Generalized Richards Model Research Articles

Generalized Richards model of slow sand filtration to remove lead pollutant from polluted waters: study of velocity effect

In Morocco, numerous regions face water scarcity. To address water shortages in the arid Marrakech region, agricultural reuse of wastewater can be a viable solution, provided that accessible and cost-effective methods, such as slow sand filtration, are employed to reduce pollutant levels. In the vicinity of the Mine Draa Lasfar, wastewater discharged into the Tensift River has led to significant contamination by metallic trace elements, including lead (Pb). This contamination spreads from the water to the soil, plant cover, and living organisms, with severe Pb contamination observed in children, leading to serious neurological diseases. Addressing this critical issue, our study explores the application of slow sand filtration for water treatment. This method is reliable and environmentally friendly, particularly effective at lower water velocities, ensuring optimal contaminant removal. We introduced the Generalized Richards model, a versatile mathematical tool, which outperformed the standard logistic model by providing a more accurate fit to our experimental data. In conclusion, slow sand filtration, when implemented in areas like Mine Draa Lasfar, offers a promising solution for mitigating metallic trace elements contamination in water sources, with the Generalized Richards model enhancing our understanding and optimization of the filtration process.

A fractional-order generalized Richards growth model and its implementation to COVID-19 data

In this article, we include the memory effect in the generalized Richards model (GRM), which is proposed in the form of a fractional-order GRM. The fractional-order GRM is developed by replacing the first-order derivative in the GRM with a fractional-order derivative and then taking care of model homogeneity of dimension. We consider two fractional-order differential operators: Caputo and Atangana–Baleanu in the Caputo sense (ABC). The proposed fractional-order models are then implemented to fit the COVID-19 data in East Java, Indonesia from March 25 until October 31, 2020. The fitting is performed by minimizing the sum of the squared residual between the numerical solutions of each fractional-order model and the daily data of a cumulative number of COVID-19 cases. The numerical solutions of both fractional-order models are determined by the predictor-corrector method. The performance of the two fractional models is measured by two performance metrics: coefficient of determination (R 2) and root mean square error (RMSE). By considering that the order of fractional derivative as an extra degree of freedom, we perform data fitting for several orders of fractional derivative and evaluate the two performance metrics. It is observed that the fractional-order model with the ABC operator generally has the best performance in calibrating and forecasting both the cumulative number of COVID-19 cases and daily new cases of COVID-19.

Probabilistic projections of granular energy technology diffusion at subnational level.

Projections of granular energy technology diffusion can support decision-making on climate mitigation policies and infrastructure investments. However, such projections often do not account for uncertainties and have low spatial resolution. S-curve models of technology diffusion are widely used to project future installations, but the results of the different models can vary significantly. We propose a method to create probabilistic projections of granular energy technology diffusion at subnational level based on historical time series data and testing how various projection models perform in terms of accuracy and uncertainty to inform the choice of models. As a case study, we investigate the growth of solar photovoltaics, heat pumps, and battery electric vehicles at municipality level throughout Switzerland in 2000-2021 (testing) and until 2050 (projections). Consistently for all S-curve models and technologies, we find that the medians of the probabilistic projections anticipate the diffusion of the technologies more accurately than the respective deterministic projections. While accuracy and probabilistic density intervals of the models vary across technologies, municipalities, and years, Bertalanffy and two versions of the generalized Richards model estimate the future diffusion with higher accuracy and sharpness than logistic, Gompertz, and Bass models. The results also highlight that all models come with trade-offs and eventually a combination of models with weights is needed. Based on these weighted probabilistic projections, we show that, given the current dynamics of diffusion in solar photovoltaics, heat pumps, and battery electric vehicles in Switzerland, the net-zero emissions target would be missed by 2050 with high certainty.

Forecasting COVID-19 Epidemic in Spain and Italy Using A Generalized Richards Model with Quantified Uncertainty

The Richards model and its generalized version are deterministic models that are often implemented to fit and forecast the cumulative number of infective cases in an epidemic outbreak. In this paper we employ a generalized Richards model to predict the cumulative number of COVID-19 cases in Spain and Italy, based on available epidemiological data. To quantify uncertainty in the parameter estimation, we use a parametric bootstrapping approach to construct a 95% confidence interval estimation for the parameter model. Here we assume that the time series data follow a Poisson distribution. It is found that the 95% confidence interval of each parameter becomes narrow with the increasing number of data. All in all, the model predicts daily new cases of COVID-19 reasonably well during calibration periods. However, the model fails to produce good forecasts when the amount of data used for parameter estimations is not sufficient. Based on our parameter estimates, it is found that the early stages of COVID-19 epidemic, both in Spain and in Italy, followed an almost exponentially growth. The epidemic peak in Spain and Italy is respectively on 2 April 2020 and 28 March 2020. The final sizes of cumulative number of COVID-19 cases in Spain and Italy are forecasted to be at 293220 and 237010, respectively.

Modeling the Epidemic Growth of Preprints on COVID-19 and SARS-CoV-2

The response of the scientific community to the global health emergency caused by the COVID-19 pandemic has produced an unprecedented number of manuscripts in a short period of time, the vast majority of which have been shared in the form of preprints posted on online preprint repositories before peer review. This surge in preprint publications has in itself attracted considerable attention, although mostly in the bibliometrics literature. In the present study we apply a mathematical growth model, known as the generalized Richards model, to describe the time evolution of the cumulative number of COVID-19 related preprints. This mathematical approach allows us to infer several important aspects concerning the underlying growth dynamics, such as its current stage and its possible evolution in the near future. We also analyze the rank-frequency distribution of preprints servers, ordered by the number of COVID-19 preprints they host, and find that it follows a power law in the low rank (high frequency) region, with the high rank (low frequency) tail being better described by a q-exponential function. The Zipf-like law in the high frequency regime indicates the presence of a cumulative advantage effect, whereby servers that already have more preprints receive more submissions.

The performance of phenomenological models in providing near-term Canadian case projections in the midst of the COVID-19 pandemic: March – April, 2020

BackgroundThe COVID-19 pandemic has had an unprecedented impact on citizens and health care systems globally. Valid near-term projections of cases are required to inform the escalation, maintenance and de-escalation of public health measures, and for short-term health care resource planning. MethodsNear-term case and epidemic growth rate projections for Canada were estimated using three phenomenological models: the logistic model, Generalized Richard’s model (GRM) and a modified Incidence Decay and Exponential Adjustment (m-IDEA) model. Throughout the COVID-19 epidemic in Canada, these models have been validated against official national epidemiological data on an ongoing basis. ResultsThe best-fit models estimated that the number of COVID-19 cases predicted to be reported in Canada as of April 1, 2020 and May 1, 2020 would be 11,156 (90 % prediction interval: 9,156−13,905) and 54,745 (90 % prediction interval: 54,252−55,239). The three models varied in their projections and their performance over the first seven weeks of their implementation. Both the logistic model and GRM under-predicted cases reported a week following the projection date in nearly all instances. The logistic model performed best at the early stages, the m-IDEA model performed best at the later stages, and the GRM performed most consistently during the full period assessed. ConclusionsAll three models have yielded qualitatively comparable near-term forecasts of cases and epidemic growth for Canada. Under or over-estimation of projected cases and epidemic growth by these models could be associated with changes in testing policies and/or public health measures. Simple forecasting models can be invaluable in projecting the changes in trajectory of subsequent waves of cases to provide timely information to support the pandemic response.

Comparison of phenomenological growth models in predicting cumulative number of COVID-19 cases in East Java Province, Indonesia

In this paper, five phenomenological (Richards, a generalized Richards, Blumberg, Tsoularis & Wallace, and Gompertz) models are implemented to predict the cumulative number of COVID-19 cases. The five phenomenological models are in the form of ordinary differential equations with a few number of model parameters. The model parameters of each model were calibrated by fitting the model with the reported cumulative number of COVID-19 cases in East Java Province from March 25 until October 31, 2020 via nonlinear least square method. We compare the performance of the five phenomenological models by measuring four performance metrics, namely the root mean square error (RMSE), the mean absolute error (MAE), the coefficient of determination (R2) and the Akaike information criterion (AIC). When calibrating the cumulative number of cases, the five models perform very well, which are indicated by their high coefficient of determination (R2>0.999). However, a comparison of the four- performance metrics shows that Tsoularis & Wallace performed the best followed by a generalized Richards model. The prediction for the final size of the COVID-19 epidemic in East Java according to the Tsoularis & Wallace model is K=78 002. Both Richards and Gompertz models tend to underestimate the final size of the epidemic, while the Blumberg model tends to overestimate. The five models estimate the peak of the COVID-19 epidemic in East Java has been occurred on August 13-14, 2020. Using the predicted cumulative number of cases, we determine the daily new cases of COVID-19 in East Java. Based on the four-performance metrics, it appears that the five phenomenological models predict new daily cases of COVID-19 equally well.

Generalized logistic growth modeling of the COVID-19 outbreak: comparing the dynamics in the 29 provinces in China and in the rest of the world

Started in Wuhan, China, the COVID-19 has been spreading all over the world. We calibrate the logistic growth model, the generalized logistic growth model, the generalized Richards model and the generalized growth model to the reported number of infected cases for the whole of China, 29 provinces in China, and 33 countries and regions that have been or are undergoing major outbreaks. We dissect the development of the epidemics in China and the impact of the drastic control measures both at the aggregate level and within each province. We quantitatively document four phases of the outbreak in China with a detailed analysis on the heterogeneous situations across provinces. The extreme containment measures implemented by China were very effective with some instructive variations across provinces. Borrowing from the experience of China, we made scenario projections on the development of the outbreak in other countries. We identified that outbreaks in 14 countries (mostly in western Europe) have ended, while resurgences of cases have been identified in several among them. The modeling results clearly show longer after-peak trajectories in western countries, in contrast to most provinces in China where the after-peak trajectory is characterized by a much faster decay. We identified three groups of countries in different level of outbreak progress, and provide informative implications for the current global pandemic.

Parameters Estimation of Generalized Richards Model for COVID-19 Cases in Indonesia Using Genetic Algorithm

In this research, genetic algorithm was implemented to estimate parameters in generalized Richards model by adjusting COVID-19 case data in Indonesia. Data collected were the daily new cases and cumulative number of COVID-19 case in Indonesia from early March to early June 2020, that was reported by databoks.katadata.co.id. The best pair of parameters was selected based on the lowest cost function value, determined from the distance between data with estimated model and real data. Next, model with estimated parameters is used to predict new cases and cumulative cases for upcoming days. Numerical simulations were carried out so that the peaks and ends of the COVID-19 pandemic can be seen easily.

Generalized Richards model for predicting COVID-19 dynamics in Saudi Arabia based on particle swarm optimization Algorithm.

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.

Some fractal thoughts about the COVID-19 infection outbreak

Some ideas are presented about a geometric motivation of the apparent capacity of generalized logistic equations to describe the outbreak of quite many epidemics, possibly including that of the COVID-19 infection. This interpretation pivots on the complex, possibly fractal, structure of the locus describing the “contagion event set”, and on what can be learnt from the models of trophic webs with “herd behaviour”.Under the hypothesis that the total number of cases, as a function of time, is fitted by a solution of the Generalized Richards Model, it is argued that the exponents appearing in that differential equation, usually determined empirically, represent the geometric signature of the non-space filling, network-like locus on which contagious contacts take place.

Forecasting the 2001 Foot-and-Mouth Disease Epidemic in the UK

Near real-time epidemic forecasting approaches are needed to respond to the increasing number of infectious disease outbreaks. In this paper, we retrospectively assess the performance of simple phenomenological models that incorporate early sub-exponential growth dynamics to generate short-term forecasts of the 2001 foot-and-mouth disease epidemic in the UK. For this purpose, we employed the generalized-growth model (GGM) for pre-peak predictions and the generalized-Richards model (GRM) for post-peak predictions. The epidemic exhibits a growth-decelerating pattern as the relative growth rate declines inversely with time. The uncertainty of the parameter estimates (r{text{ and }}p) narrows down and becomes more precise using an increasing amount of data of the epidemic growth phase. Indeed, using only the first 10–15 days of the epidemic, the scaling of growth parameter (p) displays wide uncertainty with the confidence interval for p ranging from values ~ 0.5 to 1.0, indicating that less than 15 epidemic days of data are not sufficient to discriminate between sub-exponential (i.e., p < 1) and exponential growth dynamics (i.e., p = 1). By contrast, using 20, 25, or 30 days of epidemic data, it is possible to recover estimates of p around 0.6 and the confidence interval is substantially below the exponential growth regime. Local and national bans on the movement of livestock and a nationwide cull of infected and contiguous premises likely contributed to the decelerating trajectory of the epidemic. The GGM and GRM provided useful 10-day forecasts of the epidemic before and after the peak of the epidemic, respectively. Short-term forecasts improved as the model was calibrated with an increasing length of the epidemic growth phase. Phenomenological models incorporating generalized-growth dynamics are useful tools to generate short-term forecasts of epidemic growth in near real time, particularly in the context of limited epidemiological data as well as information about transmission mechanisms and the effects of control interventions.

A primer on stable parameter estimation and forecasting in epidemiology by a problem-oriented regularized least squares algorithm.

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.

Improving epidemic size prediction through stable reconstruction of disease parameters by reduced iteratively regularized Gauss–Newton algorithm

Abstract Classical compartmental epidemic models of infectious diseases track the dynamic transition of individuals between different epidemiological states or risk groups. Reliable quantification of various transmission pathways in these models is paramount for optimal resource allocation and successful design of public health intervention programs. However, with limited epidemiological data available in the case of an emerging disease, simple phenomenological models based on a smaller number of parameters can play an important role in our quest to make forward projections of possible outbreak scenarios. In this paper, we employ the generalized Richards model for stable numerical estimation of the epidemic size (defined as the total number of infections throughout the epidemic) and its turning point using case incidence data of the early epidemic growth phase. The minimization is carried out by what we call the Reduced Iteratively Regularized Gauss–Newton (RIRGN) algorithm, a problem-oriented numerical scheme that takes full advantage of the specific structure of the non-linear operator at hand. The convergence analysis of the RIRGN method is suggested and numerical simulations are conducted with real case incidence data for the 2014–15 Ebola epidemic in West Africa. We show that the proposed RIRGN provides a stable algorithm for early estimation of turning points using simple phenomenological models with limited data.

Using phenomenological models for forecasting the 2015 Ebola challenge

BackgroundThe 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 methodsWe 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. ResultsDuring 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. ConclusionsOur 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.

Using Phenomenological Models to Characterize Transmissibility and Forecast Patterns and Final Burden of Zika Epidemics.

Background: The World Health Organization declared the ongoing Zika virus (ZIKV) epidemic in the Americas a Public Health Emergency of International Concern on February 1, 2016. ZIKV disease in humans is characterized by a “dengue-like” syndrome including febrile illness and rash. However, ZIKV infection in early pregnancy has been associated with severe birth defects, including microcephaly and other developmental issues. Mechanistic models of disease transmission can be used to forecast trajectories and likely disease burden but are currently hampered by substantial uncertainty on the epidemiology of the disease (e.g., the role of asymptomatic transmission, generation interval, incubation period, and key drivers). When insight is limited, phenomenological models provide a starting point for estimation of key transmission parameters, such as the reproduction number, and forecasts of epidemic impact.Methods: We obtained daily counts of suspected Zika cases by date of symptoms onset from the Secretary of Health of Antioquia, Colombia during January-April 2016. We calibrated the generalized Richards model, a phenomenological model that accommodates a variety of early exponential and sub-exponential growth kinetics, against the early epidemic trajectory and generated predictions of epidemic size. The reproduction number was estimated by applying the renewal equation to incident cases simulated from the fitted generalized-growth model and assuming gamma or exponentially-distributed generation intervals derived from the literature. We estimated the reproduction number for an increasing duration of the epidemic growth phase.Results: The reproduction number rapidly declined from 10.3 (95% CI: 8.3, 12.4) in the first disease generation to 2.2 (95% CI: 1.9, 2.8) in the second disease generation, assuming a gamma-distributed generation interval with the mean of 14 days and standard deviation of 2 days. The generalized-Richards model outperformed the logistic growth model and provided forecasts within 22% of the actual epidemic size based on an assessment 30 days into the epidemic, with the epidemic peaking on day 36.Conclusion: Phenomenological models represent promising tools to generate early forecasts of epidemic impact particularly in the context of substantial uncertainty in epidemiological parameters. Our findings underscore the need to treat the reproduction number as a dynamic quantity even during the early growth phase, and emphasize the sensitivity of reproduction number estimates to assumptions on the generation interval distribution.

49 Case-control design using multivariate generalization of the cox model: Application to central catheters related bacteremia

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.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.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.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.