神经放电节律转化的分岔序列模式

神经元接受到的外界信号是动态变化的,神经放电节律模式则会依据一定的规律动态转化来反映这种变化,以往确定性理论模型(如Chay模型和Rose-Hindmarsh模型)模拟出了部分神经放电模式转化的整体分岔规律。利用Chay模型仿真,通过调节具有生理学意义的参数,模拟出了神经元放电的一系列分岔序列,同时在神经起步点的实验中,应用与模型对应的参数进行调节,观察到了与仿真结果整体上一致的分岔序列,印证了数值模拟的结果,展现了真实的神经元放电整体分岔结构的基本规律,为理解具体的生理调节活动中神经放电节律的转化提供了理论基础。     

一类具有暂时免疫传染病模型的Hopf分支

本文应用Hassard的“规范形”方法,讨论了一类具有暂时免疫传染病模型的动力形态．给出了该系统发生Hopf分支的参数曲线,进一步计算出了决定分支方向及稳定性的参数条件,并给出了生态解释．     

A stochastic model for head lice infections

We investigate the dynamics of head lice infections in schools, by consideringa model for endemic infection based on a stochastic SIS (susceptible-infected-susceptible) epidemic model, with the addition of an external source of infection. We deduce a range of properties of our model, including the length of a single outbreak of infection. We use the stationary distribution of the number of infected individuals, in conjunction with data from a recent study carried out in Welsh schools on the prevalence of head lice infections, and employ maximum likelihood methods to obtain estimates of the model parameters. A complication is that, for each school, only a sample of the pupils was checked for infection. Our likelihood function takes account of the missing data by incorporating a hypergeometric sampling element. We arrive at estimates of the ratios of the “within school” and “external source” transmission rates to the recovery rate and use these to obtain estimates for various quantities of interest.      

Semi-empirical power-law scaling of new infection rate to model epidemic dynamics with inhomogeneous mixing

The expected number of new infections per day per infectious person during an epidemic has been found to exhibit power-law scaling with respect to the susceptible fraction of the population. This is in contrast to the linear scaling assumed in traditional epidemiologic modeling. Based on simulated epidemic dynamics in synthetic populations representing Los Angeles, Chicago, and Portland, we find city-dependent scaling exponents in the range of 1.7-2.06. This scaling arises from variations in the strength, duration, and number of contacts per person. Implementation of power-law scaling of the new infection rate is quite simple for SIR, SEIR, and histogram-based epidemic models. Treatment of the effects of the social contact structure through this power-law formulation leads to significantly lower predictions of final epidemic size than the traditional linear formulation.     

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     

神经起步点自发放电节律及节律转化的分岔规律

在神经起步点的实验中观察到了复杂多样的神经放电([Ca^2 ]o)节律模式,如周期簇放电、周期峰放电、混沌簇放电、混沌峰放电以及随机放电节律等。随着细胞外钙离子浓度的降低,神经放电节律从周期l簇放电,经过复杂的分岔过程(包括经倍周期分岔到混沌簇放电、混沌簇放电经激变到混沌峰放电、以及混沌峰放电经逆倍周期分岔到周期峰放电)转化为周期l峰放电。在神经放电理论模型——Chay模型中,调节与实验相关的参数(Ca^2 平衡电位),可以获得与实验相似的神经放电节律和节律转换规律。这表明复杂的神经放电节律之间存在着一定的分岔规律,它们是理解神经元信息编码的基础。     

A mathematical model for Chagas disease with infection-age-dependent infectivity

In this paper we develop a mathematical model for Chagas disease with infection-age-dependent infectivity. The effects of vector and blood transfusion transmission are considered, and the infected population is structured by the infection age (the time elapsed from infection). The authors identify the basic reproduction ratio R0 and show that the disease can invade into the susceptible population and unique endemic steady state exists if R0 > 1, whereas the disease dies out if R0 is small enough. We show that depending on parameters, backward bifurcation of endemic steady state can occur, so even if R0 < 1, there could exist endemic steady states. We also discuss local and global stability of steady states.     

The minimum effort required to eradicate infections in models with backward bifurcation

We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (μ/α) this model shows backward bifurcations, where μ is the death rate and α is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant β. This eradication effort is explicitly expressed in terms of α,r, and μ As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R ₀ is smaller than the eradication effort for r > 1 + (μ/α) and equal to the effort for other values of r. The method we present is relevant to the whole class of compartmental models with backward bifurcation.Dedicated to Karl Peter Hadeler on the occasion of his 70th birthday.     

Two-patch dispersal-linked compensatory-overcompensatory spatially discrete population models

We study the role of asynchronous and synchronous dispersals on discrete-time two-patch dispersal-linked population models, where the pre-dispersal local patch dynamics are of mixed compensatory and overcompensatory types. Single-species dispersal-linked models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and dispersal is synchronous. However, the dynamics of the corresponding two-patch population model connected by asynchronous dispersal depends on the dispersal rates. The species goes extinct on at least one patch when the asynchronous dispersal rates are high, while it persists when the rates are low. We use numerical simulations to show that in both synchronous and asynchronous mixed compensatory and overcompensatory systems, symmetric and asymmetric dispersals can control and impede the onset of cyclic population oscillations via period-doubling reversal bifurcations. Also, we show that in mixed systems both asynchronous and synchronous dispersals are capable of altering the pre-dispersal local patch dynamics from overcompensatory to compensatory dynamics. Dispersal-linked population models with ‘unstructured’ overcompensatory pre-dispersal local dynamics connected by synchronous dispersal can generate multiple attractors with fractal basin boundaries. However, mixed compensatory and overcompensatory systems appear to exhibit single attractors and not coexisting (multiple) attractors.     

Modelling the potential spread of Solenopsis invicta Buren (Hymenoptera: Formicidae) (red imported fire ant) in Australia

Abstract  Quantifying the potential spread and density of an invading organism enables decision-makers to determine the most appropriate response to incursions. We present two linked models that estimate the spread of Solenopsis invicta Buren (red imported fire ant) in Australia based on limited data gathered after its discovery in Brisbane in 2001. A stochastic cellular automaton determines spread within a location (100 km by 100 km) and this is coupled with a model that simulates human-mediated movement of S. invicta to new locations. In the absence of any control measures, the models predict that S. invicta could cover 763 000–4 066 000 km² by the year 2035 and be found at 200 separate locations around Australia by 2017–2027, depending on the rate of spread. These estimated rates of expansion (assuming no control efforts were in place) are higher than those experienced in the USA in the 1940s during the early invasion phases in that country. Active control efforts and quarantine controls in the USA (including a concerted eradication attempt in the 1960s) may have slowed spread. Further, milder winters, the presence of the polygynous social form, increased trade and human mobility in Australia in 2000s compared with the USA in 1940s could contribute to faster range expansion.     

Spiking behavior and epileptiform oscillations in a discrete model of cortical neural networks

Summary We investigate the phenomenon of epileptiform activity using a discrete model of cortical neural networks. Our model is reduced to the elementary features of neurons and assumes simplified dynamics of action potentials and postsynaptic potentials. The discrete model provides a comparably high simulation speed which allows the rendering of phase diagrams and simulations of large neural networks in reasonable time. Further the reduction to the basic features of neurons provides insight into the essentials of a possible mechanism of epilepsy. Our computer simulations suggest that the detailed dynamics of postsynaptic and action potentials are not indispensable for obtaining epileptiform behavior on the system level. The simulation results of autonomously evolving networks exhibit a regime in which the network dynamics spontaneously switch between fluctuating and oscillating behavior and produce isolated network spikes without external stimulation. Inhibitory neurons have been found to play an important part in the synchronization of neural firing: an increased number of synapses established by inhibitory neurons onto other neurons induces a transition to the spiking regime. A decreased frequency accompanying the hypersynchronous population activity has only occurred with slow inhibitory postsynaptic potentials.     

Principles of biological pattern formation: swarming and aggregation viewed as selforganization phenomena

Migration automaton models are introduced which offer the possibility to directly analyse essential selforganization properties of biological pattern formation at the cellular level. We present examples of migration automata as models of collective motion and cellular aggregation—patterns that are typical for example in the life cycle of Myxobacteria. Linear stability analysis of the corresponding automaton Boltzmann equation allows to distinguish orientation-dependent (collective motion) and density-dependent (aggregation) instabilities. Presented at the National Symposium on Evolution of Life.     

一类潜伏期和染病期均传染且具非线性传染率的流行病模型

研究了一类潜伏期和染病期都传染的具非线性传染率的SEIS流行病模型,确定了各类平衡点存在的条件阈值,讨论了各平衡点的稳定性,揭示了潜伏期传染和染病期传染对流行病发展趋势的共同影响．     

A cellular automaton model for microcarrier cultures

In order to achieve high cell densities anchoragedependent cells are commonly cultured on microcarriers, where spatial restrictions to cell growth complicates the determination of the growth kinetics. To design and operate large-scale bioreactors for microcarrier cultures, the effect of this spatial restriction to growth, referred to as contact inhibition, must be decoupled from the growth kinetics. In this article, a cellular automaton approach is recommended to model the growth of anchorage-dependent cells on microcarriers. The proposed model is simple to apply yet provides an accurate representation of contact-inhibited cell growth on microcarriers. The distribution of the number of neighboring cells per cell, microcarrier surface areas, and inoculation densities are taken into account with this model. When compared with experimental data for Vero and MRC-5 microcarrier cultures, the cellular automaton predictions were very good. Furthermore, the model can be used to generate contact-inhibition growth curves to decouple the effect of contact-inhibition from growth kinetics. With this information, the accurate determination of kinetic parameters, such as nutrient uptake rates, and the effects of other environmental factors, such as toxin levels, may be determined. (c) 1994 John Wiley & Sons, Inc.     

Epidemic Models with Heterogeneous Mixing and Treatment

We consider a two-group epidemic model with treatment and establish a final size relation that gives the extent of the epidemic. This relation can be established with arbitrary mixing between the groups even though it may not be feasible to determine the reproduction number for the model. If the mixing of the two groups is proportionate, there is an explicit expression for the reproductive number and the final size relation is expressible in terms of the components of the reproduction number. We also extend the results to a two-group influenza model with proportionate mixing. Some numerical simulations suggest that (i) the assumption of no disease deaths is a good approximation if the disease death rate is small and (ii) a one-group model is a close approximation to a two-group model but a two-group model is necessary for comparing targeted treatment strategies. This research was supported by MITACS and an NSERC Research Grant.     

Nonlinear frequency-dependent selection at a single locus with two alleles and two phenotypes

 The paper investigates the discrete frequency dynamics of two phenotype diploid models where genotypic fitness is an exponential function of the expected payoff in the matrix game. Phenotypic and genotypic equilibria are defined and their stability compared to frequency-dependent selection models based on linear fitness when there are two possible phenotypes in the population. In particular, it is shown that stable equilibria of both types can exist in the same nonlinear model. It is also shown that period-doubling bifurcations emerge when there is sufficient selection in favor of interactions between different phenotypes. Received: 22 October 1998