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Understanding how emerging viruses persist in bat populations is a fundamental step to understand the processes by which viruses are transmitted from reservoir hosts to spillover hosts. Hendra virus, which has caused fatal infections in horses and humans in eastern Australia since 1994, spills over from its natural reservoir hosts, Pteropus bats (colloquially known as flying foxes). It has been suggested that the Hendra virus maintenance mechanism in the bat populations might be implicated with their metapopulation structure. Here, we examine whether a metapopulation consisting of black flying fox (P. alecto) colonies that are smaller than the critical community size can maintain the Hendra virus. By using the Gillespie algorithm, stochastic mathematical models were used to simulate a cycle, in which viral extinction and recolonisation were repeated in a single colony within a metapopulation. Given estimated flying fox immigration rates, the simulation results showed that recolonisation occurred more frequently than extinction, which indicated that infection would not go extinct in the metapopulation. Consequently, this study suggests that a collection of transient epidemics of Hendra virus in numerous colonies of flying foxes in Australia can support the long-term persistence of the virus at the metapopulation level.

International audience; The global burden of dengue is increasing at an alarming rate and increased international travel will lead to constant importation of dengue virus into nonendemic areas. The potential for dengue epidemics in such countries during seasons with permissive temperatures has already been underlined by epidemics in Japan and Madeira. While improved surveillance can help identify clinical cases of dengue, differentiating between imported and autochthonous cases remains problematical. Implementation of a threshold criterion can help in identifying aberrant incidences of dengue. This threshold approach was applied to dengue cases reported in the Japanese surveillance system from 2011-2019. Several aberrant incidences occurring during consecutive weeks were detected, one of which was concomitant to the Yoyogi Park Tokyo epidemic but in another area, Kanagawa, and another above threshold week was coincidental with a symptomatic case of a German traveller. This indicates autochthonous transmission. Despite the occurrence of several alert periods, however, on no occasion did the spread of dengue progress into a full epidemic as was seen in Yoyogi. It thus seems likely that Yoyogi Park was a particular event and that stochastic die-out of viruses is occurring frequently without progression, perhaps reflecting the negative impact of societal infrastructure on dengue transmission despite permissive temperatures. Implementation of a dengue epidemic threshold as used for seasonal influenza may provide a basis for future seroprevalence studies to assess the true prevalence of dengue in light of the high frequency of subclinical, asymptomatic infections.

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.

The critical community size (CCS) is the minimum closed population size in which a pathogen can persist indefinitely. Below this threshold, stochastic extinction eventually causes pathogen extinction. Here we use a simulation model to explore behaviour-mediated persistence: a novel mechanism by which the population response to the pathogen determines the CCS. We model severe coronavirus 2 (SARS-CoV-2) transmission and non-pharmaceutical interventions (NPIs) in a population where both individuals and government authorities restrict transmission more strongly when SARS-CoV-2 case numbers are higher. This results in a coupled human-environment feedback between disease dynamics and population behaviour. In a parameter regime corresponding to a moderate population response, this feedback allows SARS-CoV-2 to avoid extinction in the trough of pandemic waves. The result is a very low CCS that allows long term pathogen persistence. Hence, an incomplete pandemic response represents a “sour spot” that not only ensures relatively high case incidence and unnecessarily long lockdown, but also promotes long-term persistence of the pathogen by reducing the CCS. Given the worldwide prevalence of small, isolated populations in which a pathogen with low CCS can persist, these results emphasize the need for a global approach to coronavirus disease 2019 (COVID-19) vaccination.

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.

The maintenance of infectious diseases requires a sufficient number of susceptible hosts. Host culling is a potential control strategy for animal diseases. However, the reduction in biodiversity and increasing public concerns regarding the involved ethical issues have progressively challenged the use of wildlife culling. Here, we assess the potential of wildlife culling as an epidemiologically sound management tool, by examining the host ecology, pathogen characteristics, eco-sociological contexts, and field work constraints. We also discuss alternative solutions and make recommendations for the appropriate implementation of culling for disease control., Eve Miguel et al. discuss the potential and practice of wildlife culling to control infectious disease both in the wildlife population and in transmission to humans. They identify several factors that must be considered when using culling, including host and pathogen ecology and eco-sociological contexts.

In this paper the concept of Critical Community Size (CCS) for the deterministic SIR model is introduced and its consequences for the disease dynamics are stressed. The disease can fade out after an outburst. Also the principle of competitive exclusion holds no longer true. This is exemplified for the dynamics of two competing virus strains. The virus with higher R0 can be eradicated from the population.

Summary Identifying mechanisms driving pathogen persistence is a vital component of wildlife disease ecology and control. Asymptomatic, chronically infected individuals are an oft‐cited potential reservoir of infection, but demonstrations of the importance of chronic shedding to pathogen persistence at the population‐level remain scarce.Studying chronic shedding using commonly collected disease data is hampered by numerous challenges, including short‐term surveillance that focuses on single epidemics and acutely ill individuals, the subtle dynamical influence of chronic shedding relative to more obvious epidemic drivers, and poor ability to differentiate between the effects of population prevalence of chronic shedding vs. intensity and duration of chronic shedding in individuals.We use chronic shedding of Leptospira interrogans serovar Pomona in California sea lions (Zalophus californianus) as a case study to illustrate how these challenges can be addressed. Using leptospirosis‐induced strands as a measure of disease incidence, we fit models with and without chronic shedding, and with different seasonal drivers, to determine the time‐scale over which chronic shedding is detectable and the interactions between chronic shedding and seasonal drivers needed to explain persistence and outbreak patterns.Chronic shedding can enable persistence of L. interrogans within the sea lion population. However, the importance of chronic shedding was only apparent when surveillance data included at least two outbreaks and the intervening inter‐epidemic trough during which fadeout of transmission was most likely. Seasonal transmission, as opposed to seasonal recruitment of susceptibles, was the dominant driver of seasonality in this system, and both seasonal factors had limited impact on long‐term pathogen persistence.We show that the temporal extent of surveillance data can have a dramatic impact on inferences about population processes, where the failure to identify both short‐ and long‐term ecological drivers can have cascading impacts on understanding higher order ecological phenomena, such as pathogen persistence., Chronic shedding of a pathogen by individuals is an oft‐cited but poorly understood mechanism for pathogen persistence due to difficulties in observing chronic shedders in many disease surveillance programmes. The authors show that the effect of chronic shedding is quantifiable when accounting for surveillance duration and interactions with short‐term epidemic drivers.

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.

Introduction: Bats are among the most poorly studied mammals, despite their diversity and important ecological services. The ongoing Ebola epidemic in western Africa underscores the increasingly recognized role of bats as reservoirs for emerging human diseases. An understanding of the mechanisms supporting pathogen maintenance in bat hosts is essential to informing future public health measures, as well as conservation efforts for wild bat populations. This is crucial in Africa, where the human population is burgeoning, increasing habitat loss and human contact with wildlife. The importance of natural history . The natural history and ecology of bats, especially in Africa, is poorly understood, making responsible management decisions difficult. Metrics for guiding management of pathogens and their host populations, such as critical community size and minimum viable population size, often require more natural history knowledge than we currently have available. We argue that management action without adequate research can lead to counterproductive results. In the past, fear-based management responses such as culling campaigns have counterintuitively increased the prevalence of pathogens in the target host population, as has been observed in the vampire bat-rabies system and the Egyptian fruit bat-Marburg system. Bat conservation in the time of Ebola . Many of the same anthropogenic stressors that are threatening bat populations, such as habitat loss and hunting, are also likely drivers of zoonotic pathogen emergence. Given this, and given that reducing bat populations can increase rather than decrease pathogen prevalence, we argue that conservation measures for bats should be strengthened in this time of Ebola. We offer a few potential first steps. Conclusions and Future Directions: We advocate the precautionary principle, and encourage significant and urgent increases in natural history studies, education, and conservation measures for African bats. Key words: bat conservation, Chiroptera, culling, Ebola, natural history, zoonotic pathogens

How changes in biodiversity alter the transmission of infectious diseases is presently under debate. Epidemiologists and ecologists have put a lot of effort to understand the mechanism behind biodiversity–disease relationship. Two important mechanisms, i.e. dilution and amplification theories have in some manner made it clear that biodiversity and disease outcome have an intimate relationship. The dilution effect theory seems to answer some overarching questions, but paucity of information about many disease systems is a real obstacle for its acceptance. Also, there is hardly any agreement on host population threshold and critical community size vis-à-vis wild life diseases. We suggest a multidimensional approach whereby the same disease system needs to be studied in different ecological zones and then the effect of biodiversity on disease outcome needs to be ascertained. Nonetheless, caution is to be taken while jumping to any conclusion as biodiversity–disease relationship is a multifactorial process.

Multiple subtypes of avian influenza (AI) and novel reassortants are frequently isolated from live bird markets (LBMs). However, our understanding of the drivers of persistence of multiple AI subtypes is limited. We propose a stochastic model of AI transmission within an LBM that incorporates market size, turnover rate and the balance of direct versus environmental transmissibility. We investigate the relationship between these factors and the critical community size (CCS) for the persistence of single and multiple AI strains within an LBM. We fit different models of seeding from farms to two-strain surveillance data collected from Shantou, China. For a single strain and plausible estimates for continuous turnover rates and transmissibility, the CCS was approximately 11 800 birds, only a 4.2% increase in this estimate was needed to ensure persistence of the co-infecting strains (two strains in a single host). Precise values of CCS estimates were sensitive to changes in market turnover rate and duration of the latent period. Assuming a gradual daily sell rate of birds the estimated CCS was higher than when an instantaneous selling rate was assumed. We were able to reproduce prevalence dynamics similar to observations from a single market in China with infection seeded every 5-15 days, and a maximum non-seeding duration of 80 days. Our findings suggest that persistence of co-infections is more likely to be owing to sequential infection of single strains rather than ongoing transmission of both strains concurrently. In any given system for a fixed set of ecological and epidemiological conditions, there is an LBM size below which the risk of sustained co-circulation is low and which may suggest a clear policy opportunity to reduce the frequency of influenza co-infection in poultry., published_or_final_version

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.

The field of disease ecology – the study of the spread and impact of parasites and pathogens within their host populations and communities – has a long history of using mathematical models. Dating back over 100 years, researchers have used mathematics to describe the spread of disease-causing agents, understand the relationship between host density and transmission and plan control strategies. The use of mathematical modelling in disease ecology exploded in the late 1970s and early 1980s through the work of Anderson and May (Anderson and May, 1978, 1981, 1992; May and Anderson, 1978), who developed the fundamental frameworks for studying microparasite (e.g. viruses, bacteria and protozoa) and macroparasite (e.g. helminth) dynamics, emphasizing the importance of understanding features such as the parasite's basic reproduction number (R0) and critical community size that form the basis of disease ecology research to this day. Since the initial models of disease population dynamics, which primarily focused on human diseases, theoretical disease research has expanded hugely to encompass livestock and wildlife disease systems, and also to explore evolutionary questions such as the evolution of parasite virulence or drug resistance. More recently there have been efforts to broaden the field still further, to move beyond the standard ‘one-host-one-parasite’ paradigm of the original models, to incorporate many aspects of complexity of natural systems, including multiple potential host species and interactions among multiple parasite species.

Scientific Reports, 6, ISSN:2045-2322

Bovine tuberculosis (bTB) infection in cattle is one of the most complex and persistent problems faced by the cattle industry in Great Britain today. While a number of factors have been identified as increasing the risk of infection, there has been little analysis on the causes of persistent infection within farms. In this article, we use the Cattle Tracing System to examine changes in herd size and VetNet data to correlate herd size with clearance of bTB. We find that the number of active farms fell by 16.3% between 2002 and 2007. The average farm size increased by 17.9% between 2002 and 2005. Using a measure similar to the Critical Community Size, the VetNet data reveal that herd size is positively correlated with disease persistence. Since economic policy and subsidies have been shown to influence farm size, we used a simple financial model for ideal farm size which includes disease burden to conclude that increasing herd size for efficiency gains may contribute to increased disease incidence.

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.

The dynamics of contagious bovine pleuropneumonia (CBPP) transmission vary widely between livestock production systems. This paper describes the development of a homogeneous, stochastic, compartmental model for CBPP transmission in pastoral herds of East Africa. The model was built using parameter estimates based on data published in the literature and on observations of livestock owners obtained through participatory research. The basic reproduction number for CBPP in southern Sudan was estimated to range from 3.2 to 4.6. The homogeneous model indicates that the critical community size for the persistence of CBPP falls within the typical herd sizes for pastoral communities in East Africa suggesting that individual isolated herds are capable of maintaining infection indefinitely. Vaccination alone with currently available vaccines was unlikely to eradicate the disease.

The endemicity of infectious diseases is investigated from a deterministic viewpoint. Sustained oscillation of infectives is often due to seasonal effects which may be related to climatic changes. For example the transmission of the measles virus by droplets is enhanced in cooler, more humid seasons. In many countries the onset of cooler, more humid weather coincides with the increased aggregation of people and the seasonal effect can be exacerbated. In this paper we consider non-autonomous compartmental epidemiological models and demonstrate that the critical community size phenomenon may be associated with the seasonal variation in the disease propagation. This approach is applicable to both the prevaccination phenomenon of critical community size and the current goal of worldwide elimination of measles by vaccination.

Summary 1. The population dynamics of bank voles and wood mice, and of cowpox virus infection in these two species, was studied over a 2-year period in a mainland population and in 14 nearby island populations of varying sizes. 2. For both species, there was no intrinsic variation in the pattern of host dynamics with island size: small island populations behaved as though they were small subsamples of a larger population, displaying no more than the expected random variation from the general pattern. 3. None the less, the relative numbers of bank voles to wood mice increased markedly with decreasing island size; and bank vole densities tended to be higher on smaller islands. 4. Only 22 animals were discovered to have moved either between islands or between the mainland and the islands, out of 1883 captured in all. None the less, it was apparent that males were more likely to move than females. 5. Overall patterns of cowpox virus dynamics were similar in all cases. However, on all islands there were extended periods when cowpox virus infection was apparently absent, and on the small islands the numbers of infected individuals were mostly very low and in many cases infection was never found. 6. For both host species, there was no evidence for a threshold population size for cowpox virus (critical community size) in terms of density, but clear evidence for one in terms of the numerical size of populations. This suggests little support for density-dependent transmission, despite this having been the usual default assumption for non-sexually transmitted infections. 7. There was also evidence for a separate invasion threshold (between ecological and epidemiological invasion) and persistence threshold (between epidemiological invasion and persistence). This is contrary to the output of the most-quoted (deterministic) models ‐ persistence and invasion threshold one and the same ‐ highlighting the fact that little attention has been paid in the past to the practical meaning of the theoretical concept of a threshold. 8. In the case of the wood mice, a superficial similarity to the bank vole thresholds was potentially misleading. Wood mouse thresholds were influenced at least as much by the bank vole thresholds as they were by the dynamics within the wood mouse populations themselves.

Two key linked questions in population dynamics are the relative importance of noise vs. density-dependent nonlinearities and the limits on temporal predictability of population abundance. We propose that childhood microparasitic infections, notably mea- sles, provide an unusually suitable empirical and theoretical test bed for addressing these issues. We base our analysis on a new mechanistic time series model for measles, the TSIR model, which captures the mechanistic essence of epidemic dynamics. The model, and parameter estimates based on short-term fits to prevaccination measles time series for 60 towns and cities in England and Wales, is introduced in a companion paper. Here, we explore how well the model predicts the long-term dynamics of measles and the balance between noise and determinism, as a function of population size. The TSIR model captures the basic dynamical features of the long-term pattern of measles epidemics in large cities remarkably well (based on time and frequency domain analyses). In particular, the model illustrates the impact of secular increases in birth rates, which cause a transition from biennial to annual dynamics. The model also captures the observed increase in epidemic irregularity with decreasing population size and the onset of local extinction below a critical community size. Decreased host population size is shown to be associated with an increased impact of demographic stochasticity. The interaction between nonlinearity and noise is explored using local Lyapunov exponents (LLE). These testify to the high level of stability of the biennial attractor in large cities. Irregularities are due to the limit cycle evolving with changing human birth rates and not due to complex dynamics. The geometry of the dynamics (sign and magnitude of the LLEs across phase space) is similar in the cities and the smaller urban areas. The qualitative difference in dynamics between small and large host communities is that demographic and extinction-recolonization stochasticities are much more influential in the former. The regional dynamics can therefore only be understood in terms of a core-satellite metapopulation structure for this host-enemy system. We also make a preliminary exploration of the model's ability to predict the dynamic consequences of measles vaccination.

Summary An approximation is derived for the expected time to extinction in a stochastic model for recurrent epidemics. Numerical illustrations indicate that the approximation is crude but that it has the correct order of magnitude. The quasi-stationary distribution plays an important role in the derivation. Approximations for the critical community size and of the persistence threshold are derived. Comments are made on the classical study by Bartlell (1956–1960).

1. This paper explores the concept of the critical community size for persistence of infection in wildlife populations. We use as a case study the 1988 epidemic of phocine distemper virus in the North Sea population of harbour seals, Phoca vitulina. 2. We summarize the available data on this epidemic and use it to parameterize a stochastic compartmental model for an infection spreading through a spatial array of patches coupled by nearest-neighbour mixing, with replacement of susceptibles occurring as a discrete annual event. 3. A combination of analytical and simulation techniques is used to show that the high levels of transmission between different seal subpopulations, combined with the small annual birth cohort, act to make persistence of infection impossible in this harbour seal population at realistic population levels. The well known mechanisms by which metapopulation structures may act to promote persistence can be seen to have an effect only at weaker levels of spatial coupling, and higher levels of host recruitment, than those empirically observed.

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.

The metapopulation concept provides a very powerful tool for analysing the persistence of spatially-disaggregated populations, in terms of a balance between local extinction and colonization. Exactly the same approach has been developed by epidemiologists, in order to understand patterns of diseases persistence. There is great scope for further cross-fertilization between areas. Recent work on the spatitemporal dynamics of measles illustrates that the large datasets and rich modelling literature on many infectious diseases offer great potential for developing and testing ideas about metapopulations.

Canine distemper virus (CDV) is an acute, highly immunizing pathogen that should require high densities and large populations of hosts for long-term persistence, yet CDV persists among terrestrial carnivores with small, patchily distributed groups. We used CDV in the Greater Yellowstone ecosystem's (GYE) wolves (Canis lupus) and coyotes (Canis latrans) as a case study for exploring how metapopulation structure, host demographics, and multi-host transmission affect the critical community size and spatial scale required for CDV persistence. We illustrate how host spatial connectivity and demographic turnover interact to affect both local epidemic dynamics, such as the length and variation in inter-epidemic periods, and pathogen persistence using stochastic, spatially explicit susceptible-exposed-infectious-recovered simulation models. Given the apparent absence of other known persistence mechanisms (e.g., a carrier or environmental state, densely populated host, chronic infection, or a vector), we suggest that CDV requires either large spatial scales or multi-host transmission for persistence. Current GYE wolf populations are probably too small to support endemic CDV. Coyotes are a plausible reservoir host, but CDV would still require 50000-100000 individuals for moderate persistence (> 50% over 10 years), which would equate to an area of 1-3 times the size of the GYE (60000-200000 km2). Coyotes, and carnivores in general, are not uniformly distributed; therefore, this is probably a gross underestimate of the spatial scale of CDV persistence. However, the presence of a second competent host species can greatly increase the probability of long-term CDV persistence at much smaller spatial scales. Although no management of CDV is currently recommended for the GYE, wolf managers in the region should expect periodic but unpredictable CDV-related population declines as often as every 2-5 years. Awareness and monitoring of such outbreaks will allow corresponding adjustments in management activities such as regulated public harvest, creating a smooth transition to state wolf management and conservation after > 30 years of being protected by the Endangered Species Act.

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.

Detailed studies on the spatial and temporal patterns of rubella transmission are scarce particularly in developing countries but could prove useful in improving epidemiological surveillance and intervention strategies such as vaccination. We use highly refined spatial, temporal and age-specific incidence data of Peru, a geographically diverse country, to quantify spatial-temporal patterns of incidence and transmissibility for rubella during the period 1997–2006. We estimate the basic reproduction number (R 0) based on the mean age at infection and the per capita birth rate of the population as well as the reproduction number (accounting for the fraction of the population effectively protected to infection) using the initial intrinsic growth rate of individual outbreaks and estimates of epidemiological parameters for rubella. A wavelet time series analysis is conducted to explore the periodicity of the rubella weekly time series, and the results of our analyses are compared to those carried out for time series of other childhood infectious diseases. We also identify the presence of a critical community size and quantify spatial heterogeneity across geographic regions through the use of Lorenz curves and their corresponding Gini indices. The underlying distributions of rubella outbreak attack rates and epidemic durations across Peru are characterized.

Avian influenza virus (AIV) persists in North American wild waterfowl, exhibiting major outbreaks every 2–4 years. Attempts to explain the patterns of periodicity and persistence using simple direct transmission models are unsuccessful. Motivated by empirical evidence, we examine the contribution of an overlooked AIV transmission mode: environmental transmission. It is known that infectious birds shed large concentrations of virions in the environment, where virions may persist for a long time. We thus propose that, in addition to direct fecal/oral transmission, birds may become infected by ingesting virions that have long persisted in the environment. We design a new host–pathogen model that combines within-season transmission dynamics, between-season migration and reproduction, and environmental variation. Analysis of the model yields three major results. First, environmental transmission provides a persistence mechanism within small communities where epidemics cannot be sustained by direct transmission only (i.e., communities smaller than the critical community size). Second, environmental transmission offers a parsimonious explanation of the 2–4 year periodicity of avian influenza epidemics. Third, very low levels of environmental transmission (i.e., few cases per year) are sufficient for avian influenza to persist in populations where it would otherwise vanish., Author Summary Avian influenza viruses (AIVs) in wild waterfowl constitute the historic source of human influenza viruses, having a rich pool of genetic and antigenic diversity that often leads to cross-species transmission. Although the emergence of H5N1 avian influenza virus onto the international scene has captured the most attention, we do not as yet understand the mechanisms that underpin AIV persistence and dynamics in the wild. We developed a novel host–pathogen model intended to describe the epidemiology of low pathogenic AIV in temperate environments. Our model takes into account seasonality in migration and breeding together with multiple modes of transmission. AIVs have been detected in unconcentrated lake water, soil swabs, and mud samples. Laboratory experiments show that AIVs persist and remain infectious in water for extended periods. However, so far, the possibility of environmental transmission of AIV has been largely overlooked. Our work shows that environmental transmission provides a parsimonious explanation for the patterns of persistence and outbreaks of AIV documented in the literature. In addition to their scientific importance, our conclusions impact the design of control policies for avian influenza by emphasizing the dramatic and long-term role that environmental persistence of pathogens may play at the epidemic level.

Anseriformes are the reservoir of low-pathogenicity avian influenza viruses (LPAIV). Studies have shown a high LPAIV prevalence associated with low antibody detection in a wild duck population in northern European countries, whereas in winter areas (Mediterranean basin), low viral detection and high seroprevalence were observed. In order to gain insight into the role played by both population recruitment and migration on AIV persistence, an epidemiological model was developed. A susceptible, infectious and removed (immune or dead)-individuals model coupling population and infection dynamics was developed to simulate LPAIV circulation in dabbling ducks throughout the entire year. The transmission coefficient (beta) was calculated using the original dataset of published works, whereas dabbling duck demographic parameters were obtained from the literature. The estimated host density threshold for virus persistence is 380 susceptible individuals per day whereas the critical community size needed for maintaining the virus throughout the winter has been estimated to be about 1200 individuals. The model showed peaks of viral prevalence after nesting and during the moult period because of population recruitment and high host density, respectively. During the winter and spring periods, the viruses reach the minimal endemic level and local extinction is highly probable because of stochastic phenomena, respectively 80% and 90% of probabilities. The most sensitive parameters of the model are the recruitment rate of young susceptible animals and the duration of virus shedding.

Filoviruses Ebolavirus (EBOV) and Marburgvirus (MARV) cause haemorrhagic fevers with high mortality rates, posing significant threats to public health. To understand transmission into human populations, filovirus dynamics within reservoir host populations must be understood. Studies have directly linked filoviruses to bats, but the mechanisms allowing viral persistence within bat populations are poorly understood. Theory suggests seasonal birthing may decrease the probability of pathogen persistence within populations, but data suggest MARV may persist within colonies of seasonally breeding Egyptian fruit bats, Rousettus aegyptiacus . I synthesize available filovirus and bat data in a stochastic compartmental model to explore fundamental questions relating to filovirus ecology: can filoviruses persist within isolated bat colonies; do critical community sizes exist; and how do host–pathogen relationships affect spillover transmission potential? Synchronous annual breeding and shorter incubation periods did not allow filovirus persistence, whereas bi-annual breeding and longer incubation periods, such as reported for Egyptian fruit bats and EBOV in experimental studies, allowed persistence in colony sizes often found in nature. Serological data support the findings, with bats from species with two annual birth pulses more likely to be seropositive (odds ratio (OR) 4.4, 95% confidence interval (CI) 2.5–8.7) than those with one, suggesting that biannual birthing is necessary for filovirus persistence.

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.