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Metapopulation dynamics play a critical role in driving endemic persistence and transmission of childhood infections. The endemic threshold concept, also referred to as critical community size (CCS), is a key example and is defined as the minimum population size required to sustain a continuous chain of infection transmission. The concept is fundamental to the implementation of effective vaccine-based disease control programmes. Vaccination serves to increase endemic threshold population size, promoting disease fadeout and eventual elimination of infection. To date, empirical investigations of the relationship between vaccination and endemic threshold population size have tended to focus on isolated populations in island communities. Very few studies have examined endemic threshold dynamics in 'mainland' regional populations with complex hierarchical spatial structures and varying levels of connectivity between subpopulations. The present paper provides the first spatially explicit analysis of the temporal changes in endemic threshold populations for one vaccine-preventable childhood infection (pertussis) in two dynamic regions of England and Wales: Lancashire and South Wales. Drawing upon weekly disease records of the Registrar-General of England and Wales over a 30-year period (January 1940-December 1969) regression techniques were used to estimate the endemic threshold size for pertussis in the two study regions. Survival analyses were performed to compare disease fadeout duration and probability for both regions in the pre-vaccine and vaccine eras, respectively. Our findings reveal the introduction of mass vaccination led to a considerable increase in threshold size for both Lancashire (~387,333) and South Wales (~1,460,667). Significant growth in fadeout duration was observed in the vaccine era for pertussis non-hotspots in both regions, consistent with geographical synchronisation of epidemic activity. Regional differences in endemic threshold populations reflect significant regional variations in spatial connectivity, population dispersion and level of geographical isolation.

After an epidemic outbreak, the infection persists in a community long enough to engulf the entire susceptible population. Local extinction of the disease could be possible if the susceptible population gets depleted. In large communities, the tendency of eventual damp down of recurrent epidemics is balanced by random variability. But, in small communities, the infection would die out when the number of susceptible falls below a certain threshold. Critical community size (CCS) is considered to be the mentioned threshold, at which the infection is as likely as not to die out after a major epidemic for small communities unless reintroduced from outside. The determination of CCS could aid in devising systematic control strategies to eradicate the infectious disease from small communities. In this article, we have come up with a simplified computation based approach to deduce the CCS of HIV disease dynamics. We consider a deterministic HIV model proposed by Silva and Torres, and following Nåsell, introduce stochasticity in the model through time-varying population sizes of different compartments. Besides, Metcalf’s group observed that the relative risk of extinction of some infections on islands is almost double that in the mainlands i.e. infections cease to exist at a significantly higher rate in islands compared to the mainlands. They attributed this phenomenon to the greater recolonization in the mainlands. Interestingly, the application of our method on demographic facts and figures of countries in the AIDS belt of Africa led us to expect that existing control measures and isolated locations would assist in temporary eradication of HIV infection much faster. For example, our method suggests that through systematic control strategies, after 7.36 years HIV epidemics will temporarily be eradicated from different communes of island nation Madagascar, where the population size falls below its CCS value, unless the disease is reintroduced from outside.

Online communities, which have become an integral part of the day-to-day life of people and organizations, exhibit much diversity in both size and activity level; some communities grow to a massive scale and thrive, whereas others remain small, and even wither. In spite of the important role of these proliferating communities, there is limited empirical evidence that identifies the dominant factors underlying their dynamics. Using data collected from seven large online platforms, we observe a relationship between online community size and its activity which generally repeats itself across platforms: First, in most platforms, three distinct activity regimes exist-one of low-activity and two of high-activity. Further, we find a sharp activity phase transition at a critical community size that marks the shift between the first and the second regime in six out of the seven online platforms. Essentially, we argue that it is around this critical size that sustainable interactive communities emerge. The third activity regime occurs above a higher characteristic size in which community activity reaches and remains at a constant and higher level. We find that there is variance in the steepness of the slope of the second regime, that leads to the third regime of saturation, but that the third regime is exhibited in six of the seven online platforms. We propose that the sharp activity phase transition and the regime structure stem from the branching property of online interactions.

Mycobacterium bovis is the cause of tuberculosis in animals and sometimes humans. Many developed nations have long-standing programmes to eradicate tuberculosis in livestock, principally cattle. As disease prevalence in cattle decreases these efforts are sometimes impeded by passage of M. bovis from wildlife to cattle. In epidemiological terms, disease can persist in some wildlife species, creating disease reservoirs, if the basic reproduction rate (R0) and critical community size (CCS) thresholds are achieved. Recognized wildlife reservoir hosts of M. bovis include the brushtail possum (Trichosurus vulpecula) in New Zealand, European badger (Meles meles) in Great Britain and Ireland, African buffalo (Syncerus caffer) in South Africa, wild boar (Sus scrofa) in the Iberian Peninsula and white-tailed deer (Odocoileus virginianus) in Michigan, USA. The epidemiological concepts of R0 and CCS are related to more tangible disease/pathogen characteristics such as prevalence, pathogen-induced pathology, host behaviour and ecology. An understanding of both epidemiological and disease/pathogen characteristics is necessary to identify wildlife reservoirs of M. bovis. In some cases, there is a single wildlife reservoir host involved in transmission of M. bovis to cattle. Complexity increases, however, in multihost systems where multiple potential reservoir hosts exist. Bovine tuberculosis eradication efforts require elimination of M. bovis transmission between wildlife reservoirs and cattle. For successful eradication identification of true wildlife reservoirs is critical, as disease control efforts are most effective when directed towards true reservoirs.

Scientific Reports, 6, ISSN:2045-2322

Measles epidemics in human populations exhibit what is perhaps the best empirically characterized, and certainly the most studied, stochastic persistence threshold in population biology. A critical community size (CCS) of around 250 000–500 000 separates populations where measles is predominantly persistent from smaller communities where there are frequent extinctions of measles between major epidemics. The fundamental mechanisms contributing to this pattern of persistence, which are long-lasting immunity to re-infection, recruitment of susceptibles, seasonality in transmission, age dependence of transmission and the spatial coupling between communities, have all been quantified and, to a greater or lesser level of success, captured by theoretical models. Despite these successes there has not been a consensus over whether simple models can successfully predict the value of the CCS, or indeed which mechanisms determine the persistence of measles over a broader range of population sizes. Specifically, the level of the CCS has been thought to be particularly sensitive to the detailed stochastic dynamics generated by the waiting time distribution (WTD) in the infectious and latent periods. We show that the relative patterns of persistence between models with different WTDs are highly sensitive to the criterion of comparison—in particular, the statistical measure of persistence that is employed. To this end, we introduce two new statistical measures of persitence—fade-outs post epidemic and fade-outs post invasion. Contrary to previous reports, we demonstrate that, no matter the choice of persistence measure, appropriately parametrized models of measles demonstrate similar predictions for the level of the CCS.

The critical community size (CCS) for measles, which separates persistent from extinction-prone populations, is arguably the best understood stochastic threshold in ecology. Using simple models, we explore a relatively neglected relationship of how the CCS scales with birth rate. A predominantly positive relationship of persistence with birth rate is complicated by the accompanying dynamical transitions of the underlying deterministic process. We show that these transitions imply a lower CCS for high birth rate less developed countries and contrary to the experience in lower birth rate, industrial countries, the CCS may increase after vaccination. We also consider the evolutionary implications of the CCS for the origin of measles; this analysis explores how the deterministic and stochastic thresholds for invasion and persistence set limits on the mechanism by which this highly infectious pathogen could have successfully colonized its human host.

Persistence and extinction are fundamental processes in ecological systems that are difficult to accurately measure due to stochasticity and incomplete observation. Moreover, these processes operate on multiple scales, from individual populations to metapopulations. Here we examine an extensive new dataset of measles case reports and associated demographics in pre-vaccine era U.S. cities, alongside a classic England & Wales dataset. We first infer the per-population quasi-continuous distribution of log incidence. We then use stochastic, spatially implicit metapopulation models to explore the frequency of rescue events and apparent extinctions. We show that, unlike critical community size, the inferred distributions account for observational processes, allowing direct comparisons between metapopulations. The inferred distributions scale with population size. We use these scalings to estimate extinction boundary probabilities. We compare these predictions with measurements in individual populations and random aggregates of populations, highlighting the importance of medium-sized populations in metapopulation persistence.

The onset of measles vaccination in England and Wales in 1968 coincided with a marked drop in the temporal correlation of epidemic patterns between major cities. We analyze a variety of hypotheses for the mechanisms driving this change. Straightforward stochastic models suggest that the interaction between a lowered susceptible population (and hence increased demographic noise) and nonlinear dynamics is sufficient to cause the observed drop in correlation. The decorrelation of epidemics could potentially lessen the chance of global extinction and so inhibit attempts at measles eradication.

Seasonally driven cycles of incidence have been consistently observed for a range of directly transmitted pathogens. Though frequently observed, the mechanism of seasonality for directly transmitted human pathogens is rarely well understood. Despite significant annual variation in magnitude, measles outbreaks in Niger consistently begin in the dry season and decline at the onset of the seasonal rains. We estimate the seasonal fluctuation in measles transmission rates for the 38 districts and urban centres of Niger, from 11 years of weekly incidence reports. We show that transmission rates are consistently in anti-phase to the rainfall patterns across the country. The strength of the seasonal forcing of transmission is not correlated with the latitudinal rainfall gradient, as would be expected if transmission rates were determined purely by environmental conditions. Rather, seasonal forcing is correlated with the population size, with larger seasonal fluctuation in more populous, urban areas. This pattern is consistent with seasonal variation in human density and contact rates due to agricultural cycles. The stronger seasonality in large cities drives deep inter-epidemic troughs and results in frequent local extinction of measles, which contrasts starkly to the conventional observation that large cities, by virtue of their size, act as reservoirs of measles.

Recent insights in compiling metapopulation models and infectious disease dynamics have enabled a better understanding of spatial and temporal persistence of diseases. The concept of the critical community size (CCS) has been developed with major consideration on analyses of whooping cough and measles dynamics from temperate countries for which large datasets are available. However, few studies have questioned the generality of CCS curves for other regions, i.e. tropical areas, mainly because data are very sparse. This work constitutes, to our knowledge, the first study of whooping cough persistence in a tropical area, in a rural part of Senegal (Western Africa) where the population has been followed for 20 years. In this small community, the CCS is not reached even if we observe CCS–like curves showing persistence increasing with population size. In addition, our analysis supports the idea that vaccination has impacted the spatio–temporal dynamics and persistence of the disease. These findings suggest the need for more comparative analyses of spatio–temporal data series from a wide range of contrasted countries, for example developed and developing nations, so as to have a better understanding of vaccination effects on disease dynamics and persistence.

Quasi-stationarity and time to extinction are studied for the classic endemic model. Attention is restricted to the transition region in parameter space where the quasi-stationary distribution is non-normal. A new approximation of the marginal distribution of infected individuals in quasi-stationarity is presented. It leads to a simple explicit expression for an approximation of the critical community size in terms of model parameters.

Empirical evidence shows that childhood diseases persist in large communities whereas in smaller communities the epidemic goes extinct (and is later reintroduced by immigration). The present paper treats a stochastic model describing the spread of an infectious disease giving life-long immunity, in a community where individuals die and new individuals are born. The time to extinction of the disease starting in quasi-stationarity (conditional on non-extinction) is exponentially distributed. As the population size grows the epidemic process converges to a diffusion process. Properties of the limiting diffusion are used to obtain an approximate expression for tau, the mean-parameter in the exponential distribution of the time to extinction for the finite population. The expression is used to study how tau depends on the community size but also on certain properties of the disease/community: the basic reproduction number and the means and variances of the latency period, infectious period and life-length. Effects of introducing a vaccination program are also discussed as is the notion of the critical community size, defined as the size which distinguishes between the two qualitatively different behaviours.

A basic issue in ecology is the relation between extinction and population size. One of the clearest manifestations of a population threshold for extinction is the critical community size below which infections like measles do not persist. The current generation of stochastic models overestimates the observed critical community size for measles, generating much less persistence of infection than is observed. The inclusion of a more biologically realistic model for the duration of infection produced a much closer fit to the actual critical community size and explains previously undescribed high-frequency oscillations in measles incidence.

This paper explores the relations between persistence and dynamics in measles epidemics. Most current models, including the stochastic seasonally forced and age-structured models examined here, fail to capture simultaneously the observed dynamics and persistence characteristics of epidemics in large urban populations before vaccination. Summary measures of persistence and trienniality allow us to compare epidemics in England, New York and Copenhagen with results of non-spatial and spatial stochastic models. Spatial (metapopulation) structure allows persistence and triennial dynamics to coexist in this class of models. The spatial dynamics of measles, for which detailed spatiotemporal data are available, may serve as a useful test of ideas applicable to other epidemiological and ecological systems with an important spatial component.

To the Editor: Some 80% of parasitic infections of humans are zoonoses (1). These infections are caused bymultihost parasites for which the reservoir of infection depends on hosts other than Homo sapiens. But what is a reservoir of infection? Haydon et al. (2) proposed a new series of definitions in connection with multihost pathogens, in which a target host is the host of interest in a particular context. The reservoir of infection included, for these authors, all hosts, whether incidental or not, that are epidemiologically connected to (i.e., contribute to transmission to) the target host. The availability of three terms—reservoir, reservoir of infection, and reservoir host—frequently used interchangeably, leads to confusion. This confusion is, in part, what prompted me (3) to slightly redefine a reservoir (of infection) as an ecologic system in which an infectious agent survives indefinitely. Such a system includes all the component host populations, including that of any intermediate host or vector, in the context of any environmental component, and any quantitative requisite such as critical community size, which is required to maintain the agent indefinitely. Vertebrate hosts that form an essential part of the system are reservoir hosts, though whether a whale or a fish is the reservoir host of Anisakis species can be a matter of debate. A host that becomes infected, but is not required for the maintenance of the population of a pathogen, can usefully be called an incidental host. (Accidental host is frequently used, but this is arguably a teleological term and therefore undesirable.) For incidental hosts that transmit pathogens from a reservoir to another incidental host, analogous to a pipe leading from a water reservoir, Garnham (4) coined the useful term “liaison host.” Haydon et al. dismiss my definition on two grounds. First, I exclude liaison hosts from the reservoir. This distinction is valid and could be argued either way, but I suggest that the pipes leading from a reservoir do not form part of the reservoir and that it is both conceptually and practically important to distinguish liaison hosts from reservoir hosts. The second objection is that many pathogens depend on the presence of several host species, at any given stage in the life history, for their maintenance. This concept is clearly considered in my article: together, such hosts collectively constitute part of the reservoir system, though no single one may be the reservoir host in its own right. In good scientific English, each term should have a precise definition, and synonyms should be avoided. The Oxford English Dictionary (OED) (5) definitions of reservoir generally refer to a place or container used for the collection and storage of water, other fluids, or even solid material. The OED definition of reservoir as a medical term is. “A population which is chronically infested with the causative agent of a disease and can infect other populations.” While one might argue with the terms chronically, infested, the infection of populations, this definition captures the usual sense in which reservoir host is used. The quotations given in OED are more helpful. The earliest one given for reservoir in a medical context is from 1937, “For the continuous existence of a disease there must be some reservoir of infection…” The most important reservoirs of infection are human or animal cases or carriers. Plants may be the reservoir of infection in some of the mycoses. However, according to OED, the compound term “reservoir host” was used earlier, in 1913, “The monkey is most probably the normal reservoir host [for Physaloptera mordens].” The main conceptual difference between the proposal of Haydon et al. and my own is that mine is more generalized: for a given pathogen in a given place, there is a single reservoir. The proposal of Haydon et al. is more limited to practical considerations: the reservoir for one target host may not be the same as that for another target host in the same place. The most important contribution of these two publications is that they raise an issue that has confused the literature for many years. Parasitologists, virologists, and bacteriologists should agree on a consensus set of terms for the ecologic description of multihost systems. When we all agree on what we are talking about, we will understand each other better.

Monthly measles case reports from 19 island communities covering periods up to 15 years have been analyzed. Breaks in the continuity of measles transmission were found for all communities of less than 500,000 population. The duration of epidemics was also affected by population density. It is postulated that populations sufficient to support continued propagation of this virus did not exist in primitive societies and that measles virus must have evolved since the development of early civilizations.

An infection like measles does not spread uniformly in populations from Europe and North America. Of special importance is a pronounced age-dependency in the contact rates, because of increased infection transmission within schools. Therefore an age-structured epidemiologic model is investigated here, which also pays attention to the fact that children are promoted grade-wise into and out of school. Simulation results are contrasted with pre- and post-vaccination measles data from England and Wales and the model is shown to perform better than previous global mass-action models.

Since the discovery of a power law scaling between the mean and variance of natural populations, this phenomenon has been observed for a variety of species. Here, we show that the same form of power law scaling also occurs in measles case reports in England and Wales. Remarkably this power law holds over four orders of magnitude. We consider how the natural experiment of vaccination affects the slope of the power law. By examining simple generic models, we are able to predict the effects of stochasticity and coupling and we propose a new phenomenon associated with the critical community size.