Planar radial spots in a three-component FitzHugh-Nagumo system

Localized planar patterns arise in many reaction-diffusion models. Most of the paradigm equations that have been studied so far are two-component models. While stationary localized structures are often found to be stable in such systems, travelling patterns either do not exist or are found to be unstable. In contrast, numerical simulations indicate that localized travelling structures can be stable in three-component systems. As a first step towards explaining this phenomenon, a planar singularly perturbed three-component reaction-diffusion system that arises in the context of gas-discharge systems is analysed in this paper. Using geometric singular perturbation theory, the existence and stability regions of radially symmetric stationary spot solutions are delineated and, in particular, stable spots are shown to exist in appropriate parameter regimes. This result opens up the possibility of identifying and analysing drift and Hopf bifurcations, and their criticality, from the stationary spots described here.

Pinned fronts in heterogeneous media of jump type

In this paper, we analyse the impact of a (small) heterogeneity of jump type on the most simple localized solutions of a 3-component FitzHugh–Nagumo-type system. We show that the heterogeneity can pin a 1-front solution, which travels with constant (non-zero) speed in the homogeneous setting, to a fixed, explicitly determined, distance from the heterogeneity. Moreover, we establish the stability of this heterogeneous pinned 1-front solution. In addition, we analyse the pinning of 1-pulse, or 2-front, solutions. The paper is concluded with simulations in which we consider the dynamics and interactions of N-front patterns in domains with M heterogeneities of jump type (N = 3, 4, M ≥ 1).

Front interactions in a three-component system

The three-component reaction-diffusion system introduced in [C. P. Schenk et al., Phys. Rev. Lett., 78 (1997), pp. 3781–3784] has become a paradigm model in pattern formation. It exhibits a rich variety of dynamics of fronts, pulses, and spots. The front and pulse interactions range in type from weak, in which the localized structures interact only through their exponentially small tails, to strong interactions, in which they annihilate or collide and in which all components are far from equilibrium in the domains between the localized structures. Intermediate to these two extremes sits the semistrong interaction regime, in which the activator component of the front is near equilibrium in the intervals between adjacent fronts but both inhibitor components are far from equilibrium there, and hence their concentration profiles drive the front evolution. In this paper, we focus on dynamically evolving N-front solutions in the semistrong regime. The primary result is use of a renormalization group method to rigorously derive the system of N coupled ODEs that governs the positions of the fronts. The operators associated with the linearization about the N-front solutions have N small eigenvalues, and the N-front solutions may be decomposed into a component in the space spanned by the associated eigenfunctions and a component projected onto the complement of this space. This decomposition is carried out iteratively at a sequence of times. The former projections yield the ODEs for the front positions, while the latter projections are associated with remainders that we show stay small in a suitable norm during each iteration of the renormalization group method. Our results also help extend the application of the renormalization group method from the weak interaction regime for which it was initially developed to the semistrong interaction regime. The second set of results that we present is a detailed analysis of this system of ODEs, providing a classification of the possible front interactions in the cases of $N=1,2,3,4$, as well as how front solutions interact with the stationary pulse solutions studied earlier in [A. Doelman, P. van Heijster, and T. J. Kaper, J. Dynam. Differential Equations, 21 (2009), pp. 73–115; P. van Heijster, A. Doelman, and T. J. Kaper, Phys. D, 237 (2008), pp. 3335–3368]. Moreover, we present some results on the general case of N-front interactions.

Pulse dynamics in a three-component system : existence analysis

In this article, we analyze the three-component reaction-diffusion system originally developed by Schenk et al. (PRL 78:3781–3784, 1997). The system consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a prototype three-component system that generates rich pulse dynamics and interactions, and this richness is the main motivation for the analysis we present. We demonstrate the existence of stationary one-pulse and two-pulse solutions, and travelling one-pulse solutions, on the real line, and we determine the parameter regimes in which they exist. Also, for one-pulse solutions, we analyze various bifurcations, including the saddle-node bifurcation in which they are created, as well as the bifurcation from a stationary to a travelling pulse, which we show can be either subcritical or supercritical. For two-pulse solutions, we show that the third component is essential, since the reduced bistable two-component system does not support them. We also analyze the saddle-node bifurcation in which two-pulse solutions are created. The analytical method used to construct all of these pulse solutions is geometric singular perturbation theory, which allows us to show that these solutions lie in the transverse intersections of invariant manifolds in the phase space of the associated six-dimensional travelling wave system. Finally, as we illustrate with numerical simulations, these solutions form the backbone of the rich pulse dynamics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among others.

Pulse dynamics in a three-component system : stability and bifurcations

In this article, we analyze the stability and the associated bifurcations of several types of pulse solutions in a singularly perturbed three-component reaction-diffusion equation that has its origin as a model for gas discharge dynamics. Due to the richness and complexity of the dynamics generated by this model, it has in recent years become a paradigm model for the study of pulse interactions. A mathematical analysis of pulse interactions is based on detailed information on the existence and stability of isolated pulse solutions. The existence of these isolated pulse solutions is established in previous work. Here, the pulse solutions are studied by an Evans function associated to the linearized stability problem. Evans functions for stability problems in singularly perturbed reaction-diffusion models can be decomposed into a fast and a slow component, and their zeroes can be determined explicitly by the NLEP method. In the context of the present model, we have extended the NLEP method so that it can be applied to multi-pulse and multi-front solutions of singularly perturbed reaction-diffusion equations with more than one slow component. The brunt of this article is devoted to the analysis of the stability characteristics and the bifurcations of the pulse solutions. Our methods enable us to obtain explicit, analytical information on the various types of bifurcations, such as saddle-node bifurcations, Hopf bifurcations in which breathing pulse solutions are created, and bifurcations into travelling pulse solutions, which can be both subcritical and supercritical.

Coexistence of stable spots and fronts in a three-component FitzHugh-Nagumo system

We investigate regions of bistability between different travelling and stationary structures in a planar singularly-perturbed three-component reaction-diffusion system that arises in the context of gas discharge systems. In previous work, we delineated the existence and stabil-ity regions of stationary localized spots in this system. Here, we complement this analysis by establishing the stability regions of planar travelling fronts and stationary stripes. Taken together, these results imply that stable fronts and spots can coexist in three-component systems. Numerical simulations indicate that the stable fronts never move towards stable spots but instead move away from them.

A note on a reaction-diffusion model describing the bone morphogen protein gradient in Drosophila embryonic patterning

In this article, we consider the Eldar model [3] from embryology in which a bone morphogenic protein, a short gastrulation protein, and their compound react and diffuse. We carry out a perturbation analysis in the limit of small diffusivity of the bone morphogenic protein. This analysis establishes conditions under which some elementary results of [3] are valid.

Cabin crew rostering at KLM: optimization of reserves

In this paper, we will discuss the issue of rostering jobs of cabin crew attendants at KLM. Generated schedules get easily disrupted by events such as illness of an employee. Obviously, reserve people have to be kept 'on duty' to resolve such disruptions. A lot of reserve crew requires more employees, but too few results in so-called secondary disruptions, which are particularly inconvenient for both the crew members and the planners. In this research we will discuss several modifications of the reserve scheduling policy that have a potential to reduce the number of secondary disruptions, and therefore to improve the performance of the scheduling process.

Monocular vision based autonomous navigation for a cost-effective MAV in GPS-denied environments

In this paper, we present a monocular vision based autonomous navigation system for Micro Aerial Vehicles (MAVs) in GPS-denied environments. The major drawback of monocular systems is that the depth scale of the scene can not be determined without prior knowledge or other sensors. To address this problem, we minimize a cost function consisting of a drift-free altitude measurement and up-to-scale position estimate obtained using the visual sensor. We evaluate the scale estimator, state estimator and controller performance by comparing with ground truth data acquired using a motion capture system. All resources including source code, tutorial documentation and system models are available online.

Antibiotics for bronchiectasis exacerbations in children: rationale and study protocol for a randomised placebo-controlled trial

Background Despite bronchiectasis being increasingly recognised as an important cause of chronic respiratory morbidity in both indigenous and non-indigenous settings globally, high quality evidence to inform management is scarce. It is assumed that antibiotics are efficacious for all bronchiectasis exacerbations, but not all practitioners agree. Inadequately treated exacerbations may risk lung function deterioration. Our study tests the hypothesis that both oral azithromycin and amoxicillin-clavulanic acid are superior to placebo at improving resolution rates of respiratory exacerbations by day 14 in children with bronchiectasis unrelated to cystic fibrosis. Methods We are conducting a bronchiectasis exacerbation study (BEST), which is a multicentre, randomised, double-blind, double-dummy, placebo-controlled, parallel group trial, in five centres (Brisbane, Perth, Darwin, Melbourne, Auckland). In the component of BEST presented here, 189 children fulfilling inclusion criteria are randomised (allocation-concealed) to receive amoxicillin-clavulanic acid (22.5 mg/kg twice daily) with placebo-azithromycin; azithromycin (5 mg/kg daily) with placebo-amoxicillin-clavulanic acid; or placebo-azithromycin with placebo-amoxicillin-clavulanic acid for 14 days. Clinical data and a paediatric cough-specific quality of life score are obtained at baseline, at the start and resolution of exacerbations, and at day 14. In most children, blood and deep nasal swabs are also collected at the same time points. The primary outcome is the proportion of children whose exacerbations have resolved at day 14. The main secondary outcome is the paediatric cough-specific quality of life score. Other outcomes are time to next exacerbation; requirement for hospitalisation; duration of exacerbation; and spirometry data. Descriptive viral and bacteriological data from nasal samples and blood markers will also be reported. Discussion Effective, evidence-based management of exacerbations in people with bronchiectasis is clinically important. Yet, there are few randomised controlled trials (RCTs) in the neglected area of non-cystic fibrosis bronchiectasis. Indeed, no published RCTs addressing the treatment of bronchiectasis exacerbations in children exist. Our multicentre, double-blind RCT is designed to determine if azithromycin and amoxicillin-clavulanic acid, compared with placebo, improve symptom resolution on day 14 in children with acute respiratory exacerbations. Our planned assessment of the predictors of antibiotic response, the role of antibiotic-resistant respiratory pathogens, and whether early treatment with antibiotics affects duration and time to the next exacerbation, are also all novel.

Bronchiectasis exacerbation study on azithromycin and amoxycillin-clavulanate for respiratory exacerbations in children (BEST-2) : study protocol for a randomized controlled trial

Background Bronchiectasis unrelated to cystic fibrosis (CF) is being increasingly recognized in children and adults globally, both in resource-poor and in affluent countries. However, high-quality evidence to inform management is scarce. Oral amoxycillin-clavulanate is often the first antibiotic chosen for non-severe respiratory exacerbations, because of the antibiotic-susceptibility patterns detected in the respiratory pathogens commonly associated with bronchiectasis. Azithromycin has a prolonged half-life, and with its unique anti-bacterial, immunomodulatory, and anti-inflammatory properties, presents an attractive alternative. Our proposed study will test the hypothesis that oral azithromycin is non-inferior (within a 20% margin) to amoxycillin-clavulanate at achieving resolution of non-severe respiratory exacerbations by day 21 of treatment in children with non-CF bronchiectasis. Methods This will be a multicenter, randomized, double-blind, double-dummy, placebo-controlled, parallel group trial involving six Australian and New Zealand centers. In total, 170 eligible children will be stratified by site and bronchiectasis etiology, and randomized (allocation concealed) to receive: 1) azithromycin (5 mg/kg daily) with placebo amoxycillin-clavulanate or 2) amoxycillin-clavulanate (22.5 mg/kg twice daily) with placebo azithromycin for 21 days as treatment for non-severe respiratory exacerbations. Clinical data and a parent-proxy cough-specific quality of life (PC-QOL) score will be obtained at baseline, at the start and resolution of exacerbations, and on day 21. In most children, blood and deep-nasal swabs will also be collected at the same time points. The primary outcome is the proportion of children whose exacerbations have resolved at day 21. The main secondary outcome is the PC-QOL score. Other outcomes are: time to next exacerbation; requirement for hospitalization; duration of exacerbation, and spirometry data. Descriptive viral and bacteriological data from nasal samples and blood inflammatory markers will be reported where available. Discussion Currently, there are no published randomized controlled trials (RCT) to underpin effective, evidence-based management of acute respiratory exacerbations in children with non-CF bronchiectasis. To help address this information gap, we are conducting two RCTs. The first (bronchiectasis exacerbation study; BEST-1) evaluates the efficacy of azithromycin and amoxycillin-clavulanate compared with placebo, and the second RCT (BEST-2), described here, is designed to determine if azithromycin is non-inferior to amoxycillin-clavulanate in achieving symptom resolution by day 21 of treatment in children with acute respiratory exacerbations. Trial registration Australia and New Zealand Clinical Trials Register (ANZCTR) number ACTRN12612000010897. http://www.anzctr.org.au/trial_view.aspx?id=347879

Does a 10-valent pneumococcal-Haemophilus influenzae protein D conjugate vaccine prevent respiratory exacerbations in children with recurrent protracted bacterial bronchitis, chronic suppurative lung disease and bronchiectasis : protocol for a randomised controlled trial

Background Recurrent protracted bacterial bronchitis (PBB), chronic suppurative lung disease (CSLD) and bronchiectasis are characterised by a chronic wet cough and are important causes of childhood respiratory morbidity globally. Haemophilus influenzae and Streptococcus pneumoniae are the most commonly associated pathogens. As respiratory exacerbations impair quality of life and may be associated with disease progression, we will determine if the novel 10-valent pneumococcal-Haemophilus influenzae protein D conjugate vaccine (PHiD-CV) reduces exacerbations in these children. Methods A multi-centre, parallel group, double-blind, randomised controlled trial in tertiary paediatric centres from three Australian cities is planned. Two hundred six children aged 18 months to 14 years with recurrent PBB, CSLD or bronchiectasis will be randomised to receive either two doses of PHiD-CV or control meningococcal (ACYW(135)) conjugate vaccine 2 months apart and followed for 12 months after the second vaccine dose. Randomisation will be stratified by site, age (<6 years and >= 6 years) and aetiology (recurrent PBB or CSLD/bronchiectasis). Clinical histories, respiratory status (including spirometry in children aged >= 6 years), nasopharyngeal and saliva swabs, and serum will be collected at baseline and at 2, 3, 8 and 14 months post-enrolment. Local and systemic reactions will be recorded on daily diaries for 7 and 30 days, respectively, following each vaccine dose and serious adverse events monitored throughout the trial. Fortnightly, parental contact will help record respiratory exacerbations. The primary outcome is the incidence of respiratory exacerbations in the 12 months following the second vaccine dose. Secondary outcomes include: nasopharyngeal carriage of H. influenzae and S. pneumoniae vaccine and vaccine-related serotypes; systemic and mucosal immune responses to H. influenzae proteins and S. pneumoniae vaccine and vaccine-related serotypes; impact upon lung function in children aged >= 6 years; and vaccine safety. Discussion As H. influenzae is the most common bacterial pathogen associated with these chronic respiratory diseases in children, a novel pneumococcal conjugate vaccine that also impacts upon H. influenzae and helps prevent respiratory exacerbations would assist clinical management with potential short- and long-term health benefits. Our study will be the first to assess vaccine efficacy targeting H. influenzae in children with recurrent PBB, CSLD and bronchiectasis.

Bifurcations to travelling planar spots in a three-component FitzHugh-Nagumo system

In this article, we analyse bifurcations from stationary stable spots to travelling spots in a planar three-component FitzHugh-Nagumo system that was proposed previously as a phenomenological model of gas-discharge systems. By combining formal analyses, center-manifold reductions, and detailed numerical continuation studies, we show that, in the parameter regime under consideration, the stationary spot destabilizes either through its zeroth Fourier mode in a Hopf bifurcation or through its first Fourier mode in a pitchfork or drift bifurcation, whilst the remaining Fourier modes appear to create only secondary bifurcations. Pitchfork bifurcations result in travelling spots, and we derive criteria for the criticality of these bifurcations. Our main finding is that supercritical drift bifurcations, leading to stable travelling spots, arise in this model, which does not seem possible for its two-component version.

Follow the leader if you can, hedge if you must

Follow-the-Leader (FTL) is an intuitive sequential prediction strategy that guarantees constant regret in the stochastic setting, but has poor performance for worst-case data. Other hedging strategies have better worst-case guarantees but may perform much worse than FTL if the data are not maximally adversarial. We introduce the FlipFlop algorithm, which is the first method that provably combines the best of both worlds. As a stepping stone for our analysis, we develop AdaHedge, which is a new way of dynamically tuning the learning rate in Hedge without using the doubling trick. AdaHedge refines a method by Cesa-Bianchi, Mansour, and Stoltz (2007), yielding improved worst-case guarantees. By interleaving AdaHedge and FTL, FlipFlop achieves regret within a constant factor of the FTL regret, without sacrificing AdaHedge’s worst-case guarantees. AdaHedge and FlipFlop do not need to know the range of the losses in advance; moreover, unlike earlier methods, both have the intuitive property that the issued weights are invariant under rescaling and translation of the losses. The losses are also allowed to be negative, in which case they may be interpreted as gains.

A multicenter study on chronic cough in children : burden and etiologies based on a standardized management pathway

Background While the burden of chronic cough in children has been documented, etiologic factors across multiple settings and age have not been described. In children with chronic cough, we aimed (1) to evaluate the burden and etiologies using a standard management pathway in various settings, and (2) to determine the influence of age and setting on disease burden and etiologies and etiology on disease burden. We hypothesized that the etiology, but not the burden, of chronic cough in children is dependent on the clinical setting and age. Methods From five major hospitals and three rural-remote clinics, 346 children (mean age 4.5 years) newly referred with chronic cough (> 4 weeks) were prospectively managed in accordance with an evidence-based cough algorithm. We used a priori definitions, timeframes, and validated outcome measures (parent-proxy cough-specific quality of life [PC-QOL], a generic QOL [pediatric quality of life (PedsQL)], and cough diary). Results The burden of chronic cough (PC-QOL, cough duration) significantly differed between settings (P = .014, 0.021, respectively), but was not influenced by age or etiology. PC-QOL and PedsQL did not correlate with age. The frequency of etiologies was significantly different in dissimilar settings (P = .0001); 17.6% of children had a serious underlying diagnosis (bronchiectasis, aspiration, cystic fibrosis). Except for protracted bacterial bronchitis, the frequency of other common diagnoses (asthma, bronchiectasis, resolved without specific-diagnosis) was similar across age categories. Conclusions The high burden of cough is independent of children’s age and etiology but dependent on clinical setting. Irrespective of setting and age, children with chronic cough should be carefully evaluated and child-specific evidence-based algorithms used.