ModInterv COVID-19: An online platform to monitor the evolution of epidemic curves.

We present the software ModInterv as an informatics tool to monitor, in an automated and user-friendly manner, the evolution and trend of COVID-19 epidemic curves, both for cases and deaths. The ModInterv software uses parametric generalized growth models, together with LOWESS regression analysis, to fit epidemic curves with multiple waves of infections for countries around the world as well as for states and cities in Brazil and the USA. The software automatically accesses publicly available COVID-19 databases maintained by the Johns Hopkins University (for countries as well as states and cities in the USA) and the Federal University of Viçosa (for states and cities in Brazil). The richness of the implemented models lies in the possibility of quantitatively and reliably detecting the distinct acceleration regimes of the disease. We describe the backend structure of software as well as its practical use. The software helps the user not only to understand the current stage of the epidemic in a chosen location but also to make short term predictions as to how the curves may evolve. The app is freely available on the internet (http://fisica.ufpr.br/modinterv), thus making a sophisticated mathematical analysis of epidemic data readily accessible to any interested user.

Fock-space approach to stochastic susceptible-infected-recovered models.

We investigate the stochastic susceptible-infected-recovered (SIR) model of infectious disease dynamics in the Fock-space approach. In contrast to conventional SIR models based on ordinary differential equations for the subpopulation sizes of S, I, and R individuals, the stochastic SIR model is driven by a master equation governing the transition probabilities among the system's states defined by SIR occupation numbers. In the Fock-space approach the master equation is recast in the form of a real-valued Schrödinger-type equation with a second quantization Hamiltonian-like operator describing the infection and recovery processes. We find exact analytic expressions for the Hamiltonian eigenvalues for any population size N. We present small- and large-N results for the average numbers of SIR individuals and basic reproduction number. For small N we also obtain the probability distributions of SIR states, epidemic sizes and durations, which cannot be found from deterministic SIR models. Our Fock-space approach to stochastic SIR models introduces a powerful set of tools to calculate central quantities of epidemic processes, especially for relatively small populations where statistical fluctuations not captured by conventional deterministic SIR models play a crucial role.

ModInterv: An automated online software for modeling epidemics.

The COVID-19 pandemic has proven the importance of mathematical tools to understand the evolution of epidemic outbreaks and provide reliable information to the general public and health authorities. In this perspective, we have developed ModInterv, an online software that applies growth models to monitor the evolution of the COVID-19 epidemic in locations chosen by the user among countries worldwide or states and cities in the USA or Brazil. This paper describes the software capabilities and its use both in recent research works and by technical committees assisting government authorities. Possible applications to other epidemics are also briefly discussed.

Power law behaviour in the saturation regime of fatality curves of the COVID-19 pandemic.

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.

Fock-space methods for diffusion: Capturing volume exclusion via fermionic statistics.

Volume exclusion and single-file diffusion play an important role at very small scales, such as those associated with molecular machines, ion channels, and transport in zeolites, while introducing fundamental differences compared to Brownian motion, such as changes to the power-law relation between the mean square displacement and time. In this work we map the chemical master equation for excluded diffusion onto a Schrödinger equation via annihilation and creation ladder operators with fermionic statistics, together with linear and symbolic algebra with the resulting Fock-space representation to describe the effect of volume-exclusion processes in finite one-dimensional chains. We contrast the dynamics with the nonexclusive (bosonic) diffusion for crowded, intermediate, and dilute particle populations. Motivated by shuttling in molecular machines, we proceed to investigate differences in exit time distributions introduced by volume exclusion, incorporating the presence of transport bias. More generally, this study demonstrates how one can analyze volume-excluded transport for small stochastic systems, without the need for stochastic simulation and ensemble averaging, simply by considering anticommutation relations and fermionic statistics in a Fock-space representation of the stochastic dynamics.

Situation of the Covid-19 Epidemic in Brazil in August 2020: Most Northern and Northeastern States in a Saturation Phase, while Southern States in Acceleration or Slight Deceleration / Situação da Epidemia de Covid-19 no Brasil em Agosto de 2020: Maioria do Estados do Norte-Nordeste em Saturação e Estados do Sul em Aceleração ou Leve Desaceleração

In this Technical Note we analyze the cumulative curves of deaths attributed to Covid-19 in the 26 Brazilian states and the Federal District until August 21, 2020. Mathematical growth models implemented by the ModInterv Covid-19 application, which can be accessed via internet browser or via a mobile app available at the Google Play Store, were used to investigate at which stage the epidemic is in each of these entities of the Federation. The analysis revealed that almost all states in the Northern and Northeastern regions are in the saturation phase, when the epidemic is relatively under control, while in all Southern states and in most states in the Midwest the epidemic is still accelerating or shows only a slight deceleration. The Southeastern region presents a great diversity of epidemic stages, with each state at a different stage, ranging from acceleration to saturation.

Nesta Nota Técnica nós analisamos as curvas acumuladas de mortes atribuídas à Covid-19 nos 26 estados e Distrito Federal até o dia 21 de agosto de 2020. Foram utilizados modelos matemáticos de crescimento implementados pelo aplicativo ModInterv Covid-19, que pode ser acessado via internet (http://fisica.ufpr.br/modinterv) ou através de aplicativo para celular disponível na Play Store (https://play.google.com/store/apps/details?id=com.tanxe.covid_19), para investigar em qual fase da epidemia cada um dessas unidades da federação se encontra. A análise revelou que quase todos os estados das Regiões Norte e Nordeste encontram-se em uma fase de saturação, quando a epidemia está relativamente sob controle, ao passo que em todos os estados do Sul e a maioria dos estados do Centro-Oeste a epidemia ainda está em aceleração ou apresenta apenas uma leve desaceleração. A Região Sudeste apresenta uma grande diversidade de estágios da epidemia, com cada estado em um estágio diferente, indo de acelerado à saturação.

Modelling fatality curves of COVID-19 and the effectiveness of intervention strategies.

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.

OVERVIEW OF THE COVID-19 IN NORTHEAST BRAZIL: ANALYSES AND FORECASTS VIA GROWTH MODELS / PANORAMA DA COVID-19 NO NORDESTE BRASILEIRO: ANÁLISES E PREVISÕES VIA MODELOS DE CRESCIMENTO

The Covid-19 pandemic, caused by the new coronavirus (SARS-CoV-2), is one of the gravest public health crises the world has ever faced. In this context, it is important to have effective models to describe the different stages of the epidemic, in order to offer guidance to the competent authorities regarding the adoption of public policies to contain and control the pandemic. In this work, we present a novel method to analyze epidemic curves based on growth models, using as examples the cumulative curves of deaths attributed to Covid-19 for the states of the Northeastern Region of Brazil. Depending on the case, the q-exponential model, the Richards model or the generalized Richards model were used to make the numerical fits of the respective empirical curves. The models used here describe very well the empirical curves of all the Northeastern Brazilian States, thus allowing a more precise diagnosis of the stage of the epidemic in each of the States.  Among them, only the state of Paraíba is still in the early growth phase, when the epidemic curve does not yet have an inflexion point, being in this case better described by the q-exponential model.  The other states were better described either by the Richards model or by its generalized version. The Richards model, in particular, was able to identify with reasonable reliability the emergence of the inflexion point for states that only recently have reached this stage of the epidemic, such as Piauí, Rio Grande do Norte and Sergipe. This model is also able to predict when the inflection is about to occur, as is the case in Bahia. The generalized Richards model, in turn, has proved more appropriate to describe epidemic curves in states that are in a more developed phase of the epidemic, such as Ceará and Pernambuco, when the epidemic curves already show a more consolidated trend of saturation toward the plateau.

A pandemia da Covid-19, causada pelo novo coronavírus (SARS-CoV-2), é uma das maiores crises de saúde pública que o mundo já enfrentou. Nesse contexto, é importante ter modelos eficazes para descrever os diferentes estágios da evolução da epidemia, a fim de orientar as autoridades competentes na adoção de políticas públicas para o enfrentamento e controle da pandemia. No presente trabalho, nós propomos um novo método de análise de curvas epidêmicas com base na seleção criteriosa de modelos de crescimento, tomando como exemplo as curvas acumuladas de óbitos atribuídos à Covid-19 para os estados da região Nordeste do Brasil. A depender do caso, foram utilizados o modelo q-exponencial, o modelo de Richards ou o modelo generalizado de Richards para fazer o ajuste numérico das respectivas curvas empíricas. Verificou-se que os modelos utilizados descrevem muito bem as curvas empíricas de todos os estados do Nordeste, permitindo assim diagnosticar mais precisamente o estágio da epidemia em cada um dos estados. Dentre eles, apenas o estado da Paraíba ainda encontra-se na fase inicial de crescimento, quando a curva epidêmica ainda não apresenta um ponto de inflexão, sendo nesse caso melhor descrita pelo modelo q-exponencial. Os demais estados foram mais bem descritos ou pelo modelo de Richards ou por sua versão generalizada. O modelo de Richards, em particular, foi capaz de identificar com razoável confiabilidade o surgimento do ponto de inflexão para os estados que só recentemente alcançaram esse estágio da epidemia, como foi o caso do Piauí, Rio Grande do Norte e Sergipe. Esse modelo também é capaz de prever quando a inflexão está prestes a acontecer, como é o caso da Bahia. O modelo generalizado de Richards, por sua vez, mostrou-se mais apropriado para descrever curvas epidêmicas de estados que estão em uma fase mais desenvolvida da epidemia, como Ceará e Pernambuco, quando as curvas epidêmicas já apresentam uma tendência mais consolidada de saturação em direção ao platô.

Analysis of Covid-19 epidemic curves via generalized growth models: Case study for the cities of Recife and Teresina / Análise de curvas epidêmicas da Covid-19 via modelos generalizados de crescimento: Estudo de caso para as cidades de Recife e Teresina

Introduction: The Covid-19 pandemic is one of the biggest public health crises the world has ever faced. In this context, it is important to have effective models to describe the different stages of the epidemic's evolution in order to guide the authorities in taking appropriate measures to fight the disease. Objective: To present an analysis of epidemic curves of Covid-19 based on phenomenological growth models, with applications to the curves for the cumulative numbers of confirmed cases of infection by the novel coronavirus (Sars-Cov-2) and deaths attributed to the disease (Covid-19) caused by the virus, for the Brazilian cities of Recife and Teresina. Methods: The Richards generalized model and the generalized growth model were used to make the numerical fits of the respective empirical curves. Results: The models used described very well the empirical curves against which they were tested. In particular, the generalized Richards model was able to identify the appearance of the inflexion point in the cumulative curves, which in turn represents the peak of the respective daily curves. A brief discussion is also presented on the relationship between the fitting parameters obtained from the model and the mitigation measures adopted in each of the municipalities considered. Conclusions: The generalized Richards model proved to be very effective in describing epidemic curves of Covid-19 and estimating important epidemiological parameters, such as the time of the peak of the curve for daily cases and deaths, thus allowing a practical and efficient monitoring of the epidemic evolution.

Introdução: A pandemia da Covid-19 é uma das maiores crises de saúde pública que o mundo já enfrentou. Nesse contexto, é importante ter modelos eficazes para descrever os diferentes estágios da evolução da epidemia, a fim de orientar as autoridades competen- tes na adoção de políticas públicas para o enfrentamento da mesma. Objetivo: Apresentar uma análise de curvas epidêmicas com base em modelos fenomenológicos de crescimento, tomando como exemplo as curvas acumuladas de casos confirmados de infecção pelo novo coronavírus (Sars-Cov-2) e de óbitos atribuídos à doença (Covid-19) causada pelo vírus, para as cidades do Recife e Teresina. Métodos: Foram utilizados o modelo generalizado de Richards e o modelo de crescimento generalizado para fazer o ajuste numérico das respectivas curvas empíricas. Resultados: Verificou-se que os modelos utilizados descrevem muito bem as curvas empíricas em que foram testados. Em particular, o modelo generalizado de Richards é capaz de identificar com razoável confiabilidade o surgimento do ponto de infle- xão nas curvas acumuladas, o qual corresponde ao ponto de máximo das respectivas curvas diárias. Apresenta-se ainda uma breve discussão sobre a relação entre os parâmetros obtidos no ajuste do modelo e as medidas de mitigação adotadas para retardar a propagação da Covid-19 em cada um dos municípios considerados. Conclusões: O modelo generalizado de Richards mostrou-se bastante eficaz para descrever curvas epidêmicas da Covid-19 e es- timar parâmetros epidemiológicos importantes, como o pico das curvas de casos e óbitos diários, permitindo assim realizar de modo prático e eficiente o monitoramento da evolução da epidemia.

Fighting the coronavirus: the window of opportunity for non-pharmacological interventions is narrow / La lucha contra el Coronavirus: La ventana para intervenciones no farmacológias es estrecha / Combate ao Coronavírus: a Janela para Intervenções não Farmacológicas é Estreita

In this technical note, we present a brief discussion of the main results reported in our paper "Modelling fatality curves of COVID-19 and the effectiveness of intervention strategies", MedRxiv/2020/051557 (DOI:10.1101/2020.04.02.20051557). In that paper, we applied the Richards growth model (RGM) to describe the fatality curves of the COVID-19 disease for countries that were, up to April 1, 2020, near the end or in an intermediary phase of the outbreak, such as China, Italy, Spain, and Iran. We also analyzed data from Brazil, which was still in the early growth regime, and so we used an alternative model (the generalized growth model) that is more appropriate for the early stages of the epidemic. We also used the RGM to study the effectiveness of possible intervention strategies and, within this context, we derived an analytic formula for the efficiency of non-pharmaceutical intervention strategies. Our findings show that there is only a narrow window, after the onset of the epidemic, during which effective countermeasures can be taken. Here we present a brief overview of the results obtained in the aforementioned paper, but we use more recent data to update our analysis. For more details, we refer the reader to the original article.

En esta nota técnica presentamos una breve discusión de los principales resultados de nuestro artículo "Modelado de curvas de mortalidad de COVID-19 y la efectividad de las estrategias de intervención", MedRxiv / 2020/051557 (DOI: 10.1101 / 2020.04.02.20051557). En este artículo, aplicamos el modelo de crecimiento de Richards para describir las curvas de mortalidad de COVID-19 para países que estaban, hasta el 4/1/2020, cerca del final o en la fase intermedia de la epidemia, como China, Italia, España e Irán. También analizamos datos de Brasil, aunque todavía estaba en las primeras etapas de la epidemia. Para este caso, utilizamos un modelo alternativo, el modelo de crecimiento generalizado, que es más apropiado para esa fase. También utilizamos el modelo de Richards para estudiar la efectividad de las posibles estrategias de intervención y, en este contexto, derivamos una fórmula analítica para la eficiencia de las estrategias de intervención no farmacológicas. Nuestros resultados muestran que solo hay una ventana estrecha, después de que comienza el brote, durante la cual se pueden tomar intervenciones no farmacológicas efectivas para contener la epidemia. En esta nota, también presentamos algunos resultados originales para las curvas de fatalidad de Italia y Brasil, actualizadas con datos hasta el 08/04/2020, además de una breve descripción general del trabajo mencionado anteriormente. Para más detalles, remitimos al lector al artículo original.

Nesta nota técnica apresentamos uma breve discussão dos principais resultados do nosso artigo "Modelling fatality curves of COVID-19 and the effectiveness of intervention strategies", MedRxiv/2020/051557 (DOI:10.1101/2020.04.02.20051557). Nesse artigo, aplicamos o modelo de crescimento de Richards para descrever as curvas de fatalidade da COVID-19 para países que estavam, até 01/04/2020, ou perto do fim ou na fase intermediária da epidemia, como a China, Itália, Espanha e Irã. Também analisamos dados do Brasil, embora ainda estivesse na fase inicial da epidemia, mas nesse caso usamos um modelo alternativo­o modelo de crescimento generalizado­que é mais apropriado para essa fase. Utilizamos ainda o modelo de Richards para estudar a eficácia de possíveis estratégias de intervenção e, nesse contexto, derivamos uma fórmula analítica para a eficiência das estratégias de intervenção não farmacológicas. Os nossos resultados mostram que existe apenas uma estreita janela, após o início do surto, durante a qual intervenções efetivas não farmacológicas podem ser tomadas para conter a epidemia. Nesta nota, apresentamos ainda alguns resultados originais para as curvas de fatalidade da Itália e do Brasil, atualizados com dados até 08/04/2020, além de uma breve descrição geral do trabalho acima mencionado. Para mais detalhes, remetemos o leitor para o artigo original.