform the cost‐effectiveness analysis of these strategies in order to choose the most sustainable one in terms of cost–benefit relationship. Copyright © 2016 John Wiley & Sons, Ltd.     

Qualitative analysis for a Wolbachia infection model with diffusion

We consider a reaction-diffusion model which describes the spatial Wolbachia spread dynamics for a mixed population of infected and uninfected mosquitoes. By using linearization method, comparison principle and Leray-Schauder degree theory, we investigate the influence of diffusion on the Wolbachia infection dynamics. After identifying the system parameter regions in which diffusion alters the local stability of constant steady-states, we find sufficient conditions under which the system possesses inhomogeneous steady-states. Surprisingly, our mathematical analysis, with the help of numerical simulations, indicates that diffusion is able to lower the threshold value of the infection frequency over which Wolbachia can invade the whole population.     

An optimal control problem for a prey–predator system with a general functional response

An optimal control problem is studied for a prey–predator system with a general functional response. The control functions represent the rate of mixture of the populations and the cost functional is of Mayer type. The number of switching points of the optimal control is discussed in terms of the sign of a specific constant.     

Optimal control of plant virus propagation

Plants are a food source for man and many species. But, plants are subject to diseases, many of which are caused by viruses. Usually, virus propagation is done by a vector. Insect vectors typically have a seasonal behavior, and processes have delays. To combat the vectors, chemical insecticides are commonly used as a control. Unfortunately, these chemicals not only are expensive but also have toxic effects on humans, animals, and the environment. An alternative is to introduce a predator species to prey on the insects and limit the spread of the virus. A combination of insecticide and predators can be used to control the vector population. The question is whether there is an optimal combination. We introduce a mathematical model of ordinary differential equations describing the interaction between plants, vectors, and predators. To determine the optimal amount of predators to introduce and insecticide to use, an objective function giving the total cost to the farmer of the disease is given. We find the controls that minimize the objective function subject to the population variables satisfying the differential equation model and initial conditions together with constraints. There are two main different approaches that can be used to solve the optimal control problem: indirect and direct methods. We use direct methods to solve the problem with and without seasonality and delays. From the practical side, the model can be used to help farmers determine the right balance of insecticide and predators to minimize the total cost.     

Inventory control as a discrete system control for the fixed-order quantity system

This paper shows how to model a problem to find optimal number of replenishments in the fixed-order quantity system as a basic problem of optimal control of the discrete system. The decision environment is deterministic and the time horizon is finite. A discrete system consists of the law of dynamics, control domain and performance criterion. It is primarily a simulation model of the inventory dynamics, but the performance criterion enables various order strategies to be compared. The dynamics of state variables depends on the inflow and outflow rates. This paper explicitly defines flow regulators for the four patterns of the inventory: discrete inflow – continuous/discrete outflow and continuous inflow – continuous/discrete outflow. It has been discussed how to use suggested model for variants of the fixed-order quantity system as the scenarios of the model. To find the optimal process, the simulation-based optimization is used.     

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.