Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Buruli ulcer (BU) is a neglected tropical skin disease caused by infection with Mycobacterium ulcerans. The mode of its transmission into the skin has not been fully clarified, but it is believed to involve mechanical transfer, perhaps by insect bites. M. ulcerans secretes a toxin called mycolactone which causes necrosis of the skin and formation of large skin lesions. Understanding the development of these skin lesions over time is crucial to understand the pathogenesis of BU. We formulate a mathematical model to describe the space-time dynamics of M. ulcerans bacteria, mycolactone and skin cell densities as well as fibrin deposition. We investigate the relative contributions to ulcer formation of direct cytotoxicity of mycolactone and ischaemia following the loss of coagulation control. Our results are consistent with existing biological evidence that higher mycolactone concentrations accelerate cell death. Furthermore, the model indicates that the rate of wound expansion is primarily controlled by the diffusion and net birth rate of bacteria, which sets the overall scaling for the speed of wound expansion, along with two dimensionless constants that define the shape of the trajectory of wound edge position against time, namely the ratio of the mycolactone diffusion coefficient to the bacterial diffusion coefficient, and the ratio of the mycolactone degradation rate to the bacterial net birth rate. When mycolactone diffuses significantly faster than bacteria spread or when mycolactone degrades more slowly than bacteria reproduce, the initial development of the wound is faster. In contrast, the long-term wound expansion speed appears to be relatively unchanged across a wide range of the dimensionless parameters.

Using a modified form of the SIR model, we show that, under general conditions, all pandemics exhibit certain scaling rules. Using only daily data for symptomatic, confirmed cases, these scaling rules can be used to estimate: (i) reff, the effective pandemic R-parameter; (ii) ftot, the fraction of exposed individuals that were infected (symptomatic and asymptomatic); (iii) Leff, the effective latency, the average number of days an infected individual is able to infect others in the pool of susceptible individuals; and (iv) , the probability of infection per contact between infected and susceptible individuals. We validate the scaling rules using an example and then apply our method to estimate reff, ftot, Leff and for the first phase of the SARS-Cov-2, Covid-19 pandemic for several countries where there was a well separated first peak in identified infected daily cases after the outbreak of the pandemic in early 2020. Our results are general and can be applied to any pandemic.

Clavicle fractures often exhibit markedly different clinical outcomes: some patients recover acceptable function despite shortening or displacement, whereas others with apparently similar deformity develop persistent pain, functional loss, or poor healing. To explain this distinction, we propose a minimal nonlinear mechanical model for prognostic analysis of clavicle fractures. The model describes the interaction between fracture-related shortening and compensatory shoulder-girdle posture through a reduced equilibrium equation incorporating stiffness, geometric nonlinearity, and shortening-posture coupling. Within this framework, we analyze equilibrium branches, local stability, and the emergence of critical thresholds. We show that post-fracture destabilization can be interpreted as a fold bifurcation, while more complex parameter dependence gives rise to cusp-type structures and multistability. These bifurcation mechanisms provide a mathematical explanation for sudden deterioration after injury or treatment, as well as for strong inter-individual variability. We further introduce an optimization principle based on a utility functional to guide treatment planning. The analysis predicts that the optimal safe correction should lie strictly below the bifurcation threshold, thereby generating a natural safety margin. Although the model is simplified and has not yet been calibrated against patient data, it nevertheless provides a theoretical framework for understanding why fracture prognosis may deteriorate abruptly near critical mechanical conditions and offers a dynamical-systems interpretation of empirical treatment thresholds used in clinical practice.

Epithelial monolayers are the foundation of many mammalian organs and have a significant impact on numerous biological processes, including the secretion of cytokines, absorption of waste products and barrier function. During organ development epithelia are subject to many external forces due to the growth of the surrounding tissues. They exhibit complex responses to such forces, with deformation occurring on short timescales (seconds), and relaxation and remodelling occurring on longer timescales (minutes or hours). Existing mathematical and computational models do not typically account for subcellular remodelling, assuming instead that the response to an applied force or strain acts on a single timescale. In this paper we extend an off-lattice, cell-centre modelling framework to account for subcellular and tissue remodelling by introducing a Dynamic Reference Frame (DRF). The DRF provides a discrete, cell-based analogue of morpho-elasticity: it decouples the evolution of each cell's mechanical reference configuration from its instantaneous deformation, providing a phenomenological description of subcellular remodelling processes - cytoskeletal reorganisation, myosin turnover and adhesion bond remodelling - at the cellular scale. Using this extended multicellular model, we reproduce multiscale responses to external mechanical forces, including creep and stress relaxation experiments, and demonstrate that the history of applied deformation influences tissue recovery upon release. Additionally we identify and quantify where and how these multiscale responses occur. The resulting framework allows for more detailed descriptions and analyses of the development and function of biological tissues.

AO_SCPLOWBSTRACTC_SCPLOWThe COVID-19 pandemic has brought into sharp focus the need to understand respiratory virus transmission mechanisms. In preparation for an anticipated influenza pandemic, a substantial body of literature has developed over the last few decades showing that the short-range aerosol route is an important, though often neglected transmission path. We develop a simple mathematical model for COVID-19 transmission via aerosols, apply it to known outbreaks, and present quantitative guidelines for ventilation and occupancy in the workplace.

Understanding microbial biofilm growth is important to public health, because biofilms are a leading cause of persistent clinical infections. In this paper, we develop a thin-film model for microbial biofilm growth on a solid substratum to which it adheres strongly. We model biofilms as two-phase viscous fluid mixtures of living cells and extracellular fluid. The model tracks the movement, depletion, and uptake of nutrients explicitly, and incorporates cell proliferation via a nutrient-dependent source term. Notably, our thin-film reduction is two-dimensional and includes the vertical dependence of cell volume fraction. Numerical solutions show that this vertical dependence is weak for biologically-feasible parameters, reinforcing results from previous models in which this dependence was neglected. We exploit this weak dependence by writing and solving a simplified one-dimensional model that is computationally more efficient than the full model. We use both the one and two-dimensional models to predict how model parameters affect expansion speed and biofilm thickness. This analysis reveals that expansion speed depends on cell proliferation, nutrient availability, cell-cell adhesion on the upper surface, and slip on the biofilm-substratum interface. Our numerical solutions provide a means to qualitatively distinguish between the extensional flow and lubrication regimes, and quantitative predictions that can be tested in future experiments.

Understanding the health outcomes of military exposures is a critical effort for Veterans, their health care team, and national leaders. Veterans Affairs providers receive reports of military exposure related concerns from 43% of Veterans. Understanding the causal influences of environmental exposures on health is a complex task advancement in exposure science and may require interpreting multiple data sources; particularly when exposure pathways and multi-exposure interactions are ill-defined, as is the case for complex and emerging military service-related exposures. Thus, there is a need to standardize clinically meaningful exposure metrics from different data sources to guide clinicians and researchers with a consistent model for investigating and communicating exposure risk profiles. The Linked Exposures Across Databases (LEAD) framework provides a unifying model for characterizing exposure from different exposure datatypes and databases with a focus on providing clinically relevant exposure metrics. Application of LEAD is demonstrated through comparison of different military exposure data sources: Veteran Military Occupational and Environmental Exposure Assessment Tool (VMOAT), Individual Longitudinal Exposure Record (ILER) database and a military incident report database, the Explosive Ordnance Disposal Information Management System (EODIMS). This cohesive method for evaluating military exposures leverages established information with new sources of data and has the potential to influence how military exposure data is integrated into exposure health care and investigational models.

Understanding and monitoring the major influences on SARS-CoV-2 prevalence is essential to inform policy making and devise appropriate packages of non-pharmaceutical interventions (NPIs). Through evaluating community level influences on the prevalence of SARS-CoV-2 infection and their spatiotemporal variations in England, this study aims to provide some insights into the most important risk parameters. We used spatial clusters developed in Jahanshahi and Jin, 2021 as geographical areas with distinct land use and travel patterns. We also segmented our data by time periods to control for changes in policies or development of the disease over the course of the pandemic. We then used multivariate linear regression to identify influences driving infections within the clusters and to compare the variations of those between the clusters. Our findings demonstrate the key roles that workplace and commuting modes have had on some of the sections of the working population after accounting for several interrelated influences including mobility and vaccination. We found communities of workers in care homes and warehouses and to a lesser extent textile and ready meal industries and those who rely more on public transport for commuting tend to carry a higher risk of infection across all residential area types and time periods.

The osteocyte network in bone is believed to play an important role for how bone tissues sense and respond to mechanical stimulation. Yet, bone adaptation to mechanical loads is often conceptualised as a simple response to mechanical stimuli, such as Wolffs law, which is based on mechanical variables only and takes no account of the cellular basis of mechanosensation. Wolffs law presumes the existence of a reference mechanical stimulus, the mechanical setpoint, above which bone is consolidated, and under which bone is removed. In this paper, we develop a theory of bone tissue sensing and adaptation based on osteocytes to provide new understanding of the role played by osteocyte signals in mechanical adaptation. In this theory, the mechanical setpoint of Frosts mechanostat is explicitly embodied as osteocyte properties involved in mechanotransduction. The mechanical setpoint is allowed to adapt due to the replacement of osteocytes during remodelling, making the setpoint space and time dependent. We propose a mathematical model to implement this new theory of bone adapation and present numerical simulations of this model to explore how mechanobiological response curves (effective Wolffs laws) are modulated by setpoint adaptation during remodelling. By accounting for varying osteocyte populations within bone tissue, we explore bone adaptation under osteocyte disruptions, which is particularly relevant to age-related bone loss. Our model suggests that biological disruptions of remodelling balance cannot always be compensated by mechanical feedback, and that setpoint adaptation during remodelling may have significant observable consequences, such as hysteresis in bone response signatures that resemble lazy zones.

Understanding the effectiveness of strategies such as social distancing is a central question in attempts to control the COVID-19 pandemic. A key unknown in social distancing strategies is the duration of time for which such strategies are needed. Answering this question requires an accurate model of the transmission trajectory. A challenge in fitting such a model is the limited COVID-19 case data available from a given location. To overcome this challenge, we propose fitting a model of SARS-CoV-2 transmission jointly across multiple locations. We apply the model to COVID-19 case data from Spain, UK, Germany, France, Denmark, and New York to estimate the distribution for the time needed for social distancing to end to range from May 2020 to July 2021 (95% credible interval), where the median date is October, 2020. Our method is not specific to COVID-19, and it can also be applied to future pandemics.

This work attempts to derive long-term, time-dependent cortical bone adaptation to mechanical loading. Linear control theory is used to model the adaptation process, with the stimulus defined in terms of dissipation energy density. The newly adapted area is expressed as a function of the stimulus in the form of a differential equation, which is analyzed to obtain closed-form solutions. The study explores different possibilities, such as varying the order of differential equations (including fractional order) and examining different types of responses, e.g., critically damped and overdamped. Such model diversity will help identify the most appropriate formulation that fits experimental data.

The osteocyte network embedded in bone tissues plays a central role in the control of bone adaptation to mechanical loads and micro-damage repair. However, much remains to be understood about the precise mechanisms by which the osteocyte network regulates bone formation and bone resorption based on the propagation of biochemical signals emitted in response to mechanical stimulus. In this work, we propose a simple one-dimensional computational model of bone mechanical adaptation based on the propagation of signalling molecules through a dynamic osteocyte network. The osteocyte network is extended during bone formation, and reduced during bone resorption, which affects the generation and propagation of the signalling molecules to the bone surface. We explore how this osteocyte-based model of bone mechanosensation and mechanoresponse gives rise to effective Wolffs laws, in which overloaded bone is consolidated and underloaded bone is removed. We find that the discrete addition and removal of osteocytes significantly affects signalling molecules propagating to the bone surface and leads to new bone adaptation behaviours compared to earlier models, including partial bone recovery following an unloading and reloading cycle, and the emergence of a minimum threshold of mechanical stress below which all bone is resorbed. While many extensions of this mathematical model are possible, it provides a first illustration of how the osteocyte network could act as a dynamic embedded control network for bone adaptation to mechanical loads.

Mechanical properties of epithelial tissues play essential roles in morphogenesis and physiological function. In this study, we analytically derived the in-plane bulk modulus, shear modulus, and Poissons ratio of a three-dimensional cell vertex model of epithelial monolayers. We showed that the model can robustly reproduce a near-zero in-plane Poissons ratio, a mechanical feature reported in cultured epithelial tissues. Numerical simulations further confirmed that the theoretically predicted Poissons ratio accurately describes the response of the model under finite, biologically relevant strains. In addition, the model exhibits not only morphological bistability between squamous-like and columnar-like states, but also mechanical bistability characterized by distinct elastic responses. Together, these results provide a minimal three-dimensional framework that links cell-scale mechanical interactions and epithelial morphology to tissue-scale elastic properties.

Alzheimers disease (AD) is characterized by the accumulation of amyloid-{beta} (A{beta}), yet the specific link between plaque burden and cognitive decline remains a subject of intense investigation. This paper presents a mathematical model that simulates the coupled dynamics of A{beta} monomers, soluble oligomers, and fibrillar species in the brain tissue. By modifying existing moment equations to include a dedicated conservation equation for A{beta} monomers, the model explores how various microscopic processes, such as primary nucleation, surface-catalyzed secondary nucleation, fibril elongation, and fragmentation, contribute to macroscopic disease progression. Central to this study is the concept of "accumulated neurotoxicity" as a surrogate marker of biological age, defined as the time-integrated concentration of soluble A{beta} oligomers. Unlike plaque burden, accumulated neurotoxicity cannot be reversed, and the harm it causes depends critically on the sequence of events that produced it. Numerical results demonstrate that while plaque burden and neurotoxicity both increase over time, their relationship is non-linear and highly sensitive to the efficiency of protein degradation machinery. Specifically, impaired degradation leads to a rapid advancement of biological age relative to calendar age. The model further identifies oligomer dissociation and fibril fragmentation as potential protective mechanisms that can counterintuitively reduce neurotoxic burden by diverting monomers away from the soluble oligomer pool. These findings provide a quantitative framework for understanding why individuals with similar plaque burdens may experience vastly different cognitive outcomes, underscoring the importance of targeting soluble oligomers early in therapeutic interventions.

In many biological, biomedical and industrial systems, particles are transported via fluids through confined networks, in which clogging can disrupt function. However, we lack a predictive theoretical framework that couples particle transport, suspension rheology and network flow resistance. Here, we develop a model and solution algorithm for particle suspension flow in networks based on vessel-level continuum modelling and particle distribution at nodes connecting vessels. We apply the model to study transport of dense particle suspensions in minimal and physiological biological networks. A key feature of our model is the coupling between particle volume fraction and particle flux: in line with the physics of dense suspensions, each network branch, or vessel, possesses a local carrying capacity for particle transport at an intermediate particle fraction between zero and the maximum packing fraction. If this flux capacity is reached, the vessel becomes flux-limited and particles can accumulate in upstream branches, causing them to enter a high-particle-fraction, high-resistance state that we refer to as 'clogged'. We show that these vessel flux limitations lead to network-level redistribution of particles, which can cause widespread clogging and emergent network-scale heterogeneity. By varying network topology, we find that in some regimes increasing network connectivity does not improve transport: paradoxically, additional pathways can promote clogging and reduce network-level particle flux, analogous to classic results in traffic flow networks. Our results provide a minimal mechanistic framework that links suspension physics, network topology, and transport failure in complex flow networks.

The study of tissue dynamics has been stimulated during the last decades thanks to the use of quantitative descriptions, with the development of several theoretical and computational frameworks, many of them revolving around the notion of reaction-diffusion systems, eventually with additional structure variables beyond time and space. The use of structure variables can accommodate phenotypic traits. In this work, we study a family of competition models, where a given population depends on a resource (e.g. oxygen) and several populations are competing for it. Our quantitative description incorporates phenotypic traits and heterogeneity at the level of cell cycle variations, which influence replication rates via oxygen consumption. This enables us to replicate the fitness of specific subpopulations to environmental conditions (e.g. oxygen shortage or external influences). Using numerical simulations, we show that such models display dynamical pattern formation in the form of coupled travelling wave profiles that expand or retreat at the same wave speed. The full theoretical analysis of such dynamics is quite involved; to circumvent this difficulty, we introduce a quasi-stationary approximation for the resource dynamics. We find that this approximation can reproduce the overall behaviour very accurately, with the additional benefit of allowing theoretical treatment of the reduced model. In this way, we provide estimates on the wave speed which are numerically shown to be robust across a wide range of macroscopic parameters of the full model. The wave speeds are thus found to depend strongly on the proliferation rate of the fittest population, resembling a winner-takes-all dynamics.

The tumor stroma is a tissue composed primarily of extracellular matrix, fibroblasts, immune cells, and vasculature. Its structure and functions, such as nutrient support and waste removal, are altered during malignancy. Tumor cells transform fibroblasts into cancer-associated fibroblasts, which have important immunosuppressive activity on which growth, invasion, and metastasis depend. These activated fibroblasts prevent immune cell infiltration into the tumor nest, thereby promoting cancer progression and inhibiting T-cell-based immunotherapy. To understand these complex interactions, we measure the density of different cell types in the stroma using immunohistochemistry techniques on tumor samples from lung cancer patients. We incorporate these data, and also known information on cell proliferation rates and relevant biochemical interactions, into a minimal dynamical system with few parameters. A spatio-temporal approach to the inhomogeneous environment explains the cell distribution and fate of lung carcinomas. The model reproduces that cancer-associated fibroblasts act as a barrier to tumor growth, but also reduce the efficiency of the immune response. The final outcome depends on the parameter values for each patient and leads to either tumor invasion, persistence, or eradication as a result of the interplay between cancer cell growth, T-cell cytotoxic activity, and fibroblast attraction, activation, and spatial dynamics. Our conclusion is that a wide spectrum of scenarios exists as a result of the competition between the characteristic times of cancer cell growth and the activity rates of the other species. Nevertheless, distinct trajectories and patterns allow quantitative predictions that may help in the selection of new therapies and personalized protocols. We conclude with different options for further modeling. O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=118 SRC="FIGDIR/small/575824v3_ufig1.gif" ALT="Figure 1"> View larger version (44K): org.highwire.dtl.DTLVardef@156ac4forg.highwire.dtl.DTLVardef@14a9a19org.highwire.dtl.DTLVardef@cafebeorg.highwire.dtl.DTLVardef@11aa3bf_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOGraphical AbstractC_FLOATNO C_FIG

Withdrawal StatementThe authors have withdrawn their manuscript owing to errors in the experimental design that affect the integrity of the results. Therefore, the authors do not wish this work to be cited as reference for the project. If you have any questions, please contact the corresponding author

We consider a simple model for the COVID-19 pandemic to analyse the relative effectiveness of several stages of the lockdown in Belgium, as well as of several phases of its relaxation. We also make a future projection of different types of measures relative to different stages of the already experienced lockdown.

Biological tissues grow at rates that depend on the geometry of the supporting tissue substrate. In this study, we present a novel discrete mathematical model for simulating biological tissue growth in a range of geometries. The discrete model is deterministic and tracks the evolution of the tissue interface by representing it as a chain of individual cells that interact mechanically and simultaneously generate new tissue material. To describe the collective behaviour of cells, we derive a continuum limit description of the discrete model leading to a reaction-diffusion partial differential equation governing the evolution of cell density along the evolving interface. In the continuum limit, the mechanical properties of discrete cells are directly linked to their collective diffusivity, and spatial constraints introduce curvature dependence that is not explicitly incorporated in the discrete model. Numerical simulations of both the discrete and continuum models reproduce the smoothing behaviour observed experimentally with minimal discrepancies between the models. The discrete model offers further individual-level details, including cell trajectory data, for any restoring force law and initial geometry. Where applicable, we discuss how the discrete model and its continuum description can be used to interpret existing experimental observations.