Computation of maximal local (un)stable manifold patches by the parameterization method

In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size of the image of the polynomial approximation relative to some specified constraints. More precisely we use that the manifold computations depend heavily on the scalings of the eigenvectors: indeed we study the precise effects of these scalings on the estimates which determine the validated error bounds. This relationship between the eigenvector scalings and the error estimates plays a central role in our automatic procedures. In order to illustrate the utility of these methods we present several applications, including visualization of invariant manifolds in the Lorenz and FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable manifolds in a suspension bridge problem. In the present work we treat explicitly the case where the eigenvalues satisfy a certain non-resonance condition.Comment: Revised version, typos corrected, references adde

Quality assessment of primary care for common mental disorders in isolated communities: Taking advantage of health records.

INTRODUCTION: This article is part of a research study on the organization of primary health care (PHC) for mental health in two of Quebec's remote regions. It introduces a methodological approach based on information found in health records, for assessing the quality of PHC offered to people suffering from depression or anxiety disorders. METHODS: Quality indicators were identified from evidence and case studies were reconstructed using data collected in health records over a 2-year observation period. Data collection was developed using a three-step iterative process: (1) feasibility analysis, (2) development of a data collection tool, and (3) application of the data collection method. The adaptation of quality-of-care indicators to remote regions was appraised according to their relevance, measurability and construct validity in this context. RESULTS: As a result of this process, 18 quality indicators were shown to be relevant, measurable and valid for establishing a critical quality appraisal of four recommended dimensions of PHC clinical processes: recognition, assessment, treatment and follow-up. CONCLUSIONS: There is not only an interest in the use of health records to assess the quality of PHC for mental health in remote regions but also a scientific value for the rigorous and meticulous methodological approach developed in this study. From the perspective of stakeholders in the PHC system of care in remote areas, quality indicators are credible and provide potential for transferability to other contexts. This study brings information that has the potential to identify gaps in and implement solutions adapted to the context

Continuum of care for persons with common mental health disorders in Nunavik: a descriptive study.

BACKGROUND: Changing Directions, Changing Lives, the Mental Health Strategy for Canada, prioritizes the development of coordinated continuums of care in mental health that will bridge the gap in services for Inuit populations. OBJECTIVE: In order to target ways of improving the services provided in these contexts to individuals in Nunavik with depression or anxiety disorders, this research examines delays and disruptions in the continuum of care and clinical, individual and organizational characteristics possibly associated with their occurrences. DESIGN: A total of 155 episodes of care involving a common mental disorder (CMD), incident or recurring, were documented using the clinical records of 79 frontline health and social services (FHSSs) users, aged 14 years and older, living in a community in Nunavik. Each episode of care was divided into 7 stages: (a) detection; (b) assessment; (c) intervention; (d) planning the first follow-up visit; (e) implementation of the first follow-up visit; (f) planning a second follow-up visit; (g) implementation of the second follow-up visit. Sequential analysis of these stages established delays for each one and helped identify when breaks occurred in the continuum of care. Logistic and linear regression analysis determined whether clinical, individual or organizational characteristics influenced the breaks and delays. RESULTS: More than half (62%) the episodes of care were interrupted before the second follow-up. These breaks mostly occurred when planning and completing the first follow-up visit. Episodes of care were more likely to end early when they involved anxiety disorders or symptoms, limited FHSS teams and individuals over 21 years of age. The median delay for the first follow-up visit (30 days) exceeded guideline recommendations significantly (1-2 weeks). CONCLUSION: Clinical primary care approaches for CMDs in Nunavik are currently more reactive than preventive. This suggests that recovery services for those affected are suboptimal

African-American patients with cancer Talking About Clinical Trials (TACT) with oncologists during consultations: evaluating the efficacy of tailored health messages in a randomised controlled trial—the TACT study protocol

Introduction Low rates of accrual of African-American (AA) patients with cancer to therapeutic clinical trials (CTs) represent a serious and modifiable racial disparity in healthcare that impedes the development of promising cancer therapies. Suboptimal physician–patient consultation communication is a barrier to the accrual of patients with cancer of any race, but communication difficulties are compounded with AA patients. Providing tailored health messages (THM) to AA patients and their physician about CTs has the potential to improve communication, lower barriers to accrual and ameliorate health disparities. Objective (1) Demonstrate the efficacy of THM to increase patient activation as measured by direct observation. (2) Demonstrate the efficacy of THM to improve patient outcomes associated with barriers to AA participation. (3) Explore associations among preconsultation levels of: (A) trust in medical researchers, (B) knowledge and attitudes towards CTs, (C) patient-family member congruence in decision-making, and (D) involvement/information preferences, and group assignment. Methods and analysis First, using established methods, we will develop THM materials. Second, the efficacy of the intervention is determined in a 2 by 2 factorial randomised controlled trial to test the effectiveness of (1) providing 357 AA patients with cancer with THM with 2 different ‘depths’ of tailoring and (2) either providing feedback to oncologists about the patients\u27 trial THM or not. The primary analysis compares patient engaged communication in 4 groups preconsultation and postconsultation. Ethics and dissemination This study was approved by the Virginia Commonwealth University Institutional Review Board. To facilitate use of the THM intervention in diverse settings, we will convene ‘user groups’ at 3 major US cancer centres. To facilitate dissemination, we will post all materials and the implementation guide in publicly available locations

Analytic enclosure of the fundamental matrix solution

This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form is validated in the space of analytic functions. The technique combines analytical estimates and rigorous numerical computations and no rigorous integration is needed. An application to the theory of dynamical system is presented, together with a comparison with the results obtained by computing the enclosure in the C s category

Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields

In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the Newton–Kantorovich theorem is applied to validate the existence of a so-called short connecting orbit. If the local manifolds do not intersect, a boundary value problem with boundary values in the local manifolds is rigorously solved by a contraction mapping argument on a ball centered at the numerical solution, yielding the existence of a so-called long connecting orbit. In both cases our argument yields transversality of the corresponding intersection of the manifolds. The method is applied to the Lorenz equations, where a study of a pitchfork bifurcation with saddle-to-saddle stability is done and where several proofs of existence of short and long connections are obtained

Rigorous numerics for analytic solutions of differential equations : the radii polynomial approach

Judicious use of interval arithmetic, combined with careful pen and paper estimates, leads to effective strategies for computer assisted analysis of nonlinear operator equations. The method of radii polynomials is an efficient tool for bounding the smallest and largest neighborhoods on which a Newton-like operator associated with a nonlinear equation is a contraction mapping. The method has been used to study solutions of ordinary, partial, and delay differential equations such as equilibria, periodic orbits, solutions of initial value problems, heteroclinic and homoclinic connecting orbits in the Ck category of functions. In the present work we adapt the method of radii polynomials to the analytic category. For ease of exposition we focus on studying periodic solutions in Cartesian products of infinite sequence spaces. We derive the radii polynomials for some specific application problems and give a number of computer assisted proofs in the analytic framework

Numerical computation of transverse homoclinic orbits for periodic solutions of delay differential equations

We present a computational method for studying transverse homoclinic orbits for periodic solutions of delay differential equations, a phenomenon that we refer to as the \emph{Poincar\'{e} scenario}. The strategy is geometric in nature, and consists of viewing the connection as the zero of a nonlinear map, such that the invertibility of its Fr\'{e}chet derivative implies the transversality of the intersection. The map is defined by a projected boundary value problem (BVP), with boundary conditions in the (finite dimensional) unstable and (infinite dimensional) stable manifolds of the periodic orbit. The parameterization method is used to compute the unstable manifold and the BVP is solved using a discrete time dynamical system approach (defined via the \emph{method of steps}) and Chebyshev series expansions. We illustrate this technique by computing transverse homoclinic orbits in the cubic Ikeda and Mackey-Glass systems

Shifts in Metabolic Demands in Growing Altricial Nestlings Illustrate Context-Specific Relationships between Basal Metabolic Rate and Body Composition

Basal metabolic rate (BMR) in animals is interpreted as reflecting the size and metabolic intensity of energy-consuming tissues. However, studies investigating relationships between the mass of specific organs and interindividual variation in BMR have produced inconsistent patterns with regard to which organs have the largest impact on BMR variation. Because of the known flexibility in organ mass and metabolic intensity within individual organs, relationships between BMR and body-composition variables are bound to be context specific. Altricial nestlings are excellent models to illustrate this phenomenon because of the extreme variation in body composition occurring during growth. Using European starlings at three age classes, we studied changes in body composition together with its effect on variation in resting metabolic rate (RMR) in order to highlight the context-specific nature of these relationships. Our data suggest a transition in metabolic costs during growth in starling nestlings. During the linear phase of growth, energy is mainly consumed by tissue-synthesis processes, with fast-growing organs having a large influence on RMR. In the plateau phase of growth, the energy expenditure is transferred to functional costs, with high-intensity organs having a predominant effect on RMR variation. Our data illustrates the context-specific nature of organ mass-metabolic rate correlations, which complicates inter- and intraspecific comparisons of BMR. In the future, such comparisons must be done while taking the physiological state of the study animal into account