We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Summary Background Since December, 2019, Wuhan, China, has experienced an outbreak of coronavirus disease 2019 (COVID-19), caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Epidemiological and clinical characteristics of patients with COVID-19 have been reported but risk factors for mortality and a detailed clinical course of illness, including viral shedding, have not been well described. Methods In this retrospective, multicentre cohort study, we included all adult inpatients (≥18 years old) with laboratory-confirmed COVID-19 from Jinyintan Hospital and Wuhan Pulmonary Hospital (Wuhan, China) who had been discharged or had died by Jan 31, 2020. Demographic, clinical, treatment, and laboratory data, including serial samples for viral RNA detection, were extracted from electronic medical records and compared between survivors and non-survivors. We used univariable and multivariable logistic regression methods to explore the risk factors associated with in-hospital death. Findings 191 patients (135 from Jinyintan Hospital and 56 from Wuhan Pulmonary Hospital) were included in this study, of whom 137 were discharged and 54 died in hospital. 91 (48%) patients had a comorbidity, with hypertension being the most common (58 [30%] patients), followed by diabetes (36 [19%] patients) and coronary heart disease (15 [8%] patients). Multivariable regression showed increasing odds of in-hospital death associated with older age (odds ratio 1·10, 95% CI 1·03–1·17, per year increase; p=0·0043), higher Sequential Organ Failure Assessment (SOFA) score (5·65, 2·61–12·23; p Interpretation The potential risk factors of older age, high SOFA score, and d-dimer greater than 1 μg/mL could help clinicians to identify patients with poor prognosis at an early stage. Prolonged viral shedding provides the rationale for a strategy of isolation of infected patients and optimal antiviral interventions in the future. Funding Chinese Academy of Medical Sciences Innovation Fund for Medical Sciences; National Science Grant for Distinguished Young Scholars; National Key Research and Development Program of China; The Beijing Science and Technology Project; and Major Projects of National Science and Technology on New Drug Creation and Development.

Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study

I. the origin of species by means of natural selection

BUGS is a software package for Bayesian inference using Gibbs sampling. The software has been instrumental in raising awareness of Bayesian modelling among both academic and commercial communities internationally, and has enjoyed considerable success over its 20-year life span. Despite this, the software has a number of shortcomings and a principal aim of this paper is to provide a balanced critical appraisal, in particular highlighting how various ideas have led to unprecedented flexibility while at the same time producing negative side effects. We also present a historical overview of the BUGS project and some future perspectives. Copyright © 2009 John Wiley & Sons, Ltd.

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The BUGS project: Evolution, critique and future directions

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsif.2008.0014

Random walk models in biology.

Introduction Population growth Administration of drugs Cell division Differential equations with separable variables Equations of homogeneous type Linear differential equations of the first order Numerical solution of first-order equations Symbolic computation in MATLAB Linear Ordinary Differential Equations with Constant Coefficients Introduction First-order linear differential equations Linear equations of the second order Finding the complementary function Determining a particular integral Forced oscillations Differential equations of order n Uniqueness Systems of Linear Ordinary Differential Equations First-order systems of equations with constant coefficients Replacement of one differential equation by a system The general system The fundamental system Matrix notation Initial and boundary value problems Solving the inhomogeneous differential equation Numerical solution of linear boundary value problems Modelling Biological Phenomena Introduction Heartbeat Nerve impulse transmission Chemical reactions Predator-prey models First-Order Systems of Ordinary Differential Equations Existence and uniqueness Epidemics The phase plane and the Jacobian matrix Local stability Stability Limit cycles Forced oscillations Numerical solution of systems of equations Symbolic computation on first-order systems of equations and higher-order equations Numerical solution of nonlinear boundary value problems Appendix: existence theory Mathematics of Heart Physiology The local model The threshold effect The phase plane analysis and the heartbeat model Physiological considerations of the heartbeat cycle A model of the cardiac pacemaker Mathematics of Nerve Impulse Transmission Excitability and repetitive firing Travelling waves Qualitative behavior of travelling waves Piecewise linear model Chemical Reactions Wavefronts for the Belousov-Zhabotinskii reaction Phase plane analysis of Fisher's equation Qualitative behavior in the general case Spiral waves and lambda - omega systems Predator and Prey Catching fish The effect of fishing The Volterra-Lotka model Partial Differential Equations Characteristics for equations of the first order Another view of characteristics Linear partial differential equations of the second order Elliptic partial differential equations Parabolic partial differential equations Hyperbolic partial differential equations The wave equation Typical problems for the hyperbolic equation The Euler-Darboux equation Visualization of solutions Evolutionary Equations The heat equation Separation of variables Simple evolutionary equations Comparison theorems Problems of Diffusion Diffusion through membranes Energy and energy estimates Global behavior of nerve impulse transmissions Global behavior in chemical reactions Turing diffusion driven instability and pattern formation Finite pattern forming domains Bifurcation and Chaos Bifurcation Bifurcation of a limit cycle Discrete bifurcation and period-doubling Chaos Stability of limit cycles The Poincare plane Averaging Numerical Bifurcation Analysis Fixed points and stability Path-following and bifurcation analysis Following stable limit cycles Bifurcation in discrete systems Strange attractors and chaos Stability analysis of partial differential equations Growth of Tumors Introduction Mathematical model I of tumor growth Spherical tumor growth based on model I Stability of tumor growth based on model I Mathematical model II of tumor growth Spherical tumor growth based on model II Stability of tumor growth based on model II Epidemics The Kermack-McKendrick model Vaccination An incubation model Spreading in space Answers to Selected Exercises Index

Differential equations and mathematical biology

Complex networks of interactions are ubiquitous and are particularly important in ecological communities, in which large numbers of species exhibit negative (for example, competition or predation) and positive (for example, mutualism) interactions with one another. Nestedness in mutualistic ecological networks is the tendency for ecological specialists to interact with a subset of species that also interact with more generalist species. Recent mathematical and computational analysis has suggested that such nestedness increases species richness. By examining previous results and applying computational approaches to 59 empirical data sets representing mutualistic plant–pollinator networks, we show that this statement is incorrect. A simpler metric—the number of mutualistic partners a species has—is a much better predictor of individual species survival and hence, community persistence. Nestedness is, at best, a secondary covariate rather than a causative factor for biodiversity in mutualistic communities. Analysis of complex networks should be accompanied by analysis of simpler, underpinning mechanisms that drive multiple higher-order network properties.

Disentangling nestedness from models of ecological complexity

Mast-seeding plants often produce high seed crops the year after a warm spring or summer, but the warm-temperature model has inconsistent predictive ability. Here, we show for 26 long-term data sets from five plant families that the temperature difference between the two previous summers (ΔT) better predicts seed crops. This discovery explains how masting species tailor their flowering patterns to sites across altitudinal temperature gradients; predicts that masting will be unaffected by increasing mean temperatures under climate change; improves prediction of impacts on seed consumers; demonstrates that strongly masting species are hypersensitive to climate; explains the rarity of consecutive high-seed years without invoking resource constraints; and generates hypotheses about physiological mechanisms in plants and insect seed predators. For plants, ΔT has many attributes of an ideal cue. This temperature-difference model clarifies our understanding of mast seeding under environmental change, and could also be applied to other cues, such as rainfall.

Of mast and mean: differential-temperature cue makes mast seeding insensitive to climate change

The hypothesis that the optimal search strategy is a Levy walk (LW) or Levy flight, originally suggested in 1995, has generated an explosion of interest and controversy. Long-standing empirical evidence supporting the LW hypothesis has been overturned, while new models and data are constantly being published. Statistical methods have been criticized and new methods put forward. In parallel with the empirical studies, theoretical search models have been developed. Some theories have been disproved while others remain. Here, we gather together the current state of the art on the role of LWs in optimal foraging theory. We examine the body of theory underpinning the subject. Then we present new results showing that deviations from the idealized one-dimensional search model greatly reduce or remove the advantage of LWs. The search strategy of an LW with exponent μ = 2 is therefore not as robust as is widely thought. We also review the available techniques, and their potential pitfalls, for analysing field data. It is becoming increasingly recognized that there is a wide range of mechanisms that can lead to the apparent observation of power-law patterns. The consequence of this is that the detection of such patterns in field data implies neither that the foragers in question are performing an LW, nor that they have evolved to do so. We conclude that LWs are neither a universal optimal search strategy, nor are they as widespread in nature as was once thought.

/pdf/assessing-levy-walks-as-models-of-animal-foraging-14ybdnvp7u.pdf

Michael J. Plank

Papers

Random walk models in biology.

Differential equations and mathematical biology

Disentangling nestedness from models of ecological complexity

Of mast and mean: differential-temperature cue makes mast seeding insensitive to climate change

Assessing Levy walks as models of animal foraging