Forecasting the dynamics of cumulative COVID-19 cases (confirmed, recovered and deaths) for top-16 countries using statistical machine learning models: Auto-Regressive Integrated Moving Average (ARIMA) and Seasonal Auto-Regressive Integrated Moving Average (SARIMA).

Most countries are reopening or considering lifting the stringent prevention policies such as lockdowns, consequently, daily coronavirus disease (COVID-19) cases (confirmed, recovered and deaths) are increasing significantly. As of July 25th, there are 16.5 million global cumulative confirmed cases, 9.4 million cumulative recovered cases and 0.65 million deaths. There is a tremendous necessity of supervising and estimating future COVID-19 cases to control the spread and help countries prepare their healthcare systems. In this study, time-series models - Auto-Regressive Integrated Moving Average (ARIMA) and Seasonal Auto-Regressive Integrated Moving Average (SARIMA) are used to forecast the epidemiological trends of the COVID-19 pandemic for top-16 countries where 70%-80% of global cumulative cases are located. Initial combinations of the model parameters were selected using the auto-ARIMA model followed by finding the optimized model parameters based on the best fit between the predictions and test data. Analytical tools Auto-Correlation function (ACF), Partial Auto-Correlation Function (PACF), Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) were used to assess the reliability of the models. Evaluation metrics Mean Absolute Error (MAE), Mean Square Error (MSE), Root Mean Square Error (RMSE) and Mean Absolute Percent Error (MAPE) were used as criteria for selecting the best model. A case study was presented where the statistical methodology was discussed in detail for model selection and the procedure for forecasting the COVID-19 cases of the USA. Best model parameters of ARIMA and SARIMA for each country are selected manually and the optimized parameters are then used to forecast the COVID-19 cases. Forecasted trends for confirmed and recovered cases showed an exponential rise for countries such as the United States, Brazil, South Africa, Colombia, Bangladesh, India, Mexico and Pakistan. Similarly, trends for cumulative deaths showed an exponential rise for countries Brazil, South Africa, Chile, Colombia, Bangladesh, India, Mexico, Iran, Peru, and Russia. SARIMA model predictions are more realistic than that of the ARIMA model predictions confirming the existence of seasonality in COVID-19 data. The results of this study not only shed light on the future trends of the COVID-19 outbreak in top-16 countries but also guide these countries to prepare their health care policies for the ongoing pandemic. The data used in this work is obtained from publicly available John Hopkins University's COVID-19 database.

Statistical Evaluation of Different Mathematical Models for Diffusion Weighted Imaging of Prostate Cancer Xenografts in Mice.

PURPOSE: To evaluate fitting quality and repeatability of four mathematical models for diffusion weighted imaging (DWI) during tumor progression in mouse xenograft model of prostate cancer. METHODS: Human prostate cancer cells (PC-3) were implanted subcutaneously in right hind limbs of 11 immunodeficient mice. Tumor growth was followed by weekly DWI examinations using a 7T MR scanner. Additional DWI examination was performed after repositioning following the fourth DWI examination to evaluate short term repeatability. DWI was performed using 15 and 12 b-values in the ranges of 0-500 and 0-2000 s/mm2, respectively. Corrected Akaike information criteria and F-ratio were used to evaluate fitting quality of each model (mono-exponential, stretched exponential, kurtosis, and bi-exponential). RESULTS: Significant changes were observed in DWI data during the tumor growth, indicated by ADCm, ADCs, and ADCk. Similar results were obtained using low as well as high b-values. No marked changes in model preference were present between the weeks 1-4. The parameters of the mono-exponential, stretched exponential, and kurtosis models had smaller confidence interval and coefficient of repeatability values than the parameters of the bi-exponential model. CONCLUSION: Stretched exponential and kurtosis models showed better fit to DWI data than the mono-exponential model and presented with good repeatability.

On the exponent in the Von Bertalanffy growth model.

Von Bertalanffy proposed the differential equation m'(t) = p × m(t) a  - q × m(t) for the description of the mass growth of animals as a function m(t) of time t. He suggested that the solution using the metabolic scaling exponent a = 2/3 (Von Bertalanffy growth function VBGF) would be universal for vertebrates. Several authors questioned universality, as for certain species other models would provide a better fit. This paper reconsiders this question. Based on 60 data sets from literature (37 about fish and 23 about non-fish species) it optimizes the model parameters, in particular the exponent 0 ≤ a < 1, so that the model curve achieves the best fit to the data. The main observation of the paper is the large variability in the exponent, which can vary over a very large range without affecting the fit to the data significantly, when the other parameters are also optimized. The paper explains this by differences in the data quality: variability is low for data from highly controlled experiments and high for natural data. Other deficiencies were biologically meaningless optimal parameter values or optimal parameter values attained on the boundary of the parameter region (indicating the possible need for a different model). Only 11 of the 60 data sets were free of such deficiencies and for them no universal exponent could be discerned.