How immune dynamics shape multi-season epidemics: a continuous-discrete model in one dimensional antigenic space.

We extend a previously published model for the dynamics of a single strain of an influenza-like infection. The model incorporates a waning acquired immunity to infection and punctuated antigenic drift of the virus, employing a set of coupled integral equations within a season and a discrete map between seasons. The long term behaviour of the model is demonstrated by examples where immunity to infection depends on the time since a host was last infected, and where immunity depends on the number of times that a host has been infected. The first scenario leads to complicated dynamics in some regions of parameter space, and to regions of parameter space with more than one attractor. The second scenario leads to a stable fixed point, corresponding to an identical epidemic each season. We also examine the model with both paradigms in combination, almost always but not exclusively observing a stable fixed point or periodic solution. Adding stochastic perturbations to the between season map fails to destroy the model's qualitative dynamics. Our results suggest that if the level of host immunity depends on the elapsed time since the last infection then the epidemiological dynamics may be unpredictable.

Infection dynamics in ecosystems: on the interaction between red and grey squirrels, pox virus, pine martens and trees.

Ecological and epidemiological processes and interactions influence each other, positively and negatively, directly and indirectly. The invasion potential of pathogens is influenced by the ecosystem context of their host species' populations. This extends to the capacity of (multiple) host species to maintain their (common) pathogen and the way pathogen dynamics are influenced by changes in ecosystem composition. This paper exemplifies these interactions and consequences in a study of red and grey squirrel dynamics in the UK. Differences and changes in background habitat and trophic levels above and below the squirrel species lead to different dynamic behaviour in many subtle ways. The range of outcomes of the different interactions shows that one has to be careful when drawing conclusions about the mechanisms and processes involved in explaining observed phenomena concerning pathogens in their natural environment. The dynamic behaviour also shows that planning interventions, for example for conservation purposes, benefits from understanding the complexity of interactions beyond the particular pathogen and its threatened host species.

Combining mutation and horizontal gene transfer in a within-host model of antibiotic resistance.

Antibiotics are used extensively to control infections in humans and animals, usually by injection or a course of oral tablets. There are several methods by which bacteria can develop antimicrobial resistance (AMR), including mutation during DNA replication and plasmid mediated horizontal gene transfer (HGT). We present a model for the development of AMR within a single host animal. We derive criteria for a resistant mutant strain to replace the existing wild-type bacteria, and for co-existence of the wild-type and mutant. Where resistance develops through HGT via conjugation we derive criteria for the resistant strain to be excluded or co-exist with the wild-type. Our results are presented as bifurcation diagrams with thresholds determined by the relative fitness of the bacteria strains, expressed in terms of reproduction numbers. The results show that it is possible that applying and then relaxing antibiotic control may lead to the bacterial load returning to pre-control levels, but with an altered structure with regard to the variants that comprise the population. Removing antimicrobial selection pressure will not necessarily reduce AMR and, at a population level, other approaches to infection prevention and control are required, particularly when AMR is driven by both mutation and mobile genetic elements.

Characterizing reservoirs of infection and the maintenance of pathogens in ecosystems.

We use a previously published compartmental model of the dynamics of pathogens in ecosystems to define and explore the concepts of maintenance host, maintenance community and reservoir of infection in a full ecological context of interacting host and non-host species. We show that, contrary to their current use in the literature, these concepts can only be characterized relative to the ecosystem in which the host species are embedded, and are not 'life-history traits' of (groups of) species. We give a number of examples to demonstrate that the same (group of) host species can lose or gain maintenance or reservoir capability as a result of a changing ecosystem context, even when these changes primarily affect non-hosts. One therefore has to be careful in designating host species as either maintenance or reservoir in absolute terms.

A simple influenza model with complicated dynamics.

We propose and analyse a model for the dynamics of a single strain of an influenza-like infection. The model incorporates waning acquired immunity to infection and punctuated antigenic drift of the virus, employing a set of differential equations within a season and a discrete map between seasons. We show that the between-season map displays a variety of qualitatively different dynamics: fixed points, periodic solutions, or more complicated behaviour suggestive of chaos. For some example parameters we demonstrate the existence of two distinct basins of attraction, that is the initial conditions determine the long term dynamics. Our results suggest that there is no reason to expect influenza dynamics to be regular, or to expect past epidemics to give a clear indication of future seasons' behaviour.

Modelling leptospirosis in livestock.

New Zealand has one of the highest (per capita) incidences of human leptospirosis in the world. It is the highest occurring occupational disease in New Zealand, often transmitted from livestock such as deer, sheep and cattle to humans. A cyclical model, showing the dynamics of infection of leptospirosis in farmed livestock in New Zealand, is presented. The limit cycle, bifurcation diagram and quasi-R0 value of the system are determined. Leptospire death rate is used as a control parameter. Previously published parameter values are used in a case study to produce figures demonstrating analytical results. The model is used to predict conditions under which the infection will persist in the population.

Quantifying the dilution effect for models in ecological epidemiology.

The dilution effect, where an increase in biodiversity results in a reduction in the prevalence of an infectious disease, has been the subject of speculation and controversy. Conversely, an amplification effect occurs when increased biodiversity is related to an increase in prevalence. We explore the conditions under which these effects arise, using multi species compartmental models that integrate ecological and epidemiological interactions. We introduce three potential metrics for quantifying dilution and amplification, one based on infection prevalence in a focal host species, one based on the size of the infected subpopulation of that species and one based on the basic reproduction number. We introduce our approach in the simplest epidemiological setting with two species, and show that the existence and strength of a dilution effect is influenced strongly by the choices made to describe the system and the metric used to gauge the effect. We show that our method can be generalized to any number of species and to more complicated ecological and epidemiological dynamics. Our method allows a rigorous analysis of ecological systems where dilution effects have been postulated, and contributes to future progress in understanding the phenomenon of dilution in the context of infectious disease dynamics and infection risk.

Cost-benefit analyses of supplementary measles immunisation in the highly immunized population of New Zealand.

As endemic measles is eliminated from countries through increased immunisation, the economic benefits of enhanced immunisation programs may come into question. New Zealand has suffered from outbreaks after measles introductions from abroad and we use it as a model system to understand the benefits of catch up immunisation in highly immunised populations. We provide cost-benefit analyses for measles supplementary immunisation in New Zealand. We model outbreaks based on estimates of the basic reproduction number in the vaccinated population (Rv, the number of secondary infections in a partially immunised population), based on the number of immunologically-naïve people at district and national levels, considering both pre- and post-catch up vaccination scenarios. Our analyses suggest that measles Rv often includes or exceeds one (0.18-3.92) despite high levels of population immunity. We calculate the cost of the first 187 confirmed and probable measles cases in 2014 to be over NZ$1 million (∼US$864,200) due to earnings lost, case management and hospitalization costs. The benefit-cost ratio analyses suggest additional vaccination beyond routine childhood immunisation is economically efficient. Supplemental vaccination-related costs are required to exceed approximately US$66 to US$1877 per person, depending on different scenarios, before supplemental vaccination is economically inefficient. Thus, our analysis suggests additional immunisation beyond childhood programs to target naïve individuals is economically beneficial even when childhood immunisation rates are high.

Global importation and population risk factors for measles in New Zealand: a case study for highly immunized populations.

As endemic measles is eliminated through immunization, countries must determine the risk factors for the importation of measles into highly immunized populations to target control measures. Despite eliminating endemic measles, New Zealand suffers from outbreaks after introductions from abroad, enabling us to use it as a model for measles introduction risk. We used a generalized linear model to analyze risk factors for 1137 measles cases from 2007 to June 2014, provide estimates of national immunity levels, and model measles importation risk. People of European ethnicity made up the majority of measles cases. Age is a positive risk factor, particularly 0-2-year-olds and 5-17-year-old Europeans, along with increased wealth. Pacific islanders were also at greater risk, but due to 0-2-year-old cases. Despite recent high measles, mumps, and rubella vaccine immunization coverage, overall population immunity against measles remains ~90% and is lower in people born between 1982 and 2005. Greatest measles importation risk is during December, and countries predicted to be sources have historical connections and highest travel rates (Australia and UK), followed by Asian countries with high travel rates and higher measles incidences. Our results suggest measles importation due to travel is seeding measles outbreaks, and immunization levels are insufficient to continue to prevent outbreaks because of heterogeneous immunity in the population, leaving particular age groups at risk.

An epidemic model with noisy parameters.

We analyse an SIR model where the epidemiological parameters are subject to small amplitude random fluctuations. We derive a final size equation and extend the result to an SEIR model. We use a small amplitude perturbation to estimate the expected final size of the SIR model and its variance, and compare the result with numerical simulations. We show that although individual realisations may exhibit considerable variation around solutions of the deterministic model, the mean of the final size distribution is in good agreement with the deterministic final size, and its standard deviation is small compared to the mean.

How mathematical epidemiology became a field of biology: a commentary on Anderson and May (1981) 'The population dynamics of microparasites and their invertebrate hosts'.

We discuss the context, content and importance of the paper 'The population dynamics of microparasites and their invertebrate hosts', by R. M. Anderson and R. M. May, published in the Philosophical Transactions of the Royal Society as a stand-alone issue in 1981. We do this from the broader perspective of the study of infectious disease dynamics, rather than the specific perspective of the dynamics of insect pathogens. We argue that their 1981 paper fits seamlessly in the systematic study of infectious disease dynamics that was initiated by the authors in 1978, combining effective use of simple mathematical models, firmly rooted in biology, with observable or empirically measurable ingredients and quantities, and promoting extensive capacity building. This systematic approach, taking ecology and biology rather than applied mathematics as the motivation for advance, proved essential for the maturation of the field, and culminated in their landmark textbook of 1991. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.

How population heterogeneity in susceptibility and infectivity influences epidemic dynamics.

An important concern in public health is what population group should be prioritised for vaccination. To this end, we present an epidemic model with arbitrary initial distributions for population susceptibility, and corresponding infectivity distributions. We consider four scenarios: first, a population with heterogeneous susceptibility with a uniform distribution, but homogeneous infectivity; second, a heterogeneously susceptible population with linear heterogeneous infectivity functions, where the most susceptible are either the most or least infectious; third, a bimodal distribution for susceptibility, with all combinations of infectivity functions; finally, we consider the effects of additional pre-epidemic immunity, ostensibly through vaccination, on the epidemic dynamics. For a seasonal influenza-like infectious disease, we find the smallest final size and overall number of deaths due to the epidemic to occur if the most susceptible are vaccinated, corresponding to targeting children.

Characterizing the next-generation matrix and basic reproduction number in ecological epidemiology.

We address the interaction of ecological processes, such as consumer-resource relationships and competition, and the epidemiology of infectious diseases spreading in ecosystems. Modelling such interactions seems essential to understand the dynamics of infectious agents in communities consisting of interacting host and non-host species. We show how the usual epidemiological next-generation matrix approach to characterize invasion into multi-host communities can be extended to calculate R0, and how this relates to the ecological community matrix. We then present two simple examples to illustrate this approach. The first of these is a model of the rinderpest, wildebeest, grass interaction, where our inferred dynamics qualitatively matches the observed phenomena that occurred after the eradication of rinderpest from the Serengeti ecosystem in the 1980s. The second example is a prey-predator system, where both species are hosts of the same pathogen. It is shown that regions for the parameter values exist where the two host species are only able to coexist when the pathogen is present to mediate the ecological interaction.

Epidemic models with uncertainty in the reproduction number.

One of the first quantities to be estimated at the start of an epidemic is the basic reproduction number, R0. The progress of an epidemic is sensitive to the value of R0, hence we need methods for exploring the consequences of uncertainty in the estimate. We begin with an analysis of the SIR model, with R0 specified by a probability distribution instead of a single value. We derive probability distributions for the prevalence and incidence of infection during the initial exponential phase, the peaks in prevalence and incidence and their timing, and the final size of the epidemic. Then, by expanding the state variables in orthogonal polynomials in uncertainty space, we construct a set of deterministic equations for the distribution of the solution throughout the time-course of the epidemic. The resulting dynamical system need only be solved once to produce a deterministic stochastic solution. The method is illustrated with R0 specified by uniform, beta and normal distributions. We then apply the method to data from the New Zealand epidemic of H1N1 influenza in 2009. We apply the polynomial expansion method to a Kermack-McKendrick model, to simulate a forecasting system that could be used in real time. The results demonstrate the level of uncertainty when making parameter estimates and projections based on a limited amount of data, as would be the case during the initial stages of an epidemic. In solving both problems we demonstrate how the dynamical system is derived automatically via recurrence relationships, then solved numerically.

Semi-automatic determination of detailed vertebral shape from lumbar radiographs using active appearance models.

SUMMARY: The vertebral endplates in lumbar radiographs were located by a semi-automatic annotation method using statistical shape models. INTRODUCTION: Vertebral fractures are common osteoporotic fractures, but current quantitative detection methods (morphometry) lack specificity. We have previously developed more specific quantitative classifiers of vertebral fracture using shape and appearance models. This method has only been applied to DXA vertebral fracture assessment (VFA) images and not to spinal radiographs. The classifiers require a detailed annotation of the outline of the vertebral endplate, so we investigated the application of similar semi-automated annotation methods to lumbar radiographs as the initial step in the generalisation of the statistical classifiers used in VFA images. METHODS: The vertebral body outlines in a training set of 670 lumbar radiographs were manually annotated by expert radiologists. This training set was used to build statistical models of vertebral shape and appearance using triplets of vertebrae. In order to segment vertebrae, the models were refitted using a sequence of active appearance models of vertebral triplets, using a miss-40-out train/test cross-validation experiment. The accuracy was evaluated against the manual annotation and analysed by fracture grade. RESULTS: Good accuracy was obtained for normal vertebrae (0.82 mm) and fracture grades 1 and 2 (1.19 mm), but the localisation accuracy deteriorated for grade 3 fractures to 2.12 mm. CONCLUSION: Vertebral body shape annotation can be substantially automated on lumbar radiographs. However, an occasional manual correction may be required, typically with either severe fractures, or when there is a high degree of projectional tilting or scoliosis. The located detailed shapes may enable the development of more powerful quantitative classifiers of osteoporotic vertebral fracture.

Automatic location of vertebrae on DXA images using random forest regression.

We provide a fully automatic method of segmenting vertebrae in DXA images. This is of clinical relevance to the diagnosis of osteoporosis by vertebral fracture, and to grading fractures in clinical trials. In order to locate the vertebrae we train detectors for the upper and lower vertebral endplates. Each detector uses random forest regressor voting applied to Haar-like input features. The regressors are applied at a grid of points across the image, and each tree votes for an endplate centre position. Modes in the smoothed vote image are endplate candidates, some of which are the neighbouring vertebrae of the one sought. The ambiguity is resolved by applying geometric constraints to the connections between vertebrae, although there can be some ambiguity about where the sequence starts (e.g., is the lowest vertebra L4 or L5, fig 2a). The endplate centres are used to initialise a final phase of active appearance model search for a detailed solution. The method is applied to a dataset of 320 DXA images. Accuracy is comparable to manually initialised AAM segmentation in 91% of images, but multiple grade 3 fractures can cause some edge confusion in severely osteoporotic cases.

Detection of vertebral fractures in DXA VFA images using statistical models of appearance and a semi-automatic segmentation.

SUMMARY: Morphometric methods of vertebral fracture diagnosis lack specificity. We used detailed shape and image texture model parameters to improve the specificity of quantitative fracture identification. Two radiologists visually classified all vertebrae for system training and evaluation. The vertebral endplates were located by a semi-automatic segmentation method to obtain classifier inputs. INTRODUCTION: Vertebral fractures are common osteoporotic fractures, but current quantitative detection methods (morphometry) lack specificity. We used detailed shape and texture information to develop more specific quantitative classifiers of vertebral fracture to improve the objectivity of vertebral fracture diagnosis. These classifiers require a detailed segmentation of the vertebral endplate, and so we investigated the use of semi-automated segmentation methods as part of the diagnosis. METHODS: The vertebrae in a training set of 360 dual energy X-ray absorptiometry images were manually segmented. The shape and image texture of vertebrae were statistically modelled using Appearance Models. The vertebrae were given a gold standard classification by two radiologists. Linear discriminant classifiers to detect fractures were trained on the vertebral appearance model parameters. Classifier performance was evaluated by cross-validation for manual and semi-automatic segmentations, the latter derived using Active Appearance Models (AAM). Results were compared with a morphometric algorithm using the signs test. RESULTS: With manual segmentation, the false positive rates (FPR) at 95% sensitivity were: 5% (appearance) and 18% (morphometry). With semi-automatic segmentations the sensitivities at 5% FPR were: 88% (appearance) and 79% (morphometry). CONCLUSION: Specificity and sensitivity are improved by using an appearance-based classifier compared to standard height ratio morphometry. An overall sensitivity loss of 7% occurs (at 95% specificity) when using a semi-automatic (AAM) segmentation compared to expert annotation, due to segmentation error. However, the classifier sensitivity is still adequate for a computer-assisted diagnosis system for vertebral fracture, especially if used in a triage approach.

The construction of next-generation matrices for compartmental epidemic models.

The basic reproduction number (0) is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of (0) where finitely many different categories of individuals are recognized. We clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. We present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be the NGM with large domain and the NGM with small domain. The three matrices together reflect the range of possibilities encountered in the literature for the characterization of (0). We show how they are connected and how their construction follows from the basic model ingredients, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number (0). Although we present formal recipes based on linear algebra, we encourage the construction of the NGM by way of direct epidemiological reasoning, using the clear interpretation of the elements of the NGM and of the model ingredients. We present a selection of examples as a practical guide to our methods. In the appendix we present an elementary but complete proof that (0) defined as the dominant eigenvalue of the NGM for compartmental systems and the Malthusian parameter r, the real-time exponential growth rate in the early phase of an outbreak, are connected by the properties that (0) > 1 if and only if r > 0, and (0) = 1 if and only if r = 0.

Transient Turing patterns in a neural field model.

We investigate Turing bifurcations in a neural field model with one spatial dimension. For some parameter values the resulting Turing patterns are stable, while for others the patterns appear transiently. We show that this difference is due to the relative position in parameter space of the saddle-node bifurcation of a spatially periodic pattern and the Turing bifurcation point. By varying parameters we are able to observe transient patterns whose duration scales in the same way as type-I intermittency. Similar behavior occurs in two spatial dimensions.