An epidemic model in a patchy environment

An epidemic model is proposed to describe the dynamics of disease spread among patches due to population dispersal. We establish a threshold above which the disease is uniformly persistent and below which disease-free equilibrium is locally attractive, and globally attractive when both susceptible and infective individuals in each patch have the same dispersal rate. Two examples are given to illustrate that the population dispersal plays an important role for the disease spread. The first one shows that the population dispersal can intensify the disease spread if the reproduction number for one patch is large, and can reduce the disease spread if the reproduction numbers for all patches are suitable and the population dispersal rate is strong. The second example indicates that a population dispersal results in the spread of the disease in all patches, even though the disease can not spread in each isolated patch.     

Eryngium maritimum, biology of a plant at its northernmost localities

The prospects for persistence of Eryngium maritimum in its northern distribution area was evaluated by studying historical data, matrix modelling, dispersal, seed germination and molecular variation. Historical data revealed a fragmented distribution of small populations with decrease in both population numbers and size during the last 150 years. Fruits of E. maritimum were found to have a relative low floating ability, making the prospects of long distance water dispersal more limited than would be expected from the world distribution of the species. Germination was found to be low (< 25 %). Elasticity matrices showed that survival of reproductive plants were more important than reproduction. High juvenile mortality and low germination activity emphasised the importance of rapid growth and survival. The population insurance against natural catastrophes and environmental stochasticity is suggested to be in a "root bank" of established individuals that are able to tolerate disturbances due to clonal reproduction and regrowth from root fragments. Isozyme electrophoresis revealed low molecular variation on a large geographical scale. The chromosome number is 2n = 16. A high degree of fixed heterozygosity suggests that the species is tetraploid with basic chromosome, number x = 4. The persistence prospects of E. maritimum in its northernmost distribution area are considered low considering the small population sizes, the low dispersal ability and low germination.     

Transmission Dynamics of an SIS Model with Age Structure on Heterogeneous Networks

Infection age is often an important factor in epidemic dynamics. In order to realistically analyze the spreading mechanism and dynamical behavior of epidemic diseases, in this paper, a generalized disease transmission model of SIS type with age-dependent infection and birth and death on a heterogeneous network is discussed. The model allows the infection and recovery rates to vary and depend on the age of infection, the time since an individual becomes infected. We address uniform persistence and find that the model has the sharp threshold property, that is, for the basic reproduction number less than one, the disease-free equilibrium is globally asymptotically stable, while for the basic reproduction number is above one, a Lyapunov functional is used to show that the endemic equilibrium is globally stable. Finally, some numerical simulations are carried out to illustrate and complement the main results. The disease dynamics rely not only on the network structure, but also on an age-dependent factor (for some key functions concerned in the model).     

Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model

In this paper, we introduce a basic reproduction number for a multi-group SIR model with general relapse distribution and nonlinear incidence rate. We find that basic reproduction number plays the role of a key threshold in establishing the global dynamics of the model. By means of appropriate Lyapunov functionals, a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach and LaSalle invariance principle, it is proven that if it is less than or equal to one, the disease-free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, some sufficient condition is obtained in ensuring that there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Furthermore, our results suggest that general relapse distribution are not the reason of sustained oscillations. Biologically, our model might be realistic for sexually transmitted diseases, such as Herpes, Condyloma acuminatum, etc.     

On a temporal model for the Chikungunya disease: modeling, theory and numerics

Reunion Island faced two episodes of Chikungunya, a vector-borne disease, in 2005 and in 2006. The latter was of unprecedented magnitude: one third of the population was infected. Until the severe episode of 2006, our knowledge of Chikungunya was very limited. The principal aim of our study is to propose a model, including human and mosquito compartments, that is associated to the time course of the first epidemic of Chikungunya. By computing the basic reproduction number R(0), we show there exists a disease-free equilibrium that is locally asymptotically stable if the basic reproduction number is less than 1. Moreover, we give a necessary condition for global asymptotic stability of the disease-free equilibrium. Then, we propose a numerical scheme that is qualitatively stable and present several simulations as well as numerical estimates of the basic reproduction number for some cities of Reunion Island. For the episode of 2005, R(0) was less than one, which partly explains why no outbreak appeared. Using recent entomological results, we investigate links between the episode of 2005 and the outbreak of 2006. Finally, our work shows that R(0) varied from place to place on the island, indicating that quick and focused interventions, like the destruction of breeding sites, may be effective for controlling the disease.     

Dispersal, migration, and offspring retention in saturated habitats

We examine the evolutionary stability of year-round residency in territorial populations, where breeding sites are a limiting resource. The model links individual life histories to the population-wide competition for territories and includes spatial variation in habitat quality as well as a potential parent-offspring conflict over territory ownership. The general form of the model makes it applicable to the evolution of dispersal, migration, partial migration, and delayed dispersal (offspring retention). We show that migration can be evolutionarily stable only if year-round residency in a given area would produce a sink population, where mortality exceeds reproduction. If this applies to a fraction of the breeding habitat only, partial migration is expected to evolve. In the context of delayed dispersal, habitat saturation has been argued to form an ecological constraint on independent breeding, which favors offspring retention and cooperative breeding. We show that habitat saturation must be considered as a dynamic outcome of birth, death, and dispersal rates in the population, rather than an externally determined constraint. Although delayed dispersal often associates with intense competition for territories, life-history traits have direct effects on stable dispersal strategies, which can often override the effect of habitat saturation. As an example, high survival of floaters selects against delayed dispersal, even though it increases the number of competitors for each breeding vacancy (the "habitat saturation factor"). High survival of territory owners, by contrast, generally favors natal philopatry. We also conclude that spatial variation in habitat quality only rarely selects for delayed dispersal. Within a population, however, offspring retention is more likely in high-quality territories.     

Stability and permanence in gender- and stage-structured models for the boreal toad

The boreal toad Bufo boreas boreas, once common in the western USA, is listed as an endangered species in Colorado and New Mexico, and protected in Wyoming. Populations have dramatically declined due to the presence of the fungal pathogen Batrachochytrium dendrobatidis (Bd). A gender- and stage-structured model for the boreal toad is formulated which depends on its life cycle and breeding strategies. In addition, an epizootic model for the spread of Bd is formulated. Analysis of these models provides two thresholds. The first threshold, the basic reproduction number for the population, ?(0), determines whether the population persists and the second threshold, the basic reproduction number for the fungal disease, ?(F), determines whether the disease persists. If ?(0)>1 and ?(F)<1, then the population becomes disease-free. However, if both thresholds are greater than one, the population levels are severely reduced by the fungal pathogen.     

A three-stage discrete-time population model: continuous versus seasonal reproduction

We consider a three-stage discrete-time population model with density-dependent survivorship and time-dependent reproduction. We provide stability analysis for two types of birth mechanisms: continuous and seasonal. We show that when birth is continuous there exists a unique globally stable interior equilibrium provided that the inherent net reproductive number is greater than unity. If it is less than unity, then extinction is the population's fate. We then analyze the case when birth is a function of period two and show that the unique two-cycle is globally attracting when the inherent net reproductive number is greater than unity, while if it is less than unity the population goes to extinction. The two birth types are then compared. It is shown that for low birth rates the adult average number over a one-year period is always higher when reproduction is continuous. Numerical simulations suggest that this remains true for high birth rates. Thus periodic birth rates of period two are deleterious for the three-stage population model. This is different from the results obtained for a two-stage model discussed by Ackleh and Jang (J. Diff. Equ. Appl., 13, 261-274, 2007), where it was shown that for low birth rates seasonal breeding results in higher adult averages.     

Dispersal mechanisms in a captive wild house mouse population (Mus domesticus Rutty)

The ecological implications of dispersal have been discussed in many studies of wild animals in the field but little is known about the social mechanisms leading to the emigration of certain members of a group. To study the social background of dispersal in wild house mice ( Mus domesticus Rutty) a population cage system was evaluated that allowed permanent observation of individually marked animals. It consisted of ten cages connected to a central cage by transparent plastic tubes. Two of these cages were defined as 'dispersal cages' and could be reached only by swimming through a water basin. Dispersal was defined as a permanent stay in one of these cages for at least 4 days. At the beginning of the experiment one pair of house mice with their litter was placed into the cage system. Each of six experiments lasted for 6 months during which data on spacing, social interactions, body condition, reproduction, mortality and dispersal were collected by daily observations. Results regarding this study could be summarized as follows: (1) dispersal in house mice is male-biased; (2) there are interfamiliar differences in dispersal age, dispersal rate, and in the development of the population structure; (3) after reaching sexual maturity subdominant males are evicted by the dominant one; (4) reproductive rate among females drops with increasing birth order, thus only the oldest females within a group reproduce; (5) females born under high population density conditions can only reproduce after dispersal.     

Bluetongue Disease Risk Assessment Based on Observed and Projected Culicoides obsoletus spp. Vector Densities

Bluetongue is an arboviral disease of ruminants causing significant economic losses. Our risk assessment is based on the epidemiological key parameter, the basic reproduction number. It is defined as the number of secondary cases caused by one primary case in a fully susceptible host population, in which values greater than one indicate the possibility, i.e., the risk, for a major disease outbreak. In the course of the Bluetongue virus serotype 8 (BTV-8) outbreak in Europe in 2006 we developed such a risk assessment for the University of Veterinary Medicine Vienna, Austria. Basic reproduction numbers were calculated using a well-known formula for vector-borne diseases considering the population densities of hosts (cattle and small ruminants) and vectors (biting midges of the Culicoides obsoletus spp.) as well as temperature dependent rates. The latter comprise the biting and mortality rate of midges as well as the reciprocal of the extrinsic incubation period. Most important, but generally unknown, is the spatio-temporal distribution of the vector density. Therefore, we established a continuously operating daily monitoring to quantify the seasonal cycle of the vector population by a statistical model. We used cross-correlation maps and Poisson regression to describe vector densities by environmental temperature and precipitation. Our results comprise time series of observed and simulated Culicoides obsoletus spp. counts as well as basic reproduction numbers for the period 2009–2011. For a spatio-temporal risk assessment we projected our results from the location of Vienna to the entire region of Austria. We compiled both daily maps of vector densities and the basic reproduction numbers, respectively. Basic reproduction numbers above one were generally found between June and August except in the mountainous regions of the Alps. The highest values coincide with the locations of confirmed BTV cases.     

Population regulation by dispersal over patchy environment;

A mathematical model for the population regulation is presented, which takes into account the environmental heterogeneity and the animal dispersal, and does not contain any direct density effect on reproduction or death processes. Generally speaking, there are two kinds of dispersal, one is the density independent dispersal, the other is the density dependent. It is shown that a population equilibrium can be maintained by the density dependent dispersal or threshold dispersal which occurs only when the population density exceeds a certain threshold level, but that density independent dispersal by itself can not continue to maintain a population equilibrium.     

A discrete model of avian influenza with seasonal reproduction and transmission

In this paper, we formulate a discrete-time model with the reproductive and overwintering periods to assess the impact of avian influenza transmission in poultry. It is shown that the disease is extinct if the basic reproduction number is less than one and is persistent if the basic reproductive number is greater than one. Furthermore, the model admits a closed invariant cycle, which means that avian influenza fluctuates in poultry.     

Estimation by capture-recapture of recruitment and dispersal over several sites

Dispersal in animal populations is intimately linked with accession to reproduction, i.e. recruitment, and population regulation. Dispersal processes are thus a key component of population dynamics to the same extent as reproduction or mortality processes. Despite the growing interest in spatial aspects of population dynamics, the methodology for estimating dispersal, in particular in relation with recruitment, is limited. In many animal populations, in particular vertebrates, the impossibility of following individuals over space and time in an exhaustive way leads to the need to frame the estimation of dispersal in the context of capture-recapture methodology. We present here a class of age-dependent multistate capture-recapture models for the simultaneous estimation of natal dispersal, breeding dispersal, and age-dependent recruitment. These models are suitable for populations in which individuals are marked at birth and then recaptured over several sites. Under simple constraints, they can be used in populations where non-breeders are not observed, as is often the case with colonial waterbirds monitored on their breeding grounds. Biological questions can be addressed by comparing models differing in structure, according to the generalized linear model philosophy broadly used in capture-recapture methodology. We illustrate the potential of this approach by an analysis of recruitment and dispersal in the roseate tern Sterna dougallii .     

Sex in space: population dynamic consequences

Sex, so important in the reproduction of bigametic species, is nonetheless often ignored in explorations of the dynamics of populations. Using a growth model of dispersal-coupled populations we can keep track of fluctuations in numbers of females and males. The sexes may differ from each other in their ability to disperse and their sensitivity to population density. As a further complication, the breeding system is either monogamous or polygamous. We use the harmonic mean birth function to account for sex-ratio-dependent population growth in a Moran–Ricker population renewal process. Incorporating the spatial dimension stabilizes the dynamics of populations with monogamy as the breeding system, but does not stabilize the population dynamics of polygamous species. Most notably, in populations coupled with dispersal, where the sexes differ in their dispersal ability there are rarely stable and equal sex ratios. Rather, a two-point cycle, four-point cycle and eventually complex behaviour of sex-ratio dynamics will emerge with increasing birth rates. Monogamy often leads to less noisy sex-ratio dynamics than polygamy. In our model, the sex-ratio dynamics of coupled populations differ from those of an isolated population system, where a stable 50:50 sex ratio is achievable with equal density-dependence costs for females and males. When sexes match in their dispersal ability, population dynamics and sex-ratio dynamics of coupled populations collapse to those of isolated populations.     

Mobility and lifetime fecundity in new versus old populations of the Glanville fritillary butterfly

Life history theory often assumes a trade-off between dispersal and reproduction, and such a trade-off is commonly observed in wing-dimorphic insects. The results are less consistent for wing-monomorphic species, for which it is more difficult to assess dispersal capacity and rate. Three replicate experiments were carried out in consecutive years on the Glanville fritillary butterfly in a large outdoor population cage to study the relationship between lifetime egg production and mobility. The experimental material included females originating from newly-established and old populations, as previous studies have shown dispersal capacity to depend on population age. There was a consistent and significant interaction between mobility and population age, such that in newly-established populations mobile females had higher fecundity than less mobile females, while in old populations there was no such relationship. As selection favours individuals with the highest fecundity, selection pressure on mobility is likely to be different between the two population types, which may contribute to maintenance of variation in dispersal rate in the metapopulation as a whole. Several other female traits also affected lifetime fecundity, including lifespan, number of matings and date of eclosion, although these effects were not consistent across the years. These results highlight the importance of conducting experiments in more than one year before generalizing about patterns in life history variation.     

The dilution effect of the domestic animal population on the transmission of P. vivax malaria

The diversion of disease carrying insect from humans to animals may reduce transmission of diseases such as malaria. The use of animals to mitigate mosquito bites on human is called ‘zooprophylaxis’. We introduce a mathematical model for Plasmodium vivax malaria transmission with two bloodmeal hosts (humans and domestic animals) to study the effect of zooprophylaxis. After computing the basic reproduction number from the proposed model, we explore how perturbations in the parameters, sensitive to the effects of control measures, affect its value. Zooprophylaxis is shown to determine whether a basic reproduction becomes bigger than an outbreak threshold value or not. Sensitivity analysis shows that increasing the relative animal population size works better in P. vivax malaria control than decreasing the mosquito population when the relative animal population size is larger than a threshold value.     

Sociality, individual fitness and population dynamics of yellow-bellied marmots

Social behaviour was proposed as a density-dependent intrinsic mechanism that could regulate an animal population by affecting reproduction and dispersal. Populations of the polygynous yellow-bellied marmot (Marmota flaviventris) fluctuate widely from year to year primarily driven by the number of weaned young. The temporal variation in projected population growth rate was driven mainly by changes in the age of first reproduction and fertility, which are affected by reproductive suppression. Dispersal is unrelated to population density, or the presence of the father; hence, neither of these limits population growth or acts as an intrinsic mechanism of population regulation; overall, intrinsic regulation seems unlikely. Sociality affects the likelihood of reproduction in that the annual probability of reproducing and the lifetime number of offspring are decreased by the number of older females and by the number of same-aged females present, but are increased by the number of younger adult females present. Recruitment of a yearling female is most likely when her mother is present; recruitment of philopatric females is much more important than immigration for increasing the number of adult female residents. Predation and overwinter mortality are the major factors limiting the number of resident adults. Social behaviour is not directed towards population regulation, but is best interpreted as functioning to maximize direct fitness.     

Dynamics of an SIQS epidemic model with transport-related infection and exit-entry screenings

Population dispersal, as a common phenomenon in human society, may cause the spreading of many diseases such as influenza, SARS, etc. which are easily transmitted from one region to other regions. Exit and entry screenings at the border are considered as effective ways for controlling the spread of disease. In this paper, the dynamics of an SIQS model are analyzed and the combined effects of transport-related infection enhancing and exit-entry screenings suppressing on disease spread are discussed. The basic reproduction number is computed and proved to be a threshold for disease control. If it is not greater than the unity, the disease free equilibrium is globally asymptotically stable. And there exists an endemic equilibrium which is locally asymptotically stable if the reproduction number is greater than unity. It is shown that the disease is endemic in the sense of permanence if and only if the endemic equilibrium exists. Exit screening and entry screening are shown to be helpful for disease eradication since they can always have the possibility to eradicate the disease endemic led by transport-related infection and furthermore have the possibility to eradicate disease even when the isolated cites are disease endemic.     

On the Reproduction Number of a Gut Microbiota Model

A spatially structured linear model of the growth of intestinal bacteria is analysed from two generational viewpoints. Firstly, the basic reproduction number associated with the bacterial population, i.e. the expected number of daughter cells per bacterium, is given explicitly in terms of biological parameters. Secondly, an alternative quantity is introduced based on the number of bacteria produced within the intestine by one bacterium originally in the external media. The latter depends on the parameters in a simpler way and provides more biological insight than the standard reproduction number, allowing the design of experimental procedures. Both quantities coincide and are equal to one at the extinction threshold, below which the bacterial population becomes extinct. Optimal values of both reproduction numbers are derived assuming parameter trade-offs.     

On a population pathogen model incorporating species dispersal with temporal variation in dispersal rate

In the present paper, we consider a mathematical model of ecosystem population interaction where the population suffers from a susceptible–infectious–susceptible disease. Dispersal of both the susceptible and the infective is incorporated using reaction–diffusion equations. We first study the stability criteria of the basic (non-spatial) model around the disease-free and the infected steady states. We find that the loss rate of the infective species controls disease prevalence. Also without predation pressure, the disease will continue to exist among the population. Then we analyze the spatial model with species dispersal in constant as well as in time-varying form. It is observed that though constant dispersal is unable to generate diffusion-driven instability, dispersal with sinusoidal variation in dispersion rate can generate diffusive instability when the wave number of the perturbation lies within a given range. Numerical simulations are performed to illustrate analytical studies.