A fully coupled, mechanistic model for infectious disease dynamics in a metapopulation: movement and epidemic duration

The drive to understand the invasion, spread and fade out of infectious disease in structured populations has produced a variety of mathematical models for pathogen dynamics in metapopulations. Very rarely are these models fully coupled, by which we mean that the spread of an infection within a subpopulation affects the transmission between subpopulations and vice versa. It is also rare that these models are accessible to biologists, in the sense that all parameters have a clear biological meaning and the biological assumptions are explained. Here we present an accessible model that is fully coupled without being an individual-based model. We use the model to show that the duration of an epidemic has a highly non-linear relationship with the movement rate between subpopulations, with a peak in epidemic duration appearing at small movement rates and a global maximum at large movement rates. Intuitively, the first peak is due to asynchrony in the dynamics of infection between subpopulations; we confirm this intuition and also show the peak coincides with successful invasion of the infection into most subpopulations. The global maximum at relatively large movement rates occurs because then the infectious agent perceives the metapopulation as if it is a single well-mixed population wherein the effective population size is greater than the critical community size.     

Estimating the Hidden Burden of Bovine Tuberculosis in Great Britain

The number of cattle herds placed under movement restrictions in Great Britain (GB) due to the suspected presence of bovine tuberculosis (bTB) has progressively increased over the past 25 years despite an intensive and costly test-and-slaughter control program. Around 38% of herds that clear movement restrictions experience a recurrent incident (breakdown) within 24 months, suggesting that infection may be persisting within herds. Reactivity to tuberculin, the basis of diagnostic testing, is dependent on the time from infection. Thus, testing efficiency varies between outbreaks, depending on weight of transmission and cannot be directly estimated. In this paper, we use Approximate Bayesian Computation (ABC) to parameterize two within-herd transmission models within a rigorous inferential framework. Previous within-herd models of bTB have relied on ad-hoc methods of parameterization and used a single model structure (SORI) where animals are assumed to become detectable by testing before they become infectious. We study such a conventional within-herd model of bTB and an alternative model, motivated by recent animal challenge studies, where there is no period of epidemiological latency before animals become infectious (SOR). Under both models we estimate that cattle-to-cattle transmission rates are non-linearly density dependent. The basic reproductive ratio for our conventional within-herd model, estimated for scenarios with no statutory controls, increases from 1.5 (0.26–4.9; 95% CI) in a herd of 30 cattle up to 4.9 (0.99–14.0) in a herd of 400. Under this model we estimate that 50% (33–67) of recurrent breakdowns in Britain can be attributed to infection missed by tuberculin testing. However this figure falls to 24% (11–42) of recurrent breakdowns under our alternative model. Under both models the estimated extrinsic force of infection increases with the burden of missed infection. Hence, improved herd-level testing is unlikely to reduce recurrence unless this extrinsic infectious pressure is simultaneously addressed.     

Refugia and connectivity sustain amphibian metapopulations afflicted by disease

Metapopulation persistence in fragmented landscapes depends on habitat patches that can support resilient local populations and sufficient connectivity between patches. Yet epidemiological theory for metapopulations has largely overlooked the capacity of particular patches to act as refuges from disease, and has suggested that connectivity can undermine persistence. Here, we show that relatively warm and saline wetlands are environmental refuges from chytridiomycosis for an endangered Australian frog, and act jointly with connectivity to sustain frog metapopulations. We coupled models of microclimate and infection probability to map chytrid prevalence, and demonstrate a strong negative relationship between chytrid prevalence and the persistence of frog populations. Simulations confirm that frog metapopulations are likely to go extinct when they lack environmental refuges from disease and lose connectivity between patches. This study demonstrates that environmental heterogeneity can mediate host–pathogen interactions in fragmented landscapes, and provides evidence that connectivity principally supports host metapopulations afflicted by facultative pathogens.     

A priori prediction of disease invasion dynamics in a novel environment

Directly transmitted infectious diseases spread through wildlife populations as travelling waves away from the sites of original introduction. These waves often become distorted through their interaction with environmental and population heterogeneities and by long-distance translocation of infected individuals. Accurate a priori predictions of travelling waves of infection depend upon understanding and quantifying these distorting factors. We assess the effects of anisotropies arising from the orientation of rivers in relation to the direction of disease-front propagation and the damming effect of mountains on disease movement in natural populations. The model successfully predicts the local and large-scale prevaccination spread of raccoon rabies through New York State, based on a previous spatially heterogeneous model of raccoon-rabies invasion across the state of Connecticut. Use of this model provides a rare example of a priori prediction of an epidemic invasion over a naturally heterogeneous landscape. Model predictions matched to data can also be used to evaluate the most likely points of disease introduction. These results have general implications for predicting future pathogen invasions and evaluating potential containment strategies.     

Landscape-level toxicant exposure mediates infection impacts on wildlife populations

Anthropogenic landscape modification such as urbanization can expose wildlife to toxicants, with profound behavioural and health effects. Toxicant exposure can alter the local transmission of wildlife diseases by reducing survival or altering immune defence. However, predicting the impacts of pathogens on wildlife across their ranges is complicated by heterogeneity in toxicant exposure across the landscape, especially if toxicants alter wildlife movement from toxicant-contaminated to uncontaminated habitats. We developed a mechanistic model to explore how toxicant effects on host health and movement propensity influence range-wide pathogen transmission, and zoonotic exposure risk, as an increasing fraction of the landscape is toxicant-contaminated. When toxicant-contaminated habitat is scarce on the landscape, costs to movement and survival from toxicant exposure can trap infected animals in contaminated habitat and reduce landscape-level transmission. Increasing the proportion of contaminated habitat causes host population declines from combined effects of toxicants and infection. The onset of host declines precedes an increase in the density of infected hosts in contaminated habitat and thus may serve as an early warning of increasing potential for zoonotic spillover in urbanizing landscapes. These results highlight how sublethal effects of toxicants can determine pathogen impacts on wildlife populations that may not manifest until landscape contamination is widespread.     

Signaling against the wind: modifying motion-signal structure in response to increased noise

Animal signals are optimized for particular signaling environments [1-3]. While signaling, senders often choose favorable conditions that ensure reliable detection and transmission [4-8], suggesting that they are sensitive to changes in signal efficacy. Recent evidence has also shown that animals will increase the amplitude or intensity of their acoustic signals at times of increased environmental noise [9-11]. The nature of these adjustments provides important insights into sensory processing. However, only a single piece of correlative evidence for signals defined by movement suggests that visual-signal design depends on ambient motion noise [12]. Here we show experimentally for the first time that animals communicating with movement will adjust their displays when environmental motion noise increases. Surprisingly, under sustained wind conditions, the Australian lizard Amphibolurus muricatus changed the structure and increased the duration of its introductory tail flicking, rather than increasing signaling speed. The way these lizards restructure the alerting component of their movement-based aggressive display in the presence of increased motion noise highlights the challenge we face in understanding motion-detection mechanisms under natural operating conditions.     

A comparison of fertility control and lethal control of bovine tuberculosis in badgers: the impact of perturbation induced transmission.

In this paper we use mathematical modelling to consider the broad advantages and disadvantages of fertility control over lethal control for bovine tuberculosis in badger populations. We use a deliberately simple model, attempting to capture only the key transmission processes. The model is parametrized with reference to the long-term Woodchester Park study. Estimates of mortality rate from this study suggest no significant extra mortality risk for animals with evidence of infection as indicated by the presence of anti-Mycobacterium bovis antibodies or M. bovis isolation. We find that large reductions in prevalence are sometimes the consequence of only moderate reductions in population numbers. If we assume that the act of control does not in itself affect transmission rates, then as far as eradication is concerned, both fertility control and mortality control operate through the same epidemiological mechanism, the removal of susceptibles: if one is in principle capable of keeping a population low enough to be infection free then so is the other. It is necessary to continue either form of control at regular intervals to maintain a constant level of infection in the long term. If control were to be stopped, return to precontrol levels of badger population and infection prevalence would be expected within a few years. Fertility control is less effective in reducing population density than lethal control since it can only act, at maximum, to remove one age cohort per year. It is also less effective in reducing transmission as it can only ever remove susceptibles, while lethal control also removes infectious badgers. However, if the social disturbance caused by lethal control does in fact increase contact rates for the remaining infectious badgers, the relative efficacies of the two strategies become a great deal less clear. While we have no quantitative data on the extent to which social perturbation does act to promote transmission, model simulations show that it is possible to develop plausible scenarios in which the lethal control may actually act to increase the absolute numbers of animals infected, while reducing the number of uninfected animals to very low numbers.     

Population dynamics in migratory networks

Migratory animals are comprised of a complex series of interconnected breeding and nonbreeding populations. Because individuals in any given population can arrive from a variety of sites the previous season, predicting how different populations will respond to environmental change can be challenging. In this study, we develop a population model composed of a network of breeding and wintering sites to show how habitat loss affects patterns of connectivity and species abundance. When the costs of migration are evenly distributed, habitat loss at a single site can increase the degree of connectivity (mixing) within the entire network, which then acts to buffer global populations from declines. However, the degree to which populations are buffered depends on where habitat loss occurs within the network: a site that has the potential to receive individuals from multiple populations in the opposite season will lead to smaller declines than a site that is more isolated. In other cases when there are equal costs of migration to two or more sites in the opposite season, habitat loss can result in some populations becoming segregated (disconnected) from the rest of the network. The geographic structure of the network can have a significant influence on relative population sizes of sites in the same season and can also affect the overall degree of mixing in the network, even when sites are of equal intrinsic quality. When a migratory network is widely spaced and migration costs are high, an equivalent habitat loss will lead to a larger decline in global population size than will occur in a network where the overall costs of migration are low. Our model provides an important foundation to test predictions related to habitat loss in real-world migratory networks and demonstrates that migratory networks will likely produce different dynamics from traditional metapopulations. Our results provide strong evidence that estimating population connectivity is a prerequisite for successfully predicting changes in migratory populations.     

Probability of a Disease Outbreak in Stochastic Multipatch Epidemic Models

Environmental heterogeneity, spatial connectivity, and movement of individuals play important roles in the spread of infectious diseases. To account for environmental differences that impact disease transmission, the spatial region is divided into patches according to risk of infection. A system of ordinary differential equations modeling spatial spread of disease among multiple patches is used to formulate two new stochastic models, a continuous-time Markov chain, and a system of stochastic differential equations. An estimate for the probability of disease extinction is computed by approximating the Markov chain model with a multitype branching process. Numerical examples illustrate some differences between the stochastic models and the deterministic model, important for prevention of disease outbreaks that depend on the location of infectious individuals, the risk of infection, and the movement of individuals.     

A general modeling framework for describing spatially structured population dynamics

Variation in movement across time and space fundamentally shapes the abundance and distribution of populations. Although a variety of approaches model structured population dynamics, they are limited to specific types of spatially structured populations and lack a unifying framework. Here, we propose a unified network‐based framework sufficiently novel in its flexibility to capture a wide variety of spatiotemporal processes including metapopulations and a range of migratory patterns. It can accommodate different kinds of age structures, forms of population growth, dispersal, nomadism and migration, and alternative life‐history strategies. Our objective was to link three general elements common to all spatially structured populations (space, time and movement) under a single mathematical framework. To do this, we adopt a network modeling approach. The spatial structure of a population is represented by a weighted and directed network. Each node and each edge has a set of attributes which vary through time. The dynamics of our network‐based population is modeled with discrete time steps. Using both theoretical and real‐world examples, we show how common elements recur across species with disparate movement strategies and how they can be combined under a unified mathematical framework. We illustrate how metapopulations, various migratory patterns, and nomadism can be represented with this modeling approach. We also apply our network‐based framework to four organisms spanning a wide range of life histories, movement patterns, and carrying capacities. General computer code to implement our framework is provided, which can be applied to almost any spatially structured population. This framework contributes to our theoretical understanding of population dynamics and has practical management applications, including understanding the impact of perturbations on population size, distribution, and movement patterns. By working within a common framework, there is less chance that comparative analyses are colored by model details rather than general principles.     

Regional structuring of genetic variation in short-lived rock pool populations of Branchipodopsis wolfi (Crustacea: Anostraca)

The genetic structure of three metapopulations of the southern African anostracan Branchipodopsis wolfi was compared by analysing allozyme variation at four loci (PGM, GPI, APK, AAT). In total, 17 local populations from three sites (metapopulations) were analysed from rock pools in south-eastern Botswana ranging from 0.2 to 21 m² in surface area. In three populations we found significant deviations from Hardy-Weinberg (H-W) equilibrium at one or more loci due to heterozygote deficiencies. Genetic variability at one site was significantly lower than at the other sites, which may be linked to a greater incidence of extinction and recolonisation, as the basins at this site are shallower and have shorter hydrocycles. Across all local populations, a significant level of population differentiation was revealed. More than 90% of this variation was explained by differentiation among sites (metapopulations), although this differentiation did not correlate with geographic distance, or with environmental variables. Genetic differentiation among populations within metapopulations was low, but significant at all sites. At only one of the sites was a significantly positive association measured between genetic and geographic distance among local populations. Our data suggest that persistent stochastic events and limited effective long-range dispersal appear to dominate genetic differentiation among populations of B. wolfi inhabiting desert rock pools. The lack of association between geographic distance and genetic or ecological differences between rock pool sites is indicative of historical stochastic events. Low heterozygosity, the significant deviations from H-W equilibrium, and the large inter- but low intra-site differentiation are suggestive of the importance of short-range dispersal. Gene flow between metapopulations of B. wolfi appears to be seriously constrained by distances of 2 km or even less. Received: 28 June 1999 / Accepted: 10 January 2000     

Adaptive movement and food-chain dynamics: towards food-web theory without birth–death processes

Population density can be affected by its prey [resource] and predator [consumer] abundances through two different mechanisms: the alternation of birth [or somatic growth] or death rate and inter-habitat movement. While the food-web theory has traditionally been built on the former mechanism, the latter mechanism has formed the basis of a successful theory explaining the spatial distribution of organisms in the context of behavioral and evolutionary ecology. Yet, few studies have compared these two mechanisms, leaving the question of how similar (or different) predictions derived from birth–death-based and movement-based food-web theories unanswered. Here, theoretical models of the tri-trophic (resource–consumer-top predator) food chain were used to compare food-web patterns arising from these two mechanisms. Specifically, we evaluated the response of the food-chain structure to inter-patch differences in productivity for movement-based models and birth–death-based models. Model analysis reveals that adaptive movements give rise to positively correlated responses of all trophic levels to increased productivity; however, this pattern was not observed in the corresponding birth–death-based model. The movement-based model predicts that the food chain response to productivity is determined by the sensitivity of animal movement to the environmental conditions. More specifically, increasing sensitivity of a consumer or top predator leads to smaller inter-patch variance of the resource or consumer density, while increasing inter-patch variance in the consumer or resource density. In conclusion, adaptive movement provides an alternative mechanism correlating the food-web structure to environmental conditions.     

Invasion and persistence of infectious agents in fragmented host populations

One of the important questions in understanding infectious diseases and their prevention and control is how infectious agents can invade and become endemic in a host population. A ubiquitous feature of natural populations is that they are spatially fragmented, resulting in relatively homogeneous local populations inhabiting patches connected by the migration of hosts. Such fragmented population structures are studied extensively with metapopulation models. Being able to define and calculate an indicator for the success of invasion and persistence of an infectious agent is essential for obtaining general qualitative insights into infection dynamics, for the comparison of prevention and control scenarios, and for quantitative insights into specific systems. For homogeneous populations, the basic reproduction ratio R(0) plays this role. For metapopulations, defining such an 'invasion indicator' is not straightforward. Some indicators have been defined for specific situations, e.g., the household reproduction number R*. However, these existing indicators often fail to account for host demography and especially host migration. Here we show how to calculate a more broadly applicable indicator R(m) for the invasion and persistence of infectious agents in a host metapopulation of equally connected patches, for a wide range of possible epidemiological models. A strong feature of our method is that it explicitly accounts for host demography and host migration. Using a simple compartmental system as an example, we illustrate how R(m) can be calculated and expressed in terms of the key determinants of epidemiological dynamics.     

Experimental translocation of juvenile water voles in a Scottish lowland metapopulation

Population density affects dispersal success because residents can hinder or facilitate immigration into a new site, via a “social fence effect” or “social attraction” (or “conspecific attraction”), respectively. These mechanisms can affect the dynamics of fragmented populations and the success of translocations. However, information on the settlement behaviour of dispersers is rare. We conducted a manipulative field experiment using wild water voles, which exist in metapopulations along waterways in Scotland. We translocated 17 young of dispersal age into either an occupied site or a vacant site containing good habitat, which had recently become extinct due to a feral predator (American mink) moving through. We monitored the movements of translocated voles using radio telemetry. Translocated voles were less likely to settle in occupied sites with higher densities of residents, suggesting a possible social fence effect at high density. There was evidence of a social attraction mechanism, because voles never remained at new sites unless another individual arrived soon after translocation, and they were more likely to settle in occupied or colonised sites than vacant ones. Voles remained in the transient phase of dispersal for many days, and often followed a “stepping stone” trajectory, stopping for several days at successive sites. We suggest that trajectories followed by dispersing water voles, the time scale and long dispersal distances found in this species are conducive to locating conspecifics at low density and colonising vacant habitat. These results are encouraging for prospects of metapopulation persistence and future translocation success.     

Experimental Evidence for Reduced Rodent Diversity Causing Increased Hantavirus Prevalence

Emerging and re-emerging infectious diseases have become a major global environmental problem with important public health, economic, and political consequences. The etiologic agents of most emerging infectious diseases are zoonotic, and anthropogenic environmental changes that affect wildlife communities are increasingly implicated in disease emergence and spread. Although increased disease incidence has been correlated with biodiversity loss for several zoonoses, experimental tests in these systems are lacking. We manipulated small-mammal biodiversity by removing non-reservoir species in replicated field plots in Panama, where zoonotic hantaviruses are endemic. Both infection prevalence of hantaviruses in wild reservoir (rodent) populations and reservoir population density increased where small-mammal species diversity was reduced. Regardless of other variables that affect the prevalence of directly transmitted infections in natural communities, high biodiversity is important in reducing transmission of zoonotic pathogens among wildlife hosts. Our results have wide applications in both conservation biology and infectious disease management.     

Who infects whom? Social networks and tuberculosis transmission in wild meerkats

Transmission of infectious diseases is strongly influenced by who contacts whom. Despite the global distribution of tuberculosis (TB) in free-living wild mammal populations, little is known of the mechanisms of social transmission of Mycobacterium bovis between individuals. Here, I use a network approach to examine for correlations between five distinct types of intra- and intergroup social interaction and changes in TB status of 110 wild meerkats (Suricata suricatta) in five social groups over two years. Contrary to predictions, the most socially interactive animals were not at highest risk of acquiring infection, indicating that in addition to contact frequency, the type and direction of interactions must be considered when quantifying disease risk. Within social groups, meerkats that groomed others most were more likely to become infected than individuals who received high levels of grooming. Conversely, receiving, but not initiating, aggression was associated with M. bovis infection. Incidence of intergroup roving by male meerkats was correlated with the rovers themselves subsequently testing TB-positive, suggesting a possible route for transmission of infection between social groups. Exposure time was less important than these social interactions in influencing TB risk. This study represents a novel application of social network analysis using empirical data to elucidate the role of specific interactions in the transmission of an infectious disease in a free-living wild animal population.     

Movements and source–sink dynamics of a Masai giraffe metapopulation

Spatial variation in habitat quality and anthropogenic factors, as well as social structure, can lead to spatially structured populations of animals. Demographic approaches can be used to improve our understanding of the dynamics of spatially structured populations and help identify subpopulations critical for the long-term persistence of regional metapopulations. We provide a regional metapopulation analysis to inform conservation management for Masai giraffes (Giraffa camelopardalis tippelskirchi) in five subpopulations defined by land management designations. We used data from an individual-based mark–recapture study to estimate subpopulation sizes, subpopulation growth rates, and movement probabilities among subpopulations. We assessed the source–sink structure of the study population by calculating source–sink statistics, and we created a female-based matrix metapopulation model composed of all subpopulations to examine how variation in demographic components of survival, reproduction, and movement affected metapopulation growth rate. Movement data indicated no subpopulation was completely isolated, but movement probabilities varied among subpopulations. Source–sink statistics and net flow of individuals indicated three subpopulations were sources, while two subpopulations were sinks. We found areas with higher wildlife protection efforts and fewer anthropogenic impacts were sources, and less-protected areas were identified as sinks. Our results highlight the importance of identifying source–sink dynamics among subpopulations for effective conservation planning and emphasize how protected areas can play an important role in sustaining metapopulations.     

Temperature-dependent transmission and latency of Holospora undulata, a micronucleus-specific parasite of the ciliate Paramecium caudatum

Transmission of parasites to new hosts crucially depends on the timing of production of transmission stages and their capacity to start an infection. These parameters may be influenced by genetic factors, but also by the environment. We tested the effects of temperature and host genotype on infection probability and latency in experimental populations of the ciliate Paramecium caudatum, after exposure to infectious forms of its bacterial parasite Holospora undulata. Temperature had a significant effect on the expression of genetic variation for transmission and maintenance of infection. Overall, low temperature (10 degrees C) increased levels of (multiple) infection, but arrested parasite development; higher temperatures (23 and 30 degrees C) accelerated the onset of production of infectious forms, but limited transmission success. Viability of infectious forms declined rapidly at 23 and 30 degrees C, thereby narrowing the time window for transmission. Thus, environmental conditions can generate trade-offs between transmission relevant parameters and alter levels of multiple infection or parasite-mediated selection, which may affect evolutionary trajectories of parasite life history or virulence.     

The functions of multiple visual signals in a fiddler crab

In many species, it is common for animals to have multiple signals within one channel of communication. Multiple signals may, however, be inefficient if they are redundant in nature. Identifying the functional significance of these multiple signals is therefore important if we are to understand the evolution of such elaborated behaviours. We proposed to identify the roles of movement-based multiple signals in a model animal system. Male fiddler crabs wave their sexually dimorphic enlarged claw during social interactions. Some species present multiple signals, where the level of complexity of the movement changes. Males of Austruca mjoebergi can perform a double wave consisting of a high- followed by a low-elevation lifting of the claw, or a single wave consisting of the high-elevation movement alone. We first investigated structural differences between the double and single wave types, and found that single waves were lower in elevation than double waves. We then explored the adaptive meaning of the wave types by manipulating the social context in which males wave. We found that double waves were given in all contexts and in higher proportions at long distances, suggesting a function of broadcasting male location. Single waves, on the other hand, were mainly given at close range and in the presence of conspecifics, suggesting intraspecific communication. Female presence elicited the highest number and proportion of single waves, a likely result of a female preference for higher wave rates. Finally, we point out that there is an element of interaction between wave types that deserves future attention. This paper is an important contribution to expand our understanding of the adaptive meaning of multiple visual signals and help reach a unified theory of their evolution.