An Effective Runge-Kutta Optimizer Based on Adaptive Population Size and Search Step Size

A newly proposed competent population-based optimization algorithm called RUN,which uses the principle of slope variations calculated by applying the Runge Kutta method as the key search mechanism,has gained wider interest in solving optimization problems.However,in high-dimensional problems,the search capabilities,convergence speed,and runtime of RUN deteriorate.This work aims at filling this gap by proposing an improved variant of the RUN algorithm called the Adaptive-RUN.Population size plays a vital role in both runtime efficiency and optimization effectiveness of metaheuristic algorithms.Unlike the original RUN where population size is fixed throughout the search process,Adaptive-RUN automatically adjusts population size according to two population size adaptation techniques,which are linear staircase reduction and iterative halving,during the search process to achieve a good balance between exploration and exploitation characteristics.In addition,the proposed methodology employs an adaptive search step size technique to determine a better solution in the early stages of evolution to improve the solution quality,fitness,and convergence speed of the original RUN.Adaptive-RUN performance is analyzed over 23 IEEE CEC-2017 benchmark functions for two cases,where the first one applies linear staircase reduction with adaptive search step size(LSRUN),and the second one applies iterative halving with adaptive search step size(HRUN),with the original RUN.To promote green computing,the carbon footprint metric is included in the performance evaluation in addition to runtime and fitness.Simulation results based on the Friedman andWilcoxon tests revealed that Adaptive-RUN can produce high-quality solutions with lower runtime and carbon footprint values as compared to the original RUN and three recent metaheuristics.Therefore,with its higher computation efficiency,Adaptive-RUN is a much more favorable choice as compared to RUN in time stringent applications.

Population Size, Genetic Diversity and Molecular Evidence of a Recent Population Bottleneck in Hynobius chinensis, an Endangered Salamander Species

Severe population declines can reduce species to small populations, offering permissive conditions for deleterious processes. For example, following such events, species can become prone to inbreeding and genetic drift which can lead to a loss of genetic diversity and evolutionary potentials. Hynobius chinensis is a poorly studied very rare and declining endangered amphibian species endemic to China in Changyang County. We investigated adult census population size by monitoring breeding populations from 2015 to 2018, developed microsatellite markers from the transcriptome and used them to investigate genetic diversity, and a population bottleneck in this species. We found H. chinensis in 4 different localities in a total area of 2.18 km^2 and estimated the overall adult census population size at 386–404 individuals. The adult census size（mean ± SE） per breeding pond ranged from 44 ± 6 to 141 ± 8 individuals and appeared smaller than that reported in closely related species in undisturbed habitats. We developed and characterized 13 microsatellite markers in total. Analysis of data at 7 loci（N = 118） in Hardy-Weinberg equilibrium gathered from the largest population showed that genetic diversity level was low. The average number of alleles per locus was 2.14. The observed and expected heterozygosities averaged 0.38 and 0.40, respectively. The inbreeding coefficient was –0.06. All tests performed to investigate a population bottleneck, i.e. The Garza-Williamson test, Heterozygosity excess test, Mode shift test of allele frequency, and effective population size estimates detected a population bottleneck. The contemporary and the historical effective population sizes were estimated at 36 and 234 individuals, respectively. We argue that as bottleneck effects, the studied population may have become prone to genetic drift and inbreeding, losing microsatellite alleles and heterozygosity. Our results suggest that populations of H. chinensismay have been extirpated in the study area.

Effects of effective population size on the F_2 growth and survival of bay scallop Argopecten irradians irradians (Lamarck)

In 2002, six cohorts ofbroodstock bay scallop Argopecten irradians irradians （Ne=1, 2, 10, 30, 50 and control） were randomly chosen from a population of bay scallop to produce offspring. After one year rearing, with the progeny matured, the similar experiment was done to produce the F2 generation. To determine the magnitude of Ne effects, the growth and survival rates in larvae and adult of six F2 groups were compared. Results showed that inbreeding depression existed not only in the Ne=1 group but also in the Ne=2 group. The growth and survival rates of the two groups were significantly lower than those of the other groups （Ne=10, 30, 50, control）, and there were no significant differences among the latter （P〉0.05）. At the same time, the amount of depression in the Ne=1 group was significantly higher than that of the Ne=2 group （P〈0.05）. These results indicated that the low effective population size （Ne）, which increases the possibility of inbreeding, could lead to some harmful effects on the offspring. So it is essential to maintain a high level of Ne in commercial seed production. Fta＇thermore, as the high fecundity of bay scallop might lead to increased inbreeding, selecting broodstock from different growout sites is recommended.

Population size and distribution of seabirds in the Cosmonaut Sea,Southern Ocean

The Cosmonaut Sea is one of the less studied ecosystems in the Southern Hemisphere.Unlike other seas which were near to coastal regions,however,few studies exist on the top predators in this zone.From December 2019 to January 2020,a survey of seabirds was carried out on the board icebreaker R/V Xuelong 2 in the Cosmonaut Sea and the Cooperation Sea.Twenty-three bird species were recorded.Antarctic petrel(Thalassoica antarctica),Antarctic prion(Pachyptila desolata),and Arctic tern(Sterna paradisaea)were the most abundant species.A total of about 37500 birds belonging to 23 species were recorded.Around 23%of the region had no record of birds.A large number of birds was recorded in 39°E-40°E,44°E-46°E and 59°E-60°E.Many areas,such as 33°E-35°E,39°E-41°E,44°E-46°E and 59°E-60°E show a great richness.More than two-thirds of seabirds(71%)were observed in the zone near the ocean front.The prediction of the distributions of the most dominant species Antarctic petrel also showed that the area near the ocean front region had an important ecological significance for seabirds.The results suggest that the distribution of seabirds in the Cosmonaut Sea is highly heterogenous.

Resurvey After 20 Years:Updating the Distribution,Population,and Potential Threats of the Chinting Alpine Toad

Considering the increased anthropogenic impacts,species with a limited range and low detectability often lack fundamental information and conservation actions,placing them at a high risk of endangerment.The Chinting alpine toad Scutiger chintingensis is a rare mountain amphibian endemic to the eastern margin of the Qinghai-Xizang Plateau in China.Within its whole distribution range,only three known populations(Wolong,Emei,and Wawu)exist and no recent population status report has been documented for this species over the past two decades.From 2020 to 2023,we investigated the species distribution,and assessed the risk factors for the main populations.We recorded this species in all distribution areas,and updated a new distribution site with a lower elevation limit.The relative population density was 0.024±0.012 ind./m^(2)on Mount Emei,whereas only 0.008±0.017 ind./m^(2)on Mount Wawu.No significant difference was observed in the number of individuals between the two populations;however,the relative population density was significantly different.Sewage and waste discharge resulting from the construction of scenic areas,as well as disturbances from tourism,were the primary anthropogenic factors that influenced the survival of this species.Our results provide the updated information on the distribution and population status of the Chinting alpine toad,and suggest that unrecorded populations,as well as a wider elevation range,may exist for this species.Our findings emphasise the importance of timely updates of species distribution and population information and offer a basis for the future conservation of endangered amphibians.

Effects of Population Size on Reproductive Success of the Endangered and Endemic Species Primula merrilliana

The reproductive ability related to the population size of the endangered and endemic species Primula merrilliana Schltr. was investigated. In 26 populations observed, only four contain more than 500 flowering individuals, whereas most of them （53.8%） consist of less than 100 flowering individuals. Though the ratio of pin and thrum plants keeps its balance at 1：1 for all populations, the frequency of pin and thrum flowers was significantly biased in most small populations. Population size strongly affected reproductive success; plants in small populations produced significantly fewer flowers and fruits per plant and fewer seeds per fruit, and therefore fewer seeds per plant. The floral density was another important factor that influenced the reproductive success of P. merrilliana, because four main reproductive success parameters （fruits per plant, seeds per fruit, seeds per plant, and the proportion of flowers setting fruit） were all positively correlated with floral density. The size of plants and the number of leaves per plant （measure of habitat quality） were not influenced by the variation of population size, suggesting that the reduced fecundity in small populations may not be a consequence of lower habitat quality. Inbreeding depression and pollen limitation as a result of less attractiveness in small populations are therefore likely explanations for these patterns.

Analysis of an Age Structured SEIRS Epidemic Model with Varying Total Population Size and Vaccination

This article focuses on the study of an age structured SEIRS epidemic model with a vaccination program when the total population size is not kept at constant. We first give the explicit expression of the reproduction number in the presence of vaccine ( is the exponent of growth of total population), and show that the infection-free steady state is linearly stable if and unstable if , then we apply the theoretical results to vaccination policies to determine the optimal age or ages at which an individual should be vaccinated. It is shown that the optimal strategy can be either one- or two-age strategies.

Noninvasive genetics provides insights into the population size and genetic diversity of an Amur tiger population in China

Understanding population size and genetic diversity is critical for effective conservation of endangered species.The Amur tiger(Panthera tigris altaica)is the largest felid and a flagship species for wildlife conservation.Due to habitat loss and human activities,available habitat and population size are continuously shrinking.However,little is known about the true population size and genetic diversity of wild tiger populations in China.In this study,we collected 55 fecal samples and 1 hair sample to investigate the population size and genetic diversity of wild Amur tigers in Hunchun National Nature Reserve,Jilin Province,China.From the samples,we determined that 23 fecal samples and 1 hair sample were from 7 Amur tigers:2 males,4 females and 1 individual of unknown sex.Interestingly,2 fecal samples that were presumed to be from tigers were from Amur leopards,highlighting the significant advantages of noninvasive genetics over traditional methods in studying rare and elusive animals.Analyses from this sample suggested that the genetic diversity of wild Amur tigers is much lower than that of Bengal tigers,consistent with previous findings.Furthermore,the genetic diversity of this Hunchun population in China was lower than that of the adjoining subpopulation in southwest Primorye Russia,likely due to sampling bias.Considering the small population size and relatively low genetic diversity,it is urgent to protect this endangered local subpopulation in China.

Pollen limitation in invasive populations of Solanum rostratum and its relationship to population size

Aims small plant populations may be more likely to suffer more severe pollen limitation due to the lower number of potential mates or suitable pollinators.For invasive species,this phenomenon may be more common when an invading population colonizes a new habitat.Here,we investigated whether pollen limitation occurs in invasive populations of Solanum rostratum during its invasion from North america to China and evaluated the patterns between pollen limitation and population size.Methods Pollen addition experiments were performed on six invasive populations of S.rostratum.by comparing fruit set and seed production with open pollination treatment,we calculated the index of pollen limitation and regressed it to population size and density.Important Findingsamong the six sampled invasive populations of S.rostratum,the fruit set and seed production per fruit were 0.346±0.014 and 52.38±9.29,respectively,with open pollination treatment and 0.572±0.022 and 56.28±10.79,respectively,with pollen addition treatment.Compared with open pollination,pollen addition significantly increased fruit set and seed production by 65.3 and 7.4%,respectively.The standardized index of pollen limitation ranged from 0.022 to 0.125,with an average of 0.065,suggesting that invasive populations of S.rostratum do suffer from pollen limitation.The index of pollen limitation was negatively correlated with population size,which is consistent with the pattern that smaller populations suffer from more severe pollen limitation.

Estimation of a Population Size in Large-Scale Wireless Sensor Networks

Efficient estimation of population size is a common requirement for many wireless sensor network applications. Examples include counting the number of nodes alive in the network and measuring the scale and shape of physically correlated events. These tasks must be accomplished at extremely low overhead due to the severe resource limitation of sensor nodes, which poses a challenge for large-scale sensor networks. In this article we design a novel measurement technique, FLAKE based on sparse sampling that is generic, in that it is applicable to arbitrary wireless sensor networks （WSN）. It can be used to efficiently evaluate system size, scale of event, and other global aggregating or summation information of individual nodes over the whole network in low communication cost. This functionality is useful in many applications, but hard to achieve when each node has only a limited, local knowledge of the network. Therefore, FLAKE is composed of two main components to solve this problem. One is the Injected Random Data Dissemination （Sampling） method, the other is sparse sampling algorithm based on Inverse Sampling, upon which it improves by achieving a target variance with small error and low communication cost. FLAKE uses approximately uniform random data dissemination and sparse sampling in sensor networks, which is an unstructured and localized method. At last we provide experimental results demonstrating the effectiveness of our algorithm on both small-scale and large-scale WSNs. Our measurement technique appears to be the practical and appropriate choice.

Integrative demographic study of the Iberian painted frog(Discoglossus galganoi):inter-annual variation in the effective to census population size ratio,with insights on mating system and breeding success

In the face of worldwide amphibian declines,integrative studies combining individual-based information and genetic data represent a powerful approach to produce robust,reliable,and comparable assessments of demographic dynamics.The Iberian painted frog(Discoglossus galganoi)is endemic to Spain and Portugal and shows decreasing population trends across its range,but few studies have attempted to estimate census sizes or assess genetic diversity in wild populations,and little is known about their reproductive biology.We applied an integrative approach based on the combination of capture-mark-recapture data and multilocus genotypes to monitor a breeding population of D.galganoi in central Spain during two consecutive breeding seasons,focusing on the estimation of demographic parameters and their temporal variation.Specifically,we estimated the number of adults(Na),the effective population size(Ne),and the effective number of breeders(Nb),as well as survival and migration rates.We documented a>50%decrease in the estimated number of adults of both sexes between the breeding seasons of 2018 and 2019,probably associated with reduced rainfall in the latter.Estimates of Nb and the Nb/Na ratio were low in both seasons,with a 20–30%decrease in Nb and a 47%increase in the Nb/Na ratio in 2019.Based on the reconstruction of pedigrees from larval and adult genotypes,we provide the first genetic evidence of polygamy in males and females of D.galganoi and the first estimates of breeding success in the species.

ANALYSIS OF A SUSCEPTIBLE-EXPOSEDINFECTED EPIDEMIC MODEL WITH RANDOM PERTURBATION AND VARYING POPULATION SIZE

In this paper, we study a type of susceptible-exposed-infected（SEI） epidemic model with varying population size and introduce the random perturbation of the constant contact rate into the SEI epidemic model due to the universal existence of fluctuations. Under some moderate conditions, the density of the exposed and the infected individuals exponentially approaches zero almost surely are derived. Furthermore, the stochastic SEI epidemic model admits a stationary distribution around the endemic equilibrium, and the solution is ergodic. Some numerical simulations are carried out to demonstrate the efficiency of the main results.

Genetic identification and reiterated captures suggest that the Astyanax mexicanus El Pachón cavefish population is closed and declining

The sizes of Astyanax mexicanus blind cavefish populations of North-East Mexico are demographic parameters of great importance for investigating a variety of ecological,evolutionary,and conservation issues.However,few estimates have been obtained.For these mobile animals living in an environment difficult to explore as a whole,methods based on capture-mark-recapture are appropriate,but their feasibility and interpretation of results depend on several assumptions that must be carefully examined.Here,we provide evidence that minimally invasive genetic identification from captures at different time intervals(three days and three years)can give insights into cavefish population size dynamics as well as other important demographic parameters of interest.We also provide tools to calibrate sampling and genotyping efforts necessary to reach a given level of precision.Our results suggest that the El Pachón cave population is currently very small,of an order of magnitude of a few hundreds of individuals,and is distributed in a relatively isolated area.The probable decline in population size in the El Pachón cave since the last census in 1971 raises serious conservation issues.

Evolution of the North Atlantic Current and Barents Ice Sheet as revealed by grain size populations in the northern Norwegian Sea during the last 60 ka

The grain size distribution of bulk sediment samples was decomposed in a core to reconstruct paleoceanographic evolution over the past 60 ka in the northern Norwegian Sea.The results show that sediments consisted of 3–4 grain populations derived from the North Atlantic Current(NAC)and Barents Ice Sheet(BIS).The grain size data suggest three palaeoceanographic evolution stages:(1)an environment affected by BIS and NAC and changed with the interstadial/stadial transition in phase with the Greenland ice-core record at 60–31 ka BP,during which discharge of icebergs and the content of the coarsest population containing ice-rafted debris(IRD)in the sediments increased significantly during stadial,while the fine silt population containing volcanic glasses increased with the enhancement of NAC during the interstadial;(2)an extreme environment controlled by BIS at31–13 ka BP.BIS reached to its maximum at about 31 ka BP and the turbid plumes that formed at the leading edge of BIS contributed to a significant increase in the clayey population in sediments.Icebergs drained into the northern Norwegian Sea with periodical calving of the BIS at 31–19 ka BP.Subsequently,the ablation of the BIS discharged massive floods with clayey sediments and icebergs into the Norwegian Sea at 19–13 ka BP,resulting in a constant increase in clay and IRD in sediments;and(3)a marine environment similar to the present one under the strong influence of NAC following the complete melting of the BIS after 13 ka BP,NAC is the dominant transport agent and no IRD occurred in sediments.The fine silt populations containing volcanic glasses transported by NAC significantly increased.

Patterns of change in the population and spatial distribution of oriental white storks (Ciconia boyciana) wintering in Poyang Lake

Using total counts in simultaneous annual surveys, we monitored the population size and spatial distribution of oriental white storks （Ciconia boyciana） wintering in Poyang Lake between 1998 and 2011. Results showed that Poyang Lake wetland is an important wintering ground for oriental white storks, with an annual average population number of 2 305＋326. The population sizes in 2004, 2005, 2010, and 2011 were higher than the highest-ever estimate of its global population. In 2005, we recorded 3 789 individuals, which was the maximum population number within the period of 1998-2011. The storks inhabited 52 lakes, with the greatest distance between these lakes being 180.3 km. The storks presented a clustered distribution pattern in the Poyang Lake wetland, irrespective of the number of individuals or occurrence frequencies. Shahu, Dahuchi, Banghu, and Hanchihu were most frequently used lakes and had the largest annual average numbers of storks. There was a significant positive correlation between occurrence frequency and annual average number of storks in the lakes. Most of the lakes important for storks were covered by existing nature reserves, though some lakes outside the reserves were also frequently used. About 64.9%＋ 5.5% of the storks were found in nature reserves. In addition, the storks more frequently used and clumped in significantly larger flocks in lakes within nature reserves than lakes outside.

Genetic diversity analysis of spawner and recaptured populations of Chinese shrimp(Fenneropenaeus chinensis)during stock enhancement in the Bohai Bay based on an SSR marker

Eight microsatellite markers were used to analyze genetic diversity, level of inbreeding, and effective population size of spawner and recaptured populations of Chinese shrimp（Fenneropenaeus chinensis） during stock enhancement in the Bohai Bay in 2013. A total of 254 and 238 alleles were identified in the spawner and recaptured populations, respectively, and the numbers of alleles（N_a） were 8–63 and 6–60, respectively. The numbers of effective alleles（N_e） were 2.52–21.60 and 2.67–20.72, respectively. The polymorphism information content ranged from 0.529 to 0.952. The observed heterozygosity（H_o） values（0.638–0.910 and 0.712–0.927） were lower than the expected heterozygosity（H_e） values（0.603–0.954 and 0.625–0.952）, which indicated that the two populations possessed a rich genetic diversity. In 16 tests（2 populations×8 loci）, 13 tests deviated from the HardyWeinberg equilibrium. F_（is） values were positive at seven loci and the inbreeding coefficients（F） of the two populations estimated by trio ML were 13.234% and 11.603%, suggesting that there was a relatively high degree of inbreeding. A certain level of inbreeding depression had occurred in the Chinese shrimp population. F_（st） values ranged from 0 to 0.059, with a mean of 0.028, displaying a low level of genetic differentiation in the two populations. Effective population sizes（3 060.2 and 3 842.8） were higher than the minimum number suggested for retaining the evolutionary potential to adapt to new environmental conditions. For enhancement activity in 2014,the ideal number of captured shrimp spawners should have ranged from 7 686 to 19 214 to maintain genetic diversity and effective population size. Further strategies to adjust the balance of economic cost, fishing effort and ideal number of shrimp spawners to maintain a satisfactory effective population size for ensuring the sustainability of Chinese shrimp are proposed.

African Lion Population Estimates in Tanzania’s Ruaha National Park

Tanzania is considered a country with the largest number of African lions (Panthera leo). However, the continued absence of ecological population estimates and understanding of the associated factors influencing lion distribution hinders the development of conservation planning. This is particularly true in the Ruaha-Rungwa landscape, where it was estimated that more than 10% of the global lion population currently resides. By using a call-back survey method, we aimed to provide population estimates (population size and density) of African lions in the Ruaha National Park, between wet (March 2019) and dry (October 2019) seasons. We also assessed the key factors that influenced the distribution of the observed lions towards call-back stations. Ferreira & Funston’s (2010) formula was used to calculate population size and in turn used to estimate density in the sampled area, while the Generalized Linear Model (GLMM) with zero-inflated Poisson error distribution was used to determine factors that influence the distribution of the observed lions to call-back stations. The population size we calculated for the sampled area of 3137.2 km2 revealed 286 lions (95% CI, 236 - 335) during the wet season, and 196 lions (95% CI, 192 - 200) during the dry season. The density of lions was 9.1/100 km2 during the wet season, and 6.3/100 km2 during the dry season. Distance to water source had a significant negative effect on the distribution of the observed lions to the call-back stations, while habitat had a marginal effect. Our findings show that, although lion population estimates were larger during the wet season than the dry season, the season had no effect on the distribution of the observed lions to call-back stations. We suggest that the proximity to water sources is important in study design. Further, we suggest that density and population size are useful indices in identifying conservation area priorities and lion coexistence strategies.

Role of intraspecific competition in intrinsic growth rate regulation in an Oikopleura dioica(Tunicata)population

Planktonic Oikopleura dioica respond almost instantly to opportunistic algal blooms,but it is unknown whether the population increase can change from an exponential to a logistic model to avoid a final sudden collapse.To test the hypothesis that intraspecific competition regulates the intrinsic rate of natural increase(r),density-dependent effects on growth and reproduction performance were investigated in O.dioica via laboratory incubations.Over a large food concentration range,batch maturation was observed above the per capita food supply(PFS)of 8.1µgC/ind.in 4.5 d.Somatic growth was saturated beyond this PFS value,whereas gonad length increased continuously.Below this threshold,individuals reached small body and gonad lengths,and maturation was rarely observed during the incubation period.The gonad/body volume and maturation ratios also increased with the PFS.Instead of the food concentration,the r values were regulated by competition pressure via variability in maturation duration and the proportion of mature individuals in the cohorts.When the minimum food demand was satisfied in the designated generation time,the r value tended to be regulated by the spawning proportion in the population.Otherwise,prolonged development duration and decreased r values were expected.

Population assessment and habitat associations of the Visayan Hornbill(Penelopides panini)in Northwest Panay,Philippines

Background:Seven out of ten hornbill species in the Philippines are threatened with extinction.Among these is the endangered Visayan Hornbill(Penelopides panini),found on the islands of Panay and Negros.Threatened by habitat loss and hunting,its population size is thought to have declined from 1800 individuals 20 years ago to less than 1000.However,a recent study on Negros estimated 3564 individuals across three core forest blocks.This study aims to quantify the Visayan Hornbill population size in and around the Northwest Panay Peninsula Natural Park(NWPPNP)on Panay,the largest contiguous low-elevation forest landscape remaining across its range,and its broad habitat associations across a gradient of environmental degradation.Methods:Hornbills were surveyed using 10-min distance sampling point counts(n=367)along transects(average length 1.1 km).Environmental variables were recorded along transects,while habitat was classified into primary forest,secondary forest,plantation,or open habitat.Distance software was used to estimate population densities stratified by habitat,with the overall population estimate taken as a mean of habitat density estimates weighted by habitat area.Using generalized linear mixed models,hornbill occurrence was modelled using combinations of nine environmental variables as main and two-way fixed effects.Results:Surveys covered 204.4 km^(2) of the 374.8 km^(2) Northwest Panay Peninsula.Hornbills were not recorded in plantations or open habitats.Hornbill density was significantly higher in primary forest(17.8 individuals/km~2±26.9%CV)than in secondary forest(3.7 individuals/km^(2)±33.2%CV;z=15.212,P<0.001).The overall population estimate for the NWPPNP and environs is 2109 individuals,and 2673 individuals for the entire Northwest Panay Peninsula.Hornbill presence was best explained by a model including distance from the Park boundary alongside five interaction effects and transect as a random effect.Distance,and the interaction between distance and medium-sized trees were significant predictors of hornbill presence.Conclusions:Our study evidences the habitat preference of the Visayan Hornbill,highlights the importance of the NWPPNP for the species'conservation,and provides strong evidence for re-assessing the global population size.

GLOBAL DYNAMICS OF AN SEIR EPIDEMIC MODEL WITH IMMIGRATION OF DIFFERENT COMPARTMENTS

The SEIR epidemic model studied here includes constant inflows of new susceptibles, exposeds, infectives, and recovereds. This model also incorporates a population size dependent contact rate and a disease-related death. As the infected fraction cannot be eliminated from the population, this kind of model has only the unique endemic equilibrium that is globally asymptotically stable. Under the special case where the new members of immigration are all susceptible, the model considered here shows a threshold phenomenon and a sharp threshold has been obtained. In order to prove the global asymptotical stability of the endemic equilibrium, the authors introduce the change of variable, which can reduce our four-dimensional system to a three-dimensional asymptotical autonomous system with limit equation.