An epidemiological model with a delay and a nonlinear incidence rate

An epidemiological model with both a time delay in the removed class and a nonlinear incidence rate is analysed to determine the equilibria and their stability. This model is for diseases where individuals are first susceptible, then infected, then removed with temporary immunity and then susceptible again when they lose their immunity. There are multiple equilibria for some parameter values, and, for certain of these, periodic solutions arise by Hopf bifurcation from the large nontrivial equilibrium state.Research supported in parts by Centers for Disease Control Contract 200-87-0515Research supported in part by NSERC A-8965     

Some epidemiological models with nonlinear incidence

Epidemiological models with nonlinear incidence rates can have very different dynamic behaviors than those with the usual bilinear incidence rate. The first model considered here includes vital dynamics and a disease process where susceptibles become exposed, then infectious, then removed with temporary immunity and then susceptible again. When the equilibria and stability are investigated, it is found that multiple equilibria exist for some parameter values and periodic solutions can arise by Hopf bifurcation from the larger endemic equilibrium. Many results analogous to those in the first model are obtained for the second model which has a delay in the removed class but no exposed class.Research supported in part by Centers for Disease Control Contract 200-87-0515. Support services provided at University House Research Center at the University of IowaResearch supported in part by NSERC A-8965 and the University of Victoria President's Committee on Faculty Research and Travel     

A model of a myelinated nerve axon: threshold behaviour and propagation

A model of a myelinated nerve axon is developed on the basis of FitzHugh-Nagumo dynamics under the assumption that the nodes of Ranvier are of small but finite width. It is shown that a periodic excited state may not exist if the width of the nodes is too small and the leakage across the myelin sheath is too great. The propagation of a super threshold pulse is prevented in the absence of nodes. Global stability of the resting equilibrium state is investigated as well as the propagation of wave front, type solutions.     

一类带有隔离项和分布时滞的SIQS模型的稳定性分析

研究了一类带有隔离项和分布时滞的SIQS模型,利用局部线性化方法并构造了适当的Liapunov函数,获得了该系统平衡点的稳定性结论.     

An SIS epidemic model with variable population size and a delay

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.Research Supported in part by NSERC grant A-8965 and the University of Victoria Committee on Faculty Research & Travel     

分布时滞竞争模型的周期解

研究了分布时滞竞争模型周期解,利用Gaines与Mawhin的重合度理论,通过构造恰当的Lyapunov函数得到模型正周期解的存在性、唯一性及全局稳定性。     

The reinfection threshold

Thresholds in transmission are responsible for critical changes in infectious disease epidemiology. The epidemic threshold indicates whether infection invades a totally susceptible population. The reinfection threshold indicates whether self-sustained transmission occurs in a population that has developed a degree of partial immunity to the pathogen (by previous infection or vaccination). In models that combine susceptible and partially immune individuals, the reinfection threshold is technically not a bifurcation of equilibria as correctly pointed out by Breban and Blower. However, we show that a branch of equilibria to a reinfection submodel bifurcates from the disease-free equilibrium as transmission crosses this threshold. Consequently, the full model indicates that levels of infection increase by two orders of magnitude and the effect of mass vaccination becomes negligible as transmission increases across the reinfection threshold.     

Campus drinking: an epidemiological model

The drinking behaviours of college students have posed significant public health concerns for several generations. However, the dynamics of campus drinking have not been analysed using mathematical models. An epidemiological model capturing the dynamics of campus drinking is used to study how the ‘disease’ of drinking is spread on campus. The model suggests that the reproductive numbers are not sufficient to predict whether drinking behaviour will persist on campus and that the pattern of recruiting new members plays a significant role in the reduction of campus alcohol problems. In particular, campus alcohol abuse may be reduced by minimizing the ability of problem drinkers to directly recruit non-drinkers.     

A delay reaction-diffusion model of the dynamics of botulinum in fish

A model of interaction between fish and a bacterium (Clostridium botulinum) responsible for avian botulism is introduced, considering diffusion of both fish and bacterium in water. The fish population moves randomly in water. Death fish disintegrate in water, at different locations, causing bacteria to diffuse through water and infect other fish. Existence of uniform steady states is investigated and the linearized stability of the positive uniform steady state is analyzed. A Hopf bifurcation is proved to occur from the uniform steady state when the bifurcation parameter, here the time delay, passes through a critical value and diffusion coefficients satisfy some conditions, that induces time oscillations of the populations. Comments on diffusion-driven instability are provided, and numerical simulations are carried out to illustrate the results.     

Global dynamics of a chemostat competition model with distributed delay

 We study the global dynamics of n-species competition in a chemostat with distributed delay describing the time-lag involved in the conversion of nutrient to viable biomass. The delay phenomenon is modelled by the gamma distribution. The linear chain trick and a fluctuation lemma are applied to obtain the global limiting behavior of the model. When each population can survive if it is cultured alone, we prove that at most one competitor survives. The winner is the population that has the smallest delayed break-even concentration, provided that the orders of the delay kernels are large and the mean delays modified to include the washout rate (which we call the virtual mean delays) are bounded and close to each other, or the delay kernels modified to include the washout factor (which we call the virtual delay kernels) are close in L ¹-norm. Also, when the virtual mean delays are relatively small, it is shown that the predictions of the distributed delay model are identical with the predictions of the corresponding ODEs model without delay. However, since the delayed break-even concentrations are functions of the parameters appearing in the delay kernels, if the delays are sufficiently large, the prediction of which competitor survives, given by the ODEs model, can differ from that given by the delay model. Received: 9 August 1997 / Revised version: 2 July 1998     

Threshold and stability results for an age-structured epidemic model

We study a mathematical model for an epidemic spreading in an age-structured population with age-dependent transmission coefficient. We formulate the model as an abstract Cauchy problem on a Banach space and show the existence and uniqueness of solutions. Next we derive some conditions which guarantee the existence and uniqueness for non-trivial steady states of the model. Finally the local and global stability for the steady states are examined.     

Mortality threshold for juvenile Chinook salmon Oncorhynchus tshawytscha in an epidemiological model of Ceratomyxa shasta

The myxozoan parasite Ceratomyxa shasta is a significant pathogen of juvenile Chinook salmon Oncorhynchus tshawytscha in the Klamath River, California, USA. This parasite requires 2 hosts to complete its life cycle: a freshwater polychaete (Manayunkia speciosa) and a salmonid. The complex life cycle and large geographic area where infection occurs make it difficult to monitor and manage ceratomyxosis. We present a model for ceratomyxosis-induced mortality in O. tshawytscha, from which parameters important to the persistence of C. shasta are identified. We also experimentally quantify specific parameters from the model and identify a mortality threshold (a critical parameter), by naturally exposing native O. tshawytscha to C. shasta in the Klamath River. The average percent mortality that resulted from these experimental challenges ranged from 2.5 to 98.5% over an exposure dose of 4.4 to 612 x 10(6) parasites. This experiment identified a non-linear mortality threshold of 7.7 +/- 2.1 x 10(4) actinospores fish(-1) for Chinook salmon from the Iron Gate Hatchery on the Klamath River. Below this threshold no mortality occurred and above it mortality increased dramatically, thus providing a target by which to reduce parasitism in emigrating juvenile O. tshawytscha.     

A model for dengue disease with variable human population

 A model for the transmission of dengue fever with variable human population size is analyzed. We find three threshold parameters which govern the existence of the endemic proportion equilibrium, the increase of the human population size, and the behaviour of the total number of human infectives. We prove the global asymptotic stability of the equilibrium points using the theory of competitive systems, compound matrices, and the center manifold theorem. Received: 3 November 1997 / Revised version: 3 July 1998     

具有非线性种内制约关系和分布时滞竞争模型的稳定性

构造并研究了一类具有分布时滞和非线性种内制约关系的竞争模型.得到了这类模型的边界平衡点和共存平衡点全局渐近稳定的条件,以及分布时滞、非线性种内制约关系对模型的影响.     

A mobile agent model for fault-tolerant manipulation on distributed objects

In this paper, we discuss how to realize fault-tolerant applications on distributed objects. Servers supporting objects can be fault-tolerant by taking advantage of replication and checkpointing technologies. However, there is no discussion on how application programs being performed on clients are tolerant of clients faults. For example, servers might block in the two-phase commitment protocol due to the client fault. We newly discuss how to make application programs fault-tolerant by taking advantage of mobile agent technologies where a program can move from a computer to another computer in networks. An application program to be performed on a faulty computer can be performed on another operational computer by moving the program in the mobile agent model. In this paper, we discuss a transactional agent model where a reliable and efficient application for manipulating objects in multiple computers is realized in the mobile agent model. In the transactional agent model, only a small part of the application program named routing subagent moves around computers. A routing subagent autonomously finds a computer which to visit next. We discuss a hierarchical navigation map which computer should be visited price to another computer in a transactional agent. A routing subagent makes a decision on which computer visit for the hierarchical navigation map. Programs manipulating objects in a computer are loaded to the computer on arrival of the routing subagent in order to reduce the communication overhead. This part of the transactional agent is a manipulating subagent. The manipulation subagent still exists on the computer even after the routing subagent leaves the computer in order to hold objects until the commitment. We assume every computer may stop by fault while networks are reliable. There are kinds of faulty computers for a transactional agent; current, destination, and sibling computers where a transactional agent now exists, will move, and has visited, respectively. The types of faults are detected by neighbouring manipulation subagents by communicating with each other. If some of the manipulation subagents are faulty, the routing subagent has to be aborted. However, the routing subagent is still moving. We discuss how to efficiently deliver the abort message to the moving routing subagent. We evaluate the transactional agent model in terms of how long it takes to abort the routing subagent if some computer is faulty.

A threshold theorem for the general epidemic in discrete time

Summary A theorem, analogous to the continuous time Threshold Theorem of Kermack and McKendrick, is proved for a certain discrete time epidemic model. This model, in contrast to its continuous time analogue, leads to some solutions in which the total population of susceptibles may become infected in a finite time.     

Distribution of fleas (Siphonaptera) among small mammals: mean abundance predicts prevalence via simple epidemiological model

We used data on the abundance and distribution of fleas parasitic on small mammals in Slovakia and aimed: (i) to confirm a positive relationship between abundance and distribution fleas within and across host species; and (ii) to test if prevalence of fleas can be reliably predicted from a simple epidemiological model that takes into account flea mean abundance and its variance. Prevalence of a flea species increased with an increase in its mean abundance both within and across host species. We calculated prevalences both for each flea-host association and for each flea species across all hosts. Observed prevalences did not differ significantly from those predicted by the epidemiological model using parameters of Taylor's power relationship between mean abundance of fleas and its variance. Regressions of predicted prevalences against observed prevalences produced slope values that did not differ significantly from unity and were independent of scale (within or across host species). Our results demonstrated that up to 96% of variance in flea prevalence can be explained solely by their mean abundance. We concluded that, in general, there is no need to invoke other, more complex factors for the explanation of the variation in flea prevalence.