Impact of meteorological factors and population density on COVID-19 pandemic in Saudi Arabia

Transmission and increase in cases and fatalities of coronavirus disease-2019 (COVID-19) are significantly influenced by the parameters of weather, human activities and population factors. However, study gap on the seasonality of COVID-19 and impact of environmental factors on the pandemic in Saudi Arabia is present. The main aim of the study is to evaluate the impact of environment on the COVID-19 pandemic. Data were analyzed from January 2020 to July 2021. The generalized estimating equation (GEE) was used to determine the effect of environmental variables on longitudinal outcomes. Spearman's rank correlation coefficient (r_s) was used to analyze the impact of different parameters on the outcome of the pandemic. Multiple sequence alignment was performed by using ClustalW. Vaccination and fatalities (r_s = ?0.85) had the highest association followed by vaccination with cases (r_s = ?0.81) and population density with the fatalities (r_s = 0.71). The growth rate had the highest correlation with sun hours (r_s = ?0.63). Isolates from variant of concern alpha and beta were detected. Most of the reference sequences in Saudi Arabia were closely related with B.1.427/429 variant. Clade GH (54%) was the most prevalent followed by O (27%), GR (9%), G (6%), and S (4%), respectively. Male to female patient ratio was 1.4:1. About 95% fatality and hospitalization were reported in patients aged >60 years. This study will create a comprehensive insight of the interaction of environmental factors and the pandemic and add knowledge on seasonality of COVID-19 in Saudi Arabia.     

Estimated epidemiologic parameters and morbidity associated with pandemic H1N1 influenza

Background

In the face of an influenza pandemic, accurate estimates of epidemiologic parameters are required to help guide decision-making. We sought to estimate epidemiologic parameters for pandemic H1N1 influenza using data from initial reports of laboratory-confirmed cases.

Methods

We obtained data on laboratory-confirmed cases of pandemic H1N1 influenza reported in the province of Ontario, Canada, with dates of symptom onset between Apr. 13 and June 20, 2009. Incubation periods and duration of symptoms were estimated and fit to parametric distributions. We used competing-risk models to estimate risk of hospital admission and case-fatality rates. We used a Markov Chain Monte Carlo model to simulate disease transmission.

Results

The median incubation period was 4 days and the duration of symptoms was 7 days. Recovery was faster among patients less than 18 years old than among older patients (hazard ratio 1.23, 95% confidence interval 1.06–1.44). The risk of hospital admission was 4.5% (95% CI 3.8%–5.2%) and the case-fatality rate was 0.3% (95% CI 0.1%–0.5%). The risk of hospital admission was highest among patients less than 1 year old and those 65 years or older. Adults more than 50 years old comprised 7% of cases but accounted for 7 of 10 initial deaths (odds ratio 28.6, 95% confidence interval 7.3–111.2). From the simulation models, we estimated the following values (and 95% credible intervals): a mean basic reproductive number (R₀, the number of new cases created by a single primary case in a susceptible population) of 1.31 (1.25–1.38), a mean latent period of 2.62 (2.28–3.12) days and a mean duration of infectiousness of 3.38 (2.06–4.69) days. From these values we estimated a serial interval (the average time from onset of infectiousness in a case to the onset of infectiousness in a person infected by that case) of 4–5 days.

Interpretation

The low estimates for R₀ indicate that effective mitigation strategies may reduce the final epidemic impact of pandemic H1N1 influenza.The emergence and global spread of pandemic H1N1 influenza led the World Health Organization to declare a pandemic on June 11, 2009. As the pandemic spreads, countries will need to make decisions about strategies to mitigate and control disease in the face of uncertainty.For novel infectious diseases, accurate estimates of epidemiologic parameters can help guide decision-making. A key parameter for any new disease is the basic reproductive number (R₀), defined as the average number of new cases created by a single primary case in a susceptible population. R₀ affects the growth rate of an epidemic and the final number of infected people. It also informs the optimal choice of control strategies. Other key parameters that affect use of resources, disease burden and societal costs during a pandemic are duration of illness, rate of hospital admission and case-fatality rate. Early in an epidemic, the case-fatality rate may be underestimated because of the temporal lag between onset of infection and death; the delay between initial identification of a new case and death may lead to an apparent increase in deaths several weeks into an epidemic that is an artifact of the natural history of the disease.We used data from initial reports of laboratory-confirmed pandemic H1N1 influenza to estimate epidemiologic parameters for pandemic H1N1 influenza. The parameters included R₀, incubation period and duration of illness. We also estimated risk of hospital admission and case-fatality rates, which can be used to estimate the burden of illness likely to be associated with this disease.     

Early life-history processes in benthic octopus: Relationships between temperature, feeding, food conversion, and growth in juvenile Octopus pallidus

Initial growth in cephalopods is exponential, making early life-history critical in determining growth trajectories. Few captive studies have however examined the early life-history of cephalopods in fluctuating temperatures as would be encountered in the wild. This study investigates the relationship between early growth and the significant factors affecting growth, namely food intake, food conversion and fluctuating environmental temperatures. Pale octopus (Octopus pallidus) hatchlings were reared in captivity under either a warming or cooling temperature regime. Individual variations and periodicity in feeding rates (F_r), food conversion rates (C_r), growth rates (G_r), and the relationship between these variables and temperature were examined weekly. Food conversion rates were variable between individuals but also within individual octopus, which exhibited large fluctuations in C_r over time, exceeding 100% d^− 1 in one instance. Although individual F_r, C_r and G_r displayed fluctuations over time, there was no evidence of periodicity for any of the variables. Changes in temperature were not significantly correlated to changes in F_r, C_r or G_r. Feeding rate did not appear to influence growth rate or food conversion rate. Food conversion rate was negatively correlated to feeding rate in the same week, and positively correlated to growth rate. Short periods of low or no food consumption were common and the high values obtained for food conversion rate for some individuals suggest that octopus can grow substantially with little or no food intake. Individual variability observed in octopus growth may be dependent on the growth mechanism involved, specifically a fine balance between the continuous hyperplasic and hypertrophic growth found in cephalopods.     

Epidemic modelling: Aspects where stochasticity matters

Epidemic models are always simplifications of real world epidemics. Which real world features to include, and which simplifications to make, depend both on the disease of interest and on the purpose of the modelling. In the present paper we discuss some such purposes for which a stochastic model is preferable to a deterministic counterpart. The two main examples illustrate the importance of allowing the infectious and latent periods to be random when focus lies on the probability of a large epidemic outbreak and/or on the initial speed, or growth rate, of the epidemic. A consequence of the latter is that estimation of the basic reproduction number R₀ is sensitive to assumptions about the distributions of the infectious and latent periods when using data from the early stages of an outbreak, which we illustrate with data from the H1N1 influenza A pandemic. Some further examples are also discussed as are some practical consequences related to these stochastic aspects.     

Effects of Stochasticity in Early Life History on Steepness and Population Growth Rate Estimates: An Illustration on Atlantic Bluefin Tuna

The intrinsic population growth rate (r) of the surplus production function used in the biomass dynamic model and the steepness (h) of the stock-recruitment relationship used in age-structured population dynamics models are two key parameters in fish stock assessment. There is generally insufficient information in the data to estimate these parameters that thus have to be constrained. We developed methods to directly estimate the probability distributions of r and h for the Atlantic bluefin tuna (Thunnus thynnus, Scombridae), using all available biological and ecological information. We examined the existing literature to define appropriate probability distributions of key life history parameters associated with intrinsic growth rate and steepness, paying particular attention to the natural mortality for early life history stages. The estimated probability distribution of the population intrinsic growth rate was weakly informative, with an estimated mean r = 0.77 (±0.53) and an interquartile range of (0.34, 1.12). The estimated distribution of h was more informative, but also strongly asymmetric with an estimated mean h = 0.89 (±0.20) and a median of 0.99. We note that these two key demographic parameters strongly depend on the distribution of early life history mortality rate (M₀), which is known to exhibit high year-to-year variations. This variability results in a widely spread distribution of M₀ that affects the distribution of the intrinsic population growth rate and further makes the spawning stock biomass an inadequate proxy to predict recruitment levels.     

Band reporting rates of waterfowl: does individual heterogeneity bias estimated survival rates?

In capture–recapture studies, the estimation accuracy of demographic parameters is essential to the efficacy of management of hunted animal populations. Dead recovery models based upon the reporting of rings or bands are often used for estimating survival of waterfowl and other harvested species. However, distance from the ringing site or condition of the bird may introduce substantial individual heterogeneity in the conditional band reporting rates (r), which could cause bias in estimated survival rates (S) or suggest nonexistent individual heterogeneity in S. To explore these hypotheses, we ran two sets of simulations (n = 1000) in MARK using Seber''s dead recovery model, allowing time variation on both S and r. This included a series of heterogeneity models, allowing substantial variation on logit(r), and control models with no heterogeneity. We conducted simulations using two different values of S: S = 0.60, which would be typical of dabbling ducks such as mallards (Anas platyrhynchos), and S = 0.80, which would be more typical of sea ducks or geese. We chose a mean reporting rate on the logit scale of −1.9459 with SD = 1.5 for the heterogeneity models (producing a back-transformed mean of 0.196 with SD = 0.196, median = 0.125) and a constant reporting rate for the control models of 0.196. Within these sets of simulations, estimation models where σ_S = 0 and σ_S > 0 (σ_S is SD of individual survival rates on the logit scale) were incorporated to investigate whether real heterogeneity in r would induce apparent individual heterogeneity in S. Models where σ_S = 0 were selected approximately 91% of the time over models where σ_S > 0. Simulation results showed < 0.05% relative bias in estimating survival rates except for models estimating σ_S > 0 when true S = 0.8, where relative bias was a modest 0.5%. These results indicate that considerable variation in reporting rates does not cause major bias in estimated survival rates of waterfowl, further highlighting the robust nature of dead recovery models that are being used for the management of harvested species.     

Could the intrinsic rate of increase represent the fitness in terrestrial ectotherms?

The intrinsic rate of increase (r_m) has been considered as an important indicator of fitness in terrestrial ectotherms since long. It is actually an equivalent to the instantaneous growth rate of the exponential equation for describing the density-independent population growth. In terrestrial ectotherms, r_m has been demonstrated to be temperature-dependent. The temperature at which r_m was maximal, was considered to be the “optimal” temperature for fitness in Amarasekare and Savage (2012), but this definition needs further analysis. Only r_m cannot provide thorough representation of fitness. Because body size can affect the competitive abilities in many terrestrial ectotherms, both population size and body size should be considered in measuring the fitness of ectotherms. The rule of “bigger is better” requires relatively low temperature to increase in body size, whereas relatively high temperature is required for a rapid increase in population size. Thus, there is presumably a trade-off in temperature for adjusting individual body size and population size to achieve maximum fitness. We hypothesized that this temperature could be reflected by the intrinsic optimum temperature for developmental rate in the Sharpe–Schoolfield–Ikemoto model, and it led to a temperature estimate around 20 °C. However, the traditional viewpoint based on the temperature corresponding to the maximal intrinsic rate of increase provides a temperature estimate around 30 °C. This study suggests that a low temperature around 20 °C might authentically represent the optimal ambient temperature for fitness in terrestrial ectotherms. It implies that thermal biologists who are interested in the effect of temperature on the fitness in terrestrial ectotherms should pay more attention to their performance at low temperature rather than high temperature.     

Transient Phases of the Isometric Tetanus in Frog's Striated Muscle

In an isometric tetanus in frog's sartorius muscle tension approaches the plateau exponentially with rate constant α. α a depends on sarcomere length, s, and temperature, T, according to the Arrhenius equation See PDF for Equation for temperatures between 1 and 20°C and for sarcomere lengths 2.0–2.8 µm. The energy of activation, E, does not vary significantly with s; E = 13.9 ± 2.4 kcal/mole. A(s) decreases monotonically with s; A(2.1 µm) is about three times greater than A(2.8 µm). Late in relaxation active tension approaches zero exponentially with rate constant r. r decreases exponentially with increasing duration of tetanus, D, from r₀ in a twitch to r_∞ for large D. The rate constant for decrease of r with D increases with s and with T. r₀ and r_∞ obey the Arrhenius equation and decrease with increasing s.     

Adaptation of Mycobacteria to Growth Conditions: A Theoretical Analysis of Changes in Gene Expression Revealed by Microarrays

Background

Microarray analysis is a powerful technique for investigating changes in gene expression. Currently, results (r-values) are interpreted empirically as either unchanged or up- or down-regulated. We now present a mathematical framework, which relates r-values to the macromolecular properties of population-average cells. The theory is illustrated by the analysis of published data for two species; namely, Mycobacterium bovis BCG Pasteur and Mycobacterium smegmatis mc² 155. Each species was grown in a chemostat at two different growth rates. Application of the theory reveals the growth rate dependent changes in the mycobacterial proteomes.

Principal Findings

The r-value r _(i) of any ORF (ORF_(i)) encoding protein p _(i) was shown to be equal to the ratio of the concentrations of p _(i) and so directly proportional to the ratio of the numbers of copies of p _(i) per population-average cells of the two cultures. The proportionality constant can be obtained from the ratios DNA: RNA: protein. Several subgroups of ORFs were identified because they shared a particular r-value. Histograms of the number of ORFs versus the expression ratio were simulated by combining the particular r-values of several subgroups of ORFs. The largest subgroup was ORF_(j) (r _(j)  = 1.00± SD) which was estimated to comprise respectively 59% and 49% of ORFs of M. bovis BCG Pasteur and M. smegmatis mc² 155. The standard deviations reflect the properties of the cDNA preparations investigated.

Significance

The analysis provided a quantitative view of growth rate dependent changes in the proteomes of the mycobacteria studied. The majority of the ORFs were found to be constitutively expressed. In contrast, the protein compositions of the outer permeability barriers and cytoplasmic membranes were found to be dependent on growth rate; thus illustrating the response of bacteria to their environment. The theoretical approach applies to any cultivatable bacterium under a wide range of growth conditions.     

Test system stability and natural variability of a Lemna gibba L. bioassay

Background

In ecotoxicological and environmental studies Lemna spp. are used as test organisms due to their small size, rapid predominantly vegetative reproduction, easy handling and high sensitivity to various chemicals. However, there is not much information available concerning spatial and temporal stability of experimental set-ups used for Lemna bioassays, though this is essential for interpretation and reliability of results. We therefore investigated stability and natural variability of a Lemna gibba bioassay assessing area-related and frond number-related growth rates under controlled laboratory conditions over about one year.

Methology/Principal Findings

Lemna gibba L. was grown in beakers with Steinberg medium for one week. Area-related and frond number-related growth rates (r_(area) and r_(num)) were determined with a non-destructive image processing system.To assess inter-experimental stability, 35 independent experiments were performed with 10 beakers each in the course of one year. We observed changes in growth rates by a factor of two over time. These did not correlate well with temperature or relative humidity in the growth chamber.In order to assess intra-experimental stability, we analysed six systematic negative control experiments (nontoxicant tests) with 96 replicate beakers each. Evaluation showed that the chosen experimental set-up was stable and did not produce false positive results. The coefficient of variation was lower for r_(area) (2.99%) than for r_(num) (4.27%).

Conclusions/Significance

It is hypothesised that the variations in growth rates over time under controlled conditions are partly due to endogenic periodicities in Lemna gibba. The relevance of these variations for toxicity investigations should be investigated more closely. Area-related growth rate seems to be more precise as non-destructive calculation parameter than number-related growth rate. Furthermore, we propose two new validity criteria for Lemna gibba bioassays: variability of average specific and section-by-section segmented growth rate, complementary to average specific growth rate as the only validity criterion existing in guidelines for duckweed bioassays.     

Practical considerations for measuring the effective reproductive number,Rt

Estimation of the effective reproductive number R_t is important for detecting changes in disease transmission over time. During the Coronavirus Disease 2019 (COVID-19) pandemic, policy makers and public health officials are using R_t to assess the effectiveness of interventions and to inform policy. However, estimation of R_t from available data presents several challenges, with critical implications for the interpretation of the course of the pandemic. The purpose of this document is to summarize these challenges, illustrate them with examples from synthetic data, and, where possible, make recommendations. For near real-time estimation of R_t, we recommend the approach of Cori and colleagues, which uses data from before time t and empirical estimates of the distribution of time between infections. Methods that require data from after time t, such as Wallinga and Teunis, are conceptually and methodologically less suited for near real-time estimation, but may be appropriate for retrospective analyses of how individuals infected at different time points contributed to the spread. We advise caution when using methods derived from the approach of Bettencourt and Ribeiro, as the resulting R_t estimates may be biased if the underlying structural assumptions are not met. Two key challenges common to all approaches are accurate specification of the generation interval and reconstruction of the time series of new infections from observations occurring long after the moment of transmission. Naive approaches for dealing with observation delays, such as subtracting delays sampled from a distribution, can introduce bias. We provide suggestions for how to mitigate this and other technical challenges and highlight open problems in R_t estimation.     

Age,growth, maturity and extinction risk of an exploited and endangered skate,Atlantoraja castelnaui,from off Uruguay and northern Argentina

The spotback skate Atlantoraja castelnaui (Arhynchobatidae) is a large and threatened skate species subjected to fishing pressure, endemic to the Southwest Atlantic that occurs from Rio de Janeiro, Brazil, to San Jorge Gulf, Argentina. The age, growth, age at maturity and the maximum intrinsic rate of population increase r_max of A. castelnaui were studied using 152 specimens collected from off Uruguay and north Argentina (35°–42° S), between June 2013 and February 2020. Vertebrae from 143 individuals were used for ageing (females: n = 83, size range 404–1300 mm total length, TL; males: n = 60, size range 400–1270 mm TL). Maximum ages determined for females and males were 30 and 28 years, respectively. To fit growth models, non-linear and Bayesian estimation approaches were considered. For the first approach, a set of four candidate growth (size-at-age) models were fitted: three-parameter von Bertalanffy, two-parameter von Bertalanffy with fixed L₀, Gompertz and Logistic. In the second approach, von Bertalanffy, Gompertz and Logistic were fitted. For non-linear estimation, model selection indicated that the entire set of candidate growth models were supported by the data. The von Bertalanffy was selected as the best model for Bayesian estimation. There were no differences in growth between sexes. For the sexes combined, the von Bertalanffy growth model by Bayesian method was considered the most adequate to describe the growth of A. castelnaui (growth mean parameters ± S.D. : L_∞ = 1210.29 ± 40.68 mm; k = 0.12 ± 0.01 years⁻¹; L₀ = 179.20 ± 11.62 mm). The age at maturity was estimated at 16.21 and 14.04 years for females and males, respectively. The maximum intrinsic rate of population increase r_max was estimated as 0.252 years⁻¹. Life-history traits and r_max provided in the present study suggest that this species has a relatively low productivity and may be vulnerable to an intense fishing pressure.     

Acrylonitrile Quenching of Trp Phosphorescence in Proteins: A Probe of the Internal Flexibility of the Globular Fold

Quenching of Trp phosphorescence in proteins by diffusion of solutes of various molecular sizes unveils the frequency-amplitude of structural fluctuations. To cover the sizes gap between O₂ and acrylamide, we examined the potential of acrylonitrile to probe conformational flexibility of proteins. The distance dependence of the through-space acrylonitrile quenching rate was determined in a glass at 77 K, with the indole analog 2-(3-indoyl) ethyl phenyl ketone. Intensity and decay kinetics data were fitted to a rate, k(r) = k₀ exp[−(r − r₀)/r_e], with an attenuation length r_e = 0.03 nm and a contact rate k₀ = 3.6 × 10¹⁰ s⁻¹. At ambient temperature, the bimolecular quenching rate constant (kq) was determined for a series of proteins, appositely selected to test the importance of factors such as the degree of Trp burial and structural rigidity. Relative to kq = 1.9 × 10⁹ M⁻¹s⁻¹ for free Trp in water, in proteins kq ranged from 6.5 × 10⁶ M⁻¹s⁻¹ for superficial sites to 1.3 × 10² M⁻¹s⁻¹ for deep cores. The short-range nature of the interaction and the direct correlation between kq and structural flexibility attest that in the microsecond-second timescale of phosphorescence acrylonitrile readily penetrates even compact protein cores and exhibits significant sensitivity to variations in dynamical structure of the globular fold.     

Analysis of formulae for determination of seaweed growth rate

The increasing demand for products derived from seaweeds has led to an increasing amount of research being directed towards studies related to their growth and productivity. Several investigators have attempted to develop different formulae for the estimation of growth rates and caused confusion to the readers. In this study, accuracy and reliability of the average growth rate formulae were analyzed using geometric progression theory and compared to each other. The lowest degree of error (0.023 %) and the highest matched point (61.90 %) was achieved by applying [(W _t ?/?W ₀)^1/t ???1]?×?100 % in growth rate determination. This formula has been tested and proven to be the most accurate among all the currently available ones.     

Temperature-Dependent Fecundity and Life Table of the Fennel Aphid Hyadaphis foeniculi (Passerini) (Hemiptera: Aphididae)

Hyadaphis foeniculi (Passerini) (Hemiptera: Aphididae) is a cosmopolitan species and the main pest of fennel in northeastern Brazil. Understanding the relationship between temperature variations and the population growth rates of H. foeniculi is essential to predict the population dynamics of this aphid in the fennel crop. The aim of this study was to measure the effect of constant temperature on the adult prereproductive period and the life table fertility parameters (infinitesimal increase ratio (r_m), gross reproduction rate (GRR), net reproduction rate (R₀), finite increase ratio (λ), generation time (GT), the time required for the population to double in the number of individuals (DT), and the reproduction value (RV_x)) of the fennel pest H. foeniculi. The values of lx (survival of nymphs at age x) increased as the temperature rose from 15 to 28°C and fell at 30°C, whereas mx (number of nymphs produced by each nymph of age x) increased from 15 to 25°C and fell at 28 and 30°C. The net reproduction rates (R₀) of populations of H. foeniculi increased with temperature and ranged from 1.9 at 15°C to 12.23 at 28°C for each generation. The highest population increase occurred with the apterous aphids at 28°C. The rate of population increase per unit time (r_m) (day) ranged from 0.0033 (15°C) to 0.1995 (28°C). The highest values of r_m were recorded at temperatures of 28°C and 30°C. The r_m values were a good fit to the models tested, with R² > 0.91 and R² _adj > 0.88. The models tested (Davidson, Sharpe and DeMichele modified by Schoolfield et al., Logan et al., Lamb, and Briere et al.) were very good fits for the r_m values observed, with R² > 0.91 and R² _adj > 0.88. The only exception was the Davidson model. Of the parameters studied, the reproductive capacity was higher in the apterous aphids, with the unique exception of daily fecundity at 28°C, which was higher in the alate aphids of H. foeniculi. Parameters relating to the age-specific fertility table for H. foeniculi were heavily influenced by temperature, with the highest biotic potential and population growth capacity found at 34°C. Therefore, the results obtained in this study could be of practical significance for predicting outbreaks of fennel aphids and improving the management of this aphid in fennel crops.     

Differential rate of ribosomal protein synthesis in Escherichia coli B-r

The differential rate of ribosomal protein synthesis, α_r (ribosomal protein synthesis rate/total protein synthesis rate), was measured for Escherichia coli strain B/r growing at different steady-state rates ranging from 0.67 to 2.3 doublings/hour. For growth rates above 1.2 doublings/hour, α_r was found to be proportional to the growth rate μ (doublings/h), such that α_r = 0.09 μ, and the ribosome efficiency (amino acids polymerized/second per ribosome), calculated from α_r, was found to be 14 to 18 amino acids/second per ribosome. With decreasing growth rates below 1.2 doublings/hour, α_r was found to be increasingly greater than 0.09 μ and the ribosome efficiency gradually decreased such that at μ = 0.67, α_r = 0.085, and the ribosome efficiency was reduced by 30% and was equal to 10 to 13 ammo acids/second per ribosome. These results imply that the protein to DNA ratio is constant for μ > 1.2 and equal to 4 × 10⁸ to 5 × 10⁸ amino acids/genome. For μ < 1.2, this ratio gradually decreases such that at μ = 0.67, protein to DNA = 3 × 10⁸ to 4 × 10⁸ amino acids/genome. These relationships were verified by direct measurements of the amounts of DNA, RNA and protein at different steady-state growth rates. In addition, protein accumulation was measured following a nutritional shift-up from succinate to glucose minimal medium. The results indicate that the ribosome efficiency increases by approximately 40% within the first few minutes following the shift-up.     

Molecular weight estimation using sodium dodecyl sulfate-pore gradient electrophoresis

Pore gradient electrophoresis (PGE) in the presence of sodium dodecyl sulfate (SDS) provides a means for high resolution fractionation of multicomponent protein systems and permits estimation of molecular weights for macromolecules ranging from 10³ to 10⁶. We have evaluated the performance of several methods used to construct calibration curves for estimation of molecular weights using SDS-PGE. A linear relationship between the logarithm of molecular weight, log (M_r), and the logarithm of the relative mobility, log (R_l), can be obtained for a 30-fold range of molecular weights. However, this range of linearity depends on the choice of the concentration gradient, the degree of crosslinking of the gel, and on the nature of the underlying relationship between the retardation coefficient, K_R, and the molecular weight. An empirical relationship, first introduced by Lambin et al. (1976, Anal. Biochem.74, 567) between log (M_r) and the logarithm of the gel concentration at the position reached by the protein, log (%T), provides better linearity over a wider molecular weight range than does the use of log (R_l). We have compared these relatienships by experimental analysis of 10 standard proteins and by a theoretical analysis of an idealized model system. A computer program has been developed which provides appropriate statistical estimation of the molecular weight for an unknown protein, together with its standard error and 95% confidence limits. A new method has also been developed for analysis of nonlinear calibration curves in terms of molecular weight versus distance migrated, based on a theoretically justifiable, physical-chemical model. This model implies that either the relationship between log (M_r) and log (R_l) or the one between log (M_r) and log (%T) will become nonlinear as the range of molecular weight is extended. We suggest that the use of a nonlinear least-squares curve-fitting procedure provides an optimal method for molecular weight estimation when sufficient data are available. Based on these findings, a general strategy is presented for estimation of molecular weights by polyacrylamide gel electrophoresis.     

A biological model for influenza transmission: pandemic planning implications of asymptomatic infection and immunity

Background

The clinical attack rate of influenza is influenced by prior immunity and mixing patterns in the host population, and also by the proportion of infections that are asymptomatic. This complexity makes it difficult to directly estimate R₀ from the attack rate, contributing to uncertainty in epidemiological models to guide pandemic planning. We have modelled multiple wave outbreaks of influenza from different populations to allow for changing immunity and asymptomatic infection and to make inferences about R₀.

Data and Methods

On the island of Tristan da Cunha (TdC), 96% of residents reported illness during an H3N2 outbreak in 1971, compared with only 25% of RAF personnel in military camps during the 1918 H1N1 pandemic. Monte Carlo Markov Chain (MCMC) methods were used to estimate model parameter distributions.

Findings

We estimated that most islanders on TdC were non-immune (susceptible) before the first wave, and that almost all exposures of susceptible persons caused symptoms. The median R₀ of 6.4 (95% credibility interval 3.7–10.7) implied that most islanders were exposed twice, although only a minority became ill in the second wave because of temporary protection following the first wave. In contrast, only 51% of RAF personnel were susceptible before the first wave, and only 38% of exposed susceptibles reported symptoms. R₀ in this population was also lower [2.9 (2.3–4.3)], suggesting reduced viral transmission in a partially immune population.

Interpretation

Our model implies that the RAF population was partially protected before the summer pandemic wave of 1918, arguably because of prior exposure to interpandemic influenza. Without such protection, each symptomatic case of influenza would transmit to between 2 and 10 new cases, with incidence initially doubling every 1–2 days. Containment of a novel virus could be more difficult than hitherto supposed.