A MODEL OF THE ENZOOTIOLOGY OF LYME DISEASE IN THE ATLANTIC NORTHEAST OF THE UNITED STATES

A mathematical model is presented for the dynamics of the rate of infection of the Lyme disease vector tick Ixodes dammini (Acari: Ixodidae) by the spirochete Borrelia burgdorferi, in the Atlantic Northeast of the United States. According to this model, moderate reductions in the abundance of white-tailed deer Odocoileus virginianus may either decrease or increase the spirochete infection rate in ticks, provided the deer are not reservoir hosts for Lyme disease. Expressions for the basic reproductive rate of the disease are computed analytically for special cases, and it is shown that as the basic reproductive rate increases, a proportional reduction in the tick population produces a smaller proportional reduction in the infection rate, so that vector control is less effective far above the threshold. The model also shows that control of the mouse reservoir hosts Peromyscus leucopus could reduce the infection rate if the survivorship of juvenile stages of ticks were reduced as a consequence. If the survivorship of juvenile stages does not decline as the rodent population is reduced, then rodent reduction can increase the spirochete infection rate in the ticks.     

On the asymptotic equilibrium of a population system with migration

In the analysis of economic and social issues of a country (or any larger or smaller socio-economic unit) the demographic dynamics of the considered population often play a crucial role. Very current emergencies in this respect are e.g. ageing, longevity risk, state-run healthcare etc. Over the last decade migration between EU countries also became an important issue, and in recent years the uncontrolled migration from non-EU countries is also a major concern. Therefore, the better theoretical understanding of the evolutionary mechanism of age-classified populations interacting via migration, is a timely modelling-methodological task. This paper is a preliminary demographic methodological contribution to a further research in support of socio-economic modelling and decision making concerning migration issues.It is known that in the framework of the classical age-specific Leslie model, under simple demographic conditions, a closed population in the long term tends to an equilibrium age distribution. As the main theoretical result of the paper, a similar convergence is proved for a system of several populations with migration between them, and this long-term behaviour (convergence theorem) is extended to systems of sex-structured populations. Based on the latter model, medium term projections are also analysed concerning the effect of migration among countries on the development of the old-age dependency ratio (the proportion of pensioner age classes to active ones), which is an aggregate scalar indicator of ageing, a major concern in most industrialized countries. Illustrative simulation analysis is carried out with data from three European countries.     

Multiple stability and uniqueness of the limit cycle in a Gause-type predator–prey model considering the Allee effect on prey

In this work, a bidimensional differential equation system obtained by modifying the well-known predator–prey Rosenzweig–MacArthur model is analyzed by considering prey growth influenced by the Allee effect.One of the main consequences of this modification is a separatrix curve that appears in the phase plane, dividing the behavior of the trajectories. The results show that the equilibrium in the origin is an attractor for any set of parameters. The unique positive equilibrium, when it exists, can be either an attractor or a repeller surrounded by a limit cycle, whose uniqueness is established by calculating the Lyapunov quantities. Therefore, both populations could either reach deterministic extinction or long-term deterministic coexistence.The existence of a heteroclinic curve is also proved. When this curve is broken by changing parameter values, then the origin turns out to be an attractor for all orbits in the phase plane. This implies that there are plausible conditions where both populations can go to extinction. We conclude that strong and weak Allee effects on prey population exert similar influences on the predator–prey model, thereby increasing the risk of ecological extinction.     

具有内生人口迁移的经济增长模型

本文考虑两部门的人口迁移过程，假定劳动人口迁移是由于人均资差异引起的，通过引入人口迁移函和反应函数得以一个二维微分方程组，建立起具有人口迁移的经济增长模型，文中证明模型的稳态解是稳定的。文中给出数值计算结果展示经济增长过程的人口迁移过程。     

Optimal control and sensitivity analysis of SIV→HIV dynamics with effects of infected immigrants in sub‐Saharan Africa

Regional migration has become an underlying factor in the spread of HIV transmission. In addition, immigrants with HIV status has contributed with high‐risk of sexually transmitted infection to its “destination” communities and promotes dissemination of HIV. Efforts to address HIV/AIDS among conflict‐affected populations should be properly addressed to eliminate potential role of the spread of the disease and risk of exposure to HIV. Motivated from this situation, HIV‐infected immigrants factor to HIV/SIV transmission link will be investigated in this research and examine its potential effect using optimal control method. Nonlinear deterministic mathematical model is used which is a multiple host model comprising of humans and chimpanzees. Some basic properties of the model such as invariant region and positivity of the solutions will be examined. The local stability of the disease‐free equilibrium was examined by computing the basic reproduction number, and it was found to be locally asymptotically stable when ?₀<1 and unstable otherwise. Sensitivity analysis was conducted to determine the parameters that help most in the spread of the virus. Pontryagin's maximum principle is used to obtain the optimality conditions for controlling the disease spread. Numerical simulation was conducted to obtain the analytical results. The results shows that combination of public health awareness, treatment, and culling help in controlling the HIV disease spread.     

A new epidemic model of computer viruses

This paper addresses the epidemiological modeling of computer viruses. By incorporating the effect of removable storage media, considering the possibility of connecting infected computers to the Internet, and removing the conservative restriction on the total number of computers connected to the Internet, a new epidemic model is proposed. Unlike most previous models, the proposed model has no virus-free equilibrium and has a unique endemic equilibrium. With the aid of the theory of asymptotically autonomous systems as well as the generalized Poincare–Bendixson theorem, the endemic equilibrium is shown to be globally asymptotically stable. By analyzing the influence of different system parameters on the steady number of infected computers, a collection of policies is recommended to prohibit the virus prevalence.     

Effect of diffusion on a two-species eco-epidemiological model

This paper deals with the stabilizing effect of diffusion on a prey?–?predator system where the prey population is infected by a microparasite. The predator functional response is a concave-type function. Conditions for the local as well as global stability of the model without diffusion are derived in terms of system parameters. It is also shown that an unstable equilibrium of the model without diffusion can be made stable by increasing the diffusion coefficients appropriately.     

Optimal control and stability analysis of an online game addiction model with two stages

In this paper, we establish a mathematical model of online game addiction with two stages to research the dynamic properties of it. The existence of all equilibria is obtained, and the basic reproduction number is calculated by the method of next-generation matrix. The global asymptotic stability of disease-free equilibrium (DFE) is proved by comparison principle, and the global asymptotic stability of endemic equilibrium (EE) is proved by constructing an appropriate Lyapunov function. Then we use the Pontryagin's maximum principle to find the optimal solution of the model, so that the number of infected people can be minimized. In numerical simulation, firstly, we validate the global stability of DFE and EE. Secondly, we consider three kind of control measures (treatment, isolation, and education) and divide them into four cases. The models with control and without control are solved numerically by using forward and backward sweep Runge-Kutta method. In order to achieve the best control effect, we suggest that three kind of measures should be used simultaneously according to the optimal control strategy.     

Optimal harvesting of a Gompertz population model with a marine protected area and interval‐value biological parameters

To protect fishery populations on the verge of extinction and sustain the biodiversity of the marine ecosystem, marine protected areas (MPA) are established to provide a refuge for fishery resource. However, the influence of current harvesting policies on the MPA is still unclear, and precise information of the biological parameters has yet to be conducted. In this paper, we consider a bioeconomic Gompertz population model with interval‐value biological parameters in a 2‐patch environment: a free fishing zone (open‐access) and a protected zone (MPA) where fishing is strictly prohibited. First, the existence of the equilibrium is proved, and by virtue of Bendixson‐dulac Theorem, the global stability of the nontrivial steady state is obtained. Then, the optimal harvesting policy is established by using Pontryagin's maximum principle. Finally, the results are illustrated with the help of some numerical examples. Our results show that the current harvesting policy is advantageous to the protection efficiency of an MPA on local fish populations.     

Visualization of Categorical Response Models: From Data Glyphs to Parameter Glyphs

The multinomial logit model is the most widely used model for nominal multi-category responses. One problem with the model is that many parameters are involved, and another that interpretation of parameters is much harder than for linear models because the model is nonlinear. Both problems can profit from graphical representations. We propose to visualize the effect strengths by star plots, where one star collects all the parameters connected to one term in the linear predictor. In simple models, one star refers to one explanatory variable. In contrast to conventional star plots, which are used to represent data, the plots represent parameters and are considered as parameter glyphs. The set of stars for a fitted model makes the main features of the effects of explanatory variables on the response variable easily accessible. The method is extended to ordinal models and illustrated by several datasets. Supplementary materials are available online.     

Dynamical behaviour of SIR model with coinfection: The case of finite carrying capacity

Multiple viruses are widely studied because of their negative effect on the health of host as well as on whole population. The dynamics of coinfection are important in this case. We formulated an susceptible infected recovered (SIR) model that describes the coinfection of the two viral strains in a single host population with an addition of limited growth of susceptible in terms of carrying capacity. The model describes five classes of a population: susceptible, infected by first virus, infected by second virus, infected by both viruses, and completely immune class. We proved that for any set of parameter values, there exists a globally stable equilibrium point. This guarantees that the disease always persists in the population with a deeper connection between the intensity of infection and carrying capacity of population. Increase in resources in terms of carrying capacity promotes the risk of infection, which may lead to destabilization of the population.     

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.     

OPTIMAL INSTAR PARASITIZATION IN A STAGE STRUCTURED HOST-PARASITOID MODEL

A stage structured host-parasitoid model is derived and the equilibria studied. It is shown under what conditions the parasitoid controls an exponentially growing host in the sense that a coexistence equilibrium exists. Furthermore, for host populations whose inherent growth rate is not too large it is proved that in order to minimize the adult host equilibrium level it is necessary that the parasitoids attack only one of the larval stages. It is also proved in this case that the minimum adult host equilibrium level is attained when the parasitoids attack that larval stage which also maximizes the expected number of emerging adult parasitoid per larva at equilibrium. Numerical simulations tentatively indicate that the first conclusion remains in general valid for the model. However, numerical studies also show that it is not true in general that the optimal strategy will maximize the number of emerging adult parasitoid per larva at equilibrium.     

Global stability of the virus dynamics model with intracellular delay and Crowley–Martin functional response

In this paper, a virus dynamics model with intracellular delay and Crowley–Martin functional response is discussed. By constructing suitable Lyapunov functions and using LaSalles invariance principle for delay differential equations, we established the global stability of uninfected equilibrium and infected equilibrium; it is proved that if the basic reproductive number is less than or equal to one, the uninfected equilibrium is globally asymptotically stable; if the basic reproductive number is more than one, the infected equilibrium is globally asymptotically stable. We also discuss the effects of intracellular delay on global dynamical properties by comparing the results with the stability conditions for the model without delay. Copyright © 2013 John Wiley & Sons, Ltd.     

Chaos in delay-induced Leslie–Gower prey–predator–parasite model and its control through prey harvesting

In this paper we analyze a delay-induced predator–prey–parasite model with prey harvesting, where the predator–prey interaction is represented by Leslie–Gower type model with type II functional response. Infection is assumed to spread horizontally from one infected prey to another susceptible prey following mass action law. Spreading of disease is not instantaneous but mediated by a time lag to take into account the time required for incubation process. Both the susceptible and infected preys are subjected to linear harvesting. The analysis is accomplished in two phases. First we analyze the delay-induced predator–prey–parasite system in absence of harvesting and proved the local & global dynamics of different (six) equilibrium points. It is proved that the delay has no influence on the stability of different equilibrium points except the interior one. Delay may cause instability in an otherwise stable interior equilibrium point of the system and larger delay may even produce chaos if the infection rate is also high. In the second phase, we explored the dynamics of the delay-induced harvested system. It is shown that harvesting of prey population can suppress the abrupt fluctuations in the population densities and can stabilize the system when it exceeds some threshold value.     

一类具有病毒变异的Logistic死亡率SEIR传染病模型研究分析

本文建立了一类具有病毒变异的Logistic死亡率SEIR传染病模型,借助Lyapunov函数和LaSalle''s不变原理,证明了无病平衡点全局稳定性.利用代数方法构造Lyapunov函数,证明了地方病平衡点全局稳定性.另外,通过数值模拟分析了参数对疾病传播的影响.     

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.     

Effect of toxic substance on delayed competitive allelopathic phytoplankton system with varying parameters through stability and bifurcation analysis

We have studied the combined effect of toxicant and fluctuation of the biological parameters on the dynamical behaviors of a delayed two-species competitive system with imprecise biological parameters. Due to the global increase of harmful phytoplankton blooms, the study of dynamic interactions between two competing phytoplankton species in the presence of toxic substances is an active field of research now days. The ordinary mathematical formulation of models for two competing phytoplankton species, when one or both the species liberate toxic substances, is unable to capture the oscillatory and highly variable growth of phytoplankton populations. The deterministic model never predicts the sudden localized behavior of certain species. These obstacles of mathematical modeling can be overcomed if we include interval variability of biological parameters in our modeling approach. In this investigation, we construct imprecise models of allelopathic interactions between two competing phytoplankton species as a parametric differential equation model. We incorporate the effect of toxicant on the species in two different cases known as toxic inhibition and toxic stimulatory system. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation. Analytical findings are supported through exhaustive numerical simulations.     

具有饱和发生率和免疫响应的病毒感染模型的稳定性分析

提出了具有饱和发生率和免疫响应的病毒感染数学模型,得到了基本再生数R_0的表达式.当R_01时,证明了无病平衡点是全局渐近稳定的;当R_01时,得到了免疫耗竭平衡点和持续带毒平衡点局部渐近稳定的条件.     

具有阶段结构的SIR传染病模型

研究了一类具有阶段结构的SIR传染病模型，在模型中假设种群分幼年和成年两个阶段，且只有成年种群染病，并且采用与成年易感者数量有关的一般非线性传染率，得到了系统解的有界性及无病平衡点和地方病平衡点存在的条件．通过对平衡点对应的特征方程的讨论得到了平衡点局部渐近稳定的条件，同时证明了平衡点的全局渐近稳定性，并对结论进行了数值模拟．