The time to extinction for a stochastic SIS-household-epidemic model

We analyse a Markovian SIS epidemic amongst a finite population partitioned into households. Since the population is finite, the epidemic will eventually go extinct, i.e., have no more infectives in the population. We study the effects of population size and within household transmission upon the time to extinction. This is done through two approximations. The first approximation is suitable for all levels of within household transmission and is based upon an Ornstein-Uhlenbeck process approximation for the diseases fluctuations about an endemic level relying on a large population. The second approximation is suitable for high levels of within household transmission and approximates the number of infectious households by a simple homogeneously mixing SIS model with the households replaced by individuals. The analysis, supported by a simulation study, shows that the mean time to extinction is minimized by moderate levels of within household transmission.     

Stochastic epidemics in dynamic populations: quasi-stationarity and extinction

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 τ, the mean-parameter in the exponential distribution of the time to extinction for the finite population. The expression is used to study how τ 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. Received: 14 February 2000 / Revised version: 5 June 2000 / Published online: 24 November 2000     

Harvesting can increase severity of wildlife disease epidemics

Theoretical studies of wildlife population dynamics have proved insightful for sustainable management, where the principal aim is to maximize short-term yield, without risking population extinction. Surprisingly, infectious diseases have not been accounted for in harvest models, which is a major oversight because the consequences of parasites for host population dynamics are well-established. Here, we present a simple general model for a host species subject to density dependent reproduction and seasonal demography. We assume this host species is subject to infection by a strongly immunizing, directly transmitted pathogen. In this context, we show that the interaction between density dependent effects and harvesting can substantially increase both disease prevalence and the absolute number of infectious individuals. This effect clearly increases the risk of cross-species disease transmission into domestic and livestock populations. In addition, if the disease is associated with a risk of mortality, then the synergistic interaction between hunting and disease-induced death can increase the probability of host population extinction.     

With Allee effects,life for the social carnivore is complicated

Anthropogenic modification of the landscape, resultant habitat loss, and decades of persecution have resulted in severe decline and fragmentation of large carnivore populations worldwide. Infectious disease is also identified as a primary threat to many carnivores. In wildlife species, population demography and group persistence are strongly influenced by group or population size. This is referred to as the Allee effect, in which a population or group is at an increased risk of extinction when the number or density of individuals falls below some threshold due to ecological and/or genetic factors. However, in social mammalian species, the relationship between the number of individuals and the risk of extinction is complicated because aggregation may enhance pathogen exposure and transmission. Although theoretical studies of the interaction between infectious disease transmission and Allee effects reveal important implications for carnivore management and population extinction risk, information about the interaction has yet to be synthesized. In this paper, we assess life history strategies of medium to large carnivore species (≥2.4 kg) and their influence on population dynamics, with a special focus on infectious disease. While declining population trends are observed in 73 % of all carnivores (both social and solitary species), infectious disease is identified as a significant cause of population decline in 45 % of social carnivores and 3 % of solitary carnivores. Furthermore, where carnivores suffer a combination of rapid population decline and infectious disease, Allee effects may be more likely to impact social as compared to solitary carnivore populations. These potentially additive interactions may strongly influence disease transmission dynamics and population persistence potential. Understanding the mechanisms that can result in Allee effects in endangered carnivore populations and the manner in which infectious disease interfaces at this nexus may define the outcome of developed conservation strategies.     

Intervention-Based Stochastic Disease Eradication

Disease control is of paramount importance in public health, with infectious disease extinction as the ultimate goal. Although diseases may go extinct due to random loss of effective contacts where the infection is transmitted to new susceptible individuals, the time to extinction in the absence of control may be prohibitively long. Intervention controls are typically defined on a deterministic schedule. In reality, however, such policies are administered as a random process, while still possessing a mean period. Here, we consider the effect of randomly distributed intervention as disease control on large finite populations. We show explicitly how intervention control, based on mean period and treatment fraction, modulates the average extinction times as a function of population size and rate of infection spread. In particular, our results show an exponential improvement in extinction times even though the controls are implemented using a random Poisson distribution. Finally, we discover those parameter regimes where random treatment yields an exponential improvement in extinction times over the application of strictly periodic intervention. The implication of our results is discussed in light of the availability of limited resources for control.     

Local Scale Effects of Disease on Biodiversity

To date, ecologists and conservation biologists have focused much of their attention on the population and ecosystem effects of disease at regional scales and the role that diseases play in global species extinction. Far less research has been dedicated to identifying the effects that diseases can have on local scale species assemblages. We examined the role of infectious disease in structuring local biodiversity. Our intention was to illustrate how variable outcomes can occur by focusing on three case studies: the influence of chestnut blight on forest communities dominated by chestnut trees, the influence of red-spot disease on urchin barrens and kelp forests, and the influence of sylvatic plague on grassland communities inhabited by prairie dogs. Our findings reveal that at local scales infectious disease seems to play an important, though unpredictable, role in structuring species diversity. Through our case studies, we have shown that diseases can cause drastic population declines or local extirpations in keystone species, ecosystem engineers, and otherwise abundant species. These changes in local diversity may be very important, particularly when considered alongside potentially corresponding changes in community structure and function, and we believe that future efforts to understand the importance of disease to species diversity should have an increased focus on these local scales.     

Times from Infection to Disease-Induced Death and their Influence on Final Population Sizes After Epidemic Outbreaks

For epidemic models, it is shown that fatal infectious diseases cannot drive the host population into extinction if the incidence function is upper density-dependent. This finding holds even if a latency period is included and the time from infection to disease-induced death has an arbitrary length distribution. However, if the incidence function is also lower density-dependent, very infectious diseases can lead to a drastic decline of the host population. Further, the final population size after an epidemic outbreak can possibly be substantially affected by the infection-age distribution of the initial infectives if the life expectations of infected individuals are an unbounded function of infection age (time since infection). This is the case for lognormal distributions, which fit data from infection experiments involving tiger salamander larvae and ranavirus better than gamma distributions and Weibull distributions.     

Novel bivariate moment-closure approximations

Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure schemes are used to provide approximations to these models. Existing closure approximations are often unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work has tackled this problem in the univariate case. In this study, we address this problem by introducing novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed to capture the behaviour of the stochastic SIS model with varying population size, around the threshold between persistence and extinction of disease. The idea of conditional dependence between variables of interest underlies these mixture approximations. In the first approximation, we assume that the distribution of infectives (I) conditional on population size (N) is governed by the beta-binomial and for the second form, we assume that I is governed by zero-modified beta-binomial distribution where in either case N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between population variables on the behaviour of the approximations developed. Thus, the approximations are applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture approximations are able to predict disease extinction behaviour and describe transient aspects of the process.     

Selection, subdivision and extinction and recolonization

In a subdivided population, the interaction between natural selection and stochastic change in allele frequency is affected by the occurrence of local extinction and subsequent recolonization. The relative importance of selection can be diminished by this additional source of stochastic change in allele frequency. Results are presented for subdivided populations with extinction and recolonization where there is more than one founding allele after extinction, where these may tend to come from the same source deme, where the number of founding alleles is variable or the founders make unequal contributions, and where there is dominance for fitness or local frequency dependence. The behavior of a selected allele in a subdivided population is in all these situations approximately the same as that of an allele with different selection parameters in an unstructured population with a different size. The magnitude of the quantity N(e)s(e), which determines fixation probability in the case of genic selection, is always decreased by extinction and recolonization, so that deleterious alleles are more likely to fix and advantageous alleles less likely to do so. The importance of dominance or frequency dependence is also altered by extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations.     

The invasion, persistence and spread of infectious diseases within animal and plant communities

Recent theoretical and empirical studies of the population biology of infectious diseases have helped to improve our understanding of the major factors that influence the three phases of a successful invasion, namely initial establishment, persistence in the longer term and spread to other host communities. Of central importance in all three phases is the magnitude of the basic reproductive rate or transmission potential of the parasite. The value of this parameter is determined by a variety of biological properties of the association between an individual parasite and its host and the interaction between their populations. The recent epidemic of acquired immunodeficiency syndrome (AIDS) in North America and Europe is employed to illustrate the factors that promote disease establishment and spread. The frequency distribution of the number of different sexual partners per unit of time within homosexual communities is shown to be of central importance with respect to future trends in the incidence of AIDS. Broader aspects of pathogen invasion are examined by reference to simple mathematical models of three species associations, which mirror parasite introduction into resident predator-prey, host-parasite and competitive interactions. Many outcomes are possible, depending on the values of the numerous parameters that influence multi-species population interactions. Pathogen invasion is shown to have far-reaching implications for the structure and stability of ecological communities.     

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.     

Spreading disease with transport-related infection

Transportation among regions is found as one of the main factors which affect the outbreak of diseases. It will change the disease dynamics and break infection out even if infectious diseases will go to extinction in each city without transport-related infection. In this paper, a mathematical model is proposed to demonstrate the dynamics of such disease propagation between two regions (or cities) due to the population dispersal and infection on transports. Further, our analysis shows that transport-related infection intensifies the disease spread if infectious diseases break out to cause an endemic situation in each region, in the sense that both the absolute and relative size of patients increase. This suggests that it is very essential to strengthen restrictions of passengers once we know infectious diseases appeared.     

Extinction and quasi-stationarity in the Verhulst logistic model.

We formulate and analyse a stochastic version of the Verhulst deterministic model for density-dependent growth of a single population. Three parameter regions with qualitatively different behaviours are identified. Explicit approximations of the quasi-stationary distribution and of the expected time to extinction are presented in each of these regions. The quasi-stationary distribution is approximately normal, and the time to extinction is long, in one of these regions. Another region has a short time to extinction and a quasi-stationary distribution that is approximately truncated geometric. A third region is a transition region between these two. Here the time to extinction is moderately long and the quasi-stationary distribution has a more complicated behaviour. Numerical illustrations are given.     

Plant extinction dynamics in an insular metacommunity

Changes in the composition of local communities through time (i.e. species turnover) is a common phenomenon in insular biology. However, the mechanisms promoting variation in species turnover, both among islands and among species, are poorly understood. In an effort to better understand the causes of variation in species turnover, we evaluated the colonization and extinction dynamics of plant populations on 18 small islands off the west coast of Canada. In 1997, we quantified total population sizes of 10 woody angiosperm species. A decade later, we resampled islands to test whether: 1) species turnover occurred, 2) colonization events were offset by extinction events, 2) variation in extinction rates among islands was associated with population sizes, average plant heights, island area, island isolation or each island's exposure to ocean-born disturbances, and 3) variation in extinction rates among species was associated with plant life history traits. Results showed that extinction events outnumbered colonization events, suggesting that the metacommunity is in 'disequilibrium'. Variation in extinction rates among islands was unrelated to island area and isolation. However, extinction rates increased with exposure to ocean-born disturbances and decreased with both initial population sizes and average plant heights. Species with thicker, tougher leaves (i.e. high leaf mass per area) were less prone to extinction than species with thinner, more papery leaves. Overall results indicate that species turnover is common and that it is generated primarily by extinction. Variation in extinction rates appears to result from an interaction between among-island effects (exposure, population size and plant stature) and among-species effects (leaf toughness), suggesting that ocean-born disturbances play a key role in determining metacommunity structure.     

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.     

Stochastic models of some endemic infections

Stochastic models are established and studied for several endemic infections with demography. Approximations of quasi-stationary distributions and of times to extinction are derived for stochastic versions of SI, SIS, SIR, and SIRS models. The approximations are valid for sufficiently large population sizes. Conditions for validity of the approximations are given for each of the models. These are also conditions for validity of the corresponding deterministic model. It is noted that some deterministic models are unacceptable approximations of the stochastic models for a large range of realistic parameter values.     

Selection in a subdivided population with local extinction and recolonization

In a subdivided population, local extinction and subsequent recolonization affect the fate of alleles. Of particular interest is the interaction of this force with natural selection. The effect of selection can be weakened by this additional source of stochastic change in allele frequency. The behavior of a selected allele in such a population is shown to be equivalent to that of an allele with a different selection coefficient in an unstructured population with a different size. This equivalence allows use of established results for panmictic populations to predict such quantities as fixation probabilities and mean times to fixation. The magnitude of the quantity N(e)s(e), which determines fixation probability, is decreased by extinction and recolonization. Thus deleterious alleles are more likely to fix, and advantageous alleles less likely to do so, in the presence of extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations.     

Variation Thresholds for Extinction and Their Implications for Conservation Strategies

We examine the degree to which fitting simple dynamic models to time series of population counts can predict extinction probabilities. This is both an active branch of ecological theory and an important practical topic for resource managers. We introduce an approach that is complementary to recently developed techniques for estimating extinction risks (e.g., diffusion approximations) and, like them, requires only count data rather than the detailed ecological information available for traditional population viability analyses. Assuming process error, we use four different models of population growth to generate snapshots of population dynamics via time series of the lengths commonly available to ecologists. We then ask to what extent we can identify which of several broad classes of population dynamics is evident in the time series snapshot. Along the way, we introduce the idea of "variation thresholds," which are the maximum amount of process error that a population may withstand and still have a specified probability of surviving for a given length of time. We then show how these thresholds may be useful to both ecologists and resource managers, particularly when dealing with large numbers of poorly understood species, a common problem faced by those designing biodiversity reserves.     

Natural extinction on islands

Almost all recent extinction of species or subspecies on islands comes from human activities. On the other hand, in local populations there is much natural extinction and immigration, i.e. turnover, on small islands. Most of this turnover occurs in locally rare species, and attests to the phenomenon of minimum viable population size. The MacArthur-Wilson theory is based on this turnover which, from an ecological point of view, is generally trivial. More useful theories of minimum viable population size are being developed. Rarity is the precursor of extinction, and species can be rare in several ways. Models of these phenomena are still primitive, particularly those that relate habitat availability to population density. Models of interactive communities show phenomena that may be relevant to the understanding of extinction in the geological record. Lotka-Volterra equations indicate considerable sensitivity to invasions, sometimes producing a cascade of extinction. Chemostat equations show that the behaviour of food chains can change dramatically with small changes in parameters, suggesting that small environmental effects can sometimes cause large ecological changes, including extinctions, in interactive biotic communities.     

Diseases threaten Southern Ocean albatrosses

Infectious diseases have the potential to cause rapid declines and extinction in vertebrate populations, and are likely to be spreading with increased globalisation and climate warming. In the Southern Ocean and Antarctica, no major outbreaks of infectious diseases have been reported to date, perhaps because of isolation and cold climate, although recent evidence suggests their presence. The major threat for the Southern Ocean environment is today considered to be fishing activities, and especially controversial long-lining which is assumed to be the cause of the major decreases in albatross and large petrel populations observed recently. Here we show that the worldwide spread of avian cholera is probably the major cause of the decrease on Amsterdam Island of the large yellow-nosed albatross (Diomedea chlororhynchos) population, which was previously attributed to long-line fishing. Another pathogenic bacterium, Erysipelas, was also present. The diseases affect mainly young chicks, with a cyclic pattern between years, but also kill adult birds. The outbreak of the disease probably occurred in the mid-1980s when chick mortality increased, adult survival decreased and the population started to decrease. The diseases may be currently threatening the very rare Amsterdam albatross (D. amsterdamensis) with extinction, and are probably also affecting sooty albatrosses (Phoebetria fusca). The spread of diseases to the most remote areas of the world raises major concern for the conservation of the Southern Ocean environment.