Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models

Periodic travelling waves have been reported in a number of recent spatio-temporal field studies of populations undergoing multi-year cycles. Mathematical modelling has a major role to play in understanding these results and informing future empirical studies. We review the relevant field data and summarize the statistical methods used to detect periodic waves. We then discuss the mathematical theory of periodic travelling waves in oscillatory reaction-diffusion equations. We describe the notion of a wave family, and various ecologically relevant scenarios in which periodic travelling waves occur. We also discuss wave stability, including recent computational developments. Although we focus on oscillatory reaction-diffusion equations, a brief discussion of other types of model in which periodic travelling waves have been demonstrated is also included. We end by proposing 10 research challenges in this area, five mathematical and five empirical.     

Coloration using higher order optical interference in the wing pattern of the Madagascan sunset moth

Colour patterns of animals' bodies are usually produced by the spatial distribution of pigments with different colours. However, some animals use the spatial variation of colour-producing microstructures. We have studied one distinctive example of such structurally produced colour patterns, the wing of the Madagascan sunset moth, to clarify the physical rules that underlie the colour variation. It is known that the iridescent wing scale of the sunset moth has the alternate air-cuticle multilayer structure that causes optical interference. The microscopic and optical investigations of various parts of the wing have confirmed that the thickness of the cuticle layers within the scale largely varies to produce the colour pattern. However, it varies in very different ways between the dorsal and ventral sides of the hind wing; the thickness gradually varies on the dorsal side from scale to scale, while the abrupt changes are found on the ventral side to form distinctive borders between differently coloured areas. It is also revealed that an unusual coloration mechanism is involved in the green part of the ventral hind wing: the colour is caused by higher order optical interference of the highly non-ideal multilayer structure. The physical mechanism of the colour pattern formation is briefly discussed with the several mathematical models proposed so far.     

Mechanistic movement models reveal ecological drivers of tick-borne pathogen spread

Identifying ecological drivers of tick-borne pathogen spread has great value for tick-borne disease management. However, theoretical investigations into the consequences of host movement behaviour on pathogen spread dynamics in heterogeneous landscapes remain limited because spatially explicit epidemiological models that incorporate more realistic mechanisms governing host movement are rare. We built a mechanistic movement model to investigate how the interplay between multiple ecological drivers affects the risk of tick-borne pathogen spread across heterogeneous landscapes. We used the model to generate simulations of tick dispersal by migratory birds and terrestrial hosts across theoretical landscapes varying in resource aggregation, and we performed a sensitivity analysis to explore the impacts of different parameters on the infected tick spread rate, tick infection prevalence and infected tick density. Our findings highlight the importance of host movement and tick population dynamics in explaining the infected tick spread rate into new regions. Tick infection prevalence and infected tick density were driven by predictors related to the infection process and tick population dynamics, respectively. Our results suggest that control strategies aiming to reduce tick burden on tick reproduction hosts and encounter rate between immature ticks and pathogen amplification hosts will be most effective at reducing tick-borne disease risk.     

Modelling the emergence of polarity patterns for the intercellular transport of auxin in plants

The hormone auxin is actively transported throughout plants via protein machineries including the dedicated transporter known as PIN. The associated transport is ordered with nearby cells driving auxin flux in similar directions. Here, we provide a model of both the auxin transport and of the dynamics of cellular polarization based on flux sensing. Our main findings are: (i) spontaneous intracellular PIN polarization arises if PIN recycling dynamics are sufficiently nonlinear, (ii) there is no need for an auxin concentration gradient and (iii) ordered multi-cellular patterns of PIN polarization are favoured by molecular noise.     

Modelling the spread of infectious salmon anaemia among salmon farms based on seaway distances between farms and genetic relationships between infectious salmon anaemia virus isolates

Infectious salmon anaemia (ISA) is an important infectious disease in Atlantic salmon farming causing recurrent epidemic outbreaks worldwide. The focus of this paper is on tracing the spread of ISA among Norwegian salmon farms. To trace transmission pathways for the ISA virus (ISAV), we use phylogenetic relationships between virus isolates in combination with space–time data on disease occurrences. The rate of ISA infection of salmon farms is modelled stochastically, where seaway distances between farms and genetic distances between ISAV isolates from infected farms play prominent roles. The model was fitted to data covering all cohorts of farmed salmon and the history of all farms with ISA between 2003 and summer 2009. Both seaway and genetic distances were significantly associated with the rate of ISA infection. The fitted model predicts that the risk of infection from a neighbourhood infectious farm decreases with increasing seaway distance between the two farms. Furthermore, for a given infected farm with a given ISAV genotype, the source of infection is significantly more likely to be ISAV of a small genetic distance than of moderate or large genetic distances. Nearly half of the farms with ISA in the investigated period are predicted to have been infected by an infectious farm in their neighbourhood, whereas the remaining half of the infected farms had unknown sources. For many of the neighbourhood infected farms, it was possible to point out one or a few infectious farms as the most probable sources of infection. This makes it possible to map probable infection pathways.     

Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments

Banded vegetation is a characteristic feature of semi-arid environments. It occurs on gentle slopes, with alternating stripes of vegetation and bare ground running parallel to the contours. A number of mathematical models have been proposed to investigate the mechanisms underlying these patterns, and how they might be affected by changes in environmental conditions. One of the most widely used models is due to Rietkerk and co-workers, and is based on a water redistribution hypothesis, with the key feedback being that the rate of rainwater infiltration into the soil is an increasing function of plant biomass. Here, for the first time, we present a detailed study of the existence and stability of pattern solutions of the Rietkerk model on slopes, using the software package wavetrain (www.ma.hw.ac.uk/wavetrain). Specifically, we calculate the region of the rainfall–migration speed parameter plane in which patterns exist, and the sub-region in which these patterns are stable as solutions of the model partial differential equations. We then perform a detailed simulation-based study of the way in which patterns evolve when the rainfall parameter is slowly varied. This reveals complex behaviour, with sudden jumps in pattern wavelength, and hysteresis; we show that these jumps occur when the contours of constant pattern wavelength leave the parameter region giving stable patterns. Finally, we extend our results to the case in which a diffusion term for surface water is added to the model equations. The parameter regions for pattern existence and stability are relatively insensitive to small or moderate levels of surface water diffusion, but larger diffusion coefficients significantly change the subdivision into stable and unstable patterns.     

Modelling cholera epidemics: the role of waterways,human mobility and sanitation

We investigate the role of human mobility as a driver for long-range spreading of cholera infections, which primarily propagate through hydrologically controlled ecological corridors. Our aim is to build a spatially explicit model of a disease epidemic, which is relevant to both social and scientific issues. We present a two-layer network model that accounts for the interplay between epidemiological dynamics, hydrological transport and long-distance dissemination of the pathogen Vibrio cholerae owing to host movement, described here by means of a gravity-model approach. We test our model against epidemiological data recorded during the extensive cholera outbreak occurred in the KwaZulu-Natal province of South Africa during 2000–2001. We show that long-range human movement is fundamental in quantifying otherwise unexplained inter-catchment transport of V. cholerae, thus playing a key role in the formation of regional patterns of cholera epidemics. We also show quantitatively how heterogeneously distributed drinking water supplies and sanitation conditions may affect large-scale cholera transmission, and analyse the effects of different sanitation policies.     

Is there the potential for an epidemic of variant Creutzfeldt–Jakob disease via blood transfusion in the UK?

The discovery of three individuals suspected to have contracted variant Creutzfeldt-Jakob disease (vCJD) through blood transfusions has heightened concerns that a secondary epidemic via human-to-human transmission could occur in the UK. The Department of Health responded immediately to this threat by banning those who had received blood transfusions since 1980 from donating blood. In this paper, we conduct a sensitivity analysis to explore the potential size of a blood-borne vCJD epidemic and investigate the effectiveness of public health interventions. A mathematical model was developed together with an expression for the basic reproduction number (R0). The sensitivity of model predictions to unknown parameters determining the transmission of vCJD via infected blood was assessed under pessimistic modelling assumptions. We found that the size of the epidemic (up until 2080) was bounded above by 900 cases, with self-sustaining epidemics (R0>1) also possible; but the scenarios under which such epidemics could arise were found to be biologically implausible. Under optimistic assumptions, public health interventions reduced the upper bound to 250 and further still when only biologically plausible scenarios were considered. Our results support the belief that scenarios leading to large or self-sustaining epidemics are possible but unlikely, and that public health interventions were effective.     

Modelling the dynamics of viral suppressors of RNA silencing

Virus infection in plants is limited by RNA silencing. In turn, viruses can counter RNA silencing with silencing suppressors. Viral suppressors of RNA silencing have been shown to play a role in symptom development in plants. We here study four different strategies employed by silencing suppressors: small interfering RNA (siRNA) binding, double-strand RNA (dsRNA) binding and degrading or inactivating Argonaute. We study the effect of the suppressors on viral accumulation within the cell as well as its spread on a tissue with mathematical and computational models. We find that suppressors which target Argonaute are very effective in a single cell, but that targeting dsRNA or siRNA is much more effective at the tissue level. Although targeting Argonaute can be beneficial for viral spread, it can also cause hindrance in some cases owing to raised levels of siRNAs that can spread to other cells.     

Modelling the spread of American foulbrood in honeybees

We investigate the spread of American foulbrood (AFB), a disease caused by the bacterium Paenibacillus larvae, that affects bees and can be extremely damaging to beehives. Our dataset comes from an inspection period carried out during an AFB epidemic of honeybee colonies on the island of Jersey during the summer of 2010. The data include the number of hives of honeybees, location and owner of honeybee apiaries across the island. We use a spatial SIR model with an underlying owner network to simulate the epidemic and characterize the epidemic using a Markov chain Monte Carlo (MCMC) scheme to determine model parameters and infection times (including undetected ‘occult’ infections). Likely methods of infection spread can be inferred from the analysis, with both distance- and owner-based transmissions being found to contribute to the spread of AFB. The results of the MCMC are corroborated by simulating the epidemic using a stochastic SIR model, resulting in aggregate levels of infection that are comparable to the data. We use this stochastic SIR model to simulate the impact of different control strategies on controlling the epidemic. It is found that earlier inspections result in smaller epidemics and a higher likelihood of AFB extinction.     

Experimental study of the Ca–Mg–Zn system using diffusion couples and key alloys

Nine diffusion couples and 32 key samples were prepared to map the phase diagram of the Ca–Mg–Zn system. Phase relations and solubility limits were determined for binary and ternary compounds using scanning electron microscopy, electron probe microanalysis and x-ray diffraction (XRD). The crystal structure of the ternary compounds was studied by XRD and electron backscatter diffraction. Four ternary intermetallic (IM) compounds were identified in this system: Ca₃Mg_xZn_15−x (4.6 ⩽ x ⩽ 12 at 335 °C, IM1), Ca_14.5Mg_15.8Zn_69.7 (IM2), Ca₂Mg₅Zn₁₃ (IM3) and Ca_1.5Mg_55.3Zn_43.2 (IM4). Three binary compounds were found to have extended solid solubility into ternary systems: CaZn₁₁, CaZn₁₃ and Mg₂Ca form substitutional solid solutions where Mg substitutes for Zn atoms in the first two compounds, and Zn substitutes for both Ca and Mg atoms in Mg₂Ca. The isothermal section of the Ca–Mg–Zn phase diagram at 335 °C was constructed on the basis of the obtained experimental results. The morphologies of the diffusion couples in the Ca–Mg–Zn phase diagram at 335 °C were studied. Depending on the terminal compositions of the diffusion couples, the two-phase regions in the diffusion zone have either a tooth-like morphology or contain a matrix phase with isolated and/or dendritic precipitates.     

Utility of R0 as a predictor of disease invasion in structured populations

Early theoretical work on disease invasion typically assumed large and well-mixed host populations. Many human and wildlife systems, however, have small groups with limited movement among groups. In these situations, the basic reproductive number, R0, is likely to be a poor predictor of a disease pandemic because it typically does not account for group structure and movement of individuals among groups. We extend recent work by combining the movement of hosts, transmission within groups, recovery from infection and the recruitment of new susceptibles into a stochastic model of disease in a host metapopulation. We focus on how recruitment of susceptibles affects disease invasion and how population structure can affect the frequency of superspreading events (SSEs). We show that the frequency of SSEs may decrease with the reduced movement and the group sizes due to the limited number of susceptible individuals available. Classification tree analysis of the model results illustrates the hierarchical nature of disease invasion in host metapopulations. First, the pathogen must effectively transmit within a group (R0>1), and then the pathogen must persist within a group long enough to allow for movement among the groups. Therefore, the factors affecting disease persistence--such as infectious period, group size and recruitment of new susceptibles--are as important as the local transmission rates in predicting the spread of pathogens across a metapopulation.     

Likelihood-based estimation of continuous-time epidemic models from time-series data: application to measles transmission in London

We present a new statistical approach to analyse epidemic time-series data. A major difficulty for inference is that (i) the latent transmission process is partially observed and (ii) observed quantities are further aggregated temporally. We develop a data augmentation strategy to tackle these problems and introduce a diffusion process that mimics the susceptible-infectious-removed (SIR) epidemic process, but that is more tractable analytically. While methods based on discrete-time models require epidemic and data collection processes to have similar time scales, our approach, based on a continuous-time model, is free of such constraint. Using simulated data, we found that all parameters of the SIR model, including the generation time, were estimated accurately if the observation interval was less than 2.5 times the generation time of the disease. Previous discrete-time TSIR models have been unable to estimate generation times, given that they assume the generation time is equal to the observation interval. However, we were unable to estimate the generation time of measles accurately from historical data. This indicates that simple models assuming homogenous mixing (even with age structure) of the type which are standard in mathematical epidemiology miss key features of epidemics in large populations.     

Modelling the perennial energy crop market: the role of spatial diffusion

Biomass produced from energy crops, such as Miscanthus and short rotation coppice is expected to contribute to renewable energy targets, but the slower than anticipated development of the UK market implies the need for greater understanding of the factors that govern adoption. Here, we apply an agent-based model of the UK perennial energy crop market, including the contingent interaction of supply and demand, to understand the spatial and temporal dynamics of energy crop adoption. Results indicate that perennial energy crop supply will be between six and nine times lower than previously published, because of time lags in adoption arising from a spatial diffusion process. The model simulates time lags of at least 20 years, which is supported empirically by the analogue of oilseed rape adoption in the UK from the 1970s. This implies the need to account for time lags arising from spatial diffusion in evaluating land-use change, climate change (mitigation or adaptation) or the adoption of novel technologies.     

Social network architecture and the maintenance of deleterious cultural traits

How have changes in communications technology affected the way that misinformation spreads through a population and persists? To what extent do differences in the architecture of social networks affect the spread of misinformation, relative to the rates and rules by which individuals transmit or eliminate different pieces of information (cultural traits)? Here, we use analytical models and individual-based simulations to study how a ‘cultural load’ of misinformation can be maintained in a population under a balance between social transmission and selective elimination of cultural traits with low intrinsic value. While considerable research has explored how network architecture affects percolation processes, we find that the relative rates at which individuals transmit or eliminate traits can have much more profound impacts on the cultural load than differences in network architecture. In particular, the cultural load is insensitive to correlations between an individual''s network degree and rate of elimination when these quantities vary among individuals. Taken together, these results suggest that changes in communications technology may have influenced cultural evolution more strongly through changes in the amount of information flow, rather than the details of who is connected to whom.     

The contagious nature of imprisonment: an agent-based model to explain racial disparities in incarceration rates

We build an agent-based model of incarceration based on the susceptible–infected–suspectible (SIS) model of infectious disease propagation. Our central hypothesis is that the observed racial disparities in incarceration rates between Black and White Americans can be explained as the result of differential sentencing between the two demographic groups. We demonstrate that if incarceration can be spread through a social influence network, then even relatively small differences in sentencing can result in large disparities in incarceration rates. Controlling for effects of transmissibility, susceptibility and influence network structure, our model reproduces the observed large disparities in incarceration rates given the differences in sentence lengths for White and Black drug offenders in the USA without extensive parameter tuning. We further establish the suitability of the SIS model as applied to incarceration by demonstrating that the observed structural patterns of recidivism are an emergent property of the model. In fact, our model shows a remarkably close correspondence with California incarceration data. This work advances efforts to combine the theories and methods of epidemiology and criminology.     

Modelling the effects of past and future climate on the risk of bluetongue emergence in Europe

Vector-borne diseases are among those most sensitive to climate because the ecology of vectors and the development rate of pathogens within them are highly dependent on environmental conditions. Bluetongue (BT), a recently emerged arboviral disease of ruminants in Europe, is often cited as an illustration of climate''s impact on disease emergence, although no study has yet tested this association. Here, we develop a framework to quantitatively evaluate the effects of climate on BT''s emergence in Europe by integrating high-resolution climate observations and model simulations within a mechanistic model of BT transmission risk. We demonstrate that a climate-driven model explains, in both space and time, many aspects of BT''s recent emergence and spread, including the 2006 BT outbreak in northwest Europe which occurred in the year of highest projected risk since at least 1960. Furthermore, the model provides mechanistic insight into BT''s emergence, suggesting that the drivers of emergence across Europe differ between the South and the North. Driven by simulated future climate from an ensemble of 11 regional climate models, the model projects increase in the future risk of BT emergence across most of Europe with uncertainty in rate but not in trend. The framework described here is adaptable and applicable to other diseases, where the link between climate and disease transmission risk can be quantified, permitting the evaluation of scale and uncertainty in climate change''s impact on the future of such diseases.     

Detection of an en masse and reversible B- to A-DNA conformational transition in prokaryotes in response to desiccation

The role that DNA conformation plays in the biochemistry of cells has been the subject of intensive research since DNA polymorphism was discovered. B-DNA has long been considered the native form of DNA in cells although alternative conformations of DNA are thought to occur transiently and along short tracts. Here, we report the first direct observation of a fully reversible en masse conformational transition between B- and A-DNA within live bacterial cells using Fourier transform infrared (FTIR) spectroscopy. This biospectroscopic technique allows for non-invasive and reagent-free examination of the holistic biochemistry of samples. For this reason, we have been able to observe the previously unknown conformational transition in all four species of bacteria investigated. Detection of this transition is evidence of a previously unexplored biological significance for A-DNA and highlights the need for new research into the role that A-DNA plays as a cellular defence mechanism and in stabilizing the DNA conformation. Such studies are pivotal in understanding the role of A-DNA in the evolutionary pathway of nucleic acids. Furthermore, this discovery demonstrates the exquisite capabilities of FTIR spectroscopy and opens the door for further investigations of cell biochemistry with this under-used technique.     

Analytical methods for quantifying environmental connectivity for the control and surveillance of infectious disease spread

The sustained transmission and spread of environmentally mediated infectious diseases is governed in part by the dispersal of parasites, disease vectors and intermediate hosts between sites of transmission. Functional geospatial models can be used to quantify and predict the degree to which environmental features facilitate or limit connectivity between target populations, yet typical models are limited in their geographical and analytical approach, providing simplistic, global measures of connectivity and lacking methods to assess the epidemiological implications of fine-scale heterogeneous landscapes. Here, functional spatial models are applied to problems of surveillance and control of the parasitic blood fluke Schistosoma japonicum and its intermediate snail host Oncomelania haupensis in western China. We advance functional connectivity methods by providing an analytical framework to (i) identify nodes of transmission where the degree of connectedness to other villages, and thus the potential for disease spread, is higher than is estimated using Euclidean distance alone and (ii) (re)organize transmission sites into disease surveillance units based on second-order relationships among nodes using non-Euclidean distance measures, termed effective geographical distance (EGD). Functional environmental models are parametrized using ecological information on the target organisms, and pair-wise distributions of inter-node EGD are estimated. A Monte Carlo rank product analysis is presented to identify nearby nodes under alternative distance models. Nodes are then iteratively embedded into EGD space and clustered using a k-means algorithm to group villages into ecologically meaningful surveillance groups. A consensus clustering approach is taken to derive the most stable cluster structure. The results indicate that novel relationships between nodes are revealed when non-Euclidean, ecologically determined distance measures are used to quantify connectivity in heterogeneous landscapes. These connections are not evident when analysing nodes in Euclidean space, and thus surveillance and control activities planned using Euclidean distance measures may be suboptimal. The methods developed here provide a quantitative framework for assessing the effectiveness of ecologically grounded surveillance systems and of control and prevention strategies for environmentally mediated diseases.