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      Epidemiology of COVID-19 in Brazil: using a mathematical model to estimate the outbreak peak and temporal evolution

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          Introduction Coronavirus disease 2019 (COVID-19) pandemic is vastly spreading worldwide, with more than 7.4 million confirmed cases and 418,294 deaths (5.64% mortality rate) on 12 June 2020 (WHO, 2020). Understanding the COVID-19 behaviour is fundamental for combating the dissemination of the virus while an effective vaccine or medicine is not available. At this moment, preventive measures and social distancing are the most efficient strategies to combat the COVID-19 pandemic [1]. Because this pandemic is unprecedented, models to predict the outbreak are scarce, mathematically complex [2] or not available. In Brazil, the COVID-19 pandemic started on 26 February 2020, and rapidly spread into the country, starting in Sao Paulo and Rio de Janeiro states and disseminating to other Brazilian states a few weeks later. Three months after the first COVID-19 case, several Brazilian states already are in critical condition, with their health systems overloaded, most of them above 80% occupancy or even collapsing. Nowadays, Brazil is considered the epidemic centre of Latin America, occupying the second place in total number of cases and more recently in total number of deaths. Situation in Brazil is critical and authorities require a general scenario and the developing trend of the Covid-19. By using a simple mathematical model described previously [3], we present herein the Sars-Cov-2 epidemiology in Brazil and in the five most affected Brazilian states: Sao Paulo, Rio de Janeiro, Amazonas, Ceara, and Pernambuco. We are able to predict the outbreak peak and the decreasing tendency of the pandemic with this model. Our results are important for the comprehension of the COVID-19 outbreak, estimation of the affected population size, and temporal evolution of the disease. This knowledge may help Brazilian authorities make critical decisions and direct new strategies for controlling the COVID-19 pandemic, as well as predict when life may be safely returned to normal, at least in part. The mathematical model The methodology of exponential decay proposed by Tang and Wang [3] is applied. The infected numbers, including cumulative number and daily change number, were collected from several publicly available online Brazilian sources. Then, the decay factors for each location were obtained by simulating the growth rate. Finally, the predictions of cumulative number and daily change number were calculated and the figures were plotted. Analysis and discussion of the results While no specific vaccine or treatment against COVID-19 is available, the best strategy to combat the disease is preventive measures and social distancing. The Brazilian government is making important decisions to avoid the Sars-CoV-2 spread. Different mechanisms and levels of social distancing have been imposed, including 1.5 metres of distance among people (in lines, public spaces, and transportation), quarantine, and lastly, lockdown. Most Brazilian cities adopted quarantine with only essential services allowed to work. In some critical cities, lockdown was imposed. Other preventive measures include sanitation (hands, personal objects, and public spaces), face mask usage, complete isolation of infected people, and flu vaccination. Nowadays, according to our results, Brazil and the five analysed Brazilian states are crossing by the worst moment of the COVID-19 epidemic and any easing of the preventive measures and/or social distancing will probably have a negative impact on the disease curve. Sao Paulo state, the epicentre of the COVID-19 in Brazil, was the first state to adopt several measures to avoid the fast virus spreading [4]. The occupation rate of intensive care units is ∼70% in state and 89% in Sao Paulo city (the main city and the capital of the Sao Paulo state). Quarantine started on 24 March and it was prorogated until 28 June in the last update. Transmissibility index or reproduction number (R 0) was ∼2.2 before quarantine (17–23 March), dropping to ∼1.4 after one month (14–20 April) and to ∼1.2 after 2 months (12–18 May). Isolation rate, evaluated by monitoring cell phone mobility, was 27.8%, 51.7%, and 48.8% before and after one, and two months of quarantine, respectively. Mandatory use of face masks in any public space was established on 7 May 2020. Our mathematical analysis shows that Sao Paulo state is at the peak of daily new cases (∼4000 daily cases), which would persist for some days before starting to drop. Daily new cases would drop to ∼3000 by the end of 30 June, to ∼1300 by the end of July, and to ∼450 by the end of August. Total confirmed cases would reach ∼225,000 by the end of June, ∼287,000 by the end of July, and ∼311,000 by the end of August (Figure 1). Figure 1. (A), (D), (G), (J), (M), (P): Cumulative COVID-19 cases; (B), (E), (H), (K), (N), (Q): Growth rate of COVID-19; and (C), (F), (I), (L), (O), (R): Daily change COVID-19 cases in Brazil, Sao Paulo, Rio de Janeiro, Amazonas, Ceara, and Pernambuco, respectively. The first case of Covid-19 in Rio de Janeiro was confirmed on 5 March 2020. Nowadays, Rio de Janeiro is the second state in the number of positive cases and deaths for COVID-19. Main affected cities in Rio de Janeiro state are Rio de Janeiro (capital of the state) with 40,504 confirmed cases and Niteroi with 4413 cases. Occupation rate of intensive care units is ∼90% in state and 83% in Rio de Janeiro city. Quarantine started on 24 March and it was prorogated until 10 June in the last update. Isolation rate was 41%, 53.2%, and 57.4% before and after one and two months of quarantine, respectively. Use of face masks is mandatory in any public space since 23 April 2020. Reproduction number (R 0) was around 4.5 before quarantine and decreased to 1.7 on 19 May. According to our analysis, Rio de Janeiro state is on the peak of daily new cases (∼2000 daily cases) since the last week of May it would persist for around two weeks before starting to drop. Daily new cases would drop to ∼1600, to ∼700, and to ∼250 by the end of June, July, and August, respectively. Total confirmed cases would reach ∼110,500 ∼143,700, and ∼156,400 by the end of June, July, and August, respectively. Amazonas state is located in North of Brazil and it is crossing for critical situation due to the fast spreading of COVID-19. The first case of COVID-19 in state was recorded on 13 March 2020, in a 39-year-old woman who returned from England. Occupation rate of intensive care units is 86% in state and 80% in Manaus city (most populous city and the capital of the Amazonas state). Quarantine started on 24 March, and it was prorogated until 2 July 2020. Mandatory use of face masks in any public space was established on 11 May. Lockdown was adopted in four critical cities of the State: Tefe (4 May to 22 May), Silves (11 May to 31 May), Barreirinha (5 May to 29 May), and Sao Gabriel da Cachoeira (8 May to 25 May). Reproduction number (R 0) was ∼2.83 in beginning, 1.78 after one month, and 0.82 after two months of quarantine. The mathematical model shows that Amazonas state is on the peak of daily new cases since the last week of May (∼1600) and would last approximately three weeks before starting to drop. Daily new cases would drop to ∼1200, to ∼500, and to ∼150 by the end of June, July, and August, respectively. Total confirmed cases would be around ∼86,900, ∼111,100, and ∼119,400 by the end of June, July, and August, respectively. COVID-19 cases in Ceara state started on 15 March 2020, where three people were diagnosed, all in Fortaleza city (the capital and epidemic epicentre in Ceara state). Total case number of confirmed COVID-19 patients is rapidly increasing, with the number of total cases doubling each 10 days. Rapid advance of COVID-19 in Ceara is elevating the occupancy rate of intensive care units; nowadays, this rate is ∼88% in state and ∼93% in Fortaleza. Quarantine was initiated on 23 March and prorogated until 20 July. In addition, lockdown was adopted in Fortaleza from 8 May 2020 to 31 May. Isolation rate was 28.6% one month before the quarantine, 45.7% after one month of the quarantine, and 54.9% after two months of quarantine and two weeks after lockdown restriction. On 5 May, Ceara made mandatory to wear face masks while out in public, which was recommended by public health agencies and the government, with the intension to reduce the Covid-19 transmission. At the beginning of the quarantine period, R 0 was ∼2.5, dropping to ∼1.3 after approximately two months (21 May). Our results show that Ceara state is on the peak of daily new cases (∼1900 daily cases) and it would persist until the middle of June before starting to drop. This peak would drop to ∼1200, to ∼460, and to ∼140 by the end of June, July, and August, respectively. The number of confirmed cases would reach ∼101,000, ∼125,500, and ∼133,600 by the end of June, July, and August, respectively. Pernambuco state is located in Northeast region of Brazil and it has a fast spreading of COVID-19. The first case of COVID-19 was on 12 March 2020, in a couple who returned from Rome, Italy. Occupation rate of intensive care units is 67% in state and 81% in Recife city (most populous city and the capital of the Pernambuco state). Quarantine started on 17 March and it was prorogated until 30 June 2020. Isolation rate was 32.9%, 45.3%, and 58.9% before and after one, and two months of quarantine, respectively. Reproduction number (R 0) was ∼1.49 after 2 months of quarantine (22 May) in the Pernambuco state and ∼1.43 in Recife city. Mandatory use of face masks in any public space was established on 16 May 2020, the same day of lockdown adoption in five cities of the State: Recife, Olinda, Jaboatao dos Guararapes, Camaragibe, and Sao Lourenço da Mata; lockdown was scheduled until 31 May 2020. After performing our evaluation, Pernambuco state would have already reached the peak of daily new cases by the end of May (∼1000 daily new cases), but it would persist by around three weeks. Daily new cases would drop to ∼490, to ∼150, and to ∼40 by the end of June, July, and August, respectively. Total confirmed cases would reach ∼57,300 after one month, ∼66,100 after two months, and ∼68,600 by the end of June, July and August, respectively. Some important limitations of our mathematical model have to be considered. First, because our mathematical model is based on an exponential decay curve, small variations in the system (social distancing, preventive measures, or re-opening of non-essential services and stores, etc.) can lead to high alterations in the estimated prediction. Therefore, our prediction is more accurate in the short term (weeks), as compared to long term (months). Second, we used official data from the Brazilian Ministry of Health and Municipal and State Health Secretaries, where data are released after some days of delay. In addition, it is important to consider that in Brazil, only the hospitalized people in moderate or severe conditions are tested for the diagnosis of COVID-19. The number of positive cases may be 10–15 times higher than the reported cases. However, this observation does not invalidate our analysis, since the most important sample to be considered in this analysis is exactly the patients that require hospitalization and consequently lead to the collapses of the public health systems. In addition, similarly to Brazil, several countries have performed the diagnostics only in hospitalized patients and therefore these countries would have resembling disease behaviour. Thus, our results can be used for evaluating the effects of the preventive measures, social distancing, and regulated policies on the disease evolution, as well as help country authorities make critical decisions and direct new strategies for controlling the COVID-19 pandemic.

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          Early dynamics of transmission and control of COVID-19: a mathematical modelling study

          Summary Background An outbreak of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has led to 95 333 confirmed cases as of March 5, 2020. Understanding the early transmission dynamics of the infection and evaluating the effectiveness of control measures is crucial for assessing the potential for sustained transmission to occur in new areas. Combining a mathematical model of severe SARS-CoV-2 transmission with four datasets from within and outside Wuhan, we estimated how transmission in Wuhan varied between December, 2019, and February, 2020. We used these estimates to assess the potential for sustained human-to-human transmission to occur in locations outside Wuhan if cases were introduced. Methods We combined a stochastic transmission model with data on cases of coronavirus disease 2019 (COVID-19) in Wuhan and international cases that originated in Wuhan to estimate how transmission had varied over time during January, 2020, and February, 2020. Based on these estimates, we then calculated the probability that newly introduced cases might generate outbreaks in other areas. To estimate the early dynamics of transmission in Wuhan, we fitted a stochastic transmission dynamic model to multiple publicly available datasets on cases in Wuhan and internationally exported cases from Wuhan. The four datasets we fitted to were: daily number of new internationally exported cases (or lack thereof), by date of onset, as of Jan 26, 2020; daily number of new cases in Wuhan with no market exposure, by date of onset, between Dec 1, 2019, and Jan 1, 2020; daily number of new cases in China, by date of onset, between Dec 29, 2019, and Jan 23, 2020; and proportion of infected passengers on evacuation flights between Jan 29, 2020, and Feb 4, 2020. We used an additional two datasets for comparison with model outputs: daily number of new exported cases from Wuhan (or lack thereof) in countries with high connectivity to Wuhan (ie, top 20 most at-risk countries), by date of confirmation, as of Feb 10, 2020; and data on new confirmed cases reported in Wuhan between Jan 16, 2020, and Feb 11, 2020. Findings We estimated that the median daily reproduction number (R t) in Wuhan declined from 2·35 (95% CI 1·15–4·77) 1 week before travel restrictions were introduced on Jan 23, 2020, to 1·05 (0·41–2·39) 1 week after. Based on our estimates of R t, assuming SARS-like variation, we calculated that in locations with similar transmission potential to Wuhan in early January, once there are at least four independently introduced cases, there is a more than 50% chance the infection will establish within that population. Interpretation Our results show that COVID-19 transmission probably declined in Wuhan during late January, 2020, coinciding with the introduction of travel control measures. As more cases arrive in international locations with similar transmission potential to Wuhan before these control measures, it is likely many chains of transmission will fail to establish initially, but might lead to new outbreaks eventually. Funding Wellcome Trust, Health Data Research UK, Bill & Melinda Gates Foundation, and National Institute for Health Research.
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            Scientific and ethical basis for social-distancing interventions against COVID-19

            On Dec 31, 2019, the WHO China Country Office received notice of a cluster of pneumonia cases of unknown aetiology in the Chinese city of Wuhan, Hubei province. 1 The incidence of coronavirus disease 2019 (COVID-19; caused by severe acute respiratory syndrome coronavirus 2 [SARS-CoV-2]) has since risen exponentially, now affecting all WHO regions. The number of cases reported to date is likely to represent an underestimation of the true burden as a result of shortcomings in surveillance and diagnostic capacity affecting case ascertainment in both high-resource and low-resource settings. 2 By all scientifically meaningful criteria, the world is undergoing a COVID-19 pandemic. In the absence of any pharmaceutical intervention, the only strategy against COVID-19 is to reduce mixing of susceptible and infectious people through early ascertainment of cases or reduction of contact. In The Lancet Infectious Diseases, Joel Koo and colleagues 3 assessed the potential effect of such social distancing interventions on SARS-CoV-2 spread and COVID-19 burden in Singapore. The context is worthy of study, since Singapore was among the first settings to report imported cases, and has so far succeeded in preventing community spread. During the 2003 severe acute respiratory syndrome coronavirus (SARS-CoV) outbreak in Singapore, numerous non-pharmaceutical interventions were implemented successfully, including effective triage and infection control measures in health-care settings, isolation and quarantine of patients with SARS and their contacts, and mass screening of school-aged children for febrile illness. 4 Each of these measures represented an escalation of typical public health action. However, the scale and disruptive impact of these interventions were small compared with the measures that have been implemented in China in response to COVID-19, including closure of schools, workplaces, roads, and transit systems; cancellation of public gatherings; mandatory quarantine of uninfected people without known exposure to SARS-CoV-2; and large-scale electronic surveillance.5, 6 Although these actions have been praised by WHO, 5 the possibility of imposing similar measures in other countries raises important questions. Populations for whom social-distancing interventions have been implemented require and deserve assurance that the decision to enact these measures is informed by the best attainable evidence. For a novel pathogen such as SARS-CoV-2, mathematical modelling of transmission under differing scenarios is the only viable and timely method to generate such evidence. Koo and colleagues 3 adapted an existing influenza epidemic simulation model 7 using granular data on the composition and behaviour of the population of Singapore to assess the potential consequences of specific social-distancing interventions on the transmission dynamics of SARS-CoV-2. The authors considered three infectivity scenarios (basic reproduction number [R 0] of 1·5, 2·0, or 2·5) and assumed between 7·5% and 50·0% of infections were asymptomatic. The interventions were quarantine with or without school closure and workplace distancing (whereby 50% of workers telecommute). Although the complexity of the model makes it difficult to understand the impact of each parameter, the primary conclusions were robust to sensitivity analyses. The combined intervention, in which quarantine, school closure, and workplace distancing were implemented, was the most effective: compared with the baseline scenario of no interventions, the combined intervention reduced the estimated median number of infections by 99·3% (IQR 92·6–99·9) when R 0 was 1·5, by 93·0% (81·5–99·7) when R 0 was 2·0, and by 78·2% (59·0–94·4) when R 0 was 2·5. The observation that the greatest reduction in COVID-19 cases was achieved under the combined intervention is not surprising. However, the assessment of the additional benefit of each intervention, when implemented in combination, offers valuable insight. Since each approach individually will result in considerable societal disruption, it is important to understand the extent of intervention needed to reduce transmission and disease burden. New findings emerge daily about transmission routes and the clinical profile of SARS-CoV-2, including the substantially underestimated rate of infection among children. 8 The implications of such findings with regard to the authors' conclusions about school closure remain unclear. Additionally, reproductive number estimates for Singapore are not yet available. The authors estimated that 7·5% of infections are clinically asymptomatic, although data on the proportion of infections that are asymptomatic are scarce; as shown by Koo and colleagues in sensitivity analyses with higher asymptomatic proportions, this value will influence the effectiveness of social-distancing interventions. Additionally, the analysis assumes high compliance of the general population, which is not guaranteed. Although the scientific basis for these interventions might be robust, ethical considerations are multifaceted. 9 Importantly, political leaders must enact quarantine and social-distancing policies that do not bias against any population group. The legacies of social and economic injustices perpetrated in the name of public health have lasting repercussions. 10 Interventions might pose risks of reduced income and even job loss, disproportionately affecting the most disadvantaged populations: policies to lessen such risks are urgently needed. Special attention should be given to protections for vulnerable populations, such as homeless, incarcerated, older, or disabled individuals, and undocumented migrants. Similarly, exceptions might be necessary for certain groups, including people who are reliant on ongoing medical treatment. The effectiveness and societal impact of quarantine and social distancing will depend on the credibility of public health authorities, political leaders, and institutions. It is important that policy makers maintain the public's trust through use of evidence-based interventions and fully transparent, fact-based communication. © 2020 Caia Image/Science Photo Library 2020 Since January 2020 Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on the novel coronavirus COVID-19. The COVID-19 resource centre is hosted on Elsevier Connect, the company's public news and information website. Elsevier hereby grants permission to make all its COVID-19-related research that is available on the COVID-19 resource centre - including this research content - immediately available in PubMed Central and other publicly funded repositories, such as the WHO COVID database with rights for unrestricted research re-use and analyses in any form or by any means with acknowledgement of the original source. These permissions are granted for free by Elsevier for as long as the COVID-19 resource centre remains active.
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              Mathematic modeling of COVID-19 in the United States

              ABSTRACT COVID-19, the worst pandemic in 100 years, has rapidly spread to the entire world in 2 months since its early report in January 2020. Based on the publicly available data sources, we developed a simple mathematic modeling approach to track the outbreaks of COVID-19 in the US and three selected states: New York, Michigan and California. The same approach is applicable to other regions or countries. We hope our work can stimulate more effort in understanding how an outbreak is developing and how big a scope it can be and in what kind of time framework. Such information is critical for outbreak control, resource utilization and re-opening of the normal daily life to citizens in the affected community.
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                Author and article information

                Journal
                Emerg Microbes Infect
                Emerg Microbes Infect
                Emerging Microbes & Infections
                Taylor & Francis
                2222-1751
                1 July 2020
                2020
                : 9
                : 1
                : 1453-1456
                Affiliations
                [a ]Applied NanoFemto Technologies, LLC , Lowell, MA, USA
                [b ]Interdisciplinary Program of Health Sciences, Cruzeiro do Sul University , Sao Paulo, Brazil
                [c ]Kaiser Southern California Permanente Medical Group , Riverside, CA, USA
                Author notes
                [CONTACT ] Sandro M. Hirabara sandro.hirabara@ 123456cruzeirodosul.edu.br , sandromh@ 123456yahoo.com.br Interdisciplinary Program of Health Sciences, Cruzeiro do Sul University , Rua Galvao Bueno, 868, Liberdade, Sao Paulo, SP01506-000, Brazil
                Article
                1785337
                10.1080/22221751.2020.1785337
                7473191
                32552473
                409846b7-310c-4c4e-ba70-a5036c0fe336
                © 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group, on behalf of Shanghai Shangyixun Cultural Communication Co., Ltd

                This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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