Journal of The Royal Society Interface

In insect locomotion, the transmission of energy from muscles to motion is a process within which there are many sources of dissipation. One significant but understudied source is the structural damping within the insect exoskeleton itself: the thorax and limbs. Experimental evidence suggests that exoskeletal damping shows frequency (or, rate) independence, but investigation into its nature and implications has been hampered by a lack methods for simulating the time-domain behaviour of this damping. Here, synergising and extending results across applied mathematics and seismic analysis, we provide these methods. We show that existing models of exoskeletal rate-independent damping are equivalent to an important singular integral in time: the Hilbert transform. However, these models are strongly noncausal, violating the directionality of time. We derive the unique causal analogue of these existing exoskeletal damping models, as well as an accessible approximation to them, as Hadamard finite-part integrals in time, and provide methods for simulating them. These methods are demonstrated on several current problems in insect biomechanics. Finally, we demonstrate, for the first time, that existing rate-independent damping models are not strictly dissipative: in certain circumstances they are capable of generating negative power without apparently storing energy, likely violating conservation of energy. This work resolves a key methodological impasse in the understanding of insect exoskeletal dynamics and offers new insights into the micro-structural origins of rate-independent damping as well as the directions required in order to resolve violations of causality and the conservation of energy in existing models.

The ecological and evolutionary benefits of collective behaviours are rooted in the physical principles and physiological mechanisms underpinning animal locomotion. We propose a turbulence sheltering hypothesis that collective movements of fish schools in turbulent flow can reduce the total energetic cost of locomotion by shielding individuals from the perturbation of chaotic turbulent eddies. We test this hypothesis by quantifying energetics and kinematics in schools of giant danio (Devario aequipinnatus) compared to solitary individuals swimming under control and turbulent conditions over a wide speed range. We discovered that, when swimming at high speeds and high turbulence levels, fish schools reduced their total energy expenditure (TEE, both aerobic and anaerobic energy) by 63-79% compared to solitary fish. Solitary individuals spend [~]25% more kinematic effort (tail beat amplitude*frequency) to swim in turbulence at higher speeds than in control conditions. However, fish schools swimming in turbulence reduced their three-dimensional group volume by 41-68% (at higher speeds) and did not alter their kinematic effort compared to control conditions. This substantial energy saving highlighted a [~]261% higher TEE when fish swimming alone in turbulence are compared to swimming in a school. Schooling behaviour could mitigate turbulent disturbances by sheltering fish within schools from the eddies of sufficient kinetic energy that can disrupt the locomotor gaits. Providing a more desirable internal hydrodynamic environment could be one of the ecological drivers underlying collective behaviours in a dense fluid environment. One-Sentence SummaryThe collective movement of fish schools substantially reduces the energetic cost of locomotion in turbulence compared to that of swimming alone.

Movement is integral to animal life, and most animal movement is actuated by the same engine: skeletal muscle. Muscle input is typically mediated by skeletal elements, resulting in musculoskeletal systems that are "geared": at any instant, the muscle force and velocity are related to the output force and velocity only via a proportionality constant G, the "mechanical advantage". The functional analysis of such "simple machines" has traditionally centred around this instantaneous interpretation, such that a small vs large G is thought to reflect a fast vs forceful system, respectively. But evidence is mounting that a complete analysis ought to also consider the mechanical energy output of a complete contraction. Here, we approach this task systematically, and use the theory of physiological similarity to study how gearing affects the flow of mechanical energy in a minimalist model of a musculoskeletal system. Gearing influences the flow of mechanical energy in two key ways: it can curtail muscle work output, because it determines the ratio between the characteristic muscle work and kinetic energy capacity; and it defines how each unit of muscle work is partitioned into different system energies, i. e. into kinetic vs. "parasitic" energy such as heat. As a consequence of both effects, delivering maximum work in minimum time and with maximum transmission efficiency generally requires a mechanical advantage of intermediate magnitude. This optimality condition can be expressed in terms of two dimensionless numbers, which reflect the key geometric, physiological, and physical properties of the interrogated musculoskeletal system, and the environment in which the contraction takes place. Illustrative application to exemplar musculoskeletal systems predicts plausible mechanical advantages in disparate biomechanical scenarios; yields a speculative explanation for why gearing is typically used to attenuate the instantaneous force output (Gopt < 1); and predicts how G needs to vary systematically with animal size to optimise the delivery of mechanical energy, in superficial agreement with empirical observations. A many-to-one-mapping from musculoskeletal geometry to mechanical performance is identified, such that differences in G alone do not provide a reliable indicator for specialisation for force vs speed--neither instantaneously, nor in terms of mechanical energy output. The energy framework presented here can be used to estimate an optimal mechanical advantage across variable muscle physiology, anatomy, mechanical environment and animal size, and so facilitates investigation of the extent to which selection has made efficient use of gearing as degree of freedom in musculoskeletal "design".

Mathematical models of infectious disease dynamics are routinely fitted to surveillance data to estimate epidemiological parameters and inform public health decisions. Such data are typically discrete and noisy, but before attempting estimation, it is essential to ask whether the model structure itself permits unique parameter identification at least under perfect (continuous, noise-free) observations. This mathematical property of a model with respect to observation(s), known as structural identifiability, serves as a necessary precondition for reliable inference, since a model that fails this test cannot yield unique parameter estimates even from perfect data. In this study, we systematically investigate structural identifiability in various classes of compartmental epidemic models and establish two main findings. First, we present and deploy a methodology for assessing structural identifiability of epidemiological quantities of interest and demonstrate that the basic reproduction number exhibits identifiability across diverse model structures--including models with multiple transmission pathways and host-vector dynamics--even when individual parameters are not uniquely identifiable. These findings challenge the assumption that complete model identifiability is necessary for reliable epidemiological inference and suggest reformulating the central question from "is the model identifiable?" to "are the quantities that matter for the decision-making identifiable?" Second, we prove that incorporating minimal complementary data, as little as a single time-point measurement from an additional state variable, can make otherwise nonidentifiable models globally identifiable. This result has direct implications for surveillance design: rather than putting limited resources into frequent monitoring of multiple data streams or relying on external parameter estimates that may be uncertain or context-dependent, public health systems can strategically prioritize collecting high-quality complementary measurements.

Blood Plasma Viscosity (PV) is an established biomarker for numerous diseases. While PV colloquially refers to the shear viscosity, there is a second viscosity component--the bulk viscosity--that describes the irreversible fluid compressibility on short time scales. The bulk viscosity is acutely sensitive to solid-like suspensions, and obtainable via the longitudinal viscosity from acoustic attenuation measurements. Whether it has diagnostic value remains unexplored yet may be pertinent given the association of diverse pathologies with the formation of plasma suspensions, such as fibrin-microstructures in COVID-19 and long-COVID. Here we show that the longitudinal PV measured using Brillouin Light Scattering (BLS) can serve as a proxy for the shear PV of blood plasma, and exhibits a temperature dependence consistent with increased suspension concentrations in severe COVID-patient plasma. Our results open a new avenue for PV diagnostics based on the longitudinal PV, and show that BLS can provide a means for its clinical implementation.

Many idle insects exhibit discontinuous gas exchange cycles (DCGs). During DCGs, CO2 is released in discrete bursts, followed by periods of negligible gas exchange. The standard metabolic rate (SMR) is thus determined to first order by the product between cycle frequency (fc) and burst volume (Vb, SMR {approx} fc{middle dot} Vb). The evolutionary allometry of these parameters is well studied, but it remains unclear if their static allometry, measured in individuals of the same species, sharing the same ontogenetic stage, follows the same patterns. To address this question, we investigate the static allometry of DCGs in Atta cephalotes leaf-cutter ants workers varying by two orders of magnitude in body mass. The SMR allometry significantly exceeded the standard prediction from the nutrient supply network model, and differed from the SMR allometry observed across insects. This disproportional increase was exclusively achieved by an increase in Vb, perhaps because fc is stabilised by neural and mechanical constraints. It may be necessitated by the positive allometry of the largest muscle in Atta--the mandible closer muscle--which increases with a virtually identical allometric coefficient, providing further evidence that the principles of symmorphosis may be upheld in insects.

Human mobility plays a critical role in the transmission dynamics of infectious diseases, influencing both their spread and the effectiveness of control measures. In the process of quantifying the real-time situation of an epidemic, the instantaneous reproduction number Rt appears to be one of the useful metrics widely used by public health researchers, officials, and policy makers. Since individuals can contract infections both within their region of origin and in other regions they visit, ignoring human mobility in the estimation process overlooks its impact on transmission dynamics and can lead to biased estimates of Rt, potentially misrepresenting the true epidemic situation. Our study explicitly integrates human mobility into a renewal-equation based disease transmission model to capture the mobility-driven effect on transmission. By incorporating pathogen-specific generation-time distribution, observational delay, the framework is epidemiologically informed and flexible to a wide range of diseases. We primarily validate the approach using simulated data, and demonstrate the limitations of estimating Rt without considering mobility. We then apply it to two real-world mobility settings using SARS-CoV-2 mortality data: the regions of England and the LTLAs of North East region of England, and uncover the mobility driven effect on transmission at different spatial resolutions. This framework uses non-identifiable and widely accessible publicly available datasets, demonstrating its practical applicability and supporting better-informed and more targeted public health measures. Author summaryThe real-time or instantaneous reproduction number Rt is a key metric for assessing the state of an epidemic at any given time. When estimating these numbers across multiple connected regions, human mobility plays a crucial role, as movement patterns significantly influence disease transmission. Traditional epidemic models often assume homogeneous mixing, which does not reflect real-world interactions. On the other hand, individual-based models incorporate heterogeneous mixing at individual level but demands an extremely refined data and substantial computational support. To address these challenges, we employ a renewal equation-based transmission framework, particularly useful for its effectiveness in real-time epidemic analysis, by incorporating heterogeneous mobility flows at a chosen spatial resolution. This yields the estimates of spatially connected instantaneous reproduction number for each region. This improved understanding enables better assessment of the impact of mobility on disease transmission spread, and provides valuable insights for designing targeted epidemic control and intervention strategies.

Human walking is often considered an inverted pendulum during single support, suggesting conservative dynamics. Gait consists of discrete steps connected by mechanically costly transitions. We examine how step length, walking speed, and work capacity jointly constrain walking mechanics. Using a powered simple walking model, minimum speed required to complete a step of given length is derived based on gravitational work; below this threshold, forward progression becomes mechanically infeasible, and the next heel-strike occurs early, producing shorter steps. Comparisons with empirical step length-speed relationships show that humans walk at higher speeds and require greater push-off work, indicating energy dissipation. We extend pendular dynamics by incorporating hip torque, a linearized axial force model, and muscle intervention. This framework reproduces key GRF features, including the M-shaped profile, without prescribing force trajectories a priori. Fitted parameters suggest reduced average loading (CBaseline < 1), active mid-stance unloading (Am < 0), and narrowly timed muscle action (small{sigma} m). Parameter studies show that increasing step length or speed increases transition work and peak forces, while hip torque timing indicates mechanical cost is minimized when energy modulation occurs after mid-stance. These findings indicate that preferred walking speed emerges from feasibility and work-capacity constraints, not energetic optimality alone.

The environments inhabited by flying insects demand a balance between flight efficiency and flight manoeuvrability. In structural oscillators such as the insect indirect flight motor, efficiency (arising from resonance) and manoeuvrability (arising from kinematic modulation) are typically quid pro quo, with modulation incurring penalties to efficiency. Band-type resonance is a phenomenon that offers, in theory, a strategy to lessen these penalties via careful navigation through a band of efficient kinematic states. However, identifying this band is challenging: no methods exist to identify the complete band in realistic motor models, involving elasticity distributed across thorax and wing. Nor are the effects of elasticity distribution on the band known. In this work, we address both open topics. We present a suite of numerical methods for identifying the complete resonance band in general systems. Applying them to models of the insect flight motor with distributed elasticity--thoracic and wing flexion--reveals that distributed elasticity is moderate-risk but high-reward morphological feature. Well-tuned distributions expand the resonance band over fourfold whereas poorly-tuned distributions completely extinguish the resonance band. These results indicate that distributing elasticity across the insect flight motor can have adaptive value, and motivate broader work identifying distributions across species.

A key theoretical challenge posed by single-molecule studies is the inverse problem of deducing the underlying molecular dynamics from the time evolution of low-dimensional experimental observables. Toward this goal, a variety of low-dimensional models have been proposed as descriptions of single-molecule signals, including random walks with or without conformational memory and/or with static or dynamics disorder. Differentiating among different models presents a challenge, as many distinct physical scenarios lead to similar experimentally observable behaviors such as anomalous diffusion and nonexponential relaxation. Here we show that information-theory-based analysis of single-molecule time series, inspired by Shannons work studying the information content of printed English, can differentiate between Markov (memoryless) and non-Markov single-molecule signals and between static and dynamic disorder. In particular, non-Markov time series are more predictable and thus can be compressed and transmitted within shorter messages (i.e. have a lower entropy rate) than appropriately constructed Markov approximations, and we demonstrate that in practice the LZMA compression algorithm reliably differentiates between these entropy rates across several simulated dynamical models.

Organisms and machines must use measured sensory cues to estimate unknown information about themselves or their environment. Cleverly applied sensor motion can be exploited to enrich the quality of sensory data and improve estimation. However, a major barrier to modeling such active sensing problems is the lack of empirical, yet rigorous, tools for quantifying the relationship between movement and estimation performance. Here, we introduce "BOUNDS: Bounding Observability for Uncertain Nonlinear Dynamic Systems". BOUNDS can discover patterns of sensor motion that increase information and reduce uncertainty in either real or simulated data. Crucially, it is suitable for high dimensional and partially observable nonlinear systems with sensor noise. We demonstrate BOUNDS through a case study on how flying insects estimate wind properties, showing that specific active sensing motifs improve estimation. Additionally, we present a framework to refine sporadic estimates from active sensing. When combined with an artificial neural network, we show that the information gained via active sensing in real Drosophila flight trajectories is suitable for precise wind direction estimation. Collectively, our work will help decode active sensing in organisms and inform the design of estimation algorithms for machines.

Understanding structural similarities across dynamical systems at different scales remains a central problem in nonlinear science [1, 3]. Here we propose a modeling framework for cross-scale morphogenetic dynamics, termed Generalized Morphogenesis Theory (GMT), based on a flow-inertia formulation: O_FD O_INLINEFIG[Formula 1]C_INLINEFIGM_FD(1)C_FD where S denotes system state, E environmental input, F (E, S) a driving function, and {micro}(S) an inertia function representing resistance to change. This formulation provides a structural representation that encompasses several classical dynamical models--including Newtonian relaxation, logistic growth, and reaction-diffusion systems [13]--under appropriate parameterizations. Non-dimensionalization reveals a small set of control parameters governing regime transitions. Empirical validation is performed across two independent scales. At the organism scale, crop growth time-series datasets from multiple species exhibit consistent multiplicative dynamics F (E, S) = f (E) {middle dot} S, statistically preferred over additive alternatives in 5 of 6 independently tested systems ({Delta}AIC ranging from +2 to +891; R2 up to 0.98). Independently estimated inertia time constants agree in two plant systems (cucumber:{tau} = 3.7 days, CV=3.3%; maize:{tau} = 36.8 days, CV=17.3%), with the 10-fold ratio consistent with structural complexity differences. At the molecular scale, publicly available perturbation transcriptomics datasets (Perturb-seq) show directional response structures consistent with the proposed flow-inertia decomposition (93% causal direction agreement across three independent datasets; p < 10-25). Across domains, recurrent dynamical motifs are organized into 12 canonical design patterns, derived from a 2 x 2 x 3 orthogonal structure (4 elementary operations x 3 temporal scales), associated with stability classes and bifurcation conditions. These results suggest that the flow-inertia formulation functions as a domain-independent structural modeling principle for dissipative morphogenesis.

Host-microbe interactions constitute dynamical systems that can be represented by mathematical formulations that determine their dynamic nature, and are categorized as deterministic, stochastic, or chaotic. Knowing the type of dynamical interaction is essential for understanding the system under study. Very little experimental work has been done to determine the dynamical characteristics of host-microbe interactions and its study poses significant challenges. The most straightforward experimental outcome involves an observation of time to death upon infection. However, in measuring this outcome, the internal parameters, and the dynamics of each particular host-microbe interaction in a population of interactions are hidden from the experimentalist. To investigate whether a time-to-death (time to event) dataset provides adequate information for searching for chaotic signatures, we first determined our ability to detect chaos in simulated data sets of time-to-event measurements and successfully distinguished the time-to-event distribution of a chaotic process from a comparable stochastic one. To do so, we introduced an inversion measure to test for a chaotic signature in time-to-event distributions. Next, we searched for chaos, in time-to-death of Caenorhabditis elegans and Drosophila melanogaster infected with Pseudomonas aeruginosa or Pseudomonas entomophila, respectively. We found suggestions of chaotic signatures in both systems, but caution that our results are preliminary and highlight the need for more fine-grained and larger data sets in determining dynamical characteristics. If validated, chaos in host-microbe interactions would have important implications for the occurrence and outcome of infectious diseases, the reproducibility of experiments in the field of microbial pathogenesis and the prediction of microbial threats. ImportanceIs microbial pathogenesis a predictable scientific field? At a time when we are dealing with Coronavirus Disease 2019 (COVID-19) there is intense interest in knowing about the epidemic potential of other microbial threats and new emerging infectious diseases. To know whether microbial pathogenesis will ever be a predictable scientific field requires knowing whether a host-microbe interaction follows deterministic, stochastic, or chaotic dynamics. If randomness and chaos are absent from virulence, there is the hope for prediction in the future regarding the outcome of microbe-host interactions. Chaotic systems are inherently unpredictable although it is possible to generate shortterm probabilistic models, as is done in applications of stochastic processes and machine learning to weather forecasting. Information on the dynamics of a system is also essential for understanding the reproducibility of experiments, a topic of great concern in biological sciences. Our study finds preliminary evidence for chaotic dynamics in infectious diseases.

AO_SCPLOWBSTRACTC_SCPLOWMedical practice in the intensive care unit is based on the supposition that physiological systems such as the human glucose-insulin system are reliabile. Reliability of dynamical systems refers to response to perturbation: A dynamical system is reliable if it behaves predictably following a perturbation. Here, we demonstrate that reliability fails for an archetypal physiological model, the Ultradian glucose-insulin model. Reliability failure arises because of the presence of delay. Using the theory of rank one maps from smooth dynamical systems, we precisely explain the nature of the resulting delay-induced uncertainty (DIU). We develop a recipe one may use to diagnose DIU in a general dynamical system. Guided by this recipe, we analyze DIU emergence first in a classical linear shear flow model and then in the Ultradian model. Our results potentially apply to a broad class of physiological systems that involve delay.

Patient-derived tumour spheroids are increasingly used as engineered three-dimensional tissue models for studying tumour growth, nutrient limitation, and therapeutic response. However, extracting quantitative, mechanistically interpretable information from longitudinal imaging data remains challenging. Here, we present a three-dimensional phase-field framework for modelling patient-derived tumour spheroids as continuum, self-organising tissues. The model captures the coupled evolution of viable and necrotic cell fractions through nutrient-limited growth, death, and mechanically and thermodynamically mediated motion, using seven biologically interpretable effective parameters. Key experimental observables emerge naturally from nutrient-growth coupling, without imposing explicit species interfaces or quiescent layers. The framework was quantitatively calibrated against longitudinal imaging data from melanoma spheroids across two cell lines and three initial seeding densities. Across all conditions, simulations reproduced the temporal evolution of all measured observables with low relative error ({approx} 3{sigma} of experimental data), and direct comparison with an established Greenspan-type ODE model demonstrated comparable or improved predictive accuracy. Parameter identifiability analysis revealed weak individual parameter constraints, yet model predictions remained robust, a profile consistent with biological models. We demonstrate that a general PDE-based growth framework can match or outperform a dedicated spheroid model while remaining fully biologically interpretable. Beyond predictive accuracy, the phase-field formulation naturally resolves internal mechanical structure, providing access to quantities that are not directly experimentally observable. These results establish that mechanistically grounded continuum models can be quantitatively calibrated to routine spheroid imaging data, offering a foundation for integrating spatial and mechanical information into the interpretation of organoid-based assays. Graphical Abstract O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=77 SRC="FIGDIR/small/717345v1_ufig1.gif" ALT="Figure 1"> View larger version (21K): org.highwire.dtl.DTLVardef@1cb3b45org.highwire.dtl.DTLVardef@1a053d5org.highwire.dtl.DTLVardef@dffe34org.highwire.dtl.DTLVardef@1aa0b72_HPS_FORMAT_FIGEXP M_FIG C_FIG

Many complex natural systems undergo shifts in dynamics at particular points in time. Examples include phase transitions in gene expression during the cell cycle, introduced species affecting predator-prey interactions, and disease outbreaks responding to intervention measures. Such changepoints partition timeseries into different dynamical regimes characterized by distinct parameter sets, and inference on both the changepoints and regime-specific dynamical parameters is of primary interest. Conventional approaches to analyzing switching dynamical systems first estimate changepoints, and then estimate dynamical parameters assuming the changepoints are fixed and known. Such two-stage approaches are ad-hoc, can introduce biases in the analysis, and do not fully account for uncertainty. Here, we introduce a rigorous, simulation-based inference framework that simultaneously estimates changepoints and model parameters from noisy data while admitting full uncertainty. We use simulation studies of oscillatory predator-prey dynamics and stochastic gene expression to demonstrate that our method yields accurate estimates of changepoints and model parameters together with appropriate uncertainty bounds. We then apply our approach to a real-world case study of COVID-19 intervention effects, and show that our inferred changepoints aligned closely with the actual dates of intervention implementation. Taken together, these results suggest that our framework will have broad utility in diverse scientific domains.

The structure of social networks strongly affects how different phenomena spread in human society, from the transmission of information to the propagation of contagious diseases. It is well-known that heterogeneous connectivity strongly favors spread, but a precise characterization of the redundancy present in social networks and its effect on the robustness of transmission is still lacking. This gap is addressed by the metric backbone, a weight- and connectivity-preserving subgraph that is sufficient to compute all shortest paths of weighted graphs. This subgraph is obtained via algebraically-principled axioms and does not require statistical sampling based on null-models. We show that the metric backbones of nine contact networks obtained from proximity sensors in a variety of social contexts are generally very small, 49% of the original graph for one and ranging from about 6% to 20% for the others. This reflects a surprising amount of redundancy and reveals that shortest paths on these networks are very robust to random attacks and failures. We also show that the metric backbone preserves the full distribution of shortest paths of the original contact networks--which must include the shortest inter- and intra-community distances that define any community structure--and is a primary subgraph for epidemic transmission based on pure diffusion processes. This suggests that the organization of social contact networks is based on large amounts of shortest-path redundancy which shapes epidemic spread in human populations. Thus, the metric backbone is an important subgraph with regard to epidemic spread, the robustness of social networks, and any communication dynamics that depend on complex network shortest paths. Author summaryIt is through social networks that contagious diseases spread in human populations, as best illustrated by the current pandemic and efforts to contain it. Measuring such networks from human contact data typically results in noisy and dense graphs that need to be simplified for effective analysis, without removal of their essential features. Thus, the identification of a primary subgraph that maintains the social interaction structure and likely transmission pathways is of relevance for studying epidemic spreading phenomena as well as devising intervention strategies to hinder spread. Here we propose and study the metric backbone as an optimal subgraph for sparsification of social contact networks in the study of simple spreading dynamics. We demonstrate that it is a unique, algebraically-principled network subgraph that preserves all shortest paths. We also discover that nine contact networks obtained from proximity sensors in a variety of social contexts contain large amounts of redundant interactions that can be removed with very little impact on community structure and epidemic spread. This reveals that epidemic spread on social networks is very robust to random interaction removal. However, extraction of the metric backbone subgraph reveals which interventions--strategic removal of specific social interactions--are likely to result in maximum impediment to epidemic spread.

Understanding how infectious diseases spread through populations requires models that capture real human interactions more realistically. Many classical epidemic models assume that everyone mixes randomly, overlooking the structured and clustered nature of daily contacts within households, workplaces, and schools. These simplifications can limit their predictive capability and the design of effective control strategies. To improve on this limitation, we develop an epidemic modelling framework that integrates explicitly both household and non-household interactions. Building on the semi-random mixing (SeRaMix) concept in the literature, the model captures how people move between different major contact settings (household and work/school) each day, interacting within and across clusters of contacts. We introduce a novel formulation linking contact duration and proximity to infection risk, enabling a more realistic representation of disease-specific transmission factors in an equation-based model. Analytical derivation of the basic reproduction number (R0) demonstrates how epidemic potential distinctly depends on social behaviour, mobility, and biological parameters. Validation against an individual-based model confirms that this equation-based framework reproduces epidemic dynamics within this structured network of interaction. Sensitivity analyses identify the number of contacts at home and work/school, mobility, and inter-household connections as key non-pathogen-dependent drivers of epidemic growth. This framework offers a more realistic approach to analysing the impacts of non-pharmaceutical interventions--such as reduced mobility, hybrid working, and household bubbles--on outbreak trajectories.

Millions of birds travel every year between Europe and Africa detouring ecological barriers and funnelling through migratory corridors where they face variable weather conditions. Little is known regarding the response of migrating birds to mesoscale meteorological processes during flight. Specifically, sea-breeze has a daily cycle that may directly influence the flight of diurnal migrants. We collected radar tracks of soaring migrants using modified weather radar in Latrun, central Israel, in 7 autumns between 2005 and 2016. We investigated how migrating soaring birds adjusted their flight speed and direction under the effects of daily sea-breeze circulation. We analysed the linear and, uniquely, the non-linear effects of wind on bird ground-, air- and sideways speed as function of time along the day using Generalized Additive Mixed Models and calculated first derivatives to identify when birds adjusted their response to the wind over time. Using data collected during a total of 148 days, we characterised the diel dynamics of horizontal wind flow in its two vectorial components relative to soaring migration goal (South), finding a consistent rotational movement of the wind blowing towards the East (morning) and to the South-East (late afternoon), with highest speed of crosswind component around mid-day and increasing tailwinds towards the late afternoon. We found that the airspeed of radar detected birds decreased consistently with increasing tailwind throughout the day, resulting in a rather stable groundspeed of 16-17 m/s. In addition, birds increased their sideways speed when crosswinds were at their maximum to an extent similar to that of the winds sideways component, meaning a full compensation to wind drift, which decreased after the time of crosswind maximum. Using a simple, novel and broadly applicable statistical method, we studied, for the first time, how wind influences bird flight by highlighting non-linear effects over time, providing new insights regarding the behavioural adjustments in the response of soaring birds to wind conditions. Our work enhances our understanding of how migrating birds respond to changing wind conditions during their journeys in order to exploit migratory corridors.

Strictly ultrametric matrices appear in many domains of mathematics and science; nevertheless, they can be large and dense, making them difficult to store and manipulate, unlike large but sparse matrices. In this manuscript, we exploit that strictly ultrametric matrices can be represented as binary trees to sparsify them via an orthonormal base change based on Haar-like wavelets. We show that, with overwhelmingly high probability, only an asymptotically negligible fraction of the off-diagonal entries in random but large strictly ultrametric matrices remain non-zero after the base change; and develop an algorithm to sparsify such matrices directly from their tree representation. We also identify the subclass of matrices diagonalized by the Haar-like wavelets and supply a sufficient condition to approximate the spectrum of strictly ultrametric matrices outside this subclass. Our methods give computational access to the covariance matrix of the microbiologists Tree of Life, which was previously inaccessible due to its size, and motivate introducing a new wavelet-based (beta-diversity) metric to compare microbial environments. Unlike the established (beta-diversity) metrics, the new metric may be used to identify internal nodes (i.e., splits) in the Tree that link microbial composition and environmental factors in a statistically significant manner. MSC codes05C05, 15A18, 42C40, 65F55, 92C70