On the Final Size of Epidemics with Seasonality

We first study an SIR system of differential equations with periodic coefficients describing an epidemic in a seasonal environment. Unlike in a constant environment, the final epidemic size may not be an increasing function of the basic reproduction number ℛ₀ or of the initial fraction of infected people. Moreover, large epidemics can happen even if ℛ₀<1. But like in a constant environment, the final epidemic size tends to 0 when ℛ₀<1 and the initial fraction of infected people tends to 0. When ℛ₀>1, the final epidemic size is bigger than the fraction 1−1/ℛ₀ of the initially nonimmune population. In summary, the basic reproduction number ℛ₀ keeps its classical threshold property but many other properties are no longer true in a seasonal environment. These theoretical results should be kept in mind when analyzing data for emerging vector-borne diseases (West-Nile, dengue, chikungunya) or air-borne diseases (SARS, pandemic influenza); all these diseases being influenced by seasonality.     

Genealogy with seasonality,the basic reproduction number,and the influenza pandemic

The basic reproduction number R ₀ has been used in population biology, especially in epidemiology, for several decades. But a suitable definition in the case of models with periodic coefficients was given only in recent years. The definition involves the spectral radius of an integral operator. As in the study of structured epidemic models in a constant environment, there is a need to emphasize the biological meaning of this spectral radius. In this paper we show that R ₀ for periodic models is still an asymptotic per generation growth rate. We also emphasize the difference between this theoretical R ₀ for periodic models and the “reproduction number” obtained by fitting an exponential to the beginning of an epidemic curve. This difference has been overlooked in recent studies of the H1N1 influenza pandemic.     

Mathematical Study of the Role of Gametocytes and an Imperfect Vaccine on Malaria Transmission Dynamics

A mathematical model is developed to assess the role of gametocytes (the infectious sexual stage of the malaria parasite) in malaria transmission dynamics in a community. The model is rigorously analysed to gain insights into its dynamical features. It is shown that, in the absence of disease-induced mortality, the model has a globally-asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number (denoted by ℛ₀), is less than unity. Further, it has a unique endemic equilibrium if ℛ₀>1. The model is extended to incorporate an imperfect vaccine with some assumed therapeutic characteristics. Theoretical analyses of the model with vaccination show that an imperfect malaria vaccine could have negative or positive impact (in reducing disease burden) depending on whether or not a certain threshold (denoted by ∇) is less than unity. Numerical simulations of the vaccination model show that such an imperfect anti-malaria vaccine (with a modest efficacy and coverage rate) can lead to effective disease control if the reproduction threshold (denoted by ℛ_vac) of the disease is reasonably small. On the other hand, the disease cannot be effectively controlled using such a vaccine if ℛ_vac is high. Finally, it is shown that the average number of days spent in the class of infectious individuals with higher level of gametocyte is critically important to the malaria burden in the community.     

Radial theory of the ion current onto a probe in a low-pressure plasma with allowance for volume ionization and collisions with neutrals

The ion current onto a spherical or cylindrical probe is analyzed in the cold-ion approximation with allowance for ionization and collisions with neutrals. An expression for the ion density that takes into account both ionization and collision in the finite-size perturbed region is derived. The current-voltage characteristics for the dimensionless parameters r _p/λ _D = 0.0001−10, λ _i /λ _D = 0.01 − ∞, and zλ _D/(kT _e /M)^1/2 = 0−5 are determined by numerically solving Poisson’s equation, and the corresponding approximate expressions are obtained.     

Visual pigments of marine carnivores: pinnipeds, polar bear, and sea otter

Rod and cone visual pigments of 11 marine carnivores were evaluated. Rod, middle/long-wavelength sensitive (M/L) cone, and short-wavelength sensitive (S) cone opsin (if present) sequences were obtained from retinal mRNA. Spectral sensitivity was inferred through evaluation of known spectral tuning residues. The rod pigments of all but one of the pinnipeds were similar to those of the sea otter, polar bear, and most other terrestrial carnivores with spectral peak sensitivities (λ_max) of 499 or 501 nm. Similarly, the M/L cone pigments of the pinnipeds, polar bear, and otter had inferred λ_max of 545 to 560 nm. Only the rod opsin sequence of the elephant seal had sensitivity characteristic of adaptation for vision in the marine environment, with an inferred λ_max of 487 nm. No evidence of S cones was found for any of the pinnipeds. The polar bear and otter had S cones with inferred λ_max of ∼440 nm. Flicker-photometric ERG was additionally used to examine the in situ sensitivities of three species of pinniped. Despite the use of conditions previously shown to evoke cone responses in other mammals, no cone responses could be elicited from any of these pinnipeds. Rod photoreceptor responses for all three species were as predicted by the genetic data.Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.     

Sublethal effects of fenpyroximate and pyridaben on two predatory mite species, <Emphasis Type="Italic">Neoseiulus womersleyi</Emphasis> and <Emphasis Type="Italic">Phytoseiulus persimilis</Emphasis> (Acari,Phytoseiidae)

Laboratory bioassays were conducted to evaluate the sublethal effects of fenpyroximate and pyridaben on life-table parameters of two predatory mites species, Neoseiulus (= Amblyseius) womersleyi and Phytoseiulus persimilis. In these assays, young adult females were treated with three sublethal concentrations of each acaricide. The life-table parameters were calculated at each acaricide concentration, and compared using bootstrap procedures. For each acaricide, the LC₅₀ estimates for both species were similar, yet the two species exhibited completely different susceptibility when the population growth rate was used as the endpoint. Exposure to both acaricides reduced the net reproduction rate (R _o ) in a concentration-dependent manner and their EC₅₀s were equivalent to less than LC₇. Two different scales of population-level endpoints were estimated to compare the total effect between the species and treatments: the first endpoint values were based on the net reproductive rate (fecundity λ) and the second endpoint values incorporated the mean egg hatchability into the net reproductive rate (vitality λ). The fecundity λ decreased in a concentration-dependent manner for both acaricide treatments, but the vitality λ decreased abruptly after treatment of N. womersleyi with pyridaben. The change in the patterns of λ revealed that the acaricide effects at the population level strongly depended on the life-history characteristics of the predatory mite species and the chemical mode of action. When the total effects of the two acaricides on N. womersleyi and P. persimilis were considered, fenpyroximate was found to be the most compatible acaricide for the augmentative release of N. womersleyi after treatment.     

The gas washout determination under a symmetry assumption

In an open circuit gas washout determination the output of test substance is shown to be of the form ∑ _i=0 ^∞ A _i λ _i ^k on thekth expiration whereA _i >0,i=1,2,... and 1 > λ₁ ≥ λ₂...≥ 0 provided the transition from inspiration to expiration has certain symmetry properties with the transition from expiration to inspiration. In general, no direct physical interpretation such as volume for theA _t ’s or fraction of retained gas on expiration for the λ _i is justified.     

Time delays produced by essential nonlinearity in population growth models

It is pointed out that the asymptotic general solution to the ϕ-model equation for a periodic carrying capacityK(t) andt≳r ⁻¹ is identical in form to the generalized logistic equation solution with a built-in developmental time delay τ(≲r ⁻¹) and associated parameter ranges of primary biological interest. In the case of the ϕ-model equation, the time delay is a purely dynamical consequence of the nonlinear form featured by the population growth rate.     

Chemical Characterization,Crossfeeding and Uptake Studies on Hydroxamate Siderophore of <Emphasis Type="Italic">Alcaligenes faecalis</Emphasis>

We report the production of two types of siderophores namely catecholate and hydroxamate in modified succinic acid medium (SM) from Alcaligenes faecalis. Two fractions of siderophores were purified on amberlite XAD, major fraction was hydroxamate type having a λ_max at 224 nm and minor fraction appeared as catecholate with a λ_max of 264 nm. The recovery yield obtained from major and minor fractions was 297 and 50 mg ml⁻¹ respectively. The IEF pattern of XAD-4 purified siderophore suggested the pI value of 6.5. Cross feeding studies revealed that A. faecalis accepts heterologous as well as self (hydroxamate) siderophore in both free and iron complexed forms however; the rate of siderophore uptake was more in case of siderophores complexed to iron. Siderophore iron uptake studies indicated the differences between hydroxamate siderophore of A. faecalis and Alc E, a siderophore of Alcaligenes eutrophus.     

Importance of adult survival, local recruitment and immigration in a declining boreal forest passerine, the willow tit Parus montanus

Population growth rate (λ) and its components (adult survival, local recruitment, immigration and their relative contributions to λ) were studied in the declining willow tit Parus montanus in Northern Finland. Capture–recapture models for open populations were used to estimate the population parameters and their process variation. Adult survival was fairly high with low variation (0.593, CV=0.067). As expected, local recruitment was lower and more variable (0.063, CV=0.610). During the 12-year study, the population growth rate averaged to one (0.988, CV=0.197; calculated as However, if the present processes continue, population projections show that the population is likely to decline. There was considerable temporal variation in the relative contributions of demographic parameters to λ. In all years, adult survival had the highest relative contribution (mean 64%) to the population growth rate and it was the least variable trait. Immigration had a higher relative contribution (22%) to λ than local recruitment (14%). Based on the results for the contributions to λ, the main conservation concern for willow tits is adult survival. Due to low variation, adult survival may be difficult to enhance, but at least it should be prevented from declining. High stochasticity in local recruitment and immigration is probably an inherent characteristic of highly seasonal environments, making these traits difficult to address for conservation practices.     

Reproduction,adult survival and intrinsic rate of growth ofUrolepis rufipes [Hymenoptera: Pteromalidae], a pupal parasitoid of house flies,Musca domestica

Urolepis rufipes Ashmead, a recently discovered parasitoid of house flies at New York dairies, was reared at 15, 20, 25, 30 and 34°C to measure daily fertility, fecundity and adult survivorship. Little reproduction occurred at 15°C, and only a few ♀ successfully emerged at 34°C. The intrinsic rate of growth was fastest at 30°C (0.282 ♀/day), but fecundity was highest at 25°C (165.5 hosts attacked, producing 124.5 progeny). Some reproductive statistics at 25°C were: net reproductive rate (R₀=72.1 ♀/♀, generation time =18.7 days, intrinsic rate of increase (r_m)=0.228, finite rate of increase (λ)=1.26, daily birth rate =0.302, daily death rate =0.021 and Fisher's reproductive value =418. Sex ratio (average =75.9%) did not vary significantly with temperature (between 20–30°C) nor with mother's age.      

The photoactivation energy of the visual pigment in two spectrally different populations of <Emphasis Type="Italic">Mysis relicta</Emphasis> (Crustacea,Mysida)

We report the first study of the relation between the wavelength of maximum absorbance (λ_max) and the photoactivation energy (E _a) in invertebrate visual pigments. Two populations of the opossum shrimp Mysis relicta were compared. The two have been separated for 9,000 years and have adapted to different spectral environments (“Sea” and “Lake”) with porphyropsins peaking at λ_max=529 nm and 554 nm, respectively. The estimation of E _a was based on measurement of temperature effects on the spectral sensitivity of the eye. In accordance with theory (Stiles in Transactions of the optical convention of the worshipful company of spectacle makers. Spectacle Makers’ Co., London, 1948), relative sensitivity to long wavelengths increased with rising temperature. The estimates calculated from this effect are E _a,529=47.8±1.8 kcal/mol and E _a,554=41.5±0.7 kcal/mol (different at P<0.01). Thus the red-shift of λ_max in the “Lake” population, correlating with the long-wavelength dominated light environment, is achieved by changes in the opsin that decrease the energy gap between the ground state and the first excited state of the chromophore. We propose that this will carry a cost in terms of increased thermal noise, and that evolutionary adaptation of the visual pigment to the light environment is directed towards maximizing the signal-to-noise ratio rather than the quantum catch.     

Demographic differences between two sympatric lilies (<Emphasis Type="Italic">Calochortus</Emphasis>) with contrasting distributions,as revealed by matrix analysis

Related species of similar morphology can differ greatly in distribution and abundance. Elucidating reasons for such differences can contribute to an understanding of intrinsic limiting factors and the causes of rarity. We studied sympatric populations of two terrestrial lilies with contrasting distributions: Calochortus lyallii, which is geographically restricted but locally abundant, and C. macrocarpus, which is widespread but locally sparse. Marked plants of each species were monitored for 5 years in British Columbia, Canada. Matrix projection models were used to estimate annual and stochastic population growth rates (λ and λ_s) and to compare demographic traits. Annual λ-values ranged from 0.89 to 1.04 in C. lyallii and from 0.89 to 1.01 in C. macrocarpus. Stochastic projections yielded a long-term growth rate near 1 for C. lyallii, but indicated a decline for C. macrocarpus. Elasticity analysis indicated that over the 5-year period of the study, survival of flowering plants made a larger proportional contribution to λ in C. lyallii than in C. macrocarpus. LTRE analysis showed that temporal variation in λ was driven primarily by the dynamics of flowering individuals in C. lyallii, and by the dynamics of vegetative individuals in C. macrocarpus. Similarly, higher flowering rates in C. lyallii and greater vegetative stasis in C. macrocarpus made the largest contribution to the difference in λ between species. Thus, local persistence in these two morphologically similar species appears to be achieved via different demographic pathways. Our analyses show that extrapolations about demographic processes and population dynamics based on taxonomic relatedness, morphological similarity or habitat overlap may often not be justified. Electronic Supplementary Material The online version of this article contains supplementary material, which is available to authorised users.     

Multiregional Periodic Matrix for Modeling the Population Dynamics of Sardine (<Emphasis Type="Italic">Sardina pilchardus</Emphasis>) Along the Moroccan Atlantic Coast: Management Elements for Fisheries

In this paper, we present a deterministic time discrete mathematical model based on multiregional periodic matrices to describe the dynamics of Sardina pilchardus in the Central Atlantic area of the Moroccan coast. This model deals with two stages (immature and mature) and three spatial zones where sardines are supposed to migrate from one zone to another. The population dynamics is described by an autonomous recurrence equation N(t + 1) = A.N(t), where A is a positive matrix whose entries are estimated using data collected during biannual acoustic surveys carried out from 2001 to 2003 onboard the Norwegian research vessel “Dr Fridtjof Nansen”. The dominant eigenvalue λ of A that gives the long-term growth rate of fish population is smaller than one. This agrees with the stock decrease observed in the data collected. We show that λ is highly sensitive to the recruitment rate and much less sensitive to the reproduction rate. These results can clearly be used to define an efficient scenario in order to fight for instance against a stock decrease.     

HIV spread in the San Francisco cohort: Scaling of the effective logistic rate for seropositivity

A simple scaling (semigroup) property is manifest in the functional form of the effective logistic rate for the increase in the HIV seropositive fraction in the San Francisco (City Clinic) cohort. Witht _i=4.5 years, this scaling property—r→λ^-2r undert→[λt+(λ−1)t _i] for all parameter values λ≧1—encapsulates the effects of relevant biological and sociological changes in the key epidemiological variables during the 8-year seropositive rise period, 1978–1985 inclusive.     

Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model

Multiple transmission pathways exist for many waterborne diseases, including cholera, Giardia, Cryptosporidium, and Campylobacter. Theoretical work exploring the effects of multiple transmission pathways on disease dynamics is incomplete. Here, we consider a simple ODE model that extends the classical SIR framework by adding a compartment (W) that tracks pathogen concentration in the water. Infected individuals shed pathogen into the water compartment, and new infections arise both through exposure to contaminated water, as well as by the classical SIR person–person transmission pathway. We compute the basic reproductive number (ℛ₀), epidemic growth rate, and final outbreak size for the resulting “SIWR” model, and examine how these fundamental quantities depend upon the transmission parameters for the different pathways. We prove that the endemic disease equilibrium for the SIWR model is globally stable. We identify the pathogen decay rate in the water compartment as a key parameter determining when the distinction between the different transmission routes in the SIWR model is important. When the decay rate is slow, using an SIR model rather than the SIWR model can lead to under-estimates of the basic reproductive number and over-estimates of the infectious period.     

Scattering spectrum properties and their relationship to biogeochemical parameters: a case study in Taihu Lake

The scattering spectrum properties of highly turbid and eutrophic inland case 2 water from Taihu Lake were studied during three cruises from 2006 to 2007. The scattering [b _p(λ)] and backscattering [b _bp(λ)] coefficients and the backscattering probability (B) for Taihu Lake were found to show a clear spectral dependence, and this dependence was well simulated by a power-law function. This dependence, however, became weak when algae dominated the sample points. The mean values of the power-law index for b _p(λ), v, in Oct 2006, Mar 2007 and Nov 2007 were −0.6712, −0.8129 and −0.7600, respectively. To interpret the spectral characteristics and mechanisms of b _p(λ) and b _bp(λ), water samples were collected simultaneously for the biogeochemical characterization of suspended particles. The average values of the specific scattering coefficients for total suspended matter, inorganic suspended matter (ISPM) and organic suspended matter (OSPM) were 0.634 (550 nm), 1.057 (532 nm), and 0.396 g m⁻² (532 nm), respectively. The power-law index of b _bp(λ) (Y) was significantly related to ISPM/OSPM and b _bp(532 nm), but only weakly related to the particle size distribution index. The mean (spatial and wavelength) values of B in Oct 2006, Mar 2007, and Nov 2007 were 0.0108, 0.0138, and 0.0125, respectively. B decreases with increasing ISPM concentration because of the large contribution of ISPM to b _b(λ) and the strong restraint on b _bp(λ) caused by the multi-scattering effect under high-turbidity conditions.     

Demography of a dormancy-prone geophyte: influence of spatial scale on interpretation of dynamics

Both temporal and spatial scales are important in the evaluation of population dynamics, but the latter often receives less attention in demographic analyses. We used a 5-year demographic data set for a long-lived geophyte, Calochortus lyallii, to investigate the pattern and components of spatial variation at two scales (population and microsite). We found that neither the projected population structure nor asymptotic population growth rate (λ) varied greatly across either scale, although the underlying contributors to the variation in λ, V(λ), did differ between scales. Life table response experiment analyses showed that V(λ) among populations came primarily from variation in seedling survival and progression of non-reproductive plants, whereas V(λ) among microsites was primarily due to the variable fertility of large adults. Prolonged dormancy was important in reducing V(λ) among quadrats at both the scales, partly countering fluctuations in other transitions such as recruitment. This result represents some of the first evidence that underground “bulb banks” could function to offset the effects of a spatially heterogeneous environment in a manner analogous to seed banks. Future work is needed to isolate the specific, sometimes idiosyncratic, life history phenomena acting to modulate plant population dynamics in a spatial context.     

Toward an accurate statistics of gapped alignments

Sequence alignment has been an invaluable tool for finding homologous sequences. The significance of the homology found is often quantified statistically by p-values. Theory for computing p-values exists for gapless alignments [Karlin, S., Altschul, S.F., 1990. Methods for assessing the statistical significance of molecular sequence features by using general scoring schemes. Proc. Natl. Acad. Sci. USA 87, 2264–2268; Karlin, S., Dembo A., 1992. Limit distributions of maximal segmental score among Markov-dependent partial sums. Adv. Appl. Probab. 24, 13–140], but a full generalization to alignments with gaps is not yet complete. We present a unified statistical analysis of two common sequence comparison algorithms: maximum-score (Smith-Waterman) alignments and their generalized probabilistic counterparts, including maximum-likelihood alignments and hidden Markov models. The most important statistical characteristic of these algorithms is the distribution function of the maximum score S _max, resp. the maximum free energy F _max, for mutually uncorrelated random sequences. This distribution is known empirically to be of the Gumbel form with an exponential tail P(S _max > x) ∼ exp(−λx) for maximum-score alignment and P(F _max > x) ∼ exp(−λx) for some classes of probabilistic alignment. We derive an exact expression for λ for particular probabilistic alignments. This result is then used to obtain accurate λ values for generic probabilistic and maximum-score alignments. Although the result demonstrated uses a simple match-mismatch scoring system, it is expected to be a good starting point for more general scoring functions.     

Survival and population growth of a long-lived threatened snake species, Drymarchon couperi (Eastern Indigo Snake)

Demographic data provide a basis for understanding the life history and ecology of species, factors which are vital for informing conservation efforts; however, little is known regarding the population ecology of most snake species, including the threatened Eastern Indigo Snake (Drymarchon couperi). We used 11 years (1999–2009) of capture-mark-recapture (CMR) and 2.5 years (2003–2005) of radiotelemetry data from southeastern Georgia, USA, in a CMR modeling framework to estimate apparent survival, capture and transition probabilities, and evaluate factors influencing these parameters. The model-averaged estimate of overall apparent annual survival probability was 0.700 (±0.030 SE) and is comparable to that obtained from known fate analysis (radiotelemetry) at the same site. Body size positively influenced survival, regardless of sex. Capture probability differed seasonally by sex, suggesting lower capture probability for females in fall and males in winter. There was no evidence for effect of precipitation or site-specific differences in survival. Model averaged estimate of annual adult survival estimated using multistate CMR models was 0.738 ± 0.030 and 0.515 ± 0.189 for subadults. We estimated population growth rate (λ) and elasticity (proportional sensitivity) of λ to vital rates using a stage-structured matrix population model. Population growth rate ranged from 0.96 to 1.03 depending on the value of the probability of transitioning from subadult to adult stage. The λ was proportionally most sensitive to changes in adult survival rate, followed by subadult survival. Our results suggest that protecting adult snakes and their habitats would result in the highest likelihood of long-term population stability and growth.