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BACKGROUND: Increasing crop yield per unit area by increasing planting density is essential to ensure food security. However, the optimal combination of planting density and nitrogen (N) application for high-yielding maize and its source-sink characteristics need to be more clearly understood. RESULTS: A 2-year field experiment was conducted combining three planting densities (D1: 70 000 plants ha-1 ; D2: 100 000 plants ha-1 ; D3: 130 000 plants ha-1 ) and three nitrogen rates (N1: 150 kg hm-2 ; N2: 350 kg hm-2 ; N3: 450 kg hm-2 ). The results showed that increasing planting density significantly increased leaf area index and grain yield but negatively affected ear traits. The Richards model was used to fit the dynamic changes of dry matter accumulation of maize under different treatments, and the fitting results were good. Increasing planting density increased population yield while limiting the development of individual plants, bringing the period of rapid dry matter accumulation to an early end and accelerating leaf senescence. An appropriate nitrogen rate could prolong the period of rapid accumulation of dry matter in maize, and increase the 100-kernel weight. Increasing planting density enhanced post-silking dry matter accumulation to a lesser extent, and the source-sink relationship of the maize population gradually developed from sink limitation to source limitation with increasing planting density. CONCLUSION: The decrease in yield due to the insufficient source strength to meet the sink demand at too high densities was the reason that limited further improvement of the optimal planting density. An appropriate nitrogen rate facilitated the realization of yield potential at high density. © 2023 Society of Chemical Industry.

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Background: Simple dynamic modeling tools can be useful for generating real-time short-term forecasts with quantified uncertainty of the trajectory of diverse growth processes unfolding in nature and society, including disease outbreaks. An easy-to-use and flexible toolbox for this purpose is lacking. Results: In this tutorial-based primer, we introduce and illustrate a user-friendly MATLAB toolbox for fitting and forecasting time-series trajectories using phenomenological dynamic growth models based on ordinary differential equations. This toolbox is accessible to various audiences, including students training in time-series forecasting, dynamic growth modeling, parameter estimation, parameter uncertainty and identifiability, model comparison, performance metrics, and forecast evaluation, as well as researchers and policymakers who need to conduct short-term forecasts in real-time. The models included in the toolbox capture exponential and sub-exponential growth patterns that typically follow a rising pattern followed by a decline phase, a common feature of contagion processes. Models include the 2-parameter generalized-growth model, which has proved useful to characterize and forecast the ascending phase of epidemic outbreaks, and the Gompertz model as well as the 3-parameter generalized logistic-growth model and the Richards model, which have demonstrated competitive performance in forecasting single peak outbreaks.The toolbox provides a tutorial for forecasting time-series trajectories that include the full uncertainty distribution, derived through parametric bootstrapping, which is needed to construct prediction intervals and evaluate their accuracy. Functions are available to assess forecasting performance across different models, estimation methods, error structures in the data, and forecasting horizons. The toolbox also includes functions to quantify forecasting performance using metrics that evaluate point and distributional forecasts, including the weighted interval score. Conclusions: We have developed the first comprehensive toolbox to characterize and forecast time-series data using simple phenomenological growth models. As a contagion process takes off, the tools presented in this tutorial can facilitate policymaking to guide the implementation of control strategies and assess the impact of interventions. The toolbox functionality is demonstrated through various examples, including a tutorial video, and is illustrated using weekly data on the monkeypox epidemic in the USA.

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Dengue fever is a notable vector-borne viral disease, currently becoming the most dreaded worldwide health problem in terms of the number of people affected. A data set of confirmed dengue incidences collected in the province of West Java has allowed us to explore dengue's temporal trends and spatial distributions to obtain more obvious insights into its spatial-temporal evolution. We utilized the Richards model to estimate the growth rate and detect the peak (or turning point) of the dengue infection wave by identifying the temporal progression at each location. Using spatial analysis of geo-referenced data from a local perspective, we investigated the changes in the spatial clusters of dengue cases and detected hot spots and cold spots in each weekly cycle. We found that the trend of confirmed dengue incidences significantly increases from January to March. More than two-third (70.4%) of the regions in West Java had their dengue infection turning point ranging from the first week of January to the second week of March. This trend clearly coincides with the peak of precipitation level during the rainy season. Further, the spatial analysis identified the hot spots distributed across central, northern, northeastern, and southeastern regions in West Java. The densely populated areas were likewise seen to be associated with the high-risk areas of dengue exposure. Recognizing the peak of epidemic and geographical distribution of dengue cases might provide important insights that may help local authorities optimize their dengue prevention and intervention programs.

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In the absence of reliable information about transmission mechanisms for emerging infectious diseases, simple phenomenological models could provide a starting point to assess the potential outcomes of unfolding public health emergencies, particularly when the epidemiological characteristics of the disease are poorly understood or subject to substantial uncertainty. In this study, we employ the modified Richards model to analyze the growth of an epidemic in terms of 1) the number of times cumulative cases double until the epidemic peaks and 2) the rate at which the intervals between consecutive doubling times increase during the early ascending stage of the outbreak. Our theoretical analysis of doubling times is combined with rigorous numerical simulations and uncertainty quantification using synthetic and real data for COVID-19 pandemic. The doubling-time approach allows to employ early epidemic data to differentiate between the most dangerous threats, which double in size many times over the intervals that are nearly invariant, and the least transmissible diseases, which double in size only a few times with doubling periods rapidly growing.

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BACKGROUND AND AIMS: Nitrogen is often regarded as a limiting factor to plant growth in various ecosystems. Understanding how nitrogen drives plant growth has numerous theoretical and practical applications in agriculture and ecology. In 2004, Göran I. Ågren proposed a mechanistic model of plant growth from a biochemical perspective. However, neglecting respiration and assuming stable and balanced growth made the model unrealistic for plants growing in natural conditions. The aim of the present paper is to extend Ågren's model to overcome these limitations. METHODS: We improved Ågren's model by incorporating the respiratory process and replacing the stable and balanced growth assumption with a three-parameter power function to describe the relationship between nitrogen concentration (Nc) and biomass. The new model was evaluated based on published data from three studies on corn (Zea mays) growth. KEY RESULTS: Remarkably, the mechanistic growth model derived in this study is mathematically equivalent to the classical Richards model, which is the most widely used empirical growth model. The model agrees well with empirical plant growth data. CONCLUSIONS: Our model provides a mechanistic interpretation of how nitrogen drives plant growth. It is very robust in predicting growth curves and the relationship between Nc and relative growth rate.

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Phenomenological growth models (PGMs) provide a framework for characterizing epidemic trajectories, estimating key transmission parameters, gaining insight into the contribution of various transmission pathways, and providing long-term and short-term forecasts. Such models only require a small number of parameters to describe epidemic growth patterns. They can be expressed by an ordinary differential equation (ODE) of the type C'(t)=f(t,C;Θ) for t>0, C(0)=C0, where t is time, C(t) is the total size of the epidemic (the cumulative number of cases) at time t, C0 is the initial number of cases, f is a model-specific incidence function, and Θ is a vector of parameters. The current COVID-19 pandemic is a scenario for which such models are of obvious importance. In Bürger et al. (2019) it is demonstrated that some PGMs are better at fitting data of specific epidemic outbreaks than others even when the models have the same number of parameters. This situation motivates the need to measure differences in the dynamics that two different models are capable of generating. The present work contributes to a systematic study of differences between PGMs and how these may explain the ability of certain models to provide a better fit to data than others. To this end a so-called empirical directed distance (EDD) is defined to describe the differences in the dynamics between different dynamic models. The EDD of one PGM from another one quantifies how well the former fits data generated by the latter. The concept of EDD is, however, not symmetric in the usual sense of metric spaces. The procedure of calculating EDDs is applied to synthetic data and real data from influenza, Ebola, and COVID-19 outbreaks.

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In 2020, a new type of coronavirus is in the global pandemic. Now, the number of infected patients is increasing. The trend of the epidemic has attracted global attention. Based on the traditional Richards model and the differential information principle in grey prediction model, this paper uses the modified grey action quantity to propose a new grey prediction model for infectious diseases. This model weakens the dependence of the Richards model on single-peak and saturated S-shaped data, making Richards model more applicable, and uses genetic algorithm to optimize the nonlinear terms and the background value. To illustrate the effectiveness of the model, groups of slowly growing small-sample and large-sample data are selected for simulation experiments. Results of eight evaluation indexes show that the new model is better than the traditional GM(1,1) and grey Richards model. Finally, this model is applied to China, Italy, Britain and Russia. The results show that the new model is better than the other 7 models. Therefore, this model can effectively predict the number of daily new confirmed cases of COVID-19, and provide important prediction information for the formulation of epidemic prevention policies.

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BACKGROUND: Ensemble modeling aims to boost the forecasting performance by systematically integrating the predictive accuracy across individual models. Here we introduce a simple-yet-powerful ensemble methodology for forecasting the trajectory of dynamic growth processes that are defined by a system of non-linear differential equations with applications to infectious disease spread. METHODS: We propose and assess the performance of two ensemble modeling schemes with different parametric bootstrapping procedures for trajectory forecasting and uncertainty quantification. Specifically, we conduct sequential probabilistic forecasts to evaluate their forecasting performance using simple dynamical growth models with good track records including the Richards model, the generalized-logistic growth model, and the Gompertz model. We first test and verify the functionality of the method using simulated data from phenomenological models and a mechanistic transmission model. Next, the performance of the method is demonstrated using a diversity of epidemic datasets including scenario outbreak data of the Ebola Forecasting Challenge and real-world epidemic data outbreaks of including influenza, plague, Zika, and COVID-19. RESULTS: We found that the ensemble method that randomly selects a model from the set of individual models for each time point of the trajectory of the epidemic frequently outcompeted the individual models as well as an alternative ensemble method based on the weighted combination of the individual models and yields broader and more realistic uncertainty bounds for the trajectory envelope, achieving not only better coverage rate of the 95% prediction interval but also improved mean interval scores across a diversity of epidemic datasets. CONCLUSION: Our new methodology for ensemble forecasting outcompete component models and an alternative ensemble model that differ in how the variance is evaluated for the generation of the prediction intervals of the forecasts.

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BACKGROUND: The rising burden of the ongoing COVID-19 epidemic in South Africa has motivated the application of modeling strategies to predict the COVID-19 cases and deaths. Reliable and accurate short and long-term forecasts of COVID-19 cases and deaths, both at the national and provincial level, are a key aspect of the strategy to handle the COVID-19 epidemic in the country. METHODS: In this paper we apply the previously validated approach of phenomenological models, fitting several non-linear growth curves (Richards, 3 and 4 parameter logistic, Weibull and Gompertz), to produce short term forecasts of COVID-19 cases and deaths at the national level as well as the provincial level. Using publicly available daily reported cumulative case and death data up until 22 June 2020, we report 5, 10, 15, 20, 25 and 30-day ahead forecasts of cumulative cases and deaths. All predictions are compared to the actual observed values in the forecasting period. RESULTS: We observed that all models for cases provided accurate and similar short-term forecasts for a period of 5 days ahead at the national level, and that the three and four parameter logistic growth models provided more accurate forecasts than that obtained from the Richards model 10 days ahead. However, beyond 10 days all models underestimated the cumulative cases. Our forecasts across the models predict an additional 23,551-26,702 cases in 5 days and an additional 47,449-57,358 cases in 10 days. While the three parameter logistic growth model provided the most accurate forecasts of cumulative deaths within the 10 day period, the Gompertz model was able to better capture the changes in cumulative deaths beyond this period. Our forecasts across the models predict an additional 145-437 COVID-19 deaths in 5 days and an additional 243-947 deaths in 10 days. CONCLUSIONS: By comparing both the predictions of deaths and cases to the observed data in the forecasting period, we found that this modeling approach provides reliable and accurate forecasts for a maximum period of 10 days ahead.

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ABSTRACT: The objective of this study was to compare non-linear models fitted to the growth curves of quail to determine which model best describes their growth and check the similarity between models by analyzing parameter estimates.Weight and age data of meat-type European quail (Coturnix coturnix coturnix) of three lines were used, from an experiment in a 2 × 4 factorial arrangement in a completely randomized design, consisting of two metabolizable energy levels, four crude protein levels and six replicates. The non-linear Brody, Von Bertalanffy, Richards, Logistic and Gompertz models were used. To choose the best model, the Adjusted Coefficient of Determination, Convergence Rate, Residual Mean Square, Durbin-Watson Test, Akaike Information Criterion and Bayesian Information Criterion were applied as goodness-of-fit indicators. Cluster analysis was performed to check the similarity between models based on the mean parameter estimates. Among the studied models, Richards' was the most suitable to describe the growth curves. The Logistic and Richards models were considered similar in the analysis with no distinction of lines as well as in the analyses of Lines 1, 2 and 3.

RESUMO: Objetivou-se, neste estudo, comparar modelos não lineares ajustados às curvas de crescimento de codornas para determinar qual modelo que melhor descreve o crescimento de codornas e verificar a similaridade dos modelos analisando as estimativas dos parâmetros. Para as análises foram utilizados os dados peso e idade de codornas européias de corte (Coturnix coturnix coturnix) proveniente de três linhagens, em um esquema fatorial 2x4, instalado em um delineamento inteiramente casualizado, com dois níveis de energia metabolizável e quatro níveis de proteína bruta, com seis repetições. Os modelos não lineares utilizados foram: Brody, Von Bertalanffy, Richards, Logístico e Gompertz. Para a escolha do melhor modelo utilizou-se o Coeficiente de Determinação Ajustado, o Percentual de Convergência, o Quadrado Médio do Resíduo, o Teste de Durbin-Watson, o Critério de informação Akaike e o Critério de informação Bayesiano como avaliadores da qualidade do ajuste. Utilizou-se a análise de agrupamento para verificar, baseado nas estimativas médias dos parâmetros, a similaridades entre os modelos. Entre os modelos estudados, o Richard foi o mais adequado para descrever as curvas de crescimento. Os modelos Logístico e Richards foram considerados similares nas análises sem distinção de linhagem, bem como nas análises das Linhagem 1, 2 e 3.

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COVID-19 pandemic is spreading around the world becoming thus a serious concern for health, economic and social systems worldwide. In such situation, predicting as accurately as possible the future dynamics of the virus is a challenging problem for scientists and decision-makers. In this paper, four phenomenological epidemic models as well as Suspected-Infected-Recovered (SIR) model are investigated for predicting the cumulative number of infected cases in Saudi Arabia in addition to the probable end-date of the outbreak. The prediction problem is formulated as an optimization framework and solved using a Particle Swarm Optimization (PSO) algorithm. The Generalized Richards Model (GRM) has been found to be the best one in achieving two objectives: first, fitting the collected data (covering 223 days between March 2nd and October 10, 2020) with the lowest mean absolute percentage error (MAPE = 3.2889%), the highest coefficient of determination (R2 = 0.9953) and the lowest root mean squared error (RMSE = 8827); and second, predicting a probable end date found to be around the end of December 2020 with a projected number of 378,299 at the end of the outbreak. The obtained results may help the decision-makers to take suitable decisions related to the pandemic mitigation and containment and provide clear understanding of the virus dynamics in Saudi Arabia.

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Several months into the ongoing novel coronavirus disease 2019 (COVID-19) pandemic, this work provides a simple and direct projection of the outbreak spreading potential and the pandemic cessation dates in Chinese mainland, Iran, the Philippines and Chinese Taiwan, using the generalized logistic model (GLM). The short-term predicted number of cumulative COVID-19 cases matched the confirmed reports of those who were infected across the four countries and regions, and the long-term forecasts were capable to accurately evaluate the spread of the pandemic in Chinese mainland and Chinese Taiwan, where control measures such as social distancing were fully implemented and sustained, suggesting GLM as a valuable tool for characterizing the transmission dynamics process and the trajectory of COVID-19 pandemic along with the impact of interventions.

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Objective To analyze the change in transmissibility of novel coronavirus pneumonia and predict the trend of the incidence, and to provide a reference for the government to better respond to the novel coronavirus pneumonia epidemic. Methods The EpiEstimof R language software was used to estimate the change of effective basic reproduction number, and the Richards model was run by Matlab7.0 software to fit the cumulative number of confirmed cases and the number of suspected cases. The coefficient of determination and root mean squared error were used to evaluate the fitting effect of the model. Results A total of 75 confirmed cases and 107 suspected cases were reported in Ningxia. The strict implementation of various prevention and control measures gradually reduced the effective basic reproduction number from 3.82 to less than 1, indicating that the epidemic was under control. The Richards model was used to fit the cumulative confirmed cases and suspected cases, which revealed that the natural growth rates were 0.16 and 0.23, and the coefficients of determination were 0.991 and 0.998, respectively. Conclusion Combined with the effective basic reproduction number, the Richards model fitted the trend of novel coronavirus pneumonia, which can be used to predict the trend of incidence of new coronavirus pneumonia.

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Abstract Mathematical models are often used to predict microbial growth in food products. An important class of these models involves the adaptation of classical sigmoid functions, such as the Gompertz and logistic functions. This study aimed to validate the use of the modified Richards model in various situations, which have not previously been tested. The model was obtained through solving a system of two differential equations and could be applied to both isothermal and non-isothermal environments. To test and validate this model, we used published datasets containing data for the growth of Pseudomonas spp. in fish products. The results obtained after fitting the model showed that it could be effectively used to describe and predict the Pseudomonas growth curves under various temperature regimens. However, the influence of the shape parameter on the growth curve is an issue that needs further evaluation.

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Mathematical models are often used to predict microbial growth in food products. An important class of these models involves the adaptation of classical sigmoid functions, such as the Gompertz and logistic functions. This study aimed to validate the use of the modified Richards model in various situations, which have not previously been tested. The model was obtained through solving a system of two differential equations and could be applied to both isothermal and non-isothermal environments. To test and validate this model, we used published datasets containing data for the growth of Pseudomonas spp. in fish products. The results obtained after fitting the model showed that it could be effectively used to describe and predict the Pseudomonas growth curves under various temperature regimens. However, the influence of the shape parameter on the growth curve is an issue that needs further evaluation.

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BACKGROUND: The rising number of novel pathogens threatening the human population has motivated the application of mathematical modeling for forecasting the trajectory and size of epidemics. MATERIALS AND METHODS: We summarize the real-time forecasting results of the logistic equation during the 2015 Ebola challenge focused on predicting synthetic data derived from a detailed individual-based model of Ebola transmission dynamics and control. We also carry out a post-challenge comparison of two simple phenomenological models. In particular, we systematically compare the logistic growth model and a recently introduced generalized Richards model (GRM) that captures a range of early epidemic growth profiles ranging from sub-exponential to exponential growth. Specifically, we assess the performance of each model for estimating the reproduction number, generate short-term forecasts of the epidemic trajectory, and predict the final epidemic size. RESULTS: During the challenge the logistic equation consistently underestimated the final epidemic size, peak timing and the number of cases at peak timing with an average mean absolute percentage error (MAPE) of 0.49, 0.36 and 0.40, respectively. Post-challenge, the GRM which has the flexibility to reproduce a range of epidemic growth profiles ranging from early sub-exponential to exponential growth dynamics outperformed the logistic growth model in ascertaining the final epidemic size as more incidence data was made available, while the logistic model underestimated the final epidemic even with an increasing amount of data of the evolving epidemic. Incidence forecasts provided by the generalized Richards model performed better across all scenarios and time points than the logistic growth model with mean RMS decreasing from 78.00 (logistic) to 60.80 (GRM). Both models provided reasonable predictions of the effective reproduction number, but the GRM slightly outperformed the logistic growth model with a MAPE of 0.08 compared to 0.10, averaged across all scenarios and time points. CONCLUSIONS: Our findings further support the consideration of transmission models that incorporate flexible early epidemic growth profiles in the forecasting toolkit. Such models are particularly useful for quickly evaluating a developing infectious disease outbreak using only case incidence time series of the early phase of an infectious disease outbreak.

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Public health officials are increasingly recognizing the need to develop disease-forecasting systems to respond to epidemic and pandemic outbreaks. For instance, simple epidemic models relying on a small number of parameters can play an important role in characterizing epidemic growth and generating short-term epidemic forecasts. In the absence of reliable information about transmission mechanisms of emerging infectious diseases, phenomenological models are useful to characterize epidemic growth patterns without the need to explicitly model transmission mechanisms and the natural history of the disease. In this article, our goal is to discuss and illustrate the role of regularization methods for estimating parameters and generating disease forecasts using the generalized Richards model in the context of the 2014-15 Ebola epidemic in West Africa.

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This study develops the basic idea of Pütter and Bertalanffy addressing the allometric scaling of anabolism and catabolism on somatic growth dynamics. We proposed a standardized form of the Pütter-Bertalanffy equation (PBE), which is given as the extended model of Richards function, and subsequently solved it. The analytical solution of the PBE was defined by an incomplete beta function and can take a wide range of shapes in its growth curve. The mathematical behavior of PBE due to the change in parameter values was briefly discussed. Most forms of solution consistently hold the implicit functional type with respect to the variable of body size.

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