Analysis of a disease transmission model in a population with varying size

An S I R S epidemiological model with vital dynamics in a population of varying size is discussed. A complete global analysis is given which uses a new result to establish the nonexistence of periodic solutions. Results are discussed in terms of three explicit threshold parameters which respectively govern the increase of the total population, the existence and stability of an endemic proportion equilibrium and the growth of the infective population. These lead to two distinct concepts of disease eradication which involve the total number of infectives and their proportion in the population.Partially supported by NSF Grant No. DMS-8703631. This work was done while this author was visiting the University of VictoriaResearch supported in part by NSERC A-8965     

Global dynamics of a SEIR model with varying total population size

A SEIR model for the transmission of an infectious disease that spreads in a population through direct contact of the hosts is studied. The force of infection is of proportionate mixing type. A threshold sigma is identified which determines the outcome of the disease; if sigma < or = 1, the infected fraction of the population disappears so the disease dies out, while of sigma > 1, the infected fraction persists and a unique endemic equilibrium state is shown, under a mild restriction on the parameters, to be globally asymptotically stable in the interior of the feasible region. Two other threshold parameters sigma' and sigma are also identified; they determine the dynamics of the population sizes in the cases when the disease dies out and when it is endemic, respectively.     

Effective size of a fluctuating age-structured population

Previous theories on the effective size of age-structured populations assumed a constant environment and, usually, a constant population size and age structure. We derive formulas for the variance effective size of populations subject to fluctuations in age structure and total population size produced by a combination of demographic and environmental stochasticity. Haploid and monoecious or dioecious diploid populations are analyzed. Recent results from stochastic demography are employed to derive a two-dimensional diffusion approximation for the joint dynamics of the total population size, N, and the frequency of a selectively neutral allele, p. The infinitesimal variance for p, multiplied by the generation time, yields an expression for the effective population size per generation. This depends on the current value of N, the generation time, demographic stochasticity, and genetic stochasticity due to Mendelian segregation, but is independent of environmental stochasticity. A formula for the effective population size over longer time intervals incorporates deterministic growth and environmental stochasticity to account for changes in N.     

Inbreeding coefficients for stochastically varying small population sizes-bias of calculation based on effective numbers

The commonly used procedure to calculate inbreeding coefficients by effective population numbers (Ne) by the harmonic mean of generation-by-generation population sizes involves a computational bias. If the individual population sizes are considered as realizations of a binomially distributed random variable with sample size N and probability p, this bias can be investigated for the two cases p = constant and p = variable (Markov chain). The bias is of practical relevance only for small probabilities p, short period of initial successive generations, and small population sizes. The largest values for this computational bias are in the range of 0.05-0.06. It is concluded that for most practical purposes the approximate procedure is appropriate.     

The Wright-Fisher model with temporally varying selection and population size

The Wright-Fisher model is considered in the case where the population size is random and the magnitude of the selective advantage of one of the alleles varies with time. The central question addressed is the possibility of ultimate genetic polymorphism. Partial results are obtained in the general case and complete results in the case where the population size and selective advantage are not density dependent. Bounds on the fixation probability are obtained when the selective advantage is constant.     

Difference between evolutionarily effective and germ line mutation rate due to stochastically varying haplogroup size

Within a Y-chromosome haplogroup defined by unique event mutations, variation in microsatellites can accumulate due to their rapid mutation. Estimates based on pedigrees for the Y-chromosome microsatellite mutation rate are 3 or more times greater than the same estimates from evolutionary considerations. We show by simulation that the haplogroups that survive the stochastic processes of drift and extinction accumulate microsatellite variation at a lower rate than predicted from corresponding pedigree estimates; in particular, under constant total population size, the accumulated variance is on average 3-4 times smaller.     

Dynamics of a stochastic heroin epidemic model with bilinear incidence and varying population size

We investigate a stochastic heroin epidemic model with bilinear incidence and varying population size.Sufficient criteria for the extinction of the drug abusers and the existence of ergodic stationary distribution for the model are established by constructing suitable stochastic Lyapunov functions.By analyzing the sensitivity of the threshold of spread,we obtain that prevention is better than cure.Numerical simulations are carried out to confirm the analytical results.     

Effective population size in Hymenoptera with complementary sex determination

Complementary sex determination in the haplodiploid Hymenoptera leads to the production of inviable or effectively sterile diploid males when diploid progeny are homozygous at the sex-determining locus. The production of diploid males reduces the number of females in a population and biases the effective breeding sex ratio in favor of haploid males. This in turn will reduce the effective population size (Ne) of hymenopteran populations with complementary sex determination relative to the expected reductions due to haplodiploidy alone. The effects of diploid male production on Ne in hymenopterans with complementary sex determination when diploid males are either inviable or effectively sterile are assessed theoretically. In both models, low allelic diversity at the sex locus reduces the Ne of hymenopteran populations, and this effect is largest when diploid males are effectively sterile.     

Global stability of an SEIS epidemic model with recruitment and a varying total population size

This paper considers an SEIS epidemic model that incorporates constant recruitment, disease-caused death and disease latency. The incidence term is of the bilinear mass-action form. It is shown that the global dynamics is completely determined by the basic reproduction number R(0). If R(0)1, a unique endemic equilibrium is globally stable in the interior of the feasible region and the disease persists at the endemic equilibrium.     

Effective population size and population subdivision in demographically structured populations

A fast-timescale approximation is applied to the coalescent process in a single population, which is demographically structured by sex and/or age. This provides a general expression for the probability that a pair of alleles sampled from the population coalesce in the previous time interval. The effective population size is defined as the reciprocal of twice the product of generation time and the coalescence probability. Biologically explicit formulas for effective population size with discrete generations and separate sexes are derived for a variety of different modes of inheritance. The method is also applied to a nuclear gene in a population of partially self-fertilizing hermaphrodites. The effects of population subdivision on a demographically structured population are analyzed, using a matrix of net rates of movement of genes between different local populations. This involves weighting the migration probabilities of individuals of a given age/sex class by the contribution of this class to the leading left eigenvector of the matrix describing the movements of genes between age/sex classes. The effects of sex-specific migration and nonrandom distributions of offspring number on levels of genetic variability and among-population differentiation are described for different modes of inheritance in an island model. Data on DNA sequence variability in human and plant populations are discussed in the light of the results.     

The genetic balance between varying population size and selective neutrality

This note extends to an arbitrary offspring distribution the generalized model for random fluctuation of allele frequency, where population size is permitted to fluctuate randomly from generation to generation. Martingale methods analogous to those of Seneta (1974) and Heyde and Seneta (1975) are applied to discuss conditions for Pr(Y(1–Y)>0)>0, where Y is the (almost sure) limiting frequency of one allele. An overlapping generation study of the same phenomenon has recently been made by Heyde (1981).     

Effective size of a wild salmonid population is greatly reduced by hatchery supplementation

Many declining and commercially important populations are supplemented with captive-born individuals that are intentionally released into the wild. These supplementation programs often create large numbers of offspring from relatively few breeding adults, which can have substantial population-level effects. We examined the genetic effects of supplementation on a wild population of steelhead (Oncorhynchus mykiss) from the Hood River, Oregon, by matching 12 run-years of hatchery steelhead back to their broodstock parents. We show that the effective number of breeders producing the hatchery fish (broodstock parents; N(b)) was quite small (harmonic mean N(b)=25 fish per brood-year vs 373 for wild fish), and was exacerbated by a high variance in broodstock reproductive success among individuals within years. The low N(b) caused hatchery fish to have decreased allelic richness, increased average relatedness, more loci in linkage disequilibrium and substantial levels of genetic drift in comparison with their wild-born counterparts. We also documented a substantial Ryman-Laikre effect whereby the additional hatchery fish doubled the total number of adult fish on the spawning grounds each year, but cut the effective population size of the total population (wild and hatchery fish combined) by nearly two-thirds. We further demonstrate that the Ryman-Laikre effect is most severe in this population when (1) >10% of fish allowed onto spawning grounds are from hatcheries and (2) the hatchery fish have high reproductive success in the wild. These results emphasize the trade-offs that arise when supplementation programs attempt to balance disparate goals (increasing production while maintaining genetic diversity and fitness).     

Bacterial foraging algorithm with varying population

Most of evolutionary algorithms (EAs) are based on a fixed population. However, due to this feature, such algorithms do not fully explore the potential of searching ability and are time consuming. This paper presents a novel nature-inspired heuristic optimization algorithm: bacterial foraging algorithm with varying population (BFAVP), based on a more bacterially-realistic model of bacterial foraging patterns, which incorporates a varying population framework and the underlying mechanisms of bacterial chemotaxis, metabolism, proliferation, elimination and quorum sensing. In order to evaluate its merits, BFAVP has been tested on several benchmark functions and the results show that it performs better than other popularly used EAs, in terms of both accuracy and convergency.     

Loss of alleles in a haploid population with varying environment

The evolution of populations may be affected by a number of factors. The basic forces of migration, mutation, and selection are self-explanatory. However, finite populations are also known to be subject to the fundamental undirected force of genetic drift—the random fluctuation of gene frequencies. It is this random effect which will be investigated via the consideration of discrete stochastic models.     

Effective population size is positively correlated with levels of adaptive divergence among annual sunflowers

The role of adaptation in the divergence of lineages has long been a central question in evolutionary biology, and as multilocus sequence data sets have become available for a wide range of taxa, empirical estimates of levels of adaptive molecular evolution are increasingly common. Estimates vary widely among taxa, with high levels of adaptive evolution in Drosophila, bacteria, and viruses but very little evidence of widespread adaptive evolution in hominids. Although estimates in plants are more limited, some recent work has suggested that rates of adaptive evolution in a range of plant taxa are surprisingly low and that there is little association between adaptive evolution and effective population size in contrast to patterns seen in other taxa. Here, we analyze data from 35 loci for six sunflower species that vary dramatically in effective population size. We find that rates of adaptive evolution are positively correlated with effective population size in these species, with a significant fraction of amino acid substitutions driven by positive selection in the species with the largest effective population sizes but little or no evidence of adaptive evolution in species with smaller effective population sizes. Although other factors likely contribute as well, in sunflowers effective population size appears to be an important determinant of rates of adaptive evolution.