The COVID-19 pandemics challenges governments across the world.To develop adequate responses,they need accurate models for the spread of the disease.Using least squares,we fitted Bertalanffy-Pütter(BP)trend curves to data about the first wave of the COVID-19 pandemic of 2020 from 49 countries and provinces where the peak of the first wave had been passed.BP-models achieved excellent fits(R-squared above 99%)to all data.Using them to smoothen the data,in the median one could forecast that the final count(asymptotic limit)of infections and fatalities would be 2.48 times(95%confidence limits 2.42-2.6)and 2.67 times(2.39-2.765)the total count at the respective peak(inflection point).By comparison,using logistic growth would evaluate this ratio as 2.00 for all data.The case fatality rate,defined as the quotient of the asymptotic limits of fatalities and confirmed infections,was in the median 4.85%(confidence limits 4.4%e6.5%).Our result supports the strategies of governments that kept the epidemic peak low,as then in the median fewer infections and fewer fatalities could be expected.
The Bertalanffy-Pütter (BP) five-parameter growth model provides a versatile framework for the modeling of growth. Using data from a growth experiment in literature about the average size-at-age of 24 species of tropical trees over ten years in the same area, we identified their best-fit BP-model parameters. While different species had different best-fit exponent-pairs, there was a model with a good fit to 21 (87.5%) of the data (“Good fit” means a normalized root-mean-squared-error NRMSE below 2.5%. This threshold was the 95% quantile of the lognormal distribution that was fitted to the NRMSE values for the best-fit models for the data). In view of the sigmoidal character of this model despite the early stand we discuss whether the setting of the growth experiment may have impeded growth.
This paper mainly addresses maximum likelihood estimation for a response-selective stratified sampling scheme, the basic stratified sampling (BSS), in which the maximum subsample size in each stratum is fixed. We derived the complete-data likelihood for BSS, and extended it as a full-data likelihood by incorporating incomplete data. We also similarly extended the empirical proportion likelihood approach for consistent and efficient estimation. We conducted a simulation study to compare these two new approaches with the existing estimation methods in BSS. Our result indicates that they perform as well as the standard full information likelihood approach. Methods were illustrated using a growth model for fish size at age, including between-individual variability. One of our major conclusions is that the fully observed BSS data, the partially observed data used for stratification, and the sampling strategy are all important in constructing a consistent and efficient estimator.