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.

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.

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.

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 (R_v, 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 R_v 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.

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.

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.

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.

Modelling physical activity: a multi-state life-table approach

OBJECTIVE

To develop a consistent set of epidemiological estimates (incidence, prevalence, remission, mortality) for physical activity in New Zealand; project these estimates in the light of demographic trends; and predict the effectiveness of different health promotion strategies.

METHOD

Multi-state life tables were constructed using physical inactivity prevalence data from the 1996/97 New Zealand Health Survey, and estimates of the relative risk of mortality, and of remission rates, from the literature. Statistics New Zealand population projections were used to forecast these multi-state life tables to 2021. Two physical activity health promotion strategies -uptake (remission enhancement) and maintenance (incidence or relapse reduction)--were simulated by changing the relevant epidemiological variables.

RESULTS

The current fatal burden of physical inactivity in New Zealand is estimated to be 2,600 deaths per year (9% of all deaths). By 2021, the prevalence of physical inactivity will rise 4% as a result of demographic trends. Relapse reduction (enabling active people to remain active) is about 50% more effective than uptake enhancement (enabling inactive people to become active) as a physical activity health promotion strategy, but the two approaches are additive. Maximum realistic changes in relapse prevention and uptake enhancement could reduce the prevalence of physical inactivity by about 30%.

CONCLUSIONS AND IMPLICATIONS

Multi-state life table methods can be used to model health risks (such as behaviours), as well as (chronic) diseases. The model has provided valuable insights for policy makers into the burden of physical inactivity in New Zealand, the impact of demographic trends, and the relative effectiveness of different health promotion strategies.

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.

The pluses and minuses of R0

The concept of the basic reproduction number (R0) occupies a central place in epidemic theory. The value of R0 determines the proportion of the population that becomes infected over the course of a (modelled) epidemic. In many models, (i) an endemic infection can persist only if R0>1, (ii) the value of R0 provides a direct measure of the control effort required to eliminate the infection, and (iii) pathogens evolve to maximize their value of R0. These three statements are not universally true. In this paper, some exceptions to them are discussed, based on the extensions of the SIR model.

Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection

We investigate the merit of deriving an estimate of the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \mathcal{R}_0 $$ \end{document} early in an outbreak of an (emerging) infection from estimates of the incidence and generation interval only. We compare such estimates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \mathcal{R}_0 $$ \end{document} with estimates incorporating additional model assumptions, and determine the circumstances under which the different estimates are consistent. We show that one has to be careful when using observed exponential growth rates to derive an estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \mathcal{R}_0 $$ \end{document} , and we quantify the discrepancies that arise.

A model for the spread and control of pandemic influenza in an isolated geographical region

In the event of an influenza pandemic, the most probable way in which the virus would be introduced to an isolated geographical area is by an infected traveller. We use a mathematical model, structured on the location at which infection occurs and based on published parameters for influenza, to describe an epidemic in a community of one million people. The model is then modified to reflect a variety of control strategies based on social distancing measures, targeted antiviral treatment and antiviral prophylaxis and home quarantine, and the effectiveness of the strategies is compared. The results suggest that the only single strategy that would be successful in preventing an epidemic (with R0=2.0) is targeted antiviral treatment and prophylaxis, and that closing schools combined with either closing work places or home quarantine would only prevent such an epidemic if these strategies were combined with a modest level of antiviral coverage.

The type-reproduction number T in models for infectious disease control

A ubiquitous quantity in epidemic modelling is the basic reproduction number R(0). This became so popular in the 1990s that 'All you need know is R(0)!' became a familiar catch-phrase. The value of R(0) defines, among other things, the control effort needed to eliminate the infection from a homogeneous host population, but can be misleading when applied to a heterogeneous population for the same purpose. We have defined the type-reproduction number T for an infectious disease, and shown that this not only has the required threshold behaviour, but also correctly determines the critical control effort for heterogeneous populations. The two quantities coincide for homogeneous populations. In this paper we further develop the new threshold quantity as an indicator of control effort required in a system where multiple types of individuals are recognised when control targets a specific type.

An integral equation model for the control of a smallpox outbreak

An integral equation model of a smallpox epidemic is proposed. The model structures the incidence of infection among the household, the workplace, the wider community and a health-care facility; and incorporates a finite incubation period and plausible infectivity functions. Linearisation of the model is appropriate for small epidemics, and enables analytic expressions to be derived for the basic reproduction number and the size of the epidemic. The effects of control interventions (vaccination, isolation, quarantine and public education) are explored for a smallpox epidemic following an imported case. It is found that the rapid identification and isolation of cases, the quarantine of affected households and a public education campaign to reduce contact would be capable of bringing an epidemic under control. This could be used in conjunction with the vaccination of healthcare workers and contacts. Our results suggest that prior mass vaccination would be an inefficient method of containing an outbreak.

Modelling strategies for minimizing the impact of an imported exotic infection

The global epidemic of severe acute respiratory syndrome (SARS) in 2003 demonstrated the need to determine control strategies for exotic infections. The prior determination of such strategies, and the use of mathematical models to assist this, is hampered by the obvious lack of data. We propose an integral equation model of Kermack-McKendrick type that may be used to compare strategies based on the isolation of infectious individuals. The model structures the incidence of infection according to the location of an infected individual at exposure, and requires knowledge of the infectivity kernel and the initial rate of exponential increase of cases. The model's use in the design of strategies to minimize the risk of SARS in a previously unexposed community is demonstrated.

Vertebral shape: automatic measurement with dynamically sequenced active appearance models

The shape and appearance of vertebrae on lateral dual x-ray absorptiometry (DXA) scans were statistically modelled. The spine was modelled by a sequence of overlapping triplets of vertebrae, using Active Appearance Models (AAMs). To automate vertebral morphometry, the sequence of trained models was matched to previously unseen scans. The dataset includes a significant number of pathologies. A new dynamic ordering algorithm was assessed for the model fitting sequence, using the best quality of fit achieved by multiple sub-model candidates. The accuracy of the search was improved by dynamically imposing the best quality candidate first. The results confirm the feasibility of substantially automating vertebral morphometry measurements even with fractures or noisy images.

A new method for estimating the effort required to control an infectious disease

We propose a new threshold quantity for the analysis of the epidemiology of infectious diseases. The quantity is similar in concept to the familiar basic reproduction ratio, R0, but it singles out particular host types instead of providing a criterion that is uniform for all host types. Using this methodology we are able to identify the long-term effects of disease-control strategies for particular subgroups of the population, to estimate the level of control necessary when targeting control effort at a subset of host types, and to identify host types that constitute a reservoir of infection. These insights cannot be obtained by using R0 alone.

The metapopulation dynamics of an infectious disease: tuberculosis in possums

An SEI metapopulation model is developed for the spread of an infectious agent by migration. The model portrays two age classes on a number of patches connected by migration routes which are used as host animals mature. A feature of this model is that the basic reproduction ratio may be computed directly, using a scheme that separates topography, demography, and epidemiology. We also provide formulas for individual patch basic reproduction numbers and discuss their connection with the basic reproduction ratio for the system. The model is applied to the problem of spatial spread of bovine tuberculosis in a possum population. The temporal dynamics of infection are investigated for some generic networks of migration links, and the basic reproduction ratio is computed-its value is not greatly different from that for a homogeneous model. Three scenarios are considered for the control of bovine tuberculosis in possums where the spatial aspect is shown to be crucial for the design of disease management operations.

Does pet helminth prophylaxis increase the rate of selection for drug resistance?

There is a growing tendency to control helminths in pets by the prophylactic use of broad-spectrum drug combinations (Allwormers), some of which are of low efficacy. If similar treatment regimes were applied to livestock, parasite strains resistant to chemotherapy would be expected to evolve. The rate of selection for resistance depends significantly on epidemiological parameters and strategic recommendations based on experience with farm animals might not be applicable to pets without critical examinations. Also, the routine use of Allwormers reduces the level of interaction between veterinarians and pet owners and the valuable contribution by veterinarians towards educating pet owners about animal and public health issues in addition to parasite treatment.

The use of multistate life-table models for improving population health

We demonstrate how incidence, prevalence, remission, mortality (IPRM) models may be constructed on population life-tables, how the incidence of a condition may be calculated, and how the consequences of demographic changes and public health interventions may be predicted. We illustrate the methodology by applying it to the epidemiology of diabetes, physical inactivity and obesity in New Zealand.

Nematode parasites of sheep: a survey of epidemiological parameters and their application in a simple model

We review the literature on parameter values relevant to the epidemiology of strongyle nematode infections of domestic sheep. Information is subdivided by parasite genus, country of origin and climate type. While field observations have been made in a large number of countries, the bulk of studies under controlled conditions have been conducted in Australia, New Zealand and the UK. For these countries, experiments and parameters are interpreted in terms of a previously published model of nematode dynamics, and are used to calculate the basic reproduction number. Average values range from less than 6 for Haemonchus contortus in New Zealand and a winter rainfall region of Australia, to more than 16 for Ostertagia circumcincta in New Zealand and the UK. Additional considerations of the effects of climate and the annual replacement of host stock show that for conditions favourable for parasite transmission this is a robust indicator of parasite epidemiology. When climate variation and annual replacement are added to the model, it is shown to reasonably describe the qualitative behaviour of an experimental data set, indicating it to be a useful tool for further investigation of some of the underlying assumptions of sheep-nematode dynamics.

Predicting and preventing measles epidemics in New Zealand: application of a mathematical model

A mathematical model of the dynamics of measles in New Zealand was developed in 1996. The model successfully predicted an epidemic in 1997 and was instrumental in the decision to carry out an intensive MMR (measles-mumps rubella) immunization campaign in that year. While the epidemic began some months earlier than anticipated, it was rapidly brought under control, and its impact on the population was much reduced. In order to prevent the occurrence of further epidemics in New Zealand, an extended version of the model has since been developed and applied to the critical question of the optimal timing of MMR immunization.

A Kermack-McKendrick model applied to an infectious disease in a natural population

The dynamics of a fatal infectious disease in a population regulated by density-dependent constraints are represented as a system of nonlinear integral equations. Survival probabilities and disease transmission coefficients may vary with the time elapsed since infection, and horizontal and vertical modes of transmission are allowed for. Criteria for the existence and stability of steady states are derived, and an example based on the dynamics of tuberculosis is presented. Finally, the relative merits of this approach, and the familiar compartmental models based on differential equations are discussed.

The immunoepidemiology of nematode parasites of farmed animals: a mathematical approach

The population dynamics of farmed animals are controlled by humans, and often involve high host densities, which encourage higher parasite burdens than would be usual in wild animals. As a result, the immunity to reinfection acquired by the host is an important determinant of parasite population dynamics. For example, lambs are highly susceptible to gastrointestinal nematodes as they begin to graze, but develop an immunity that accounts for the observed within-year variation in parasite load and pasture contamination. In the longer term, control measures are compromised by the development of parasite strains resistant to chemotherapy, focusing attention on the development of 'natural' measures, including the selection for resistant hosts and the development of antiparasite vaccines. Mick Roberts here considers the immunoepidemiology of parasites of farmed animals on three levels: the interaction between the parasite and the host's immune system determining the individual's level of protection; the development of acquired immunity determining the within-year parasite population dynamics; and the long-term effects of control measures on the between-year parasite population dynamics.

A comparison of wildlife control and cattle vaccination as methods for the control of bovine tuberculosis

The Australian brushtail possum is the major source of infection for new cases of bovine tuberculosis in cattle in New Zealand. Using hypothetical values for the cost of putative cattle and possum Tb vaccines, the relative efforts required to eradicate Tb in cattle using possum culling, possum vaccination or cattle vaccination are compared. For realistic assumed costs for 1080 poison bait, possum culling is found to be a cost-effective strategy compared to cattle vaccination if the required control area is below 13 ha per cattle herd, while possum vaccination is cost-effective for control areas of less than 3 ha per herd. Examination of other considerations such as the possible roles of possum migration and heterogeneities in possum population density suggest that each control strategy may be superior under different field conditions. Finally, the roles of the possum in New Zealand, and the Eurasian badger in Great Britain and Ireland in the transmission of bovine tuberculosis to cattle are compared.

A simple parasite model with complicated dynamics

During their first year of life sheep acquire parasites through grazing, and simultaneously build up an immunity to infection. At the beginning of each year non-immune lambs are introduced onto contaminated pasture. We represent this process by differential equations describing the within-year dynamics, and defining a difference equation that describes the between-year dynamics. An example with two system parameters is analysed in detail. It is shown that regions exist in parameter space where periodic (between-year) or aperiodic solutions occur. Parasite control schemes could change the system dynamics from a stable equilibrium to complicated long-term fluctuations.

The dynamics of an infectious disease in a population with birth pulses

In most models of population dynamics increases in population due to births are assumed to be time-independent, but many species of wild animal give birth only during a single period of the year. We propose a model for the dynamics of a fatal infectious disease in a wild animal population for which births occur in a single pulse once per time period. Periodic solutions are found and criteria for their stability determined. A simple example applied to tuberculosis in the possum is used to illustrate the effect of the birth pulse on critical population parameters.

A model of bovine tuberculosis control in domesticated cattle herds

A typical strategy for disease control in domesticated animals involves regular field tests and quarantine of infected herds. This prevents disease spread beyond the herd, while slaughter of diseased animals removes the infection from within the herd. A model of bovine tuberculosis (Tb) control in cattle is examined, which includes 'test and slaughter' combined with herd isolation and vaccination. Herd status is represented by an integral equation expressing the duration of herd isolation. The current Tb situation in New Zealand is used as an example, and vaccination strategy discussed. Extrapolation of existing management strategies indicate that a vaccine of efficacy greater than 96% would be required, reaching 95% of target Tb levels within six years. These results suggest that a complementary strategy of vaccination and vector control may be more promising than vaccination alone.

An algorithm for the detection of surface-active alpha helices with the potential to anchor proteins at the membrane interface

MOTIVATION

Surface-active peptides are amphiphilic in nature and have been shown to have the potential to interact at the membrane interface, possibly by lying parallel to the membrane surface. Present methodology for the identification of these helices uses a fixed window size, is based on a two-dimensional sum of hydrophobicity vectors and gives no measure of the statistical significance for any region identified as amphiphilic. Identification of weakly surface-active structures is difficult and here we have attempted to remedy this by introducing an algorithm which considers three-dimensional geometries and variable window size.

RESULTS

A new measure of membrane-interactive potential is proposed, called the depth-weighted inserted hydrophobicity (DWIH), which is based on the sequestration of hydrophobic residues within a hydrophobic compartment, such as that produced by a membrane bilayer. A statistical significance for this measure has been derived using Monte Carlo techniques. The algorithm is applied to a set of proteins which are thought to anchor to the membrane via C-terminal amphiphilic alpha helices. The DWIH measure appears to allow the identification of this category of membrane-interactive helices which lie near the boundary of the hydrophobic moment plot and which have previously been hard to classify.

An SEI model with density-dependent demographics and epidemiology

Two compartmental models for the dynamics of an infectious disease in a wild-animal population are analysed. The models feature density-dependent birth, death, and contact rates, and pseudovertical disease transmission. For the SI model it is shown that up to four steady states may exist, depending on the parameter values, and that one of these states is globally stable. For the SEI model four equivalent steady states exist, but periodic solutions are now possible for some parameter values.

A pocket guide to host-parasite models

Mathematical models have been used to describe the population dynamics of a wide range of host-parasite interactions. Mick Roberts here discusses mathematical models for the dynamics of helminth endoparasites of non-human mammalian hosts, paying particular attention to the density-dependent factors that regulate the parasite populations, and the interaction between parasite and wild or feral animal host populations.

The dynamics of nematode infections of farmed ruminants

In this paper the dynamics and control of nematode parasites of farmed ruminants are discussed via a qualitative analysis of a differential equation model. To achieve this a quantity, 'the basic reproduction quotient' (Q0), whose definition coincides with previous definitions of R0 for macroparasites, but extends to models with periodic time-varying transition rates between parasite stages or management interventions, is introduced. This quantity has the usual threshold property: if Q0 is less than one the parasite population cannot maintain itself in the host population, and in the long term becomes extinct; but if Q0 is greater than one the parasite can invade the host population. An alternative quantity, R(E), that is often easier to calculate is also introduced, and shown to have the same threshold property. The use of these two quantities in analysing models for the dynamics of nematodes in complex situations is then demonstrated, with reference to the dynamics of mixed parasite species in one host; the effects of breeding host animals for resistance to parasitism; and the development of parasite strains that are resistant to chemotherapy. Five examples are discussed using parameters for the dynamics of nematode infections in sheep, and some statements on control policies are derived.

The population dynamics of communities of parasitic helminths

This paper considers the dynamics of a host (animal) species that would grow exponentially in the absence of parasitism, and a community of parasite species that may regulate this growth. The model consists of a single differential equation for the host and one for each of the parasite species. This level of simplicity is achieved by assuming that each parasite species has a negative binomial distribution within the host population, with either zero covariance between the species (exploitation competition), or a specified covariance structure (interference competition). Conditions on the model parameters that determine the abundance of the different species are formulated, as are conditions that determine when a parasite species can invade a community and when a species is likely to be squeezed out. The results show that highly aggregated parasite species are more likely to coexist, but are less able to regulate their host population. A negative correlation between the distributions of the parasite species enhances both their ability to coexist and their ability to regulate the host population. The results of this analysis apply more generally to other systems where communities of exploiter species coexist on discretely distributed hosts, for example, insects on plants.

Threshold quantities for helminth infections

For parasites with a clearly defined life-cycle we give threshold quantities that determine the stability of the parasite-free steady state for autonomous and periodic deterministic systems formulated in terms of mean parasite burdens. We discuss the biological interpretations of the quantities, how to deal with heterogeneity in both parasite and host populations, how to incorporate the effects of periodic discontinuities, and the relation of the threshold quantities to the basic reproduction ratio Ro. Examples from the literature are given. The analysis of the periodic case extends easily to 'micro-parasitic' systems.

A model for the control of Echinococcus multilocularis in France

In some areas of France the prevalence of Echinococcus multilocularis in foxes is as high as 50%, whereas less than one in a thousand voles (principally Microtus arvalis) are infected. In these regions the control of rabies in foxes is achieved by using helicopters to spread bait containing oral vaccine in capsules. A mathematical model has been constructed in an attempt to determine if the addition of praziquantel to bait would be effective in eradicating E. multilocularis, or at least achieve a useful measure of control. It has been shown that the qualitative population dynamics of E. multilocularis are not affected by the detail of its epidemiology in the intermediate host population. The model is, however, sensitive to assumptions about the distribution and longevity of the adult worm in the definitive host. Given these assumptions, a method is provided that determines the feasibility of eradication conditional on the pre-control prevalence in foxes, or predicts the post-control prevalence if eradication is not feasible. If experiments could be designed to provide better information about the biological factors that determine the epidemiology of this parasite in the definitive host, a more reliable assessment of the feasibility of control would be achieved.

Modelling of parasitic populations: cestodes

The philosophy of mathematical modelling as it applies to the epidemiology of cestode populations is reviewed. A model provides, via the "threshold theorem", a criterion for deciding in advance if a control programme can succeed in eradicating the parasite. In order to use this criterion it is necessary to have an estimate of the basic reproduction ratio, R0, which can only be obtained if reliable epidemiological data are available before the control programme is started. A model has been used to describe the population dynamics of Echinococcus granulosus, Taenia hydatigena and Taenia ovis in sheep and dogs in New Zealand. For these parasites, data from a 40-year longitudinal study, as well as short-term field and laboratory studies, were available. A model has also been used to evaluate a proposed control programme directed against Echinococcus multilocularis in foxes and voles in France. Here the type and extent of control intervention is predetermined by the existing rabies control programme. These two examples, which demonstrate the different techniques required to model cestodes in domestic and wild-animal populations, are reviewed, and the use of a model as the basis for a benefit/cost analysis of control options is discussed. These techniques could, in principal, be used to design control programmes for Taenia saginata or Taenia solium in humans.