Long-term effects of non-pharmaceutical interventions on total disease burden in parsimonious epidemiological models.

The recent global COVID-19 pandemic resulted in governments enacting non-pharmaceutical interventions (NPIs) targeted at reducing transmission of SARS-CoV-2. But the NPIs also affected the transmission of viruses causing non-target seasonal respiratory diseases, including influenza and respiratory syncytial virus (RSV). In many countries, the NPIs were found to reduce cases of such seasonal respiratory diseases, but there is also evidence that subsequent relaxation of NPIs led to outbreaks of these diseases that were larger than pre-pandemic ones, due to the accumulation of susceptible individuals prior to relaxation. Therefore, the net long-term effects of NPIs on the total disease burden of non-target diseases remain unclear. Knowledge of this is important for infectious disease management and maintenance of public health. In this study, we shed light on this issue for the simplified scenario of a set of NPIs that prevent or reduce transmission of a seasonal respiratory disease for about a year and are then removed, using mathematical analyses and numerical simulations of a suite of four epidemiological models with varying complexity and generality. The model parameters were estimated using empirical data pertaining to seasonal respiratory diseases and covered a wide range. Our results showed that NPIs reduced the total disease burden of a non-target seasonal respiratory disease in the long-term. Expressed as a percentage of population size, the reduction was greater for larger values of the basic reproduction number and the immunity loss rate, reflecting larger outbreaks and hence more infections averted by imposition of NPIs. Our study provides a foundation for exploring the effects of NPIs on total disease burden in more-complex scenarios.

Single Diode Solar Cells-Improved Model and Exact Current-Voltage Analytical Solution Based on Lambert's W Function.

There are three standard equivalent circuit models of solar cells in the literature-single-diode, double-diode, and triple-diode models. In this paper, first, a modified version of the single diode model, called the Improved Single Diode Model (ISDM), is presented. This modification is realized by adding resistance in series with the diode to enable better power loss dissipation representation. Second, the mathematical expression for the current-voltage relation of this circuit is derived in terms of Lambert's W function and solved by using the special trans function theory. Third, a novel hybrid algorithm for solar cell parameters estimation is proposed. The proposed algorithm, called SA-MRFO, is used for the parameter estimation of the standard single diode and improved single diode models. The proposed model's accuracy and the proposed algorithm's efficiency are tested on a standard RTC France solar cell and SOLAREX module MSX 60. Furthermore, the experimental verification of the proposed circuit and the proposed solar cell parameter estimation algorithm on a solar laboratory module is also realized. Based on all the results obtained, it is shown that the proposed circuit significantly improves current-voltage solar cell representation in comparison with the standard single diode model and many results in the literature on the double diode and triple diode models. Additionally, it is shown that the proposed algorithm is effective and outperforms many literature algorithms in terms of accuracy and convergence speed.

Evaluation of the paraoxonase-1 kinetic parameters of the lactonase activity by nonlinear fit of progress curves.

Although paraoxonase-1 (PON1) activity has been demonstrated to be a reliable biomarker of various diseases, clinical studies have been based only on relative comparison of specific enzyme activities, which capture differences mainly due to (usually unknown) PON1 concentration. Hence, the aim of this report is to present for the first time the simple evaluation method for determining autonomous kinetic parameter of PON1 that could be also associated with polymorphic forms and diseases; i.e. the Michaelis constant which is enzyme concentration independent quantity. This alternative approach significantly reduces the number of experiments needed, and it yields the results with great accuracy.

Analytical Parameter Estimation of the SIR Epidemic Model. Applications to the COVID-19 Pandemic.

The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epidemic outbreaks, yet for decades it evaded the efforts of the mathematical community to derive an explicit solution. The present paper reports novel analytical results and numerical algorithms suitable for parametric estimation of the SIR model. Notably, a series solution of the incidence variable of the model is derived. It is proven that the explicit solution of the model requires the introduction of a new transcendental special function, which is a solution of a non-elementary integral equation. The paper introduces iterative algorithms approximating the incidence variable, which allows for estimation of the model parameters from the numbers of observed cases. The approach is applied to the case study of the ongoing coronavirus disease 2019 (COVID-19) pandemic in five European countries: Belgium, Bulgaria, Germany, Italy and the Netherlands. Incidence and case fatality data obtained from the European Centre for Disease Prevention and Control (ECDC) are analysed and the model parameters are estimated and compared.

When does seed limitation matter for scaling up reforestation from patches to landscapes?

Restoring forest to hundreds of millions of hectares of degraded land has become a centerpiece of international plans to sequester carbon and conserve biodiversity. Forest landscape restoration will require scaling up ecological knowledge of secondary succession from small-scale field studies to predict forest recovery rates in heterogeneous landscapes. However, ecological field studies reveal widely divergent times to forest recovery, in part due to landscape features that are difficult to replicate in empirical studies. Seed rain can determine reforestation rate and depends on landscape features that are beyond the scale of most field studies. We develop mathematical models to quantify how landscape configuration affects seed rain and forest regrowth in degraded patches. The models show how landscape features can alter the successional trajectories of otherwise identical patches, thus providing insight into why some empirical studies reveal a strong effect of seed rain on secondary succession, while others do not. We show that seed rain will strongly limit reforestation rate when patches are near a threshold for arrested succession, when positive feedbacks between tree canopy cover and seed rain occur during early succession, and when directed dispersal leads to between-patch interactions. In contrast, seed rain has weak effects on reforestation rate over a wide range of conditions, including when landscape-scale seed availability is either very high or very low. Our modeling framework incorporates growth and survival parameters that are commonly estimated in field studies of reforestation. We demonstrate how mathematical models can inform forest landscape restoration by allowing land managers to predict where natural regeneration will be sufficient to restore tree cover. Translating quantitative forecasts into spatially targeted interventions for forest landscape restoration could support target goals of restoring millions of hectares of degraded land and help mitigate global climate change.

Non-empirical peak shape methods based on the physical model of the one trap one recombination center model.

Any experimental Thermoluminescent (TL) glow-peak contains the activation energy information of its corresponding energy level within the band gap in insulating materials. The theory of peak shape methods (PSM) correlates the macroscopic geometrical characteristics of a single TL peak with activation energy of the level responsible for the TL peak by assuming that the area under a TL peak can be approximated by the area of a triangle. In this way the geometrical characteristics becomes the measure of the activation energy. In the present work new PSM expressions are derived, which are not empirical as the existing ones but are based of the physical model of one trap one recombination (OTOR) center. Three cases are considered. (I) Delocalized OTOR for re-trapping probability smaller than the recombination probability. (II) Delocalized OTOR for re-trapping probability greater than recombination probability. (III) Localized transitions OTOR model. The system of differential equations of each case model were solved analytically using the Lambert W function (or equivalently the Wright ω function). Then the resulted analytical expressions of TL intensity as a function of temperature were used to derive new PSM. The new PSM from all cases are formally exactly the same, having, however, strong differentiation in their coefficients. The functionality of the new expressions is tested and its comparison with pre-existing PSM is performed.

A mathematical model of COVID-19 transmission.

Disease transmission is studied through disciplines like epidemiology, applied mathematics, and statistics. Mathematical simulation models for transmission have implications in solving public and personal health challenges. The SIR model uses a compartmental approach including dynamic and nonlinear behavior of transmission through three factors: susceptible, infected, and removed (recovered and deceased) individuals. Using the Lambert W Function, we propose a framework to study solutions of the SIR model. This demonstrates the applications of COVID-19 transmission data to model the spread of a real-world disease. Different models of disease including the SIR, SIRmp and SEIRρqr model are compared with respect to their ability to predict disease spread. Physical distancing impacts and personal protection equipment use are discussed with relevance to the COVID-19 spread.

Explicit solution of a Lotka-Sharpe-McKendrick system involving neutral delay differential equations using the r-Lambert W function.

Structured population models, which account for the state of individuals given features such as age, gender, and size, are widely used in the fields of ecology and biology. In this paper, we consider an age-structured population model describing the population of adults and juveniles. The model consists of a system of ordinary and neutral delay differential equations. We present an explicit solution to the model using a generalization of the Lambert W function called the r-Lambert W function. Numerical simulations with varying parameters and initial conditions are done to illustrate the obtained solution. The proposed method is also applied to an insect population model with long larval and short adult phases.

GENERATING SURVIVAL TIMES WITH TIME-VARYING COVARIATES USING THE LAMBERT W FUNCTION.

Simulation studies provide an important statistical tool in evaluating survival methods, requiring an appropriate data-generating process to simulate data for an underlying statistical model. Many studies with time-to-event outcomes use the Cox proportional hazard model. While methods for simulating such data with time-invariant predictors have been described, methods for simulating data with time-varying covariates are sorely needed. Here, we describe an approach for generating data for the Cox proportional hazard model with time-varying covariates when event times follow an Exponential or Weibull distribution. For each distribution, we derive a closed-form expression to generate survival times and link the time-varying covariates with the hazard function. We consider a continuous time-varying covariate measured at regular intervals over time, as well as time-invariant covariates, in generating time-to-event data under a number of scenarios. Our results suggest this method can lead to simulation studies with reliable and robust estimation of the association parameter in Cox-Weibull and Cox-Exponential models.

The h-index as an almost-exact function of some basic statistics.

As is known, the h-index, h, is an exact function of the citation pattern. At the same time, and more generally, it is recognized that h is "loosely" related to the values of some basic statistics, such as the number of publications and the number of citations. In the present study we introduce a formula that expresses the h-index as an almost-exact function of some (four) basic statistics. On the basis of an empirical study-in which we consider citation data obtained from two different lists of journals from two quite different scientific fields-we provide evidence that our ready-to-use formula is able to predict the h-index very accurately (at least for practical purposes). For comparative reasons, alternative estimators of the h-index have been considered and their performance evaluated by drawing on the same dataset. We conclude that, in addition to its own interest, as an effective proxy representation of the h-index, the formula introduced may provide new insights into "factors" determining the value of the h-index, and how they interact with each other.

A theoretical model of the relationship between the h-index and other simple citation indicators.

Of the existing theoretical formulas for the h-index, those recently suggested by Burrell (J Informetr 7:774-783, 2013b) and by Bertoli-Barsotti and Lando (J Informetr 9(4):762-776, 2015) have proved very effective in estimating the actual value of the h-index Hirsch (Proc Natl Acad Sci USA 102:16569-16572, 2005), at least at the level of the individual scientist. These approaches lead (or may lead) to two slightly different formulas, being based, respectively, on a "standard" and a "shifted" version of the geometric distribution. In this paper, we review the genesis of these two formulas-which we shall call the "basic" and "improved" Lambert-W formula for the h-index-and compare their effectiveness with that of a number of instances taken from the well-known Glänzel-Schubert class of models for the h-index (based, instead, on a Paretian model) by means of an empirical study. All the formulas considered in the comparison are "ready-to-use", i.e., functions of simple citation indicators such as: the total number of publications; the total number of citations; the total number of cited paper; the number of citations of the most cited paper. The empirical study is based on citation data obtained from two different sets of journals belonging to two different scientific fields: more specifically, 231 journals from the area of "Statistics and Mathematical Methods" and 100 journals from the area of "Economics, Econometrics and Finance", totaling almost 100,000 and 20,000 publications, respectively. The citation data refer to different publication/citation time windows, different types of "citable" documents, and alternative approaches to the analysis of the citation process ("prospective" and "retrospective"). We conclude that, especially in its improved version, the Lambert-W formula for the h-index provides a quite robust and effective ready-to-use rule that should be preferred to other known formulas if one's goal is (simply) to derive a reliable estimate of the h-index.

The kinetics of regeneration of rhodopsin under enzyme-limited availability of 11-cis retinoid.

In order to describe the regeneration of rhodopsin and the recovery of visual sensitivity following exposure of the eye to intense bleaching illumination, two models have been proposed, in which there is either a "resistive" or an "enzymatic" limit to the supply of retinoid. A solution has previously been derived for the resistive model, and here we derive an analytical solution for the enzymatic model and we investigate the form of this solution as a function of parameter values. We demonstrate that this enzymatic model provides a good fit to human post-bleach recovery, for four cases: for rhodopsin regeneration in normal subjects; for psychophysical scotopic dark adaptation in normal subjects; for rhodopsin regeneration and scotopic dark adaptation in fundus albipunctatus patients; and for cone pigment regeneration in normal subjects. Finally, we present arguments favouring the enzymatic model as the cellular basis for normal human rod and cone pigment regeneration.