Individual-based vs deterministic models for macroparasites: host cycles and extinction

Host-pathogen metapopulation dynamics suggest high elevation refugia for boreal toads

Emerging infectious diseases are an increasingly common threat to wildlife. Chytridiomycosis, caused by the fungal pathogen Batrachochytrium dendrobatidis (Bd), is an emerging infectious disease that has been linked to amphibian declines around the world. Few studies exist that explore amphibian-Bd dynamics at the landscape scale, limiting our ability to identify which factors are associated with variation in population susceptibility and to develop effective in situ disease management. Declines of boreal toads (Anaxyrus boreas boreas) in the southern Rocky Mountains are largely attributed to chytridiomycosis but variation exists in local extinction of boreal toads across this metapopulation. Using a large-scale historic data set, we explored several potential factors influencing disease dynamics in the boreal toad-Bd system: geographic isolation of populations, amphibian community richness, elevational differences, and habitat permanence. We found evidence that boreal toad extinction risk was lowest at high elevations where temperatures may be suboptimal for Bd growth and where small boreal toad populations may be below the threshold needed for efficient pathogen transmission. In addition, boreal toads were more likely to recolonize high elevation sites after local extinction, again suggesting that high elevations may provide refuge from disease for boreal toads. We illustrate a modeling framework that will be useful to natural resource managers striving to make decisions in amphibian-Bd systems. Our data suggest that in the southern Rocky Mountains high elevation sites should be prioritized for conservation initiatives like reintroductions.

When can we infer mechanism from parasite aggregation? A constraint-based approach to disease ecology

Few hosts have many parasites while many hosts have few parasites. This axiom of macroparasite aggregation is so pervasive it is considered a general law in disease ecology, with important implications for the dynamics of host-parasite systems. Because of these dynamical implications, a significant amount of work has explored both the various mechanisms leading to parasite aggregation patterns and how to infer mechanism from these patterns. However, as many disease mechanisms can produce similar aggregation patterns, it is not clear whether aggregation itself provides any additional information about mechanism. Here we apply a "constraint-based" approach developed in macroecology that allows us to explore whether parasite aggregation contains any additional information beyond what is provided by mean parasite load. We tested two constraint-based null models, both of which were constrained on the total number of parasites P and hosts H found in a sample, using data from 842 observed amphibian host-trematode parasite distributions. We found that constraint-based models captured ~85% of the observed variation in host-parasite distributions, suggesting that the constraints P and H contain much of the information about the shape of the host-parasite distribution. However, we also found that extending the constraint-based null models can identify the potential role of known aggregating mechanisms (such as host heterogeneity) and disaggregating mechanisms (such as parasite-induced host mortality) in constraining host-parasite distributions. Thus, by providing robust null models, constraint-based approaches can help guide investigations aimed at detecting biological processes that directly affect parasite aggregation above and beyond those that indirectly affect aggregation through P and H.

Integral Projection Models for host-parasite systems with an application to amphibian chytrid fungus

Host parasite models are typically constructed under either a microparasite or macroparasite paradigm. However, this has long been recognized as a false dichotomy because many infectious disease agents, including most fungal pathogens, have attributes of both microparasites and macroparasites.We illustrate how Integral Projection Models (IPM)s provide a novel, elegant modeling framework to represent both types of pathogens. We build a simple host-parasite IPM that tracks both the number of susceptible and infected hosts and the distribution of parasite burdens in infected hosts.The vital rate functions necessary to build IPMs for disease dynamics share many commonalities with classic micro and macroparasite models and we discuss how these functions can be parameterized to build a host-parasite IPM. We illustrate the utility of this IPM approach by modeling the temperature-dependent epizootic dynamics of amphibian chytrid fungus in Mountain yellow-legged frogs (Rana muscosa).The host-parasite IPM can be applied to other diseases such as facial tumor disease in Tasmanian devils and white-nose syndrome in bats. Moreover, the host-parasite IPM can be easily extended to capture more complex disease dynamics and provides an exciting new frontier in modeling wildlife disease.

Impact of host sex and group composition on parasite dynamics in experimental populations

To better understand the spread of disease in nature, it is fundamentally important to have broadly applicable model systems with readily available species which can be replicated and controlled in the laboratory. Here we used an experimental model system of fish hosts and monogenean parasites to determine whether host sex, group size and group composition (single-sex or mixed-sex) influenced host-parasite dynamics at an individual and group level. Parasite populations reached higher densities and persisted longer in groups of fish compared with isolated hosts and reached higher densities on isolated females than on isolated males. However, individual fish within groups had similar burdens to isolated males regardless of sex, indicating that females may benefit more than males by being in a group. Relative condition was positively associated with high parasite loads for isolated males, but not for isolated females or grouped fish. No difference in parasite dynamics between mixed-sex groups and single-sex groups was detected. Overall, these findings suggest that while host sex influences dynamics on isolated fish, individual fish in groups have similar parasite burdens, regardless of sex. We believe our experimental results contribute to a mechanistic understanding of host-parasite dynamics, although we are cautious about directly extrapolating these results to other systems.

Bayesian calibration of simulation models for supporting management of the elimination of the macroparasitic disease, Lymphatic Filariasis

Background

Mathematical models of parasite transmission can help integrate a large body of information into a consistent framework, which can then be used for gaining mechanistic insights and making predictions. However, uncertainty, spatial variability and complexity, can hamper the use of such models for decision making in parasite management programs.

Methods

We have adapted a Bayesian melding framework for calibrating simulation models to address the need for robust modelling tools that can effectively support management of lymphatic filariasis (LF) elimination in diverse endemic settings. We applied this methodology to LF infection and vector biting data from sites across the major LF endemic regions in order to quantify model parameters, and generate reliable predictions of infection dynamics along with credible intervals for modelled output variables. We used the locally calibrated models to estimate breakpoint values for various indicators of parasite transmission, and simulate timelines to parasite extinction as a function of local variations in infection dynamics and breakpoints, and effects of various currently applied and proposed LF intervention strategies.

Results

We demonstrate that as a result of parameter constraining by local data, breakpoint values for all the major indicators of LF transmission varied significantly between the sites investigated. Intervention simulations using the fitted models showed that as a result of heterogeneity in local transmission and extinction dynamics, timelines to parasite elimination in response to the current Mass Drug Administration (MDA) and various proposed MDA with vector control strategies also varied significantly between the study sites. Including vector control, however, markedly reduced the duration of interventions required to achieve elimination as well as decreased the risk of recrudescence following stopping of MDA.

Conclusions

We have demonstrated how a Bayesian data-model assimilation framework can enhance the use of transmission models for supporting reliable decision making in the management of LF elimination. Extending this framework for delivering predictions in settings either lacking or with only sparse data to inform the modelling process, however, will require development of procedures to estimate and use spatio-temporal variations in model parameters and inputs directly, and forms the next stage of the work reported here.

Electronic supplementary material

The online version of this article (doi:10.1186/s13071-015-1132-7) contains supplementary material, which is available to authorized users.

Allee effect from parasite spill-back

The exchange of native pathogens between wild and domesticated animals can lead to novel disease threats to wildlife. However, the dynamics of wild host-parasite systems exposed to a reservoir of domesticated hosts are not well understood. A simple mathematical model reveals that the spill-back of native parasites from domestic to wild hosts may cause a demographic Allee effect in the wild host population. A second model is tailored to the particulars of pink salmon (Oncorhynchus gorbuscha) and salmon lice (Lepeophtheirus salmonis), for which parasite spill-back is a conservation and fishery concern. In both models, parasite spill-back weakens the coupling of parasite and wild host abundance-particularly at low host abundance-causing parasites per host to increase as a wild host population declines. These findings show that parasites shared across host populations have effects analogous to those of generalist predators and can similarly cause an unstable equilibrium in a focal host population that separates persistence and extirpation. Allee effects in wildlife arising from parasite spill-back are likely to be most pronounced in systems where the magnitude of transmission from domestic to wild host populations is high because of high parasite abundance in domestic hosts, prolonged sympatry of domestic and wild hosts, a high transmission coefficient for parasites, long-lived parasite larvae, and proximity of domesticated populations to wildlife migration corridors.

MAXIMUM POPULATION SIZES IN HOST–PARASITOID MODELS

We are concerned with the problem of characterizing the distribution of the maximum number of individuals alive during a fixed time interval in host–parasitoid models, which is shown to have a matrix exponential form. We present simple conditions on the rates of change of population sizes for the matrix exponential solution to be explicit or algorithmically tractable. A particularly appealing feature of our solution based on splitting methods is that it allows us to obtain global error control.

Modelling the Dynamics of Host-Parasite Interactions: Basic Principles

Mathematical modelling is a valuable tool for the analysis of the infectious diseases spread. Dynamical models may help to represent and summarize available knowledge on transmission and disease evolution, to test assumptions and analyse scenarios, and to predict outcomes of the host-pathogen interactions. This chapter aims at introducing basic concepts and methods of epidemiological modelling, in order to provide a starting point for further developments. After positioning modelling in the process of disease investigation, we first present the main principles of model building and analysis, using simple biological and also mathematical systems. We then provide an overview of the methods that can be employed to describe more complex systems. Last, we illustrate how the modelling approach may help for different practical purposes, including evaluation of control strategies. A brief conclusion discusses the challenge of including genetic and molecular variability in epidemiological modelling.

Analytical models approximating individual processes: a validation method

Upscaling population models from fine to coarse resolutions, in space, time and/or level of description, allows the derivation of fast and tractable models based on a thorough knowledge of individual processes. The validity of such approximations is generally tested only on a limited range of parameter sets. A more general validation test, over a range of parameters, is proposed; this would estimate the error induced by the approximation, using the original model's stochastic variability as a reference. This method is illustrated by three examples taken from the field of epidemics transmitted by vectors that bite in a temporally cyclical pattern, that illustrate the use of the method: to estimate if an approximation over- or under-fits the original model; to invalidate an approximation; to rank possible approximations for their qualities. As a result, the application of the validation method to this field emphasizes the need to account for the vectors' biology in epidemic prediction models and to validate these against finer scale models.

Modelling stochastic transmission processes in helminth infections

The number of helminths within a host can only increase by the host encountering additional infectious stages, so it is important to consider not only whether a host is infected, but also the severity of its infection. Stochastic models consider explicitly the number of parasites within the host and treat infection, death and other demographic events as random processes. I discuss stochastic helminth population models of increasing degrees of complexity, starting with the infection dynamics within a single host and finishing with the full parasite lifecycle among a population of hosts. I demonstrate the mathematical techniques that can help to analyse these models and discuss the insights into parasite population biology that these methods can bring.

A Mathematical model of Schistosoma mansoni in Biomphalaria glabrata with control strategies

We describe and analyze a mathematical model for schistosomiasis in which infected snails are distinguished from susceptible through increased mortality and no reproduction. We based the model on the same derivation as Anderson and May (J. Anim. Ecol. 47:219-247, 1978), Feng and Milner (A New Mathematical Model of Schistosomiasis, Mathematical Models in Medical and Health Science, Nashville, TN, 1997. Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, pp. 117-128, 1998), and May and Anderson (J. Anim. Ecol. 47:249-267, 1978), but used logistic growth both in human and snail hosts. We introduce a parameter r, the effective coverage of medical treatment/prevention to control the infection. We determine a reproductive number for the disease directly related to its persistence and extinction. Finally, we obtain a critical value for r that indicates the minimum treatment effort needed in order to clear out the disease from the population.

Seasonality, cohort-dependence and the development of immunity in a natural host-nematode system

Acquired immunity is known to be a key modulator of the dynamics of many helminth parasites in domestic and human host populations, but its relative importance in natural populations is more controversial. A detailed long-term dataset on the gastrointestinal nematode Trichostrongylus retortaeformis in a wild population of European rabbits (Oryctolagus cuniculus) shows clear evidence of seasonal acquired immunity in the age-structured infection profiles. By fitting a hierarchy of demographic infection-immunity models to the observed age-structured infection patterns, we are able to quantify the importance of different components (seasonality, immunity and host age structure) of the parasite dynamics. We find strong evidence that the hosts' immunocompetence waxes and wanes with the seasons, but also contains a lifelong cohort factor, possibly acting through a maternal effect dependent on the host's month of birth. These observations have important and broad implications for the ecology of parasite infection in seasonal natural herbivore systems.

Ruminating on complexity: macroparasites of wildlife and livestock

Recent advances in ecology have improved our understanding of the role of parasites in the dynamics of wildlife populations. However, conditions that prevail in many wildlife systems, such as host movement, contact with livestock, and heterogeneity in the environment of the parasite outside of the host, have largely been ignored in existing models of macroparasite transmission. We need to refine these models if we are to stand a chance of developing effective parasite control strategies. New quantitative approaches enable us to address key complexities and make better use of scarce data, and these should enhance our efforts to understand and control emerging problems of interspecific parasite transmission.

The dynamical consequences of developmental variability and demographic stochasticity for host-parasitoid interactions

Few age-structured models of species dynamics incorporate variability and uncertainty in population processes. Motivated by laboratory data for an insect and its parasitoid, we investigate whether such assumptions are appropriate when considering the population dynamics of a single species and its interaction with a natural enemy. Specifically, we examine the effects of developmental variability and demographic stochasticity on different types of cyclic dynamics predicted by traditional models. We show that predictions based on the deterministic fixed-development approach are differentially sensitive to variability and noise in key life stages. In particular, we find that the demonstration of half-generation cycles in the single-species model and the multigeneration cycles in the host-parasitoid model are sensitive to the introduction of developmental variability and noise, whereas generation cycles are robust to the intrinsic variability and uncertainty that may be found in nature.