Reproduction Number and Asymptotic Stability for the Dynamics of a Honey Bee Colony with Continuous Age Structure

A system of partial differential equations is derived as a model for the dynamics of a honey bee colony with a continuous age distribution, and the system is then extended to include the effects of a simplified infectious disease. In the disease-free case, we analytically derive the equilibrium age distribution within the colony and propose a novel approach for determining the global asymptotic stability of a reduced model. Furthermore, we present a method for determining the basic reproduction number \(R_0\) of the infection; the method can be applied to other age-structured disease models with interacting susceptible classes. The results of asymptotic stability indicate that a honey bee colony suffering losses will recover naturally so long as the cause of the losses is removed before the colony collapses. Our expression for \(R_0\) has potential uses in the tracking and control of an infectious disease within a bee colony.     

Control of emerging infectious diseases using responsive imperfect vaccination and isolation

This paper is concerned with a stochastic model for the spread of an SEIR (susceptible --> exposed (= latent) --> infective --> removed) epidemic among a population partitioned into households, featuring different rates of infection for within and between households. The model incorporates responsive vaccination and isolation policies, based upon the appearance of diagnosed cases in households. Different models for imperfect vaccine response are considered. A threshold parameter R*, which determines whether or not a major epidemic can occur, and the probability of a major epidemic are obtained for different infectious and latent period distributions. Simpler expressions for these quantities are obtained in the limiting case of infinite within-household infection rate. Numerical studies suggest that the choice of infectious period distribution and whether or not latent individuals are vaccine-sensitive have a material influence on the spread of the epidemic, while, for given vaccine efficacy, the choice of vaccine action model is less influential. They also suggest that an effective isolation policy has a more significant impact than vaccination. The results show that R* alone is not sufficient to summarise the potential for an epidemic.     

Stability properties of pulse vaccination strategy in SEIR epidemic model

The problem of the applicability of the pulse vaccination strategy (PVS) for the stable eradication of some relevant general class of infectious diseases is analyzed in terms of study of local asymptotic stability (LAS) and global asymptotic stability (GAS) of the periodic eradication solution for the SEIR epidemic model in which is included the PVS. Demographic variations due or not to diseased-related fatalities are also considered. Due to the non-triviality of the Floquet's matrix associate to the studied model, the LAS is studied numerically and in this way it is found a simple approximate (but analytical) sufficient criterion which is an extension of the LAS constraint for the stability of the trivial equilibrium in SEIR model without vaccination. The numerical simulations also seem to suggest that the PVS is slightly more efficient than the continuous vaccination strategy. Analytically, the GAS of the eradication solutions is studied and it is demonstrated that the above criteria for the LAS guarantee also the GAS.     

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     

Stabilization of telomeres in nonlinear models of proliferating cell lines

We analyse an age-structured model of telomere loss in a proliferating cell population. The cell population is divided into telomere classes, which shorten each round of division. The model consists of a nonlinear system of partial differential equations for the telomere classes. We prove that if the highest telomere class is exempted from mortality, then all the classes stabilize to a nontrivial equilibrium dependent on the initial state of cells in the highest telomere class.     

Casein Micelle Dispersions under Osmotic Stress

Casein micelles dispersions have been concentrated and equilibrated at different osmotic pressures using equilibrium dialysis. This technique measured an equation of state of the dispersions over a wide range of pressures and concentrations and at different ionic strengths. Three regimes were found. i), A dilute regime in which the osmotic pressure is proportional to the casein concentration. In this regime, the casein micelles are well separated and rarely interact, whereas the osmotic pressure is dominated by the contribution from small residual peptides that are dissolved in the aqueous phase. ii), A transition range that starts when the casein micelles begin to interact through their κ-casein brushes and ends when the micelles are forced to get into contact with each other. At the end of this regime, the dispersions behave as coherent solids that do not fully redisperse when osmotic stress is released. iii), A concentrated regime in which compression removes water from within the micelles, and increases the fraction of micelles that are irreversibly linked to each other. In this regime the osmotic pressure profile is a power law of the residual free volume. It is well described by a simple model that considers the micelle to be made of dense regions separated by a continuous phase. The amount of water in the dense regions matches the usual hydration of proteins.     

Epidemiological effects of seasonal oscillations in birth rates

Seasonal oscillations in birth rates are ubiquitous in human populations. These oscillations might play an important role in infectious disease dynamics because they induce seasonal variation in the number of susceptible individuals that enter populations. We incorporate seasonality of birth rate into the standard, deterministic susceptible-infectious-recovered (SIR) and susceptible-exposed-infectious-recovered (SEIR) epidemic models and identify parameter regions in which birth seasonality can be expected to have observable epidemiological effects. The SIR and SEIR models yield similar results if the infectious period in the SIR model is compared with the "infected period" (the sum of the latent and infectious periods) in the SEIR model. For extremely transmissible pathogens, large amplitude birth seasonality can induce resonant oscillations in disease incidence, bifurcations to stable multi-year epidemic cycles, and hysteresis. Typical childhood infectious diseases are not sufficiently transmissible for their asymptotic dynamics to be likely to exhibit such behaviour. However, we show that fold and period-doubling bifurcations generically occur within regions of parameter space where transients are phase-locked onto cycles resembling the limit cycles beyond the bifurcations, and that these phase-locking regions extend to arbitrarily small amplitude of seasonality of birth rates. Consequently, significant epidemiological effects of birth seasonality may occur in practice in the form of transient dynamics that are sustained by demographic stochasticity.     

一个具非单调传染率的SEIR传染病模型的全局稳定性

本文研究一类描述某种严重疾病的传染数目变大时在心理上产生影响的非单调传染率的SEIR传染病模型.研究表明模型的动力行为和疾病的爆发完全由基本再生数R0决定.当R0≤1时,无病平衡点是全局稳定的,疾病消亡;当R0〉1时,地方病平衡点是全局稳定的,疾病持续且发展成地方病.     

Bayesian clustering with AutoClass explicitly recognises uncertainties in landscape classification

Clustering of multivariate data is a commonly used technique in ecology, and many approaches to clustering are available. The results from a clustering algorithm are uncertain, but few clustering approaches explicitly acknowledge this uncertainty. One exception is Bayesian mixture modelling, which treats all results probabilistically, and allows comparison of multiple plausible classifications of the same data set. We used this method, implemented in the AutoClass program, to classify catchments (watersheds) in the Murray Darling Basin (MDB), Australia, based on their physiographic characteristics (e.g. slope, rainfall, lithology). The most likely classification found nine classes of catchments. Members of each class were aggregated geographically within the MDB. Rainfall and slope were the two most important variables that defined classes. The second-most likely classification was very similar to the first, but had one fewer class. Increasing the nominal uncertainty of continuous data resulted in a most likely classification with five classes, which were again aggregated geographically. Membership probabilities suggested that a small number of cases could be members of either of two classes. Such cases were located on the edges of groups of catchments that belonged to one class, with a group belonging to the second-most likely class adjacent. A comparison of the Bayesian approach to a distance-based deterministic method showed that the Bayesian mixture model produced solutions that were more spatially cohesive and intuitively appealing. The probabilistic presentation of results from the Bayesian classification allows richer interpretation, including decisions on how to treat cases that are intermediate between two or more classes, and whether to consider more than one classification. The explicit consideration and presentation of uncertainty makes this approach useful for ecological investigations, where both data and expectations are often highly uncertain.     

Using virtual 3-D plant architecture to assess fungal pathogen splash dispersal in heterogeneous canopies: a case study with cultivar mixtures and a non-specialized disease causal agent

Background and Aims

Recent developments in plant disease management have led to a growing interest in alternative strategies, such as increasing host diversity and decreasing the use of pesticides. Use of cultivar mixtures is one option, allowing the spread of plant epidemics to be slowed down. As dispersal of fungal foliar pathogens over short distances by rain-splash droplets is a major contibutor to the spread of disease, this study focused on modelling the physical mechanisms involved in dispersal of a non-specialized pathogen within heterogeneous canopies of cultivar mixtures, with the aim of optimizing host diversification at the intra-field level.

Methods

Virtual 3-D wheat-like plants (Triticum aestivum) were used to consider interactions between plant architecture and disease progression in heterogeneous canopies. A combined mechanistic and stochastic model, taking into account splash droplet dispersal and host quantitative resistance within a 3-D heterogeneous canopy, was developed. It consists of four sub-models that describe the spatial patterns of two cultivars within a complex canopy, the pathway of rain-splash droplets within this canopy, the proportion of leaf surface area impacted by dispersal via the droplets and the progression of disease severity after each dispersal event.

Key Results

Different spatial organization, proportions and resistance levels of the cultivars of two-component mixtures were investigated. For the eight spatial patterns tested, the protective effect against disease was found to vary by almost 2-fold, with the greatest effect being obtained with the smallest genotype unit area, i.e. the ground area occupied by an independent unit of the host population that is genetically homogeneous. Increasing both the difference between resistance levels and the proportion of the most resistant cultivar often resulted in a greater protective effect; however, this was not observed for situations in which the most resistant of the two cultivars in the mixture had a relatively low level of resistance.

Conclusions

The results show agreement with previous data obtained using experimental approaches. They demonstrate that in order to maximize the potential mixture efficiency against a splash-dispersed pathogen, optimal susceptible/resistant cultivar proportions (ranging from 1/9 to 5/5) have to be established based on host resistance levels. The results also show that taking into account dispersal processes in explicit 3-D plant canopies can be a key tool for investigating disease progression in heterogeneous canopies such as cultivar mixtures.     

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     

Semi-supervised learning via penalized mixture model with application to microarray sample classification

MOTIVATION: It is biologically interesting to address whether human blood outgrowth endothelial cells (BOECs) belong to or are closer to large vessel endothelial cells (LVECs) or microvascular endothelial cells (MVECs) based on global expression profiling. An earlier analysis using a hierarchical clustering and a small set of genes suggested that BOECs seemed to be closer to MVECs. By taking advantage of the two known classes, LVEC and MVEC, while allowing BOEC samples to belong to either of the two classes or to form their own new class, we take a semi-supervised learning approach; for high-dimensional data as encountered here, we propose a penalized mixture model with a weighted L1 penalty to realize automatic feature selection while fitting the model. RESULTS: We applied our penalized mixture model to a combined dataset containing 27 BOEC, 28 LVEC and 25 MVEC samples. Analysis results indicated that the BOEC samples appeared to form their own new class. A simulation study confirmed that, compared with the standard mixture model with or without initial variable selection, the penalized mixture model performed much better in identifying relevant genes and forming corresponding clusters. The penalized mixture model seems to be promising for high-dimensional data with the capability of novel class discovery and automatic feature selection.     

Assessment of mechanistic proposals for the binding of agonists to cardiac muscarinic receptors

N-[3H]Methylscopolamine has been used to characterize muscarinic receptors in crude homogenates prepared from hearts of Syrian golden hamsters. The Hill coefficient is one for specific binding of the radioligand itself and for its inhibition by muscarinic antagonists; markedly lower values are obtained for its inhibition by muscarinic agonists. The binding patterns of agonists have been analyzed in terms of a mixture of sites differing in affinity for the drug and reveal the following. All agonists discern at least two classes of receptor in atrial and ventricular homogenates. The number of classes and the relative size of each differ for different agonists in the same region and for the same agonist in different regions. Atrial and ventricular affinities are in good agreement for some agonists but differ for others. Guanylyl imidodiphosphate (GMP-PNP) is without effect on the specific binding of the radioligand but alters the binding of carbachol via an apparent redistribution of receptors from one class to another; the apparent affinity at either class remains unchanged. Carbachol reveals two classes of sites in ventricular preparations, and the nucleotide mediates an interconversion from higher to lower affinity; three classes are revealed in atrial preparations, and the nucleotide eliminates the sites of highest affinity with a concomitant increase in the number of sites of lowest affinity. Taken together, the data are incompatible with the notion of different, noninterconverting sites; rather, there appear to be several possible states of affinity such that the equilibrium distribution of receptors among the various states is determined by the tissue, by the agonist, and by neurohumoral modulators such as guanylyl nucleotides. The effects of agonists and GMP-PNP cannot be rationalized in terms of a ternary complex model in which the low Hill coefficients arise from a spontaneous equilibrium between receptor (R) and G protein (G) and in which agonists bind preferentially to the RG complex.     

Soybean and Maize Cropping Models for the Management of Meloidogyne incognita in the Coastal Plain

Models are presented to describe the influence of rotations of Meloidogyne incognita-susceptible cultivars, resistant cultivars, and maize on postharvest abundance of M. incognita juveniles in the soil. Depending on initial densities of juveniles, monocultured regimes reached equilibrium densities after a few years of 287, 40, and 10 juveniles per 10 cm³ soil for susceptible soybean, resistant soybean, and maize, respectively. Yearly changes in the population density of juveniles due to rotation of these crops were simulated by iterative substitution of the model equations for each crop. A maximum density of 319 per 10 cm³ soil was reached following a susceptible cultivar in a susceptible-resistant soybean rotation. Soybean yield loss estimates are presented for monocultured regimes and for various rotations with maize.     

Bayesian Semiparametric Frailty Selection in Multivariate Event Time Data

Biomedical studies often collect multivariate event time data from multiple clusters (either subjects or groups) within each of which event times for individuals are correlated and the correlation may vary in different classes. In such survival analyses, heterogeneity among clusters for shared and specific classes can be accommodated by incorporating parametric frailty terms into the model. In this article, we propose a Bayesian approach to relax the parametric distribution assumption for shared and specific‐class frailties by using a Dirichlet process prior while also allowing for the uncertainty of heterogeneity for different classes. Multiple cluster‐specific frailty selections rely on variable selection‐type mixture priors by applying mixtures of point masses at zero and inverse gamma distributions to the variance of log frailties. This selection allows frailties with zero variance to effectively drop out of the model. A reparameterization of log‐frailty terms is performed to reduce the potential bias of fixed effects due to variation of the random distribution and dependence among the parameters resulting in easy interpretation and faster Markov chain Monte Carlo convergence. Simulated data examples and an application to a lung cancer clinical trial are used for illustration.     

A Bayesian mixture model for across-site heterogeneities in the amino-acid replacement process

Most current models of sequence evolution assume that all sites of a protein evolve under the same substitution process, characterized by a 20 x 20 substitution matrix. Here, we propose to relax this assumption by developing a Bayesian mixture model that allows the amino-acid replacement pattern at different sites of a protein alignment to be described by distinct substitution processes. Our model, named CAT, assumes the existence of distinct processes (or classes) differing by their equilibrium frequencies over the 20 residues. Through the use of a Dirichlet process prior, the total number of classes and their respective amino-acid profiles, as well as the affiliations of each site to a given class, are all free variables of the model. In this way, the CAT model is able to adapt to the complexity actually present in the data, and it yields an estimate of the substitutional heterogeneity through the posterior mean number of classes. We show that a significant level of heterogeneity is present in the substitution patterns of proteins, and that the standard one-matrix model fails to account for this heterogeneity. By evaluating the Bayes factor, we demonstrate that the standard model is outperformed by CAT on all of the data sets which we analyzed. Altogether, these results suggest that the complexity of the pattern of substitution of real sequences is better captured by the CAT model, offering the possibility of studying its impact on phylogenetic reconstruction and its connections with structure-function determinants.     

The evolutionary consequences of alternative types of imperfect vaccines

The emergence and spread of mutant pathogens that evade the effects of prophylactic interventions, including vaccines, threatens our ability to control infectious diseases globally. Imperfect vaccines (e.g. those used against influenza), while not providing life-long immunity, confer protection by reducing a range of pathogen life-history characteristics; conversely, mutant pathogens can gain an advantage by restoring the same range of traits in vaccinated hosts. Using an SEIR model motivated by equine influenza, we investigate the evolutionary consequences of alternative types of imperfect vaccination, by comparing the spread rate of three types of mutant pathogens, in response to three types of vaccines. All mutant types spread faster in response to a transmission-blocking vaccine, relative to vaccines that reduce the proportion of exposed vaccinated individuals becoming infectious, and to vaccines that reduce the length of the infectious period; this difference increases with increasing vaccine efficacy. We interpret our results using the first published Price equation formulation for an SEIR model, and find that our main result is explained by the effects of vaccines on the equilibrium host distribution across epidemiological classes. In particular, the proportion of vaccinated infectious individuals among all exposed and infectious hosts, which is relatively higher in the transmission-blocking vaccine scenario, is important in explaining the faster spread of mutant strains in response to that vaccine. Our work illustrates the connection between epidemiological and evolutionary dynamics, and the need to incorporate both in order to explain and interpret findings of complicated infectious disease dynamics.     

Lodgepole pine forest age class dynamics and susceptibility to mountain pine beetle attack

Because mountain pine beetle attack mature pine stands, an understanding of forest age class dynamics is important to managing forests within the distribution of the beetle. The assumed theoretical negative exponential forest age distribution provides an estimate when ecosystem dynamics are in equilibrium. This study investigates the dynamics of forest age distribution for non-equilibrium ecosystem dynamics, which result primarily from large and irregular stand-replacement fire disturbances that alter the forest age distribution. A model experiment using the SEM-LAND model on a 1 million ha lodgepole pine forest landscape was conducted to estimate how the proportion of susceptible area could be influenced by different fire regimes. The results of the simulation suggest that the temporal dynamics of the area susceptible to mountain pine beetle attack are complex and depend on the fire history of the study area, if the area is experiencing large and irregular stand-replacement fires. The age range of the lodgepole pine forest stands susceptible to mountain pine beetle attack might significantly affect the estimate of the area susceptible to attack.     

Year-class coexistence in biennial plants

I extend the well known and biologically well motivated Skellam model of plant population dynamics to biennial plants. The model has two attractors: either one year class competitively excludes the other, resulting in 2-cycles with only vegetative vs only flowering plants in alternating years, or the two year classes coexist at an interior equilibrium. Contrary to earlier models, these two attractors can exist also simultaneously. I investigate the robustness of the model by including delayed flowering, a common phenomenon in plants, and provide a full numerical bifurcation analysis of the generalized model. High fecundity implies strong competition within year classes and promotes coexistence, whereas high survival results in strong competition between year classes and promotes competitive exclusion. Delayed flowering tends to stabilize the interior equilibrium, but (unlike in density-independent matrix models) the population cycles are robust with respect to some delay in flowering.     

Ecological conditions that favor the evolution of intermediate-virulence in an environmentally transmitted parasite

In this paper we develop and analyze several populaion-dynamic models of an environmentally transmitted symbiotic parasite infecting an isolated population of susceptible hosts. In our most basic model infection acts only to decrease the average lifetime of the infected host, parasites are only transmitted to uninfected hosts, there is no recovery from infection, and the rate of parasite transmission is an increasing function of the level of parasite virulence. It is shown that invasion of the parasite-free equilibrium cannot occur for virulence levels that are either too high or too low. We then incorporate a number of modifications to the model, among them the possibility that host fertility is reduced by infection, and that transmission rate depends additionally on susceptible host density. It is shown that the essential nature of the conditions for invasion are preserved. Thus, natural selection for intermediate virulence is a generic property of a broad class of population models.