Novel epidemiological model of gastrointestinal nematode infection to assess grazing cattle resilience by integrating host growth, parasite, grass and environmental dynamics

Gastrointestinal nematode (GIN) infections are ubiquitous and often cause morbidity and reduced performance in livestock. Emerging anthelmintic resistance and increasing change in climate patterns require evaluation of alternatives to traditional treatment and management practices. Mathematical models of parasite transmission between hosts and the environment have contributed towards the design of appropriate control strategies in ruminants, but have yet to account for relationships between climate, infection pressure, immunity, resources, and growth. Here, we develop a new epidemiological model of GIN transmission in a herd of grazing cattle, including host tolerance (body weight and feed intake), parasite burden and acquisition of immunity, together with weather-dependent development of parasite free-living stages, and the influence of grass availability on parasite transmission. Dynamic host, parasite and environmental factors drive a variable rate of transmission. Using literature sources, the model was parametrised for Ostertagia ostertagi, the prevailing pathogenic GIN in grazing cattle populations in temperate climates. Model outputs were validated on published empirical studies from first season grazing cattle in northern Europe. These results show satisfactory qualitative and quantitative performance of the model; they also indicate the model may approximate the dynamics of grazing systems under co-infection by O. ostertagi and Cooperia oncophora, a second GIN species common in cattle. In addition, model behaviour was explored under illustrative anthelmintic treatment strategies, considering impacts on parasitological and performance variables. The model has potential for extension to explore altered infection dynamics as a result of management and climate change, and to optimise treatment strategies accordingly. As the first known mechanistic model to combine parasitic and free-living stages of GIN with host feed-intake and growth, it is well suited to predict complex system responses under non-stationary conditions. We discuss the implications, limitations and extensions of the model, and its potential to assist in the development of sustainable parasite control strategies.

Microfilarial distribution of Loa loa in the human host: population dynamics and epidemiological implications

Severe adverse events (SAEs) following ivermectin treatment may occur in people harbouring high Loa loa microfilarial (mf) densities. In the context of mass ivermectin distribution for onchocerciasis control in Africa, it is crucial to define precisely the geographical distribution of L. loa in relation to that of Onchocerca volvulus and predict the prevalence of heavy infections. To this end, we analysed the distribution of mf loads in 4183 individuals living in 36 villages of central Cameroon. Mf loads were assessed quantitatively by calibrated blood smears, collected prior to ivermectin distribution. We explored the pattern of L. loa mf aggregation by fitting the (zero-truncated) negative binomial distribution and estimating its overdispersion parameter k by maximum likelihood. The value of k varied around 0.3 independently of mf intensity, host age, village and endemicity level. Based on these results, we developed a semi-empirical model to predict the prevalence of heavy L. loa mf loads in a community given its overall mf prevalence. If validated at the continental scale and linked to predictive spatial models of loiasis distribution, this approach would be particularly useful for optimizing the identification of areas at risk of SAEs and providing estimates of populations at risk in localities where L. loa and O. volvulus are co-endemic.

Co-infection withOnchocerca volvulusandLoa loamicrofilariae in central Cameroon: are these two species interacting?

Ivermectin treatment may induce severe adverse reactions in some individuals heavily infected withLoa loa. This hampers the implementation of mass ivermectin treatment against onchocerciasis in areas whereOnchocerca volvulusandL. loaare co-endemic. In order to identify factors, including co-infections, which may explain the presence of highL. loamicrofilaraemia in some individuals, we analysed data collected in 19 villages of central Cameroon. Two standardized skin snips and 30 μl of blood were obtained from each of 3190 participants and the microfilarial (mf) loads of bothO. volvulusandL. loawere quantified. The data were analysed using multivariate hierarchical models. Individual-level variables were: age, sex, mf presence, and mf load; village-related variables included the endemicity levels for each infection. The two species show a certain degree of ecological separation in the study area. However, for a given individual host, the presence of microfilariae of one species was positively associated with the presence of microfilariae of the other (OR=1·79, 95% CI [1·43–2·24]). Among individuals harbouringLoamicrofilariae, there was a slight positive relationship between theL. loaandO. volvulusmf loads which corresponded to an 11% increase inL. loamf load per 100O. volvulusmicrofilariae. Co-infection withO. volvulusis not sufficient to explain the very highL. loamf loads harboured by some individuals.

A model for radiation-induced bystander effects, with allowance for spatial position and the effects of cell turnover

Bystander effects, whereby cells that are not directly exposed to ionizing radiation exhibit adverse biological effects, have been observed in a number of experimental systems. A novel stochastic model of the radiation-induced bystander effect is developed that takes account of spatial location, cell killing and repopulation. The ionizing radiation dose- and time-responses of this model are explored, and it is shown to exhibit pronounced downward curvature in the high dose-rate region, similar to that observed in many experimental systems, reviewed in the paper. It is also shown to predict the augmentation of effect after fractionated delivery of dose that has been observed in certain experimental systems. It is shown that the generally intractable solution of the full stochastic system can be considerably simplified by assumption of pairwise conditional dependence that varies exponentially over time.

On ?Analytical models for the patchy spread of plant disease?

Epidemiologists are interested in using models that incorporate the effects of clustering in the spatial pattern of disease on epidemic dynamics. Bolker (1999, Bull. Math. Biol. 61, 849-874) has developed an approach to study such models based on a moment closure assumption. We show that the assumption works above a threshold initial level of disease that depends on the spatial dispersal of the pathogen. We test an alternative assumption and show that it does not have this limitation. We examine the relation between lattice and continuous-medium implementations of the approach.

Inferring the dynamics of a spatial epidemic from time-series data

Spatial interactions are key determinants in the dynamics of many epidemiological and ecological systems; therefore it is important to use spatio-temporal models to estimate essential parameters. However, spatially-explicit data sets are rarely available; moreover, fitting spatially-explicit models to such data can be technically demanding and computationally intensive. Thus non-spatial models are often used to estimate parameters from temporal data. We introduce a method for fitting models to temporal data in order to estimate parameters which characterise spatial epidemics. The method uses semi-spatial models and pair approximation to take explicit account of spatial clustering of disease without requiring spatial data. The approach is demonstrated for data from experiments with plant populations invaded by a common soilborne fungus, Rhizoctonia solani. Model inferences concerning the number of sources of disease and primary and secondary infections are tested against independent measures from spatio-temporal data. The applicability of the method to a wide range of host-pathogen systems is discussed.

Effects of dispersal mechanisms on spatio-temporal development of epidemics

The nature of pathogen transport mechanisms strongly determines the spatial pattern of disease and, through this, the dynamics and persistence of epidemics in plant populations. Up to recently, the range of possible mechanisms or interactions assumed by epidemic models has been limited: either independent of the location of individuals (mean-field models) or restricted to local contacts (between nearest neighbours or decaying exponentially with distance). Real dispersal processes are likely to lie between these two extremes, and many are well described by long-tailed contact kernels such as power laws. We investigate the effect of different spatial dispersal mechanisms on the spatio-temporal spread of disease epidemics by simulating a stochastic Susceptible-infective model motivated by previous data analyses. Both long-term stationary behaviour (in the presence of a control or recovery process) and transient behaviour (which varies widely within and between epidemics) are examined. We demonstrate the relationship between epidemic size and disease pattern (characterized by spatial autocorrelation), and its dependence on dispersal and infectivity parameters. Special attention is given to boundary effects, which can decrease disease levels significantly relative to standard, periodic geometries in cases of long-distance dispersal. We propose and test a definition of transient duration which captures the dependence of transients on dispersal mechanisms. We outline an analytical approach that represents the behaviour of the spatially-explicit model, and use it to prove that the epidemic size is predicted exactly by the mean-field model (in the limit of an infinite system) when dispersal is sufficiently long ranged (i.e. when the power-law exponent a</=2).

Exact solution of a generalized model for surface deposition

We consider a model for surface deposition in one dimension, in the presence of both precursor-layer diffusion and desorption. The model is a generalization that includes random sequential adsorption (RSA), accelerated RSA, and growth-and-coalescence models as special cases. Exact solutions are obtained for the model for both its lattice and continuum versions. Expressions are obtained for physically important quantities such as the surface coverage, average island size, mass-adsorption efficiency, and the process efficiency. The connection between a limiting case of the model and epidemic models is discussed.

Analytical methods for predicting the behaviour of population models with general spatial interactions

Many biologists use population models that are spatial, stochastic and individual based. Analytical methods that describe the behaviour of these models approximately are attracting increasing interest as an alternative to expensive computer simulation. The methods can be employed for both prediction and fitting models to data. Recent work has extended existing (mean field) methods with the aim of accounting for the development of spatial correlations. A common feature is the use of closure approximations for truncating the set of evolution equations for summary statistics. We investigate an analytical approach for spatial and stochastic models where individuals interact according to a generic function of their distance; this extends previous methods for lattice models with interactions between close neighbours, such as the pair approximation. Our study also complements work by Bolker and Pacala (BP) [Theor. Pop. Biol. 52 (1997) 179; Am. Naturalist 153 (1999) 575]: it treats individuals as being spatially discrete (defined on a lattice) rather than as a continuous mass distribution; it tests the accuracy of different closure approximations over parameter space, including the additive moment closure (MC) used by BP and the Kirkwood approximation. The study is done in the context of an susceptible-infected-susceptible epidemic model with primary infection and with secondary infection represented by power-law interactions. MC is numerically unstable or inaccurate in parameter regions with low primary infection (or density-independent birth rates). A modified Kirkwood approximation gives stable and generally accurate transient and long-term solutions; we argue it can be applied to lattice and to continuous-space models as a substitute for MC. We derive a generalisation of the basic reproduction ratio, R(0), for spatial models.

Solution of epidemic models with quenched transients

We consider a model for single-season disease epidemics, with a delay (latent period) in the onset of infectivity and a decay ("quenching") in host susceptibility described by time-varying rates of primary and secondary infections. The classical susceptible-exposed-infected (SEI) model of epidemiology is a special case with constant rates. The decaying rates force the epidemics to slow down, and eventually stop in a "quenched transient" state that depends on the full history of the epidemic including its initial state. This equilibrium state is neutrally stable (i.e., has zero-value eigenvalues), and cannot be studied using standard equilibrium analysis. We introduce a method that gives an approximate analytical solution for the quenched state. The method uses an interpolation between two exactly solvable limits and applies to the whole, five-dimensional parameter space of the model. Some applications of the solutions for analysis of epidemics are given.

Studying and approximating spatio–temporal models for epidemic spread and control

A class of simple spatio–temporal stochastic models for the spread and control of plant disease is investigated. We consider a lattice–based susceptible–infected model in which the infection of a host occurs through two distinct processes: a background infective challenge representing primary infection from external sources, and a short–range interaction representing the secondary infection of susceptibles by infectives within the population. Recent data–modelling studies have suggested that the above model may describe the spread of aphid–borne virus diseases in orchards. In addition, we extend the model to represent the effects of different control strategies involving replantation (or recovery). The Contact Process is a particular case of this model. The behaviour of the model has been studied using Cellular–Automata simulations. An alternative approach is to formulate a set of deterministic differential equations that captures the essential dynamics of the stochastic system. Approximate solutions to this set of equations, describing the time evolution over the whole parameter range, have been obtained using the pairwise approximation (PA) as well as the most commonly used mean–field approximation (MF). Comparison with simulation results shows that PA is significantly superior to MF, predicting accurately both transient and long–run, stationary behaviour over relevant parts of the parameter space. The conditions for the validity of the approximations to the present model and extensions thereof are discussed.