Understanding the persistence of measles: reconciling theory, simulation and observation

Ever since the pattern of localized extinction associated with measles was discovered by Bartlett in 1957, many models have been developed in an attempt to reproduce this phenomenon. Recently, the use of constant infectious and incubation periods, rather than the more convenient exponential forms, has been presented as a simple means of obtaining realistic persistence levels. However, this result appears at odds with rigorous mathematical theory; here we reconcile these differences. Using a deterministic approach, we parameterize a variety of models to fit the observed biennial attractor, thus determining the level of seasonality by the choice of model. We can then compare fairly the persistence of the stochastic versions of these models, using the 'best-fit' parameters. Finally, we consider the differences between the observed fade-out pattern and the more theoretically appealing 'first passage time'.

An unlikely partnership: parasites, concomitant immunity and host defence

Concomitant immunity (CI) against macroparasites describes a state of effective anti-larval immunity coupled with persistent adult infection. Experimental studies indicate that immunologically concealed adult worms might promote anti-larval immunity via the release of cross-reactive antigens, thus creating a barrier against continual infection and restricting burden size within the host. CI offers an important potential benefit to established worms by preventing overcrowding within the host. Thus, CI may be interpreted as akin to vaccination; relatively long-lived adult worms 'vaccinate' their host with larval surface antigens and so benefit from reduced conspecific competition. The shared responsibility for host vaccination among adult worms leads to a problem of collective action. Here, we build on earlier analytical findings about the evolutionary forces that shape cooperation among parasites in order to produce a stochastic simulation model of macroparasite social evolution. First, we theoretically investigate a parasite adaptation hypothesis of CI and demonstrate its plausibility under defined conditions, despite the possibility of evolutionary 'cheats'. Then we derive a set of predictions for testing the hypothesis that CI is partly a host-manipulative parasite adaptation. Evidence in support of this model would present an unusual case of adaptive population regulation.

Travelling waves and spatial hierarchies in measles epidemics

Spatio-temporal travelling waves are striking manifestations of predator-prey and host-parasite dynamics. However, few systems are well enough documented both to detect repeated waves and to explain their interaction with spatio-temporal variations in population structure and demography. Here, we demonstrate recurrent epidemic travelling waves in an exhaustive spatio-temporal data set for measles in England and Wales. We use wavelet phase analysis, which allows for dynamical non-stationarity--a complication in interpreting spatio-temporal patterns in these and many other ecological time series. In the pre-vaccination era, conspicuous hierarchical waves of infection moved regionally from large cities to small towns; the introduction of measles vaccination restricted but did not eliminate this hierarchical contagion. A mechanistic stochastic model suggests a dynamical explanation for the waves-spread via infective 'sparks' from large 'core' cities to smaller 'satellite' towns. Thus, the spatial hierarchy of host population structure is a prerequisite for these infection waves.

Quantifying mosquito biting patterns on humans by DNA fingerprinting of bloodmeals

A major debate in infectious disease epidemiology concerns the relative importance of exposure and host factors, such as sex and acquired immunity, in determining observed age patterns of parasitic infection in endemic communities. Nonhomogeneous contact between hosts and vectors is also expected to increase the reproductive rate, and hence transmission, of mosquito-borne infections. Resolution of these questions for human parasitic diseases has been frustrated by the lack of a quantitative tool for quantifying the exposure rate of people in communities. Here, we show that the polymerase chain reaction (PCR) technique for amplifying and fingerprinting human DNA from mosquito bloodmeals can address this problem for mosquito-borne diseases. Analysis of parallel human and mosquito (resting Culex quinquefasciatus) samples from the same households in an urban endemic focus for bancroftian filariasis in South India demonstrates that a 9-locus radioactive short-tandem repeat system is able to identify the source of human DNA within the bloodmeals of nearly 80% of mosquitoes. The results show that a person's exposure rate, and hence the age and sex patterns of exposure to bites in an endemic community, can be successfully quantified by this method. Out of 276 bloodmeal PCR fingerprints, we also found that on average, 27% of the mosquitoes caught resting within individual households had fed on people outside the household. Additionally, 13% of mosquitoes biting within households contained blood from at least 2 people, with the rate of multiple feeding depending on the density of humans in the household. These complex vector feeding behaviors may partly account for the discrepancies in estimates of the infection rates of mosquito-borne diseases calculated parasitologically and entomologically, and they underline the potential of this tool for investigating the transmission dynamics of infection.

Host manipulation by Ligula intestinalis: accident or adaptation?

Numerous studies have demonstrated that parasites with complex life-cycles can cause phenotypic modifications in their hosts that lead to an increased rate of transmission, and suggest that these modifications are the result of parasitic adaptations to manipulate the host. Little attention is paid, however, to separating the possibility of adaptive host manipulation from incidental (if fortuitous) side-effects of infection. In this study we combine statistical and analytical tools to interpret the impact of the macroparasite Ligula intestinalis L. (Cestoda, Pseudophyllidea) on the behaviour of its intermediate fish host (the roach, Rutilus rutilus L.), using field data on a natural system. Two distinct sets of generalized linear models agree that both the presence and the intensity of infection contribute to a modified behavioural response in the host. This was illustrated by a preference for the lake-edge in infected fish during autumn. Furthermore, the effect of parasites upon their host is heterogeneous with respect to parasite size, with larger parasite individuals having a disproportionate impact. A series of game-theoretic models of adaptive host manipulation illustrate a potential rationale for a size-dependent manipulation strategy in parasites. These findings illustrate the potential complexity and functionality of the impact of L. intestinalis upon its fish host, which together reduce the parsimony of the alternative 'incidental effect' hypothesis.

Dynamics of the 2001 UK foot and mouth epidemic: stochastic dispersal in a heterogeneous landscape

Foot-and-mouth is one of the world's most economically important livestock diseases. We developed an individual farm-based stochastic model of the current UK epidemic. The fine grain of the epidemiological data reveals the infection dynamics at an unusually high spatiotemporal resolution. We show that the spatial distribution, size, and species composition of farms all influence the observed pattern and regional variability of outbreaks. The other key dynamical component is long-tailed stochastic dispersal of infection, combining frequent local movements with occasional long jumps. We assess the history and possible duration of the epidemic, the performance of control strategies, and general implications for disease dynamics in space and time.

Noisy clockwork: time series analysis of population fluctuations in animals

Both biotic interactions and abiotic random forcing are crucial influences on population dynamics. This frequently leads to roughly equal importance of deterministic and stochastic forces. The resulting tension between noise and determinism makes ecological dynamics unique, with conceptual and methodological challenges distinctive from those in other dynamical systems. The theory for stochastic, nonlinear ecological dynamics has been developed alongside methods to test models. A range of dynamical components has been considered-density dependence, environmental and demographic stochasticity, and climatic forcing-as well as their often complex interactions. We discuss recent advances in understanding ecological dynamics and testing theory using long-term data and review how dynamical forces interact to generate some central field and laboratory time series.

Age, sex, density, winter weather, and population crashes in Soay sheep

Quantifying the impact of density, extrinsic climatic fluctuations, and demography on population fluctuations is a persistent challenge in ecology. We analyzed the effect of these processes on the irregular pattern of population crashes of Soay sheep on the St. Kilda archipelago, United Kingdom. Because the age and sex structure of the population fluctuates independently of population size, and because animals of different age and sex respond in different ways to density and weather, identical weather conditions can result in different dynamics in populations of equal size. In addition, the strength of density-dependent processes is a function of the distribution of weather events. Incorporating demographic heterogeneities into population models can influence dynamics and their response to climate change.

Drug-resistant parasites and aggregated infection--early-season dynamics

We investigate the effect of spatial aggregation in the infection dynamics of nematode parasites in ruminants. We show that a high degree of spatial aggregation is likely to lead to a dramatically enhanced rate of invasion by drug-resistant strains.

Individual-based perspectives on R(0)

Without doubt the basic reproductive ratio, R(0), is the most widely used quantity in epidemic theory. Standard compartmental models show how R(0)is related to the average age of infection, vaccination thresholds for eradication and equilibrium solutions. However, many of the basic formulae for R(0)break down when we consider transmission of infection to be a stochastic process involving discrete individuals. This paper clarifies why and when these differences arise and predicts when individual-based considerations are likely to be important in modelling infection dynamics.

A simple model for complex dynamical transitions in epidemics

Dramatic changes in patterns of epidemics have been observed throughout this century. For childhood infectious diseases such as measles, the major transitions are between regular cycles and irregular, possibly chaotic epidemics, and from regionally synchronized oscillations to complex, spatially incoherent epidemics. A simple model can explain both kinds of transitions as the consequences of changes in birth and vaccination rates. Measles is a natural ecological system that exhibits different dynamical transitions at different times and places, yet all of these transitions can be predicted as bifurcations of a single nonlinear model.

Impact of immunisation on pertussis transmission in England and Wales

Pertussis immunisation reduces disease frequency, but is not thought to prevent transmission. We show that vaccination has substantially reduced transmission in England and Wales.

Opposite patterns of synchrony in sympatric disease metapopulations

Measles epidemics in UK cities, which were regular and highly synchronous before vaccination, are known to have become irregular and spatially uncorrelated in the vaccine era. Whooping cough shows the reverse pattern, namely a shift from spatial incoherence and irregularity before vaccination to regular, synchronous epidemics afterward. Models show that these patterns can arise from disease-specific responses to dynamical noise. This analysis has implications for vaccination strategies and illustrates the power of comparative dynamical studies of sympatric metapopulations.

The effect of pregnancy on Wuchereria bancrofti microfilarial load in humans

As part of a drug trial against bancroftian filariasis in the East Sepik Province of Papua New Guinea we measured the pre-treatment microfilarial densities of 2219 individuals. Mean levels generally increased with age in both sexes, with a tendency to plateau at the highest ages. However, there was a reduction among women of approximately reproductive age. Allowing for the tendency for aggregation to decrease with age, this reduction was statistically significant. However, a comparison of pregnant women and controls showed no evidence that the reduction is specifically related to pregnancy. Moreover, a simple differential equation model of microfilarial acquisition and loss suggests that age-specific patterns of exposure are also unlikely to be solely responsible. Therefore, we suggest that the observed reduction in microfilarial intensity may result from hormonal changes associated with female reproduction, possibly in combination with other factors.

Anthelmintic resistance revisited: under-dosing, chemoprophylactic strategies, and mating probabilities

Deterministic and stochastic models are used to examine the evolution of anthelmintic resistance among trichostrongylid parasites of domestic ruminants. We find that the relative selection pressures exerted by chemoprophylactic (preventive) control strategies, chemotherapeutic (salvage) control strategies, and regimens involving "under-dosing" are critically dependent on a variety of host and parasite parameters (particularly host immunity and grazing behaviour, parasite fecundity, and the survival of the free-living stages on the pasture). Chemoprophylactic strategies are not necessarily more likely to exert a stronger selection pressure than chemotherapeutic strategies. Similarly, as one reduces dosage levels, there is a range of dose levels where under-dosing promotes resistance and a range of dose levels where under-dosing impedes resistance. The most dangerous dose is either that necessary to kill all the susceptible homozygotes, or that necessary to kill all the susceptible homozygotes and all the heterozygotes. Which one prevails depends upon model parameters. The stochastic formulation indicates that spatial heterogeneity in transmission may be a significant force in promoting the spread of resistant genotypes--at least when infection is at low levels.

Patterns of macroparasite aggregation in wildlife host populations

Frequency distributions from 49 published wildlife host-macroparasite systems were analysed by maximum likelihood for goodness of fit to the negative binomial distribution. In 45 of the 49 (90%) data-sets, the negative binomial distribution provided a statistically satisfactory fit. In the other 4 data-sets the negative binomial distribution still provided a better fit than the Poisson distribution, and only 1 of the data-sets fitted the Poisson distribution. The degree of aggregation was large, with 43 of the 49 data-sets having an estimated k of less than 1. From these 49 data-sets, 22 subsets of host data were available (i.e. host data could be divided by either host sex, age, where or when hosts were sampled). In 11 of these 22 subsets there was significant variation in the degree of aggregation between host subsets of the same host-parasite system. A common k estimate was always larger than that obtained with all the host data considered together. These results indicate that lumping host data can hide important variations in aggregation between hosts and can exaggerate the true degree of aggregation. Wherever possible common k estimates should be used to estimate the degree of aggregation. In addition, significant differences in the degree of aggregation between subgroups of host data, were generally associated with significant differences in both mean parasite burdens and the prevalence of infection.

Population dynamic interference among childhood diseases

Epidemiologists usually study the interaction between a host population and one parasitic infection. However, different parasite species effectively compete, in an ecological sense, for the same finite group of susceptible hosts, so there may be an indirect effect on the population dynamics of one disease due to epidemics of another. In human populations, recovery from any serious infection is normally preceded by a period of convalescence, during which infected individuals stay at home and are effectively shielded from exposure to other infectious diseases. We present a model for the dynamics of two infectious diseases, incorporating a temporary removal of susceptibles. We use this model to explore population-level consequences of a temporary insusceptibility in childhood diseases, the dynamics of which are partly driven by differences in contact rates in and out of school terms. Significant population dynamic interference is predicted and cannot be dismissed in the limited case-study data available for measles and whooping cough in England before the vaccination era.

Modelling variability in lymphatic filariasis: macrofilarial dynamics in the Brugia pahangi--cat model

A striking feature of lymphatic filariasis is the considerable heterogeneity in infection burden observed between hosts, which greatly complicates the analysis of the population dynamics of the disease. Here, we describe the first application of the moment closure equation approach to model the sources and the impact of this heterogeneity for macrofilarial population dynamics. The analysis is based on the closest laboratory equivalent of the life cycle and immunology of infection in humans--cats chronically infected with the filarial nematode Brugia pahangi. Two sets of long-term experiments are analysed: hosts given either single primary infections or given repeat infections. We begin by quantifying changes in the mean and aggregation of adult parasites (inversely measured by the negative binomial parameter, kappa in cohorts of hosts using generalized linear models. We then apply simple stochastic models to interpret observed patterns. The models and empirical data indicate that parasite aggregation tracks the decline in the mean burden with host age in primary infections. Conversely, in repeat infections, aggregation increases as the worm burden declines with experience of infection. The results show that the primary infection variability is consistent with heterogeneities in parasite survival between hosts. By contrast, the models indicate that the reduction in parasite variability with time in repeat infections is most likely due to the 'filtering' effect of a strong, acquired immune response, which gradually acts to remove the initial variability generated by heterogeneities in larval mortality. We discuss this result in terms of the homogenizing effect of host immunity-driven density-dependence on macrofilarial burden in older hosts.

Effect of variability in infection period on the persistence and spatial spread of infectious diseases

The effects of the mean infection (incubation plus infectious) period on the dynamics of infectious diseases are well understood. We examine the dynamics and persistence of epidemics when the distribution of the infection period also is modelled, using the well-documented childhood disease measles as a test case. We pay particular attention to the differences between exponentially distributed and constant periods. The use of constant periods increases the persistence of epidemics by reducing the individual stochasticity. The infection period distribution is also shown to have a significant effect on the spatial spread of disease.

Persistence, chaos and synchrony in ecology and epidemiology

The decline of species in natural habitats concerns ecologists, who view extinction as a danger and conservation of biological diversity as a goal. In contrast, the proliferation of 'undesirable' species is the principal concern of epidemiologists, who view persistence as a problem and eradication as an achievement. While ecologists and epidemiologists have essentially opposite goals, the mathematical structure of the population dynamics that they study is very similar. We briefly review the similarities and differences between these two fields, emphasizing recent work in both areas on the effects of spatial synchrony and dynamical chaos. We hope to stimulate further cross-fertilization of ideas between the disciplines.