Rhythmicity of patient flow in an acute medical unit: relationship to hospital occupancy, 7-day working and the effect of COVID-19.

BACKGROUND: The Acute Medical Unit (AMU) provides care for unscheduled hospital admissions. Seven-day consultant presence and morning AMU discharges have been advocated to improve hospital bed management. AIMS: To determine whether a later time of daily peak AMU occupancy correlates with measures of hospital stress; whether 7-day consultant presence, for COVID-19, abolished weekly periodicity of discharges. DESIGN: Retrospective cohort analysis. METHODS: : Anonymised AMU admission and discharge times were retrieved from the Profile Information Management System (PIMS), at a large, urban hospital from 14 April 2014 to 31 December 2018 and 20 March to 2 May 2020 (COVID-19 peak). Minute-by-minute admission and discharge times were combined to construct a running total of AMU bed occupancy. Fourier transforms were used to determine periodicity. We tested association between (i) average AMU occupancy and (ii) time of peak AMU occupancy, with measures of hospital stress (total medical bed occupancy and 'medical outliers' on non-medical wards). RESULTS: : Daily, weekly and seasonal patterns of AMU bed occupancy were evident. Timing of AMU peak occupancy was unrelated to each measure of hospital stress: total medical inpatients (Spearman's rho, rs = 0.04, P = 0.24); number of medical outliers (rs = -0.06, P = 0.05). During COVID-19, daily bed occupancy was similar, with continuation of greater Friday and Monday discharges than the weekend. CONCLUSIONS: : Timing of peak AMU occupancy did not alter with hospital stress. Efforts to increase morning AMU discharges are likely to have little effect on hospital performance. Seven-day consultant presence did not abolish weekly periodicity of discharges-other factors influence weekend discharges.

Creation of discontinuities in circle maps.

Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and 'threshold' systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particular, we describe how maps evolve near the creation of a discontinuity. We show that the natural way to create discontinuities in the maps associated with both threshold systems and Cherry flows results in a singularity in the derivative of the map as the discontinuity is approached from either one or both sides. For the threshold systems, the associated maps have square root singularities and we analyse the generic properties of such maps with gaps, showing how border collisions and saddle-node bifurcations are interspersed. This highlights how the Arnold tongue picture for tongues bordered by saddle-node bifurcations is amended once gaps are present. We also show that a loss of injectivity naturally results in the creation of multiple gaps giving rise to a novel codimension two bifurcation.

Three-wave interactions and spatiotemporal chaos.

Three-wave interactions form the basis of our understanding of many pattern-forming systems because they encapsulate the most basic nonlinear interactions. In problems with two comparable length scales, it is possible for two waves of the shorter wavelength to interact with one wave of the longer, as well as for two waves of the longer wavelength to interact with one wave of the shorter. Consideration of both types of three-wave interactions can generically explain the presence of complex patterns and spatiotemporal chaos. Two length scales arise naturally in the Faraday wave experiment, and our results enable some previously unexplained experimental observations of spatiotemporal chaos to be interpreted in a new light. Our predictions are illustrated with numerical simulations of a model partial differential equation.

Scaling properties of weakly nonlinear coefficients in the Faraday problem.

Interesting and exotic surface wave patterns have regularly been observed in the Faraday experiment. Although symmetry arguments provide a qualitative explanation for the selection of some of these patterns (e.g., superlattices), quantitative analysis is hindered by mathematical difficulties inherent in a time-dependent, free-boundary Navier-Stokes problem. More tractable low viscosity approximations are available, but these do not necessarily capture the moderate viscosity regime of the most interesting experiments. Here we focus on weakly nonlinear behavior and compare the scaling results derived from symmetry arguments in the low viscosity limit with the computed coefficients of appropriate amplitude equations using both the full Navier-Stokes equations and a reduced set of partial differential equations due to Zhang and Vinãls. We find the range of viscosities over which one can expect "low viscosity" theories to hold. We also find that there is an optimal viscosity range for locating superlattice patterns experimentally-large enough that the region of parameters giving stable patterns is not impracticably small, yet not so large that crucial resonance effects are washed out. These results help explain some of the discrepancies between theory and experiment.

Parametrically excited surface waves: two-frequency forcing, normal form symmetries, and pattern selection.

Motivated by experimental observations of exotic free surface standing wave patterns in the two-frequency Faraday experiment, we investigate the role of normal form symmetries in the associated pattern-selection problem. With forcing frequency components in ratio m/n, where m and n are coprime integers that are not both odd, there is the possibility that both harmonic waves and subharmonic waves may lose stability simultaneously, each with a different wave number. We focus on this situation and compare the case where the harmonic waves have a longer wavelength than the subharmonic waves with the case where the harmonic waves have a shorter wavelength. We show that in the former case a normal form transformation can be used to remove all quadratic terms from the amplitude equations governing the relevant resonant triad interactions. Thus the role of resonant triads in the pattern-selection problem is greatly diminished in this situation. We verify our general bifurcation theoretic results within the example of one-dimensional surface wave solutions of the Zhang-Viñals model [J. Fluid Mech. 341, 225 (1997)] of the two-frequency Faraday problem. In one-dimension, a 1:2 spatial resonance takes the place of a resonant triad in our investigation. We find that when the bifurcating modes are in this spatial resonance, it dramatically effects the bifurcation to subharmonic waves in the case that the forcing frequencies are in ratio 1/2; this is consistent with the results of Zhang and Viñals. In sharp contrast, we find that when the forcing frequencies are in a ratio 2/3, the bifurcation to (sub)harmonic waves is insensitive to the presence of another spatially resonant bifurcating mode. This is consistent with the results of our general analysis.