Disease‐structured N‐mixture models: A practical guide to model disease dynamics using count data

Obtaining inferences on disease dynamics (e.g., host population size, pathogen prevalence, transmission rate, host survival probability) typically requires marking and tracking individuals over time. While multistate mark–recapture models can produce high‐quality inference, these techniques are difficult to employ at large spatial and long temporal scales or in small remnant host populations decimated by virulent pathogens, where low recapture rates may preclude the use of mark–recapture techniques. Recently developed N‐mixture models offer a statistical framework for estimating wildlife disease dynamics from count data. N‐mixture models are a type of state‐space model in which observation error is attributed to failing to detect some individuals when they are present (i.e., false negatives). The analysis approach uses repeated surveys of sites over a period of population closure to estimate detection probability. We review the challenges of modeling disease dynamics and describe how N‐mixture models can be used to estimate common metrics, including pathogen prevalence, transmission, and recovery rates while accounting for imperfect host and pathogen detection. We also offer a perspective on future research directions at the intersection of quantitative and disease ecology, including the estimation of false positives in pathogen presence, spatially explicit disease‐structured N‐mixture models, and the integration of other data types with count data to inform disease dynamics. Managers rely on accurate and precise estimates of disease dynamics to develop strategies to mitigate pathogen impacts on host populations. At a time when pathogens pose one of the greatest threats to biodiversity, statistical methods that lead to robust inferences on host populations are critically needed for rapid, rather than incremental, assessments of the impacts of emerging infectious diseases.     

Making use of multiple surveys: Estimating breeding probability using a multievent‐robust design capture–recapture model

Increased environmental stochasticity due to climate change will intensify temporal variance in the life‐history traits, and especially breeding probabilities, of long‐lived iteroparous species. These changes may decrease individual fitness and population viability and is therefore important to monitor. In wild animal populations with imperfect individual detection, breeding probabilities are best estimated using capture–recapture methods. However, in many vertebrate species (e.g., amphibians, turtles, seabirds), nonbreeders are unobservable because they are not tied to a territory or breeding location. Although unobservable states can be used to model temporary emigration of nonbreeders, there are disadvantages to having unobservable states in capture–recapture models. The best solution to deal with unobservable life‐history states is therefore to eliminate them altogether. Here, we achieve this objective by fitting novel multievent‐robust design models which utilize information obtained from multiple surveys conducted throughout the year. We use this approach to estimate annual breeding probabilities of capital breeding female elephant seals (Mirounga leonina). Conceptually, our approach parallels a multistate version of the Barker/robust design in that it combines robust design capture data collected during discrete breeding seasons with observations made at other times of the year. A substantial advantage of our approach is that the nonbreeder state became “observable” when multiple data sources were analyzed together. This allowed us to test for the existence of state‐dependent survival (with some support found for lower survival in breeders compared to nonbreeders), and to estimate annual breeding transitions to and from the nonbreeder state with greater precision (where current breeders tended to have higher future breeding probabilities than nonbreeders). We used program E‐SURGE (2.1.2) to fit the multievent‐robust design models, with uncertainty in breeding state assignment (breeder, nonbreeder) being incorporated via a hidden Markov process. This flexible modeling approach can easily be adapted to suit sampling designs from numerous species which may be encountered during and outside of discrete breeding seasons.     

Pathogen interactions, population cycles, and phase shifts

Interspecific pathogen interactions can profoundly affect pathogen population dynamics and the efficacy of control strategies. However, many pathogens exhibit cyclic abundance patterns (e.g., seasonality), and temporal asynchrony between interacting pathogens could reduce the impact of those interactions. Here we use an extension of our previously published model to investigate the effects of cycles on pathogen interaction. We demonstrate that host immune memory can maintain the impact of an interaction, even when the effector pathogen abundance is low or the pathogen is absent. Paradoxically, immune memory can result in pathogens interacting more strongly when temporally out of phase. We find that interactions between species can result in changes to the temporal pattern of the affected species. We further demonstrate that this may be observed in a natural host-pathogen system. Given the continuing debate regarding the relevance of pathogen interactions in natural systems and increasing concern about treatment strategies for coinfections, both the discovery of a shift in cycle in empirical data and the mechanism by which we identified it are important. Finally, because the model structure used here is analogous to models of a simple predator-prey system, we also consider the consequences of these findings in the context of that system.     

Ecosystem dynamics, biological diversity and emerging infectious diseases

In this article, we summarize the major scientific developments of the last decade on the transmission of infectious agents in multi-host systems. Almost sixty percent of the pathogens that have emerged in humans during the last 30-40 years are of animal origin and about sixty percent of them show an important variety of host species besides humans (3 or more possible host species). In this review, we focus on zoonotic infections with vector-borne transmission and dissect the contrasting effects that a multiplicity of host reservoirs and vectors can have on their disease dynamics. We discuss the effects exerted by host and vector species richness and composition on pathogen prevalence (i.e., reduction, including the dilution effect, or amplification). We emphasize that, in multiple host systems and for vector-borne zoonotic pathogens, host reservoir species and vector species can exert contrasting effect locally. The outcome on disease dynamics (reduced pathogen prevalence in vectors when the host reservoir species is rich and increased pathogen prevalence when the vector species richness increases) may be highly heterogeneous in both space and time. We then ask briefly how a shift towards a more systemic perspective in the study of emerging infectious diseases, which are driven by a multiplicity of hosts, may stimulate further research developments. Finally, we propose some research avenues that take better into account the multi-host species reality in the transmission of the most important emerging infectious diseases, and, particularly, suggest, as a possible orientation, the careful assessment of the life-history characteristics of hosts and vectors in a community ecology-based perspective.     

Chronic disease in the Mojave desert tortoise: Host physiology and recrudescence obscure patterns of pathogen transmission

A seminatural, factorial‐design experiment was used to quantify dynamics of the pathogen Mycoplasma agassizii and upper respiratory tract disease in the Mojave desert tortoise (Gopherus agassizii) over 2 years. Groups of initially healthy animals were separated into serologically positive (seropositive), seronegative, and artificially infected groups and paired into 23 pens. We found no evidence of long‐term immune protection to M. agassizii or of immunological memory. Initially seronegative, healthy tortoises experienced an equal amount of disease when paired with other seronegative groups as when paired with seropositive and artificially infected groups—suggesting that recrudescence is as significant as transmission in introducing disease in individuals in this host–pathogen system. Artificially infected groups of tortoises showed reduced levels of morbidity when paired with initially seronegative animals—suggesting either a dilution effect or a strong effect of pathogen load in this system. Physiological dynamics within the host appear to be instrumental in producing morbidity, recrudescence, and infectiousness, and thus of population‐level dynamics. We suggest new avenues for studying diseases in long‐lived ectothermic vertebrates and a shift in modeling such diseases.     

Evolution of virulence driven by predator-prey interaction: Possible consequences for population dynamics

The evolution of pathogen virulence in natural populations has conventionally been considered as a result of selection caused by the interactions of the host with its pathogen(s). The host population, however, is generally embedded in complex trophic interactions with other populations in the community, in particular, intensive predation on the infected host can increase its mortality, and this can affect the course of virulence evolution. Reciprocally, in the long run, the evolution of virulence within an infected host can affect the patterns of population dynamics of a predator consuming the host (e.g. resulting in large amplitude oscillations, causing a severe drop in the population size, etc.). Surprisingly, neither the effect of predation on the evolution of virulence within a host, nor the influence of the evolution of virulence upon the consumer's dynamics has been addressed in the literature yet. In this paper, we consider a classical S-I ecoepidemiological model in which the infected host is consumed by a predator. We are particularly interested in the evolutionarily stable virulence of the pathogen in the model and its dependence upon ecologically relevant parameters. We show that predation can prominently shift the evolutionarily stable virulence towards more severe strains as compared to the same system without predation. We demonstrate that the evolution of virulence can result in a succession of dynamical regimes and can even lead to the extinction of the predator in the long run. The presence of a predator can indirectly affect the evolution within its prey since the evolutionarily stable virulence becomes a function of the prey growth rate, which would not be the case in a predator-free system. We find that the evolutionarily stable virulence largely depends on the carrying capacity K of the prey in a non-monotonous way. The model also predicts that in an eutrophic environment the shift of virulence towards evolutionarily stable benign strains can cause demographically stochastic evolutionary suicide, resulting in the extinction of both species, thus artificially maintaining severe strains of pathogen can enhance the persistence of both species.     

Characterizing the next-generation matrix and basic reproduction number in ecological epidemiology

We address the interaction of ecological processes, such as consumer-resource relationships and competition, and the epidemiology of infectious diseases spreading in ecosystems. Modelling such interactions seems essential to understand the dynamics of infectious agents in communities consisting of interacting host and non-host species. We show how the usual epidemiological next-generation matrix approach to characterize invasion into multi-host communities can be extended to calculate $\mathcal{R _{0}}$ , and how this relates to the ecological community matrix. We then present two simple examples to illustrate this approach. The first of these is a model of the rinderpest, wildebeest, grass interaction, where our inferred dynamics qualitatively matches the observed phenomena that occurred after the eradication of rinderpest from the Serengeti ecosystem in the 1980s. The second example is a prey-predator system, where both species are hosts of the same pathogen. It is shown that regions for the parameter values exist where the two host species are only able to coexist when the pathogen is present to mediate the ecological interaction.     

Movement can mediate temporal mismatches between resource availability and biological events in host–pathogen interactions

Global change is shifting the timing of biological events, leading to temporal mismatches between biological events and resource availability. These temporal mismatches can threaten species’ populations. Importantly, temporal mismatches not only exert strong pressures on the population dynamics of the focal species, but can also lead to substantial changes in pairwise species interactions such as host–pathogen systems. We adapted an established individual‐based model of host–pathogen dynamics. The model describes a viral agent in a social host, while accounting for the host''s explicit movement decisions. We aimed to investigate how temporal mismatches between seasonal resource availability and host life‐history events affect host–pathogen coexistence, that is, disease persistence. Seasonal resource fluctuations only increased coexistence probability when in synchrony with the hosts’ biological events. However, a temporal mismatch reduced host–pathogen coexistence, but only marginally. In tandem with an increasing temporal mismatch, our model showed a shift in the spatial distribution of infected hosts. It shifted from an even distribution under synchronous conditions toward the formation of disease hotspots, when host life history and resource availability mismatched completely. The spatial restriction of infected hosts to small hotspots in the landscape initially suggested a lower coexistence probability due to the critical loss of susceptible host individuals within those hotspots. However, the surrounding landscape facilitated demographic rescue through habitat‐dependent movement. Our work demonstrates that the negative effects of temporal mismatches between host resource availability and host life history on host–pathogen coexistence can be reduced through the formation of temporary disease hotspots and host movement decisions, with implications for disease management under disturbances and global change.     

Modelling transmission of vector-borne pathogens shows complex dynamics when vector feeding sites are limited

The relationship between species richness and the prevalence of vector-borne disease has been widely studied with a range of outcomes. Increasing the number of host species for a pathogen may decrease infection prevalence (dilution effect), increase it (amplification), or have no effect. We derive a general model, and a specific implementation, which show that when the number of vector feeding sites on each host is limiting, the effects on pathogen dynamics of host population size are more complex than previously thought. The model examines vector-borne disease in the presence of different host species that are either competent or incompetent (i.e. that cannot transmit the pathogen to vectors) as reservoirs for the pathogen. With a single host species present, the basic reproduction ratio R(0) is a non-monotonic function of the population size of host individuals (H), i.e. a value [Formula: see text] exists that maximises R(0). Surprisingly, if [Formula: see text] a reduction in host population size may actually increase R(0). Extending this model to a two-host species system, incompetent individuals from the second host species can alter the value of [Formula: see text] which may reverse the effect on pathogen prevalence of host population reduction. We argue that when vector-feeding sites on hosts are limiting, the net effect of increasing host diversity might not be correctly predicted using simple frequency-dependent epidemiological models.     

Disease evolution across a range of spatio-temporal scales

Traditional explorations of infectious disease evolution have considered the competition between two cross-reactive strains within the standard framework of disease models. Such techniques predict that diseases should evolve to be highly transmissible, benign to the host and possess a long infectious period: in general, diseases do not conform to this ideal. Here we consider a more holistic approach, suggesting that evolution is a trade-off between adaptive pressures at different scales: within host, between hosts and at the population level. We present a model combining within-host pathogen dynamics and transmission between individuals governed by an explicit contact network, where transmission dynamics between hosts are a function of the interaction between the pathogen and the hosts' immune system, though ultimately constrained by the contacts each infected host possesses. Our results show how each of the scales places constraints on the evolutionary behavior, and that complex dynamics may emerge due to the feedbacks between epidemiological and evolutionary dynamics. In particular, multiple stable states can occur with switching between them stochastically driven.     

The role of initial inoculum on epidemic dynamics

Transient dynamics are important in many epidemics in agricultural and ecological systems that are prone to regular disturbance, cyclical and random perturbations. Here, using a simple host-pathogen model for a sessile host and a pathogen that can move by diffusion and advection, we use a range of mathematical techniques to examine the effect of initial spatial distribution of inoculum of the pathogen on the transient dynamics of the epidemic. We consider an isolated patch and a group of patches with different boundary conditions. We first determine bounds on the host population for the full model, then non-dimensionalizing the model allows us to obtain approximate solutions for the system. We identify two biologically intuitive groups of parameters to analyse transient behaviour using perturbation techniques. The first parameter group is a measure of the relative strength of initial primary to secondary infection. The second group is derived from the ratio of host removal rate (via infection) to pathogen removal rate (by decay and natural mortality) and measures the infectivity of initial inoculum on the system. By restricting the model to mimic primary infection only (in which all infections arise from initial inoculum), we obtain exact solutions and demonstrate how these depend on initial conditions, boundary conditions and model parameters. Finally, we suggest that the analyses on the balance of primary and secondary infection provide the epidemiologist with some simple rules to predict the transient behaviours.     

Nest use dynamics of an undisturbed population of bald eagles

Management or conservation targets based on demographic rates should be evaluated within the context of expected population dynamics of the species of interest. Wild populations can experience stable, cyclical, or complex dynamics, therefore undisturbed populations can provide background needed to evaluate programmatic success. Many raptor species have recovered from large declines caused by environmental contaminants, making them strong candidates for ongoing efforts to understand population dynamics and ecosystem processes in response to human‐caused stressors. Dynamic multistate occupancy models are a useful tool for analyzing species dynamics because they leverage the autocorrelation inherent in long‐term monitoring datasets to obtain useful information about the dynamic properties of population or reproductive states. We analyzed a 23‐year bald eagle monitoring dataset in a dynamic multistate occupancy modeling framework to assess long‐term nest occupancy and reproduction in Lake Clark National Park and Preserve, Alaska. We also used a hierarchical generalized linear model to understand changes in nest productivity in relation to environmental factors. Nests were most likely to remain in the same nesting state between years. Most notably, successful nests were likely to remain in use (either occupied or successful) and had a very low probability of transitioning to an unoccupied state in the following year. There was no apparent trend in the proportion of nests used by eagles through time, and the probability that nests transitioned into or out of the successful state was not influenced by temperature or salmon availability. Productivity was constant over the course of the study, although warm April minimum temperatures were associated with increased chick production. Overall our results demonstrate the expected nesting dynamics of a healthy bald eagle population that is largely free of human disturbance and can be used as a baseline for the expected dynamics for recovering bald eagle populations in the contiguous 48 states.     

Primates and Cameras

Field-based primate studies often make population inferences using count-based indices (e.g., individuals/plot) or distance sampling; the first does not account for the probability of detection and thus can be biased, while the second requires large sample sizes to obtain precise estimates, which is difficult for many primate studies. We discuss photographic sampling and occupancy modeling to correct for imperfect detection when estimating system states and dynamics at the landscape level, specifically in relation to primate ecology. We highlight the flexibility of the occupancy framework and its many applications to studying low-density primate populations or species that are difficult to detect. We discuss relevant sampling and estimation procedures with special attention to data collection via photographic sampling. To provide tangible meaning to terminology and clarify subtleties, we use illustrative examples. Photographic sampling can have many advantages over observer-based sampling, especially when studying rare or elusive species. Combining photographic sampling with an occupancy framework allows inference to larger scales than is common in primate studies, addresses uncertainty due to the observation process, and allows researchers to examine questions of how landscape-level anthropogenic changes affect primate distributions.     

Populations and infectious diseases: Dynamics and evolution

The host-parasite or host-pathogen system was analyzed from dynamical and evolutionary viewpoints using simple mathematical models incorporating vertical transmission, immunity and its loss. We first analyzed a model without density regulation of host population. In the analysis on dynamics, the condition for the pathogen to work as a density regulating factor was obtained. In the analysis on evolution, criteria for the evolution of host and pathogen were proposed. These criteria implies that the evolution of hosts should result in an increase in infected host density, whereas the evolution of pathogens a decrease in susceptible host density. The direction of evolution at some parameters of host and that of pathogen were examined when the parameters were independently and freely changeable. Among the parameters, only reduction in additional mortality due to infection was the evolutionary trend common to both host and pathogen. In all the other parameters examined, trend of evolution predicted in host is reversed in pathogen. We then analyzed whether the obtained criteria still hold in models with density regulation of hosts. Using randomly generated parameter sets, we obtained the result that the criteria should hold very likely though they do not always hold. We discussed evolution of virulence when there is a constraint between the traits.     

Cryptic population dynamics: rapid evolution masks trophic interactions

Trophic relationships, such as those between predator and prey or between pathogen and host, are key interactions linking species in ecological food webs. The structure of these links and their strengths have major consequences for the dynamics and stability of food webs. The existence and strength of particular trophic links has often been assessed using observational data on changes in species abundance through time. Here we show that very strong links can be completely missed by these kinds of analyses when changes in population abundance are accompanied by contemporaneous rapid evolution in the prey or host species. Experimental observations, in rotifer-alga and phage-bacteria chemostats, show that the predator or pathogen can exhibit large-amplitude cycles while the abundance of the prey or host remains essentially constant. We know that the species are tightly linked in these experimental microcosms, but without this knowledge, we would infer from observed patterns in abundance that the species are weakly or not at all linked. Mathematical modeling shows that this kind of cryptic dynamics occurs when there is rapid prey or host evolution for traits conferring defense against attack, and the cost of defense (in terms of tradeoffs with other fitness components) is low. Several predictions of the theory that we developed to explain the rotifer-alga experiments are confirmed in the phage-bacteria experiments, where bacterial evolution could be tracked. Modeling suggests that rapid evolution may also confound experimental approaches to measuring interaction strength, but it identifies certain experimental designs as being more robust against potential confounding by rapid evolution.     

Evolution of Pathogen Specialisation in a Host Metapopulation: Joint Effects of Host and Pathogen Dispersal

Metapopulation processes are important determinants of epidemiological and evolutionary dynamics in host-pathogen systems, and are therefore central to explaining observed patterns of disease or genetic diversity. In particular, the spatial scale of interactions between pathogens and their hosts is of primary importance because migration rates of one species can affect both spatial and temporal heterogeneity of selection on the other. In this study we developed a stochastic and discrete time simulation model to specifically examine the joint effects of host and pathogen dispersal on the evolution of pathogen specialisation in a spatially explicit metapopulation. We consider a plant-pathogen system in which the host metapopulation is composed of two plant genotypes. The pathogen is dispersed by air-borne spores on the host metapopulation. The pathogen population is characterised by a single life-history trait under selection, the infection efficacy. We found that restricted host dispersal can lead to high amount of pathogen diversity and that the extent of pathogen specialisation varied according to the spatial scale of host-pathogen dispersal. We also discuss the role of population asynchrony in determining pathogen evolutionary outcomes.     

Infection with Batrachochytrium dendrobatidis is common in tropical lowland habitats: Implications for amphibian conservation

Numerous species of amphibians declined in Central America during the 1980s and 1990s. These declines mostly affected highland stream amphibians and have been primarily linked to chytridiomycosis, a deadly disease caused by the chytrid fungus Batrachochytrium dendrobatidis (Bd). Since then, the majority of field studies on Bd in the Tropics have been conducted in midland and highland environments (>800 m) mainly because the environmental conditions of mountain ranges match the range of ideal abiotic conditions for Bd in the laboratory. This unbalanced sampling has led researchers to largely overlook host–pathogen dynamics in lowlands, where other amphibian species declined during the same period. We conducted a survey testing for Bd in 47 species (n = 348) in four lowland sites in Costa Rica to identify local host–pathogen dynamics and to describe the abiotic environment of these sites. We detected Bd in three sampling sites and 70% of the surveyed species. We found evidence that lowland study sites exhibit enzootic dynamics with low infection intensity and moderate to high prevalence (55% overall prevalence). Additionally, we found evidence that every study site represents an independent climatic zone, where local climatic differences may explain variations in Bd disease dynamics. We recommend more detection surveys across lowlands and other sites that have been historically considered unsuitable for Bd occurrence. These data can be used to identify sites for potential disease outbreaks and amphibian rediscoveries.     

Modeling host-pathogen interactions: H. sapiens as a host and C. difficile as a pathogen

Many complex mechanisms in immunological studies cannot be measured by experiments, but can be analyzed by mathematical simulations. Using theoretical modeling techniques, general principles of host–pathogen system interactions can be explored and clinical treatment schedules can be optimized to lower the microbial toxin burden and side effects in the host system. In this study, we use a computational modeling technique that aims to explain the host–pathogen interactions and suggests how the host system tries to survive from the pathogen attack. The method generates data on reaction fluxes in a pathway at steady state. A set of constraints is incorporated and an objective function for the minimization of toxin expression, with respect to some parameters such as concentration of signaling molecules, is formulated. We have integrated the toxin expression regulatory pathway in Clostridium difficile, apoptosis and mitogen‐activated protein kinase pathways in an infected host (Homo sapiens). We have found that due to the minimization of the toxin expression, the signal flow values for most of the survival genes are at the higher side, whereas it is the reverse for most of the proapoptotic genes. We have observed increased signal flow values of the molecules for extracellular regulated kinase as compared with the molecules present in c‐Jun NH2‐terminal kinase/p38 pathways. In light of these observations, we can hypothesize that lower toxin level in a pathogen implies higher chance of host survival. Copyright © 2012 John Wiley & Sons, Ltd.     

Host species composition influences infection severity among amphibians in the absence of spillover transmission

Wildlife epidemiological outcomes can depend strongly on the composition of an ecological community, particularly when multiple host species are affected by the same pathogen. However, the relationship between host species richness and disease risk can vary with community context and with the degree of spillover transmission that occurs among co‐occurring host species. We examined the degree to which host species composition influences infection by Batrachochytrium dendrobatidis (Bd), a widespread fungal pathogen associated with amphibian population declines around the world, and whether transmission occurs from one highly susceptible host species to other co‐occurring host species. By manipulating larval assemblages of three sympatric amphibian species in the laboratory, we characterized the relationship between host species richness and infection severity, whether infection mediates growth and survivorship differently across various combinations of host species, and whether Bd is transmitted from experimentally inoculated tadpoles to uninfected tadpoles. We found evidence of a dilution effect where Bd infection severity was dramatically reduced in the most susceptible of the three host species (Anaxyrus boreas). Infection also mediated survival and growth of all three host species such that the presence of multiple host species had both positive (e.g., infection reduction) and negative (e.g., mortality) effects on focal species. However, we found no evidence that Bd infection is transmitted by this species. While these results demonstrate that host species richness as well as species identity underpin infection dynamics in this system, dilution is not the product of reduced transmission via fewer infectious individuals of a susceptible host species. We discuss various mechanisms, including encounter reduction and antagonistic interactions such as competition and opportunistic cannibalism that may act in concert to mediate patterns of infection severity, growth, and mortality observed in multihost communities.