Modeling Tick-Borne Disease: A Metapopulation Model

Recent increases in reported outbreaks of tick-borne diseases have led to increased interest in understanding and controlling epidemics involving these transmission vectors. Mathematical disease models typically assume constant population size and spatial homogeneity. For tick-borne diseases, these assumptions are not always valid. The disease model presented here incorporates non-constant population sizes and spatial heterogeneity utilizing a system of differential equations that may be applied to a variety of spatial patches. We present analytical results for the one patch version and find parameter restrictions under which the populations and infected densities reach equilibrium. We then numerically explore disease dynamics when parameters are allowed to vary spatially and temporally and consider the effectiveness of various tick-control strategies.     

Covariation between the physiological and behavioral components of pathogen transmission: host heterogeneity determines epidemic outcomes

Although heterogeneity in contact rate, physiology, and behavioral response to infection have all been empirically demonstrated in host–pathogen systems, little is known about how interactions between individual variation in behavior and physiology scale‐up to affect pathogen transmission at a population level. The objective of this study is to evaluate how covariation between the behavioral and physiological components of transmission might affect epidemic outcomes in host populations. We tested the consequences of contact rate covarying with susceptibility, infectiousness, and infection status using an individual‐based, dynamic network model where individuals initiate and terminate contacts with conspecifics based on their behavioral predispositions and their infection status. Our results suggest that both heterogeneity in physiology and subsequent covariation of physiology with contact rate could powerfully influence epidemic dynamics. Overall, we found that 1) individual variability in susceptibility and infectiousness can reduce the expected maximum prevalence and increase epidemic variability; 2) when contact rate and susceptibility or infectiousness negatively covary, it takes substantially longer for epidemics to spread throughout the population, and rates of epidemic spread remained suppressed even for highly transmissible pathogens; and 3) reductions in contact rate resulting from infection‐induced behavioral changes can prevent the pathogen from reaching most of the population. These effects were strongest for theoretical pathogens with lower transmissibility and for populations where the observed variation in contact rate was higher, suggesting that such heterogeneity may be most important for less infectious, more chronic diseases in wildlife. Understanding when and how variability in pathogen transmission should be modelled is a crucial next step for disease ecology.     

Selective predation and rapid evolution can jointly dampen effects of virulent parasites on Daphnia populations

Parasites are ubiquitous and often highly virulent, yet clear examples of parasite-driven changes in host density in natural populations are surprisingly scarce. Here, we illustrate an example of this phenomenon and offer a theoretically reasonable resolution. We document the effects of two parasites, the bacterium Spirobacillus cienkowskii and the yeast Metschnikowia bicuspidata, on a common freshwater invertebrate, Daphnia dentifera. We show that while both parasites were quite virulent to individual hosts, only bacterial epidemics were associated with significant changes in host population dynamics and density. Our theoretical results may help explain why yeast epidemics did not significantly affect population dynamics. Using a model parameterized with data we collected, we argue that two prominent features of this system, rapid evolution of host resistance to the parasite and selective predation on infected hosts, both decrease peak infection prevalence and can minimize decline in host density during epidemics. Taken together, our results show that understanding the outcomes of host-parasite interactions in this Daphnia-microparasite system may require consideration of ecological context and evolutionary processes and their interaction.     

Effects of host heterogeneity on pathogen diversity and evolution

Phenotypic variation is common in most pathogens, yet the mechanisms that maintain this diversity are still poorly understood. We asked whether continuous host variation in susceptibility helps maintain phenotypic variation, using experiments conducted with a baculovirus that infects gypsy moth (Lymantria dispar) larvae. We found that an empirically observed tradeoff between mean transmission rate and variation in transmission, which results from host heterogeneity, promotes long‐term coexistence of two pathogen types in simulations of a population model. This tradeoff introduces an alternative strategy for the pathogen: a low‐transmission, low‐variability type can coexist with the high‐transmission type favoured by classical non‐heterogeneity models. In addition, this tradeoff can help explain the extensive phenotypic variation we observed in field‐collected pathogen isolates, in traits affecting virus fitness including transmission and environmental persistence. Similar heterogeneity tradeoffs might be a general mechanism promoting phenotypic variation in any pathogen for which hosts vary continuously in susceptibility.     

A generalized stochastic model for the analysis of infectious disease final size data.

A stochastic infectious disease model was developed by Ball (1986, Advances in Applied Probability 18, 289-310) in which the distribution of the length of the infectious period is allowed to have any distribution that can be described by its Laplace transform. We extend this model such that the infection can be transmitted within the population or from an unspecified source outside the population. Also, discrete heterogeneity in the population can be modeled to incorporate variable susceptibility, variable infectivity, and/or mixing behaviors. The model is fitted to serologic data from two influenza epidemics in Tecumseh, Michigan, using maximum likelihood estimation procedures. The estimates show a clustering pattern by age groups.     

Analytic models for the patchy spread of plant disease

Plant epidemiologists have long been concerned with the patchy nature of plant disease epidemics. This paper presents a new analytical model for patchy plant epidemics (and patchy dynamics in general), using a second-order approximation to capture the spatial dynamics in terms of the densities and spatial covariances of healthy and infected hosts. Using these spatial moment equations helps us to explain the dynamic growth of patchiness during the early phase of the epidemic, and how the patchiness feeds back on the growth rate of the epidemic. Both underlying heterogeneity in the host spatial arrangement and dynamically generated heterogeneity in the spatial arrangement of infected plants initially accelerate but later decelerate the epidemic.     

Epidemiological and evolutionary consequences of targeted vaccination

Recent theory has examined the way in which vaccination strategies are expected to influence the evolution of parasite virulence. Most of this work has assumed that vaccination is imposed on a homogeneous host population. However, host populations are typically composed of different types of individuals, with each type responding differently to infection. Moreover, actual interventions often focus treatment on those hosts that are likely to suffer the most ill effects of a particular disease. Here we consider the epidemiological and evolutionary consequences of interventions that focus vaccination on individuals expressing the greatest susceptibility to infection and/or the greatest vulnerability to mortality once infected. Our results indicate that predictions are very sensitive to the nature and degree of heterogeneity in susceptibility and vulnerability. They further suggest that accounting for realistic kinds of heterogeneity when contemplating targeted treatment plans and policies might provide a new tool in the design of more effective virulence management strategies.     

Unhealthy herds: Some epidemiological consequences of host heterogeneity in predator-host-parasite systems

Models in which all hosts respond in the same fashion to challenge by disease make a number of clear predictions regarding the ameliorating effect of predation on disease burden in prey populations. However, natural populations are typically exposed to a broad spectrum of stressors, some of which can induce changes in an individual's susceptibility to infection and transmission, as well as vulnerability to mortality once infected. When only a subset of the population is exposed to these other factors, host populations will express some heterogeneity in resistance to disease. Here I investigate the influence that such heterogeneity can have on the predicted beneficial epidemiological effect of predators on certain homogeneous prey populations. Results show that, under some conditions, predation can exacerbate disease burden in the heterogeneous prey population. I conclude that such a possibility might have implications for wild and domesticated animal management programs.     

Host-pathogen interactions, insect outbreaks, and natural selection for disease resistance

The theory of insect population dynamics has shown that heterogeneity in natural-enemy attack rates is strongly stabilizing. We tested the usefulness of this theory for outbreaking insects, many of which are attacked by infectious pathogens. We measured heterogeneity among gypsy moth larvae in their risk of infection with a nucleopolyhedrovirus, which is effectively heterogeneity in the pathogen's attack rate. Our data show that heterogeneity in infection risk in this insect is so high that it leads to a stable equilibrium in the models, which is inconsistent with the outbreaks seen in North American gypsy moth populations. Our data further suggest that infection risk declines after epidemics, in turn suggesting that the model assumption of constant infection risk is incorrect. We therefore constructed an alternative model in which natural selection drives fluctuations in infection risk, leading to reductions after epidemics because of selection for resistance and increases after epidemics because of a cost of resistance. This model shows cycles even for high heterogeneity, and experiments confirm that infection risk is indeed heritable. The model is very general, and so we argue that natural selection for disease resistance may play a role in many insect outbreaks.     

Parasite strain coexistence in a heterogeneous host population

Heterogeneity in host susceptibility and transmissibility to parasite attack allows a lower transmission rate to sustain an epidemic than is required in homogeneous host populations. However, this heterogeneity can leave some hosts with little susceptibility to disease, and at high transmission rates, epidemic size can be smaller than for diseases where the host population is homogeneous. In a heterogeneous host population, we model natural selection in a parasite population where host heterogeneity is exploited by different strains to varying degrees. This partitioning of the host population allows coexistence of competing parasite strains, with the heterogeneity-exploiting strains infecting the more susceptible hosts, in the absence of physiological tradeoffs and spatial heterogeneity, and even for markedly different transmission rates. In our model, intermediate-strategy parasites were selected against: should coexistence occur, an equilibrium is reached where strains occupied only the extreme ends of trait space, under appropriate conditions selecting for lower R₀.     

Linking field epidemiological and individual plant resistance patterns in the Betula pubescens–Melampsoridium betulinum host–pathogen interaction

Over a period of seven seasons (1995–2001 inclusively) four distinct epidemics of the rust pathogen Melampsoridium betulinum were recorded in stands of Betula pubescens . During those epidemics, host plants incurred varying levels of disease severity. Some individuals suffered high levels on all occasions; some low; while yet others were either intermediately affected or showed variable disease severity from season to season. Tests of pathotypic differences among 33 isolates of M. betulinum collected from six sites found a broad range of pathotypes ranging from highly virulent to highly avirulent. Similarly a random sample of 40 B. pubescens lines from one site showed a wide range of resistance phenotypes, although individuals that were either susceptible to all but one pathotype or resistant to all pathotypes were commonest. A strong relationship existed between the cumulative level of disease incurred by B. pubescens individuals in the field and the mean susceptibility of each host line as determined by their reaction to infection by each of the 33 different isolates of M. betulinum individually. Resistance in this B. pubescens population to M. betulinum is postulated to be based on a mixture of quantitative and qualitative traits, selection for which has resulted from an interplay of life history attributes of both host and pathogen.     

Pathogen-Driven Outbreaks in Forest Defoliators Revisited: Building Models from Experimental Data

Models of outbreaks in forest-defoliating insects are typically built from a priori considerations and tested only with long time series of abundances. We instead present a model built from experimental data on the gypsy moth and its nuclear polyhedrosis virus, which has been extensively tested with epidemic data. These data have identified key details of the gypsy moth-virus interaction that are missing from earlier models, including seasonality in host reproduction, delays between host infection and death, and heterogeneity among hosts in their susceptibility to the virus. Allowing for these details produces models in which annual epidemics are followed by bouts of reproduction among surviving hosts and leads to quite different conclusions than earlier models. First, these models suggest that pathogen-driven outbreaks in forest defoliators occur partly because newly hatched insect larvae have higher average susceptibility than do older larvae. Second, the models show that a combination of seasonality and delays between infection and death can lead to unstable cycles in the absence of a stabilizing mechanism; these cycles, however, are stabilized by the levels of heterogeneity in susceptibility that we have observed in our experimental data. Moreover, our experimental estimates of virus transmission rates and levels of heterogeneity in susceptibility in gypsy moth populations give model dynamics that closely approximate the dynamics of real gypsy moth populations. Although we built our models from data for gypsy moth, our models are, nevertheless, quite general. Our conclusions are therefore likely to be true, not just for other defoliator-pathogen interactions, but for many host-pathogen interactions in which seasonality plays an important role. Our models thus give qualitative insight into the dynamics of host-pathogen interactions, while providing a quantitative interpretation of our gypsy moth-virus data.     

Attack rates of seasonal epidemics

We study the attack rate, that is the total fraction of the population infected each year, for a disease with seasonally varying transmission rate. The attack rate is shown to be governed by both the reproductive number, reflecting the transmissibility of the disease, and the birth rate, which provides a source of new susceptibles. For the case of epidemics which have an annual period (like the seasonality), we prove inequalities which show that the attack rate is close to that of the non-seasonal model, so that it is nearly independent of the strength of the forcing, despite the fact that the shape of the epidemic curve depends strongly on the degree of seasonality of the forcing. Numerical simulations show that this holds to an even stronger extent than is implied by our rigorous results. When the system has subharmonic or chaotic solutions, we show that similar results hold when the attack rate is replaced by the average attack rate over several years. Consequences of these findings for analyzing the effect of vaccination in seasonally-forced models are noted.     

Lions and Prions and Deer Demise

Background

Contagious prion diseases – scrapie of sheep and chronic wasting disease of several species in the deer family – give rise to epidemics that seem capable of compromising host population viability. Despite this prospect, the ecological consequences of prion disease epidemics in natural populations have received little consideration.

Methodology/Principal Findings

Using a cohort study design, we found that prion infection dramatically lowered survival of free-ranging adult (>2-year-old) mule deer (Odocoileus hemionus): estimated average life expectancy was 5.2 additional years for uninfected deer but only 1.6 additional years for infected deer. Prion infection also increased nearly fourfold the rate of mountain lions (Puma concolor) preying on deer, suggesting that epidemics may alter predator–prey dynamics by facilitating hunting success. Despite selective predation, about one fourth of the adult deer we sampled were infected. High prevalence and low survival of infected deer provided a plausible explanation for the marked decline in this deer population since the 1980s.

Conclusion

Remarkably high infection rates sustained in the face of intense predation show that even seemingly complete ecosystems may offer little resistance to the spread and persistence of contagious prion diseases. Moreover, the depression of infected populations may lead to local imbalances in food webs and nutrient cycling in ecosystems in which deer are important herbivores.     

A diffusive SI model with Allee effect and application to FIV

A minimal reaction-diffusion model for the spatiotemporal spread of an infectious disease is considered. The model is motivated by the Feline Immunodeficiency Virus (FIV) which causes AIDS in cat populations. Because the infected period is long compared with the lifespan, the model incorporates the host population growth. Two different types are considered: logistic growth and growth with a strong Allee effect. In the model with logistic growth, the introduced disease propagates in form of a travelling infection wave with a constant asymptotic rate of spread. In the model with Allee effect the spatiotemporal dynamics are more complicated and the disease has considerable impact on the host population spread. Most importantly, there are waves of extinction, which arise when the disease is introduced in the wake of the invading host population. These waves of extinction destabilize locally stable endemic coexistence states. Moreover, spatially restricted epidemics are possible as well as travelling infection pulses that correspond either to fatal epidemics with succeeding host population extinction or to epidemics with recovery of the host population. Generally, the Allee effect induces minimum viable population sizes and critical spatial lengths of the initial distribution. The local stability analysis yields bistability and the phenomenon of transient epidemics within the regime of disease-induced extinction. Sustained oscillations do not exist.     

How do pathogen evolution and host heterogeneity interact in disease emergence?

Heterogeneity in the parameters governing the spread of infectious diseases is a common feature of real-world epidemics. It has been suggested that for pathogens with basic reproductive number R(0)>1, increasing heterogeneity makes extinction of disease more likely during the early rounds of transmission. The basic reproductive number R(0) of the introduced pathogen may, however, be less than 1 after the introduction, and evolutionary changes are then required for R(0) to increase to above 1 and the pathogen to emerge. In this paper, we consider how host heterogeneity influences the emergence of both non-evolving pathogens and those that must undergo adaptive changes to spread in the host population. In contrast to previous results, we find that heterogeneity does not always make extinction more likely and that if adaptation is required for emergence, the effect of host heterogeneity is relatively small. We discuss the application of these ideas to vaccination strategies.     

Use of multivariate linkage analysis for dissection of a complex cognitive trait

Replication of linkage results for complex traits has been exceedingly difficult, owing in part to the inability to measure the precise underlying phenotype, small sample sizes, genetic heterogeneity, and statistical methods employed in analysis. Often, in any particular study, multiple correlated traits have been collected, yet these have been analyzed independently or, at most, in bivariate analyses. Theoretical arguments suggest that full multivariate analysis of all available traits should offer more power to detect linkage; however, this has not yet been evaluated on a genomewide scale. Here, we conduct multivariate genomewide analyses of quantitative-trait loci that influence reading- and language-related measures in families affected with developmental dyslexia. The results of these analyses are substantially clearer than those of previous univariate analyses of the same data set, helping to resolve a number of key issues. These outcomes highlight the relevance of multivariate analysis for complex disorders for dissection of linkage results in correlated traits. The approach employed here may aid positional cloning of susceptibility genes in a wide spectrum of complex traits.     

Infection of the fittest: devil facial tumour disease has greatest effect on individuals with highest reproductive output

Emerging infectious diseases rarely affect all members of a population equally and determining how individuals’ susceptibility to infection is related to other components of their fitness is critical to understanding disease impacts at a population level and for predicting evolutionary trajectories. We introduce a novel state‐space model framework to investigate survival and fecundity of Tasmanian devils (Sarcophilus harrisii) affected by a transmissible cancer, devil facial tumour disease. We show that those devils that become host to tumours have otherwise greater fitness, with higher survival and fecundity rates prior to disease‐induced death than non‐host individuals that do not become infected, although high tumour loads lead to high mortality. Our finding that individuals with the greatest reproductive value are those most affected by the cancer demonstrates the need to quantify both survival and fecundity in context of disease progression for understanding the impact of disease on wildlife populations.     

Some properties of a simple stochastic epidemic model of SIR type

We investigate the properties of a simple discrete time stochastic epidemic model. The model is Markovian of the SIR type in which the total population is constant and individuals meet a random number of other individuals at each time step. Individuals remain infectious for R time units, after which they become removed or immune. Individual transition probabilities from susceptible to diseased states are given in terms of the binomial distribution. An expression is given for the probability that any individuals beyond those initially infected become diseased. In the model with a finite recovery time R, simulations reveal large variability in both the total number of infected individuals and in the total duration of the epidemic, even when the variability in number of contacts per day is small. In the case of no recovery, R=infinity, a formal diffusion approximation is obtained for the number infected. The mean for the diffusion process can be approximated by a logistic which is more accurate for larger contact rates or faster developing epidemics. For finite R we then proceed mainly by simulation and investigate in the mean the effects of varying the parameters p (the probability of transmission), R, and the number of contacts per day per individual. A scale invariant property is noted for the size of an outbreak in relation to the total population size. Most notable are the existence of maxima in the duration of an epidemic as a function of R and the extremely large differences in the sizes of outbreaks which can occur for small changes in R. These findings have practical applications in controlling the size and duration of epidemics and hence reducing their human and economic costs.