A new look at the critical community size for childhood infections

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.     

Density dependence and stochastic variation in a newly established population of a small songbird

Models describing fluctuations in population size should include both density dependence and stochastic effects. We examine the relative contribution of variation in parameters of the expected dynamics as well as demographic and environmental stochasticity to fluctuations in a population of a small passerine bird, the pied flycatcher, that was newly established in a Dutch study area. Using the theta-logistic model of density regulation, we demonstrate that the estimated quasi-stationary distribution including demographic stochasticity is close to the stationary distribution ignoring demographic stochasticity, indicating a long expected time to extinction. We also show that the variance in the estimated quasi-stationary distribution is especially sensitive to variation in the density regulation function. Reliable population projections must therefore account for uncertainties in parameter estimates which we do by using the population prediction interval (PPI). After 2 years the width of the 90% PPI was already larger than the corresponding estimated range of variation in the quasi-stationary distribution. More precise prediction of future population size than can be derived from the quasi-stationary distribution could only be made for a time span less than about five years.     

On the quasi-stationary distribution of the stochastic logistic epidemic

An approximation is derived for the quasi-stationary distribution of the stochastic logistic epidemic in the intricate case where the transmission factor R0 lies in the transition region near the deterministic threshold value 1. An approximation for the expected time to extinction from quasi-stationarity in the same parameter region is also given.     

On the time to extinction for a two-type version of Bartlett's epidemic model

We are interested in how the addition of type heterogeneities affects the long time behaviour of models for endemic diseases. We do this by analysing a two-type version of a model introduced by Bartlett under the restriction of proportionate mixing. This model is used to describe diseases for which individuals switch states according to susceptible-->infectious-->recovered and immune, where the immunity is life-long. We describe an approximation of the distribution of the time to extinction given that the process is started in the quasi-stationary distribution, and we analyse how the variance and the coefficient of variation of the number of infectious individuals depends on the degree of heterogeneity between the two types of individuals. These are then used to derive an approximation of the time to extinction. From this approximation we conclude that if we increase the difference in infectivity between the two types the expected time to extinction decreases, and if we instead increase the difference in susceptibility the effect on the expected time to extinction depends on which part of the parameter space we are in, and we can also obtain non-monotonic behaviour. These results are supported by simulations.     

Comparison of deterministic and stochastic SIS and SIR models in discrete time

The dynamics of deterministic and stochastic discrete-time epidemic models are analyzed and compared. The discrete-time stochastic models are Markov chains, approximations to the continuous-time models. Models of SIS and SIR type with constant population size and general force of infection are analyzed, then a more general SIS model with variable population size is analyzed. In the deterministic models, the value of the basic reproductive number R0 determines persistence or extinction of the disease. If R0 < 1, the disease is eliminated, whereas if R0 > 1, the disease persists in the population. Since all stochastic models considered in this paper have finite state spaces with at least one absorbing state, ultimate disease extinction is certain regardless of the value of R0. However, in some cases, the time until disease extinction may be very long. In these cases, if the probability distribution is conditioned on non-extinction, then when R0 > 1, there exists a quasi-stationary probability distribution whose mean agrees with deterministic endemic equilibrium. The expected duration of the epidemic is investigated numerically.     

Stochastic disease dynamics of a hospital infection model

A stochastic model for hospital infection incorporating both direct transmission and indirect transmission via free-living bacteria in the environment is investigated. We examine the long term behavior of the model by calculating a stationary distribution and normal approximation of the distribution. The quasi-stationary distribution of the model is studied to investigate the models’ behavior before extinction and the time to extinction. Numerical results show agreement between the calculated distributions and results of event-driven simulations. Hand hygiene of volunteers is more effective in terms of reducing the mean (or standard deviation) of the stationary distribution of colonized patients and the expected time to extinction compared to hand hygiene of health care workers (HCWs), on the basis of our parameter values. However, the indirect (or direct) transmission rate can lead to either increase or decrease in the standard deviation of the stationary distribution, but the impact of the indirect transmission is much greater than that of the direct transmission. The findings suggest that isolation of new admitted colonized patients is most effective in reducing both the mean and standard deviation of the stationary distribution and measures related to indirect transmission are secondary in their effects compared to other interventions.     

The expected extinction time of a population within a system of interacting biological populations

The quasi-stationary distribution of a population within a system of interacting populations is approximated by a stochastic logistic process. The parameters of this process can be expressed in the parameters of the full system. Using the diffusion approximation, an expression for the expected extinction time is derived from this logistic process. Since the expected extinction time is expressed in the parameters of the full system, the effect of these parameters on the extinction risk can be easily evaluated, which may be of use for studies in ecology, conservation biology and epidemiology. The outcome is compared with simulation results for the case of a prey-predator system.     

Moment closure and the stochastic logistic model

The quasi-stationary distribution of the stochastic logistic model is studied in the parameter region where its body is approximately normal. Improved asymptotic approximations of its first three cumulants are derived. It is shown that the same results can be derived with the aid of the moment closure method. This indicates that the moment closure method leads to expressions for the cumulants that are asymptotic approximations of the cumulants of the quasi-stationary distribution.     

On predicting extinction in simple population models. I. Stochastic linearization

The technique of stochastic linearization derived by Bartlett is used to give an approximate solution to each of three stochastic models of predation. By defining the existence of a quasi-stationary equilibrium distribution of population sizes, the approximate variances and covariances of the joint distribution of population sizes are calculated. The results are used to predict whether prey or predators are likely to become extinct first and the predictions are tested against simulation data. The linearization technique is a good predictor of the outcome of extinction but not of how long it takes.     

Equilibrium and extinction in stochastic population dynamics

Stochastic models of interacting biological populations, with birth and death rates depending on the population size are studied in the quasi-stationary state. Confidence regions in the state space are constructed by a new method for the numerical, solution of the ray equations. The concept of extinction time, which is closely related to the concept of stability for stochastic systems, is discussed. Results of numerical calculations for two-dimensional stochastic population models are presented.     

Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction

We are interested in the long time behavior of a two-type density-dependent biological population conditioned on non-extinction, in both cases of competition or weak cooperation between the two species. This population is described by a stochastic Lotka–Volterra system, obtained as limit of renormalized interacting birth and death processes. The weak cooperation assumption allows the system not to blow up. We study the existence and uniqueness of a quasi-stationary distribution, that is convergence to equilibrium conditioned on non-extinction. To this aim we generalize in two-dimensions spectral tools developed for one-dimensional generalized Feller diffusion processes. The existence proof of a quasi-stationary distribution is reduced to the one for a d-dimensional Kolmogorov diffusion process under a symmetry assumption. The symmetry we need is satisfied under a local balance condition relying the ecological rates. A novelty is the outlined relation between the uniqueness of the quasi-stationary distribution and the ultracontractivity of the killed semi-group. By a comparison between the killing rates for the populations of each type and the one of the global population, we show that the quasi-stationary distribution can be either supported by individuals of one (the strongest one) type or supported by individuals of the two types. We thus highlight two different long time behaviors depending on the parameters of the model: either the model exhibits an intermediary time scale for which only one type (the dominant trait) is surviving, or there is a positive probability to have coexistence of the two species.     

Global diversity of mayflies (Ephemeroptera, Insecta) in freshwater

The extant global Ephemeroptera fauna is represented by over 3,000 described species in 42 families and more than 400 genera. The highest generic diversity occurs in the Neotropics, with a correspondingly high species diversity, while the Palaearctic has the lowest generic diversity, but a high species diversity. Such distribution patterns may relate to how long evolutionary processes have been carrying on in isolation in a bioregion. Over an extended period, there may be extinction of species, but evolution of more genera. Dramatic extinction events such as the K-T mass extinction have affected current mayfly diversity and distribution. Climatic history plays an important role in the rate of speciation in an area, with regions which have been climatically stable over long periods having fewer species per genus, when compared to regions subjected to climatic stresses, such as glaciation. A total of 13 families are endemic to specific bioregions, with eight among them being monospecific. Most of these have restricted distributions which may be the result of them being the relict of a previously more diverse, but presently almost completely extinct family, or may be the consequence of vicariance events, resulting from evolution due to long-term isolation. Guest editors: E. V. Balian, C. Lévêque, H. Segers & K. Martens Freshwater Animal Diversity Assessment     

The intrinsic mean time to extinction: a unifying approach to analysing persistence and viability of populations

Analysing the persistence and viability of small populations is a key issue in extinction theory and population viability analysis. However, there is still no consensus on how to quantify persistence and viability. We present an approach to evaluate any simulation model concerned with extinction. The approach is devised from general Markov models of stochastic population dynamics. From these models, we distil insights into the general mathematical structure of the risk of extinction by time t, P₀(t). From this mathematical structure, we devise a simple but effective protocol – the ln(1−P₀)-plot – which is applicable for situations including environmental noise or catastrophes. This plot delivers two quantities which are fundamental to the assessment of persistence and viability: the intrinsic mean time to extinction, T_m, and the probability c₁ of the population reaching the established phase. The established phase is characterized by typical fluctuations of the population's state variable which can be described by quasi-stationary probability distributions. The risk of extinction in the established phase is constant and given by 1/T_m. We show that T_m is the basic currency for the assessment of persistence and viability because T_m is independent of initial conditions and allows the risk of extinction to be calculated for any time horizon. For situations where initial conditions are important, additionally c₁ has to be considered.     

Extinction times and size of the surviving species in a two-species competition process

We investigate a stochastic model for the competition between two species. Based on percentiles of the maximum number of individuals in the ecosystem, we present an approximating model for which the extinction time can be thought of as a phase-type random variable. We determine formulae for the probabilities of extinction and the moments of the extinction time. We discuss the use of several quasi-stationary assumptions. We include a comparative study between existing asymptotic results, results obtained from a simulation of the process, and our solution.     

Calculating the time to extinction of a reactivating virus, in particular bovine herpes virus

The expected time to extinction of a herpes virus is calculated from a rather simple population-dynamical model that incorporates transmission, reactivation and fade-out of the infectious agent. We also derive the second and higher moments of the distribution of the time to extinction. These quantities help to assess the possibilities to eradicate a reactivating infection. The key assumption underlying our calculations is that epidemic outbreaks are fast relative to the time scale of demographic turnover. Four parameters influence the expected time to extinction: the reproduction ratio, the reactivation rate, the population size, and the demographic turn-over in the host population. We find that the expected time till extinction is very long when the reactivation rate is high (reactivation is expected more than once in a life time). Furthermore, the infectious agent will go extinct much more quickly in small populations. This method is applied to bovine herpes virus (BHV) in a cattle herd. The results indicate that without vaccination, BHV will persist in large herds. The use of a good vaccine can induce eradication of the infection from a herd within a few decades. Additional measures are needed to eradicate the virus from a whole region within a similar time-span.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

Early evolutionary history of the flowering plant family Annonaceae: steady diversification and boreotropical geodispersal

Aim Rain forest‐restricted plant families show disjunct distributions between the three major tropical regions: South America, Africa and Asia. Explaining these disjunctions has become an important challenge in biogeography. The pantropical plant family Annonaceae is used to test hypotheses that might explain diversification and distribution patterns in tropical biota: the museum hypothesis (low extinction leading to steady accumulation of species); and dispersal between Africa and Asia via Indian rafting versus boreotropical geodispersal. Location Tropics and boreotropics. Methods Molecular age estimates were calculated using a Bayesian approach based on 83% generic sampling representing all major lineages within the family, seven chloroplast markers and two fossil calibrations. An analysis of diversification was carried out, which included lineage‐through‐time (LTT) plots and the calculation of diversification rates for genera and major clades. Ancestral areas were reconstructed using a maximum likelihood approach that implements the dispersal–extinction–cladogenesis model. Results The LTT plots indicated a constant overall rate of diversification with low extinction rates for the family during the first 80 Ma of its existence. The highest diversification rates were inferred for several young genera such as Desmopsis, Uvariopsis and Unonopsis. A boreotropical migration route was supported over Indian rafting as the best fitting hypothesis to explain present‐day distribution patterns within the family. Main conclusions Early diversification within Annonaceae fits the hypothesis of a museum model of tropical diversification, with an overall steady increase in lineages possibly due to low extinction rates. The present‐day distribution of species within the two largest clades of Annonaceae is the result of two contrasting biogeographic histories. The ‘long‐branch clade’ has been diversifying since the beginning of the Cenozoic and underwent numerous geodispersals via the boreotropics and several more recent long‐distance dispersal events. In contrast, the ‘short‐branch clade’ dispersed once into Asia via the boreotropics during the Early Miocene and further dispersal was limited.     

Precise Estimates of Persistence Time for SIS Infections in Heterogeneous Populations

For a susceptible–infectious–susceptible infection model in a heterogeneous population, we derive simple and precise estimates of mean persistence time, from a quasi-stationary endemic state to extinction of infection. Heterogeneity may be in either individuals’ levels of infectiousness or of susceptibility, as well as in individuals’ infectious period distributions. Infectious periods are allowed to follow arbitrary non-negative distributions. We also obtain a new and accurate approximation to the quasi-stationary distribution of the process, as well as demonstrating the use of our estimates to investigate the effects of different forms of heterogeneity. Our model may alternatively be interpreted as describing an infection spreading through a heterogeneous directed network, under the annealed network approximation.     

Patterns in spontaneous mutation revealed by human-baboon sequence comparison

We have analyzed the alignment of a long homologous region of the human and baboon genomes (approximately 1.5 Mb). We show that the frequency of gaps between aligned segments decreases slowly with gap length, indicating that several successive nucleotides are often deleted or inserted in one event. By contrast, runs of consecutive mismatches decrease rapidly in frequency with increasing length, following an exponential distribution, indicating that nucleotides are mostly substituted one at a time. Nucleotide substitutions are clumped at the scales of <10 and 1000-10,000 nucleotides, but show almost no aggregation at the scales of <10-100 and over approximately 50,000 nucleotides. Apparently, two rather different factors make the substitution rate not exactly uniform along the DNA sequence. Comparison of regions of very similar genomes that are approximately selectively neutral makes it possible to study spontaneous mutation at a new level of resolution.     

A discrete-time host–parasitoid model with an Allee effect

We introduce a discrete-time host–parasitoid model with a strong Allee effect on the host. We adapt the Nicholson–Bailey model to have a positive density dependent factor due to the presence of an Allee effect, and a negative density dependence factor due to intraspecific competition. It is shown that there are two scenarios, the first with no interior fixed points and the second with one interior fixed point. In the first scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and an exclusion region in which the host survives and tends to its carrying capacity. In the second scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and a coexistence region where both species survive.