Analysis of model for macroparasitic infection with variable aggregation and clumped infections

Integrating Infection Intensity into Within- and Between-Host Pathogen Dynamics: Implications for Invasion and Virulence Evolution

AbstractInfection intensity can dictate disease outcomes but is typically ignored when modeling infection dynamics of microparasites (e.g., bacteria, virus, and fungi). However, for a number of pathogens of wildlife typically categorized as microparasites, accounting for infection intensity and within-host infection processes is critical for predicting population-level responses to pathogen invasion. Here, we develop a modeling framework we refer to as reduced-dimension host-parasite integral projection models (reduced IPMs) that we use to explore how within-host infection processes affect the dynamics of pathogen invasion and virulence evolution. We find that individual-level heterogeneity in pathogen load-a nearly ubiquitous characteristic of host-parasite interactions that is rarely considered in models of microparasites-generally reduces pathogen invasion probability and dampens virulence-transmission trade-offs in host-parasite systems. The latter effect likely contributes to widely predicted virulence-transmission trade-offs being difficult to observe empirically. Moreover, our analyses show that intensity-dependent host mortality does not always induce a virulence-transmission trade-off, and systems with steeper than linear relationships between pathogen intensity and host mortality rate are significantly more likely to exhibit these trade-offs. Overall, reduced IPMs provide a useful framework to expand our theoretical and data-driven understanding of how within-host processes affect population-level disease dynamics.

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.

Fundamental factors determining the nature of parasite aggregation in hosts

The distribution of parasites in hosts is typically aggregated: a few hosts harbour many parasites, while the remainder of hosts are virtually parasite free. The origin of this almost universal pattern is central to our understanding of host-parasite interactions; it affects many facets of their ecology and evolution. Despite this, the standard statistical framework used to characterize parasite aggregation does not describe the processes generating such a pattern. In this work, we have developed a mathematical framework for the distribution of parasites in hosts, starting from a simple statistical description in terms of two fundamental processes: the exposure of hosts to parasites and the infection success of parasites. This description allows the level of aggregation of parasites in hosts to be related to the random variation in these two processes and to true host heterogeneity. We show that random variation can generate an aggregated distribution and that the common view, that encounters and success are two equivalent filters, applies to the average parasite burden under neutral assumptions but it does not apply to the variance of the parasite burden, and it is not true when heterogeneity between hosts is incorporated in the model. We find that aggregation decreases linearly with the number of encounters, but it depends non-linearly on parasite success. We also find additional terms in the variance of the parasite burden which contribute to the actual level of aggregation in specific biological systems. We have derived the formal expressions of these contributions, and these provide new opportunities to analyse empirical data and tackle the complexity of the origin of aggregation in various host-parasite associations.

Modelling multi-species parasite transmission

Some models are presented for the dynamics of a host population with two parasite species. The models differ in two main aspects: whether they include direct competition among parasites and whether the analysis is based on some approximation and which one. If the analysis is not constrained by a priori assumptions about parasite distributions, it is found that species coexistence is very unlikely without some kind of direct competition among parasites; on the other hand, coexistence generally occurs when inter-specific competition is lower than intraspecific, similarly to standard theory for free-living species. If hosts differ in their predisposition to infection, but not in an identical way towards the two parasite species, then species coexistence becomes feasible even if inter-specific competition is as strong as intraspecific; in this case, coexistence becomes easier as the variance in predisposition increases. These models do not yield universal predictions for patterns of parasite distributions; an analysis of the mechanisms of interaction in each specific system is necessary for that.

Compound processes as models for clumped parasite data

Compound processes are proposed as models for the acquisition of hydatid cysts in sheep, caused by the parasite Echinococcus granulosus. The hypothesis of a clumped infection process against single ingestions is tested and it is shown that the clump-based approach provides a more accurate description of the two data sets investigated. Models with simple and mixed Poisson incidence processes and different clump size distributions are compared. A mixed Poisson incidence process with a zero-truncated negative binomial distribution for the clump sizes is shown to give an adequate description, suggesting that the acquisition of hydatid cysts in the sheep population is heterogeneous, and that the clump sizes are aggregated. The estimates of the parameters derived from the data take plausible values. The average infection rate and the clump size distribution are comparable in both data sets. Goodness-of-fit measures indicate that the model fits the data reasonably well.

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.

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.

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.

A schistosomiasis model with an age-structure in human hosts and its application to treatment strategies

We study a system of partial differential equations which models the disease transmission dynamics of schistosomiasis. The model incorporates both the definitive human hosts and the intermediate snail hosts. The human hosts have an age-dependent infection rate and the snail hosts have an infection-age-dependent cercaria releasing rate. The parasite reproduction number R is computed and is shown to determine the disease dynamics. Stability results are obtained via both analytic and numerical studies. Results of the model are used to discuss age-targeted drug treatment strategies for humans. Sensitivity and uncertainty analysis is conducted to determine the role of various parameters on the variation of R. The effects of various drug treatment programs on disease control are compared in terms of both R and the mean parasite load within the human hosts.

Mathematical models and lymphatic filariasis control: endpoints and optimal interventions

The current global initiative to eliminate lymphatic filariasis is a major renewed commitment to reduce or eliminate the burden of one of the major helminth infections from resource-poor communities of the world. Mathematical models of filariasis transmission can serve as an effective tool for guiding the scientific development and management of successful community-level intervention programmes by acting as analytical frameworks for integrating knowledge regarding parasite transmission dynamics with programmatic factors. However, the power of these tools for supporting control interventions will be realized fully only if researchers address the current uncertainties and gaps in data and knowledge of filarial population dynamics and the effectiveness of currently proposed filariasis intervention options.

Sex ratio variation in gastrointestinal nematodes of Svalbard reindeer; density dependence and implications for estimates of species composition

Estimates of the intensity and abundance of species provide essential data for ecological, evolutionary and epidemiological studies of gastrointestinal nematode communities. These estimates are typically derived from the species composition of adult males when only males have readily scorable species-specific morphological traits. Such estimation assumes that all species in the community have the same adult sex ratio. We evaluated this assumption for the trichostrongyle nematodes Ostertagia gruehneri and Marshallagia marshalli in infracommunities in Svalbard reindeer by identifying to species adult females using a polymerase chain reaction assay. The proportion of males was found to be slightly higher in O. gruehneri than in M. marshalli. Evidence for seasonal variation and density dependence in the adult sex ratio was only found for O. gruehneri. Possible demographic mechanisms for such sex ratio variation are discussed, and stochastic models that generate density-dependent sex ratios proposed. Sex ratio variation caused substantial bias in some male-based estimates of intensity of infection, while substantial and consistent bias in estimates of abundances was only evident in late winter samples. Our results suggest that estimating sex ratios can be particularly important in individual host level studies of nematode species of low abundance.

Stochastic and spatial dynamics of nematode parasites in farmed ruminants

Host-parasite systems provide powerful opportunities for the study of spatial and stochastic effects in ecology; this has been particularly so for directly transmitted microparasites. Here, we construct a fully stochastic model of the population dynamics of a macroparasite system: trichostrongylid gastrointestinal nematode parasites of farmed ruminants. The model subsumes two implicit spatial effects: the host population size (the spatial extent of the interaction between hosts) and spatial heterogeneity ('clumping') in the infection process. This enables us to investigate the roles of several different processes in generating aggregated parasite distributions. The necessity for female worms to find a mate in order to reproduce leads to an Allee effect, which interacts nonlinearly with the stochastic population dynamics and leads to the counter-intuitive result that, when rare, epidemics can be more likely and more severe in small host populations. Clumping in the infection process reduces the strength of this Allee effect, but can hamper the spread of an epidemic by making infection events too rare. Heterogeneity in the hosts' response to infection has to be included in the model to generate aggregation at the level observed empirically.

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

Our understanding of the qualitative dynamics of host-macroparasite systems is mainly based on deterministic models. We study here an individual-based stochastic model that incorporates the same assumptions as the classical deterministic model. Stochastic simulations, using parameter values based on some case studies, preserve many features of the deterministic model, like the average value of the variables and the approximate length of the cycles. An important difference is that, even when deterministic models yield damped oscillations, stochastic simulations yield apparently sustained oscillations. The amplitude of such oscillations may be so large to threaten parasites' persistence.With density-dependence in parasite demographic traits, persistence increases somewhat. Allowing instead for infections from an external parasite reservoir, we found that host extinction may easily occur. However, the extinction probability is almost independent of the level of external infection over a wide intermediate parameter region.

Aggregation, stability, and oscillations in different models for host-macroparasite interactions

Aggregation is generally recognized as an important factor in the dynamics of host-macroparasite interactions and it has been found relevant in stabilizing the dynamics toward an equilibrium coexistence. In this paper we review the models of Anderson and May (1978, J. Anim. Ecol. 47, 219-247, 249-267) and compare them with some more recently developed models, which incorporate explicit mechanisms (multiple infections or host heterogeneity) for generating aggregation and different degrees of mathematical accuracy. We found that the stabilization yielded by aggregation depends strongly on the mechanism producing the aggregation: multiple infections are much less stabilizing than when aggregation is assumed to be fixed from the outside, while the opposite holds for host heterogeneity. We also give analytical estimates of the period of oscillations occurring when the equilibrium is unstable. Finally, we explore in these models the role of aggregation in host regulation and in determining a threshold value for parasite establishment.

Patterns of parasite aggregation in the wild European rabbit (Oryctolagus cuniculus)

Understanding the factors controlling the distribution of parasites within their host population is fundamental to the wider understanding of parasite epidemiology and ecology. To explore changes in parasite aggregation, Taylor's power law was used to examine the distributions of five gut helminths of the wild rabbit. Aggregation was found to be a dynamic process that varied with year, season, host sex, age class, and myxomatosis. Yearly and seasonal changes are thought, in the main, to be the result of variations in weather conditions acting upon infectious stages (or intermediate hosts). Evidence in support of this was the comparatively low degree of fluctuation in the aggregation of the pinworm, Passalurus ambiguus, as the infectious stage of this parasite is likely to be less susceptible to environmental variation. Host age had a marked effect on the level of aggregation of all parasites, but this effect varied between parasite species. P. ambiguus, Trichostrongylus retortaeformis and Cittotaenia denticulata aggregation were lower in adult than juvenile rabbits whilst Graphidium strigosum and Mosgovoyia pectinata aggregation tended to increase with age. Host immunity is thought to be responsible for these differences. Differences in aggregation for different parasites were also seen when the rabbit population was split into males and females. Myxomatosis had a marked effect on helminth distribution with substantially less aggregation in rabbits showing clinical signs of the disease.

A study of the role of the transmission mechanism in macroparasite aggregation

The dynamics of host-macroparasite infections pose considerable challenges for stochastic modelling because of the need to take into account a large number of relevant factors and many nonlinear interactions between them. This paper focuses attention on the infection transmission process and the effects of specific modelling assumptions about the mechanisms involved. Some dramatically simplified linear models are considered; they are based on multidimensional linear birth and death processes, and are designed to illuminate qualitative effects of interest. Both single and compound infections are allowed. It is shown that such simple models can generate and increase dispersion of parasite counts, even among homogeneous hosts.

A study of the role of the transmission mechanism in macroparasite aggregation

The dynamics of host-macroparasite infections pose considerable challenges for stochastic modelling because of the need to take into account a large number of relevant factors and many nonlinear interactions between them. This paper focuses attention on the infection transmission process and the effects of specific modelling assumptions about the mechanisms involved. Some dramatically simplified linear models are considered; they are based on multidimensional linear birth and death processes, and are designed to illuminate qualitative effects of interest. Both single and compound infections are allowed. It is shown that such simple models can generate and increase dispersion of parasite counts, even among homogeneous hosts.

Coexistence of macroparasites without direct interactions

Coexistence of macroparasites is studied by extending the infinite-dimensional model considered by Anderson and May (1978, J. Anim. Ecol. 47, 219-247, 249-267) to several species of parasites that are assumed to interact only by causing the death of a common host. An exact invadability condition is found for this model. By studying when mutual invasibility is possible, the region where two parasite species can coexist is found. The result is that, if there is a trade-off between virulence and transmissibility, then coexistence of two species of parasites is possible, but only when the parameters of the model fall into a very narrow parameter region. If, on the other hand, one parasite is more virulent and less transmissible, then it will be competitively excluded. This latter result, though expected in terms of competition theory, is in contrast with what found in the approximate models so far used for studying interacting macroparasites. The effect of parasite aggregation on coexistence is studied by considering two modifications of the basic model (clumped infections and host population heterogeneity in predisposition to infections) that allow for higher aggregation. It appears that the width of the coexistence region is insensitive to these modifications.

Parasite-mediated and direct competition in a two-host shared macroparasite system

This paper investigates the local dynamical behaviour of a deterministic model describing two host species experiencing three forms of competition: direct competition, apparent competition mediated by macroparasites, and intra-specific (density-dependent) competition. The problem of algebraic intractability is sidestepped by adopting a geometric approach, in which an array of maps is constructed in parameter space, each structured by bifurcation surfaces which mark qualitative changes in system behaviour. The maps provide both a succinct and a comprehensive overview of the stability and feasibility structure of the system equilibria, from which can be deduced the possible modes of local dynamical behaviour. A detailed examination of these maps shows that (i) the system is highly sensitive to the effect of infection on fecundity with synchronous sustained cycles readily generated by Hopf bifurcations; (ii) for a broad range of parameter values, pertinent to actual biological systems, apparent competition mediated by macroparasites is sufficient, on its own, to explain host exclusion; (iii) direct competition reinforces parasite-mediated competition to expand the host exclusion region; and (iv) the condition for host exclusion can be expressed simply in a form which holds for both micro- and macroparasite models and which involves just two key indices, measuring tolerance to the infection and the strength of direct competition. The techniques used in this paper are not restricted to the analysis of host-parasite systems but can be applied to a wide range of nonlinear population models. They are therefore as relevant to the analysis of such general issues as exploitative competition and trophic interactions as they are to specific epidemiological problems.