Contributions to the mathematical theory of epidemics—II. The problem of endemicity

(1) A mathematical investigation has been made of the prevalence of a disease in a population from which certain individuals are being removed as the result of the disease, whilst fresh individuals are being introduced as the result of birth or immigration. Allowance is made for the effects of the immunity produced as the result of an attack of the disease, but the effect of deaths from other causes is not taken into account, and the action of the disease is supposed to be independent of the age of the individual. (2) As a special case of the above, results have been obtained for a closed population in which no deaths occur and to which no fresh individuals are added, but in which the individuals after being infected acquire immunity, and then may be again infected. A threshold density of population exists analogous to that described in the previous paper, which is such that no disease can exist in a population, the density of which is below the threshold. (3) In other special cases investigated when either immigration or birth is operative in the supply of fresh individuals, as well as in the general case, only one steady state of disease is possible. To reach this state the population must be of a certain density which will be determined by the functions characterizing the infectivity, morbidity, etc., of the disease. (4) Increase of the immigration rate or of the birth-rate results in an increase in the rate of infection of the healthy individuals and also in the percentage rate of infection, the percentage of sick, and in the percentage of mortality from the disease. This result is, of course, a necessary consequence of our assumption that the disease is the only cause of death. (5) More particular results have been obtained by substituting constants in the place of the undetermined functions assumed in the general theory. Further, under these conditions the nature of the steady states has been more fully investigated and it has been shown that in all cases, except one, the steady states are stable ones. In the exception, a disturbance would result in purely periodic oscillations about the steady state.     

Fatal disease and demographic Allee effect: population persistence and extinction

If a healthy stable host population at the disease-free equilibrium is subject to the Allee effect, can a small number of infected individuals with a fatal disease cause the host population to go extinct? That is, does the Allee effect matter at high densities? To answer this question, we use a susceptible–infected epidemic model to obtain model parameters that lead to host population persistence (with or without infected individuals) and to host extinction. We prove that the presence of an Allee effect in host demographics matters even at large population densities. We show that a small perturbation to the disease-free equilibrium can eventually lead to host population extinction. In addition, we prove that additional deaths due to a fatal infectious disease effectively increase the Allee threshold of the host population demographics.     

Fatal disease and demographic Allee effect: population persistence and extinction

If a healthy stable host population at the disease-free equilibrium is subject to the Allee effect, can a small number of infected individuals with a fatal disease cause the host population to go extinct? That is, does the Allee effect matter at high densities? To answer this question, we use a susceptible-infected epidemic model to obtain model parameters that lead to host population persistence (with or without infected individuals) and to host extinction. We prove that the presence of an Allee effect in host demographics matters even at large population densities. We show that a small perturbation to the disease-free equilibrium can eventually lead to host population extinction. In addition, we prove that additional deaths due to a fatal infectious disease effectively increase the Allee threshold of the host population demographics.     

Spatial heterogeneity, social structure and disease dynamics of animal populations

Social groupings, population dynamics and population movements of animals all give rise to spatio-temporal variations in population levels. These variations may be of crucial importance when considering the spread of infectious diseases since infection levels do not increase unless there is a sufficient pool of susceptible individuals. This paper explores the impact of social groupings on the potential for an endemic disease to develop in a spatially explicit model system. Analysis of the model demonstrates that the explicit inclusion of space allows asymmetry between groups to arise when this was not possible in the equivalent spatially homogeneous system. Moreover, differences in movement behaviours for susceptible and infected individuals gives rise to different spatial profiles for the populations. These profiles were not observed in previous work on an epidemic system. The results are discussed in an ecological context with reference to furious and dumb strains of infectious diseases.     

Immunity in society: diverse solutions to common problems

Understanding how organisms fight infection has been a central focus of scientific research and medicine for the past couple of centuries, and a perennial object of trial and error by humans trying to mitigate the burden of disease. Vaccination success relies upon the exposure of susceptible individuals to pathogen constituents that do not cause (excessive) pathology and that elicit specific immune memory. Mass vaccination allows us to study how immunity operates at the group level; denser populations are more prone to transmitting disease between individuals, but once a critical proportion of the population becomes immune, "herd immunity" emerges. In social species, the combination of behavioural control of infection--e.g., segregation of sick individuals, disposal of the dead, quality assessment of food and water--and aggregation of immune individuals can protect non-immune members from disease. While immune specificity and memory are well understood to underpin immunisation in vertebrates, it has been somewhat surprising to find similar phenomena in invertebrates, which lack the vertebrate molecular mechanisms deemed necessary for immunisation. Indeed, reports showing alternative forms of immune memory are accumulating in invertebrates. In this issue of PLoS Biology, Konrad et al. present an example of fungus-specific immune responses in social ants that lead to the active immunisation of nestmates by infected individuals. These findings join others in showing how organisms evolved diverse mechanisms that fulfil common functions, namely the discrimination between pathogens, the transfer of immunity between related individuals, and the group-level benefits of immunisation.     

Individual variations in infectiousness explain long-term disease persistence in wildlife populations

Viral disease persistence in species without a reservoir host is of importance for public health and disease management. But how can disease persistence be explained? We developed a spatially‐explicit individual‐based model that takes into account both ecological and viral traits as well as variable space to test disease persistence hypotheses under debate. We introduce a novel concept of modeling alternative disease courses at the individual level, causing transient infections or killing infected animals, with the lethally infected having a variable life‐expectancy. We systematically distinguish between disease invasion and persistence. We use classical swine fever (CSF), an economically very important livestock disease in a social host, the wild boar, as a reference system to test and rank the persistence hypotheses under debate. Parameter values for host population demographics and CSF epidemiology reflect current knowledge. Sensitivity analysis of the model parameters revealed that the most important factor for disease persistence is a disease profile with mostly transient, i.e. surviving individuals requiring immunity, and some chronically, long‐term infected animals. Immune individuals can constantly produce susceptible offspring, while some chronically infected individuals act as ‘super spreaders’ in time. Thus, variations in the course of the disease at the individual level are important factors determining persistence, which is usually not taken into account in the prominent measure of epidemiology, i.e. the basic reproductive number R₀, which reflects the ‘reproductive potential’ of the infected sub‐population. We discuss our results with regard to the general issues of modeling epidemics and disease management issues.     

Are we underestimating the diversity and incidence of insect bacterial symbionts? A case study in ladybird beetles

Vertically transmitted bacterial symbionts are common in arthropods. However, estimates of their incidence and diversity are based on studies that test for a single bacterial genus and often only include small samples of each host species. Focussing on ladybird beetles, we collected large samples from 21 species and tested them for four different bacterial symbionts. Over half the species were infected, and there were often multiple symbionts in the same population. In most cases, more females than males were infected, suggesting that the symbionts may be sex ratio distorters. Many of these infections would have been missed in previous studies as they only infect a small proportion of the population. Furthermore, 11 out of the 17 symbionts discovered by us were either in the genus Rickettsia or Spiroplasma, which are rarely sampled. Our results suggest that the true incidence and diversity of bacterial symbionts in insects may be far greater than previously thought.     

Population size influences amphibian detection probability: implications for biodiversity monitoring programs

Monitoring is an integral part of species conservation. Monitoring programs must take imperfect detection of species into account in order to be reliable. Theory suggests that detection probability may be determined by population size but this relationship has not yet been assessed empirically. Population size is particularly important because it may induce heterogeneity in detection probability and thereby cause bias in estimates of biodiversity. We used a site occupancy model to analyse data from a volunteer-based amphibian monitoring program to assess how well different variables explain variation in detection probability. An index to population size best explained detection probabilities for four out of six species (to avoid circular reasoning, we used the count of individuals at a previous site visit as an index to current population size). The relationship between the population index and detection probability was positive. Commonly used weather variables best explained detection probabilities for two out of six species. Estimates of site occupancy probabilities differed depending on whether the population index was or was not used to model detection probability. The relationship between the population index and detectability has implications for the design of monitoring and species conservation. Most importantly, because many small populations are likely to be overlooked, monitoring programs should be designed in such a way that small populations are not overlooked. The results also imply that methods cannot be standardized in such a way that detection probabilities are constant. As we have shown here, one can easily account for variation in population size in the analysis of data from long-term monitoring programs by using counts of individuals from surveys at the same site in previous years. Accounting for variation in population size is important because it can affect the results of long-term monitoring programs and ultimately the conservation of imperiled species.     

Infection with cowpox virus decreases female maturation rates in wild populations of woodland rodents

Sublethal effects of parasitic infection, such as reductions in reproductive rate, can significantly affect host population dynamics. Here we show that in wild populations of both Clethrionomys glareolus (bank vole) and Apodemus sylvaticus (wood mouse), females infected with cowpox virus are likely to delay maturation and therefore reproduction – in most cases until the following breeding season. Some infected bank voles do mature in their year of birth but still take longer than uninfected females. Together with our previous demonstration that individuals infected with cowpox virus in the summer survive better than uninfected individuals, these results support the prediction that hosts that develop an acute infection may best optimise their fitness by decreasing current reproduction to maximise the probability of surviving infection. Moreover, as the proportion of individuals infected increases with density, the reduction in host fecundity may have significant consequences for host dynamics.     

How many species are infected with Wolbachia? Cryptic sex ratio distorters revealed to be common by intensive sampling.

Inherited bacterial symbionts from the genus Wolbachia have attracted much attention by virtue of their ability to manipulate the reproduction of their arthropod hosts. The potential importance of these bacteria has been underlined by surveys, which have estimated that 17% of insect species are infected. We examined whether these surveys have systematically underestimated the proportion of infected species through failing to detect the low-prevalence infections that are expected when Wolbachia distorts the sex ratio. We estimated the proportion of species infected with Wolbachia within Acraea butterflies by testing large collections of each species for infection. Seven out of 24 species of Acraea were infected with Wolbachia. Four of these were infected with Wolbachia at high prevalence, a figure compatible with previous broad-scale surveys, whilst three carried low-prevalence infections that would have had a very low likelihood of being detected by previous sampling methods. This led us to conclude that sex-ratio-distorting Wolbachia may be common in insects that have an ecology and/or genetics that permit the invasion of these parasites and that previous surveys may have seriously underestimated the proportion of species that are infected.     

Modeling Inter‐Subject Variability in fMRI Activation Location: A Bayesian Hierarchical Spatial Model

Summary The aim of this article is to develop a spatial model for multi‐subject fMRI data. There has been extensive work on univariate modeling of each voxel for single and multi‐subject data, some work on spatial modeling of single‐subject data, and some recent work on spatial modeling of multi‐subject data. However, there has been no work on spatial models that explicitly account for inter‐subject variability in activation locations. In this article, we use the idea of activation centers and model the inter‐subject variability in activation locations directly. Our model is specified in a Bayesian hierarchical framework which allows us to draw inferences at all levels: the population level, the individual level, and the voxel level. We use Gaussian mixtures for the probability that an individual has a particular activation. This helps answer an important question that is not addressed by any of the previous methods: What proportion of subjects had a significant activity in a given region. Our approach incorporates the unknown number of mixture components into the model as a parameter whose posterior distribution is estimated by reversible jump Markov chain Monte Carlo. We demonstrate our method with a fMRI study of resolving proactive interference and show dramatically better precision of localization with our method relative to the standard mass‐univariate method. Although we are motivated by fMRI data, this model could easily be modified to handle other types of imaging data.     

Endemic disease, awareness, and local behavioural response

The spread of a contagious disease is often accompanied by a rise in awareness of those in the social vicinity of infected individuals, and a subsequent change in behaviour. Such reactions can manifest themselves in lower susceptibility as people try to prevent themselves from catching the disease, but also in lower infectivity because of self-imposed quarantine or better hygiene, shorter durations of infectiousness or longer immunity. We here focus on the scenario of an endemic disease of which members of the population can be either aware or unaware, and consider a broad set of possible reactions. We quantify the impact on the endemicity of a disease in a well-mixed population under the variation of different disease parameters as a consequence of growing awareness in the population. Applying a pair-closure scheme allows us to analyse the effect of local correlations if aware individuals tend to occur near infected cases, and to link this to the amount of overlap between the networks underlying the spread of awareness and disease, respectively. Lastly, we study the consequences on the dynamics when the pathogen and awareness spread at different velocities.     

The effect of density-dependent treatment and behavior change on the dynamics of HIV transmission

In this work, we propose a model for heterosexual transmission of HIV/AIDS in a population of varying size with an intervention program in which treatment and/or behavior change of the infecteds occur as an increasing function of the density of the infected class in the population. This assumption has socio-economic implications which is important for public health considerations since density-dependent treatment/behavior change may be more cost-saving than a program where treatment/behavior change occurs linearly with respect to the number of infecteds. We will make use of the conservation law of total sexual contacts which enables us to reduce the two-sex model to a simpler one-sex formulation. Analytical results will be given. Unlike a similar model with linear treatment/behavior change in Hsieh (1996) where conditions were obtained for the eradication of disease, we will show that density-dependent treatment/behavior change cannot eradicate the disease if the disease is able to persist without any treatment/behavior change. This work demonstrates the need to further understand how treatment/behavior change occurs in a society with varying population.     

Autophagy plays an important role in the containment of HIV-1 in nonprogressor-infected patients

Recent in vitro studies have suggested that autophagy may play a role in both HIV-1 replication and disease progression. In this study we investigated whether autophagy protects the small proportion of HIV-1 infected individuals who remain clinically stable for years in the absence of antiretroviral therapy, these named long-term nonprogressors (LTNP) and elite controllers (EC). We found that peripheral blood mononuclear cells (PBMC) of the HIV-1 controllers present a significantly higher amount of autophagic vesicles associated with an increased expression of autophagic markers with respect to normal progressors. Of note, ex vivo treatment of PBMC from the HIV-1 controllers with the MTOR inhibitor rapamycin results in a more efficient autophagic response, leading to a reduced viral production. These data lead us to propose that autophagy contributes to limiting viral pathogenesis in HIV-1 controllers by targeting viral components for degradation.     

Dynamics of Specific Anti-Mycobacterium avium Subsp. paratuberculosis Antibody Response through Age

Mycobacterium avium subsp. paratuberculosis (MAP) causes a chronic infection in cattle. MAP infected cattle with humoral immune (HI) reactions with IgG antibodies are usually those where latency of infection has ceased and their infection is progressing towards reduced milk yield, weight loss and significant bacterial excretion in feces. The proportion of detectable infections among all infected animals that will develop disease is often referred to as ‘the tip of the iceberg’. The purpose of this study was to estimate this proportion. Test-records from 18,972 Danish dairy cows with MAP specific IgG antibodies on their final test-record were used to estimate age-specific sensitivities (Se). These cows were the infected ones considered to develop disease in a population with a representative age-distribution and were defined as cases. The specificity (Sp) of the test was estimated based on test-results from 166,905 cows, which had no MAP IgG antibodies in their final four test-records. The Sp, age-specific Se and maximum Se were used to estimate the probability of having HI at a given age resulting in the proportion of infected cows with HI at a given age. For cows 2 years of age, the proportion of detectable cases was 0.33, while it was 0.94 for cows 5 years of age. Thus, there was a significant shift in the tip of the iceberg with aging. This study provided a model for estimating the proportion of latent chronic infections that would progress to disease, and the results can be used to model infection dynamics.     

Increasing sequence correlation limits the efficiency of recombination in a multisite evolution model

The accumulation of preexisting beneficial alleles in a haploid population, under selection and infrequent recombination, and in the absence of new mutation events is studied numerically by means of detailed Monte Carlo simulations. On the one hand, we confirm our previous work, in that the accumulation rate follows modified single-site kinetics, with a timescale set by an effective selection coefficient s(eff) as shown in a previous work, and we confirm the qualitative features of the dependence of s(eff) on the population size and the recombination rate reported therein. In particular, we confirm the existence of a threshold population size below which evolution stops before the emergence of best-fit individuals. On the other hand, our simulations reveal that the population dynamics is essentially shaped by the steady accumulation of pairwise sequence correlation, causing sequence congruence in excess of what one would expect from a uniformly random distribution of alleles. By sequence congruence, we understand here the opposite of genetic distance, that is, the fraction of monomorphic sites of specified allele type in a pair of genomes (individual sequences). The effective selection coefficient changes more rapidly with the recombination rate and has a higher threshold in this parameter than found in the previous work, which neglected correlation effects. We examine this phenomenon by monitoring the time dependence of sequence correlation based on a set of sequence congruence measures and verify that it is not associated with the development of linkage disequilibrium. We also discuss applications to HIV evolution in infected individuals and potential implications for drug therapy.     

The effects of population structure on the spread of the HIV infection

A model for the spread of human immunodeficiency virus (HIV) in a population of male homosexuals is presented. The population is divided into five groups on the basis of degree of sexual activity. Within each group, the individuals are classified as 1) susceptible; 2) infective; or 3) removed because of a lack of sexual activity associated with advanced acquired immunodeficiency disease (AIDS). The infective individuals are further subdivided into four stages of infection. Analyses of the model address two questions with regard to the spread of HIV: (1) What is the effect of level of sexual activity on an individual's risk for infection, and (2) What is the effect that assumptions about mixing between groups have on both individual risk and transmission throughout a population? Results from analyses using a number of different parameter estimates show that increased levels of sexual activity increase the likelihood that an individual will become infected. In addition, the initial spread of the disease is markedly affected by variation in the amount of contact among individuals from different subpopulations. The steady-state incidence of the disease is not markedly affected by variation in the contact patterns, but the size of the steady-state population and therefore the proportion of infected individuals in the population does vary significantly with changes in the degree of mixing among subpopulations. These results show clearly the sensitivity of model outcomes to variation in the patterns of contact among individuals and the need for better data on such interactions to aid in understanding and predicting the spread of HIV.     

Quantifying the disease transmission function: effects of density on Batrachochytrium dendrobatidis transmission in the mountain yellow-legged frog Rana muscosa

1. Chytridiomycosis is an emerging infectious disease of amphibians, caused by the fungal pathogen Batrachochytrium dendrobatidis, which has been implicated recently in population declines and possible extinctions throughout the world. 2. The transmission rate of this pathogen was quantified in the mountain yellow-legged frog Rana muscosa through laboratory and field experiments, and a maximum likelihood approach was used to determine the form of the transmission function that was best supported by the experimental data. 3. The proportion of R. muscosa tadpole hosts that became infected increased with the number of previously infected R. muscosa tadpoles to which they were exposed, as would be expected in an infectious disease. 4. The laboratory experiment revealed some support for a transmission function in which the transmission rate levels off as the density of infected individuals increases. However, there was not enough power to distinguish between a frequency-dependent form and several other asymptotic forms of the transmission function. 5. The impacts of crowding and temperature on transmission were also investigated; however, neither of these factors significantly affected the transmission rate.     

The effects of spatial movement and group interactions on disease dynamics of social animals

The effects of spatial movements of infected and susceptible individuals on disease dynamics is not well understood. Empirical studies on the spatial spread of disease and behaviour of infected individuals are few and theoretical studies may be useful to explore different scenarios. Hence due to lack of detail in empirical studies, theoretical models have become necessary tools in investigating the disease influence in host-pathogen systems. In this paper we developed and analysed a spatially explicit model of two interacting social groups of animals of the same species. We investigated how the movement scenarios of susceptible and infected individuals together with the between-group contact parameter affect the survival rate of susceptible individuals in each group. This work can easily be applied to various host-pathogen systems. We define bounds on the number of susceptibles which avoid infection once the disease has died out as a function of the initial conditions and other model parameters. For example, once disease has passed through the populations, a larger diffusion coefficient for each group can result in higher population levels when there is no between-group interaction but in lower levels when there is between-group interaction. Numerical simulations are used to demonstrate these bounds and behaviours and to describe the different outcomes in ecological terms.     

Whole-Exome Sequencing Reveals a Rapid Change in the Frequency of Rare Functional Variants in a Founding Population of Humans

Whole-exome or gene targeted resequencing in hundreds to thousands of individuals has shown that the majority of genetic variants are at low frequency in human populations. Rare variants are enriched for functional mutations and are expected to explain an important fraction of the genetic etiology of human disease, therefore having a potential medical interest. In this work, we analyze the whole-exome sequences of French-Canadian individuals, a founder population with a unique demographic history that includes an original population bottleneck less than 20 generations ago, followed by a demographic explosion, and the whole exomes of French individuals sampled from France. We show that in less than 20 generations of genetic isolation from the French population, the genetic pool of French-Canadians shows reduced levels of diversity, higher homozygosity, and an excess of rare variants with low variant sharing with Europeans. Furthermore, the French-Canadian population contains a larger proportion of putatively damaging functional variants, which could partially explain the increased incidence of genetic disease in the province. Our results highlight the impact of population demography on genetic fitness and the contribution of rare variants to the human genetic variation landscape, emphasizing the need for deep cataloguing of genetic variants by resequencing worldwide human populations in order to truly assess disease risk.