We apply a versatile growth model, whose growth rate is given by a generalised beta distribution, to describe the complex behaviour of the fatality curves of the COVID-19 disease for several countries in Europe and North America. We show that the COVID-19 epidemic curves not only may present a subexponential early growth but can also exhibit a similar subexponential (power-law) behaviour in the saturation regime. We argue that the power-law exponent of the latter regime, which measures how quickly the curve approaches the plateau, is directly related to control measures, in the sense that the less strict the control, the smaller the exponent and hence the slower the diseases progresses to its end. The power-law saturation uncovered here is an important result, because it signals to policymakers and health authorities that it is important to keep control measures for as long as possible, so as to avoid a slow, power-law ending of the disease. The slower the approach to the plateau, the longer the virus lingers on in the population, and the greater not only the final death toll but also the risk of a resurgence of infections.

The main objective of the present article is twofold: first, to model the fatality curves of the COVID-19 disease, as represented by the cumulative number of deaths as a function of time; and second, to use the corresponding mathematical model to study the effectiveness of possible intervention strategies. We applied the Richards growth model (RGM) to the COVID-19 fatality curves from several countries, where we used the data from the Johns Hopkins University database up to May 8, 2020. Countries selected for analysis with the RGM were China, France, Germany, Iran, Italy, South Korea, and Spain. The RGM was shown to describe very well the fatality curves of China, which is in a late stage of the COVID-19 outbreak, as well as of the other above countries, which supposedly are in the middle or towards the end of the outbreak at the time of this writing. We also analysed the case of Brazil, which is in an initial sub-exponential growth regime, and so we used the generalised growth model which is more appropriate for such cases. An analytic formula for the efficiency of intervention strategies within the context of the RGM is derived. Our findings show that there is only a narrow window of opportunity, after the onset of the epidemic, during which effective countermeasures can be taken. We applied our intervention model to the COVID-19 fatality curve of Italy of the outbreak to illustrate the effect of several possible interventions.

The main objective of the present paper is twofold: first, to model the fatality curves of the COVID-19 disease, as represented by the cumulative number of deaths as a function of time; and second, to use the corresponding mathematical model to study the effectiveness of possible intervention strategies. We applied the Richards growth model (RGM) to the COVID-19 fatality curves from several countries, where we used the data from the Johns Hopkins University database up to April 1, 2020. Countries selected for analysis were China, Italy, Spain, Iran, and Brazil. The RGM was shown to describe very well the fatality curves of China, which is in a late stage of the COVID-19 outbreak, as well as of Italy, Spain, and Iran, which supposedly are in the middle of the outbreak at the time of this writing. As for Brazil, which is still in the so-called exponential growth regime, we used the generalized growth model which is more appropriate for such cases. An analytic formula for the efficiency of intervention strategies within the context of the RGM is derived. Our findings show that there is only a narrow window of opportunity, after the onset of the epidemic, during which effective countermeasures can be taken. We applied our intervention model to the COVID-19 fatality curve of Italy to illustrate the effect of several possible interventions.All rights reserved. No reuse allowed without permission.(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

In this work we introduce a novel methodology to classify the dynamical stages of an epidemic, based on the different acceleration regimes of the corresponding growth curve. Our classification scheme is implemented by fitting the empirical data with a general class of mathematical growth models, from which we compute not only the growth acceleration but also its jerk and jounce (i.e., the first and second derivatives of the acceleration, respectively), thus allowing for a finer distinction of the epidemic stages. Using this methodology, we analyze the cumulative curves of deaths attributed to COVID-19 in the 26 Brazilian States and the Federal District, up until August 21, 2020. The online application ModInterv COVID-19, which automatically implements the classification scheme and which can be accessed via an internet browser or a mobile app, was used to investigate the epidemic stages in each of the Brazilian federal units. The analysis revealed that almost all states in the Northern and Northeastern regions were already in the saturation phase, meaning that the epidemic was relatively under control, whereas in all Southern states and in most states in the Midwest the epidemic was still accelerating or showed only a slight deceleration. The Southeastern region presented a great diversity of epidemic stages, with each state being found at a different stage, ranging from acceleration to saturation. It is argued that understanding this heterogeneous geographical distribution of the epidemic is relevant for public health authorities, as it may help in devising more effective strategies against the COVID-19 pandemic in a continental country like Brazil.

We apply a versatile growth model, whose rate of change is given by a generalised beta distribution, to describe the complex behaviour of the fatality curves of the COVID-19 disease for several countries in Europe and Asia. We show that the COVID-19 epidemic curves not only may present a subexponential early growth but can also exhibit a similar subexponential (power-law) behaviour in the saturation regime. We argue that the power-law exponent of the latter regime, which measures how quickly the curve approaches the plateau, is directly related to control measures, in the sense that the less strict the control, the smaller the exponent and hence the slower the diseases progresses to its end. The power-law saturation uncovered here is an important result, because it signals to health authorities that it is important to keep control measures for as long as possible, so as to avoid a slow, power-law ending of the disease. The slower the approach to the plateau, the longer the virus lingers on in the population, and the greater not only the death toll but also the risk of resurgence.

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We apply a generalized logistic growth model, with time-dependent parameters, to describe the fatality curves of the COVID-19 disease for several countries that exhibit multiple waves of infections. In the case of two waves only, the model parameters vary as a function of time according to a logistic function, whose two extreme values, i.e., for early and late times, characterize the first and second waves, respectively. For the multiple-wave model, the time-dependency of the parameters is likewise described by a multi-step logistic function with N intermediate plateaus, representing the N waves of the epidemic. We show that the theoretical curves are in excellent agreement with the empirical data for all countries considered here, namely: Brazil, Canada, Germany, Iran, Italy, Japan, Mexico, South Africa, Sweden, and the USA. The model also allows for predictions about the time of occurrence and severity of the subsequent waves. It is shown furthermore that the subsequent waves of COVID-19 can be generically classified into two main types, namely, standard and anomalous waves, according as to whether a given wave starts well after or well before the preceding one has subsided, respectively.

We apply a generalised logistic growth model, with time dependent parameters, to describe the fatality curves of the COVID-19 disease for several countries that exhibit a second wave of infections. The model parameters vary as a function of time according to a logistic function, whose two extreme values, i.e., for early and late times, characterise the first and second waves, respectively. We show that the theoretical curves are in excellent agreement with the empirical data for all cases considered. The model also allows for predictions about the time of occurrence and relative severity of the second wave, in comparison to the first wave. It is shown furthermore that the COVID-19 second waves can be generically classified in two main types, namely, standard and anomalous second waves, according as to whether the second wave starts well after or still during the first wave, respectively. We have also observed that the standard second waves tend, in their majority, to be more severe than the corresponding first wave, whereas for anomalous second waves the opposite occurs.