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Lima HS, Tupinambás U, Guimarães FG. Estimating time-varying epidemiological parameters and underreporting of Covid-19 cases in Brazil using a mathematical model with fuzzy transitions between epidemic periods. PLoS One 2024; 19:e0305522. [PMID: 38885221 PMCID: PMC11182538 DOI: 10.1371/journal.pone.0305522] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2023] [Accepted: 06/01/2024] [Indexed: 06/20/2024] Open
Abstract
Our study conducts a comprehensive analysis of the Covid-19 pandemic in Brazil, spanning five waves over three years. We employed a novel Susceptible-Infected-Recovered-Dead-Susceptible (SIRDS) model with a fuzzy transition between epidemic periods to estimate time-varying parameters and evaluate case underreporting. The initial basic reproduction number (R0) is identified at 2.44 (95% Confidence Interval (CI): 2.42-2.46), decreasing to 1.00 (95% CI: 0.99-1.01) during the first wave. The model estimates an underreporting factor of 12.9 (95% CI: 12.5-13.2) more infections than officially reported by Brazilian health authorities, with an increasing factor of 5.8 (95% CI: 5.2-6.4), 12.9 (95% CI: 12.5-13.3), and 16.8 (95% CI: 15.8-17.5) in 2020, 2021, and 2022 respectively. Additionally, the Infection Fatality Rate (IFR) is initially 0.88% (95% CI: 0.81%-0.94%) during the initial phase but consistently reduces across subsequent outbreaks, reaching its lowest value of 0.018% (95% CI: 0.011-0.033) in the last outbreak. Regarding the immunity period, the observed uncertainty and low sensitivity indicate that inferring this parameter is particularly challenging. Brazil successfully reduced R0 during the first wave, coinciding with decreased human mobility. Ineffective public health measures during the second wave resulted in the highest mortality rates within the studied period. We attribute lower mortality rates in 2022 to increased vaccination coverage and the lower lethality of the Omicron variant. We demonstrate the model generalization by its application to other countries. Comparative analyses with serological research further validate the accuracy of the model. In forecasting analysis, our model provides reasonable outbreak predictions. In conclusion, our study provides a nuanced understanding of the Covid-19 pandemic in Brazil, employing a novel epidemiological model. The findings contribute to the broader discourse on pandemic dynamics, underreporting, and the effectiveness of health interventions.
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Affiliation(s)
- Hélder Seixas Lima
- Instituto Federal do Norte de Minas Gerais, Januária, MG, Brazil
- Graduate Program in Electrical Engineering, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
| | - Unaí Tupinambás
- Department of Medical Clinic, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
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Rickards CG, Kilpatrick AM. Age-specific SARS-CoV-2 infection fatality rates derived from serological data vary with income and income inequality. PLoS One 2023; 18:e0285612. [PMID: 37196049 DOI: 10.1371/journal.pone.0285612] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Accepted: 04/26/2023] [Indexed: 05/19/2023] Open
Abstract
The ongoing COVID-19 pandemic has killed at least 1.1 million people in the United States and over 6.7 million globally. Accurately estimating the age-specific infection fatality rate (IFR) of SARS-CoV-2 for different populations is crucial for assessing and understanding the impact of COVID-19 and for appropriately allocating vaccines and treatments to at-risk groups. We estimated age-specific IFRs of wild-type SARS-CoV-2 using published seroprevalence, case, and death data from New York City (NYC) from March to May 2020, using a Bayesian framework that accounted for delays between key epidemiological events. IFRs increased 3-4-fold with every 20 years of age, from 0.06% in individuals between 18-45 years old to 4.7% in individuals over 75. We then compared IFRs in NYC to several city- and country-wide estimates including England, Switzerland (Geneva), Sweden (Stockholm), Belgium, Mexico, and Brazil, as well as a global estimate. IFRs in NYC were higher for individuals younger than 65 years old than most other populations, but similar for older individuals. IFRs for age groups less than 65 decreased with income and increased with income inequality measured using the Gini index. These results demonstrate that the age-specific fatality of COVID-19 differs among developed countries and raises questions about factors underlying these differences, including underlying health conditions and healthcare access.
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Affiliation(s)
- Chloe G Rickards
- Department of Ecology and Evolutionary Biology, University of California Santa Cruz, Santa Cruz, CA, United States of America
| | - A Marm Kilpatrick
- Department of Ecology and Evolutionary Biology, University of California Santa Cruz, Santa Cruz, CA, United States of America
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Colnago M, Benvenuto GA, Casaca W, Negri RG, Fernandes EG, Cuminato JA. Risk Factors Associated with Mortality in Hospitalized Patients with COVID-19 during the Omicron Wave in Brazil. Bioengineering (Basel) 2022; 9:bioengineering9100584. [PMID: 36290552 PMCID: PMC9598428 DOI: 10.3390/bioengineering9100584] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/13/2022] [Revised: 10/03/2022] [Accepted: 10/18/2022] [Indexed: 11/16/2022] Open
Abstract
Considering the imminence of new SARS-CoV-2 variants and COVID-19 vaccine availability, it is essential to understand the impact of the disease on the most vulnerable groups and those at risk of death from the disease. To this end, the odds ratio (OR) for mortality and hospitalization was calculated for different groups of patients by applying an adjusted logistic regression model based on the following variables of interest: gender, booster vaccination, age group, and comorbidity occurrence. A massive number of data were extracted and compiled from official Brazilian government resources, which include all reported cases of hospitalizations and deaths associated with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in Brazil during the “wave” of the Omicron variant (BA.1 substrain). Males (1.242; 95% CI 1.196–1.290) aged 60–79 (3.348; 95% CI 3.050–3.674) and 80 years or older (5.453; 95% CI 4.966–5.989), and hospitalized patients with comorbidities (1.418; 95% CI 1.355–1.483), were more likely to die. There was a reduction in the risk of death (0.907; 95% CI 0.866–0.951) among patients who had received the third dose of the anti-SARS-CoV-2 vaccine (booster). Additionally, this big data investigation has found statistical evidence that vaccination can support mitigation plans concerning the current scenario of COVID-19 in Brazil since the Omicron variant and its substrains are now prevalent across the entire country.
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Affiliation(s)
- Marilaine Colnago
- Institute of Chemistry, São Paulo State University (UNESP), Araraquara 14800-060, Brazil
| | - Giovana A. Benvenuto
- Faculty of Science and Technology (FCT), São Paulo State University (UNESP), Presidente Prudente 19060-900, Brazil
| | - Wallace Casaca
- Institute of Biosciences, Letters and Exact Sciences, São Paulo State University (UNESP), São José do Rio Preto 15054-000, Brazil
- Correspondence:
| | - Rogério G. Negri
- Science and Technology Institute, São Paulo State University (UNESP), São José dos Campos 12247-004, Brazil
| | - Eder G. Fernandes
- Immunization Division—Centre of Epidemiology Surveillance of the São Paulo State Health Department, São Paulo 01246-000, Brazil
| | - José A. Cuminato
- Institute of Mathematics and Computer Science, São Paulo University (USP), São Carlos 13566-590, Brazil
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Lone KS, Khan SMS, Qurieshi MA, Majid S, Pandit MI, Haq I, Ahmad J, Bhat AA, Bashir K, Bilquees S, Fazili AB, Hassan M, Jan Y, Kaul RUR, Khan ZA, Mushtaq B, Nazir F, Qureshi UA, Raja MW, Rasool M, Asma A, Bhat AA, Chowdri IN, Ismail S, Jeelani A, Kawoosa MF, Khan MA, Khan MS, Kousar R, Lone AA, Nabi S, Qazi TB, Rather RH, Sabah I, Sumji IA. Seroprevalence of SARS-CoV-2-specific anti-spike IgM, IgG, and anti-nucleocapsid IgG antibodies during the second wave of the pandemic: A population-based cross-sectional survey across Kashmir, India. Front Public Health 2022; 10:967447. [PMID: 36276377 PMCID: PMC9582950 DOI: 10.3389/fpubh.2022.967447] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2022] [Accepted: 09/12/2022] [Indexed: 01/25/2023] Open
Abstract
Background Within Kashmir, which is one of the topographically distinct areas in the Himalayan belt of India, a total of 2,236 cumulative deaths occurred by the end of the second wave. We aimed to conduct this population-based study in the age group of 7 years and above to estimate the seropositivity and its attributes in Kashmir valley. Methods We conducted a community-based household-level cross-sectional study, with a multistage, population-stratified, probability-proportionate-to-size, cluster sampling method to select 400 participants from each of the 10 districts of Kashmir. We also selected a quota of healthcare workers, police personnel, and antenatal women from each of the districts. Households were selected from each cluster and all family members with age 7 years or more were invited to participate. Information was collected through a standardized questionnaire and entered into Epicollect 5 software. Trained healthcare personnel were assigned for collecting venous blood samples from each of the participants which were transferred and processed for immunological testing. Testing was done for the presence of SARS-CoV-2-specific anti-spike IgM, IgG antibodies, and anti-nucleocapsid IgG antibodies. Weighted seropositivity was estimated along with the adjustment done for the sensitivity and specificity of the test used. Findings The data were collected from a total of 4,229 participants from the general population within the 10 districts of Kashmir. Our results showed that 84.84% (95% CI 84.51-85.18%) of the participants were seropositive in the weighted imputed data among the general population. In multiple logistic regression, the variables significantly affecting the seroprevalence were the age group 45-59 years (odds ratio of 0.73; 95% CI 0.67-0.78), self-reported history of comorbidity (odds ratio of 1.47; 95% CI 1.33-1.61), and positive vaccination history (odds ratio of 0.85; 95% CI 0.79-0.90) for anti-nucleocapsid IgG antibodies. The entire assessed variables showed a significant role during multiple logistic regression analysis for affecting IgM anti-spike antibodies with an odds ratio of 1.45 (95% CI 1.32-1.57) for age more than 60 years, 1.21 (95% CI 1.15-1.27) for the female gender, 0.87 (95% CI 0.82-0.92) for urban residents, 0.86 (95% CI 0.76-0.92) for self-reported comorbidity, and an odds ratio of 1.16 (95% CI 1.08-1.24) for a positive history of vaccination. The estimated infection fatality ratio was 0.033% (95% CI: 0.034-0.032%) between 22 May and 31 July 2021 against the seropositivity for IgM antibodies. Interpretation During the second wave of the SARS-CoV-2 pandemic, 84.84% (95% CI 84.51-85.18%) of participants from this population-based cross-sectional sample were seropositive against SARS-CoV-2. Despite a comparatively lower number of cases reported and lower vaccination coverage in the region, our study found such high seropositivity across all age groups, which indicates the higher number of subclinical and less severe unnoticed caseload in the community.
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Affiliation(s)
- Kouser Sideeq Lone
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | | | - Mariya Amin Qurieshi
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Sabhiya Majid
- Department of Biochemistry, Government Medical College Srinagar, Srinagar, India
| | - Mohammad Iqbal Pandit
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Inaamul Haq
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India,*Correspondence: Inaamul Haq
| | - Javid Ahmad
- Department of Community Medicine, Sher-i-Kashmir Institute of Medical Sciences, Srinagar, India
| | - Ashfaq Ahmad Bhat
- Department of Community Medicine, SKIMS Medical College Srinagar, Srinagar, India
| | - Khalid Bashir
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Sufoora Bilquees
- Department of Community Medicine, Government Medical College Baramulla, Baramulla, India
| | - Anjum Bashir Fazili
- Department of Community Medicine, Sher-i-Kashmir Institute of Medical Sciences, Srinagar, India
| | - Muzamil Hassan
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Yasmeen Jan
- Department of Community Medicine, SKIMS Medical College Srinagar, Srinagar, India
| | - Rauf-ur Rashid Kaul
- Department of Community Medicine, Sher-i-Kashmir Institute of Medical Sciences, Srinagar, India
| | - Zahid Ali Khan
- Department of Community Medicine, Government Medical College Baramulla, Baramulla, India
| | - Beenish Mushtaq
- Department of Community Medicine, SKIMS Medical College Srinagar, Srinagar, India
| | - Fouzia Nazir
- Department of Community Medicine, Government Medical College Anantnag, Anantnag, India
| | - Uruj Altaf Qureshi
- Department of Community Medicine, Government Medical College Baramulla, Baramulla, India
| | - Malik Waseem Raja
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Mahbooba Rasool
- Department of Community Medicine, Government Medical College Anantnag, Anantnag, India
| | - Anjum Asma
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Arif Akbar Bhat
- Department of Biochemistry, Government Medical College Srinagar, Srinagar, India
| | - Iqra Nisar Chowdri
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Shaista Ismail
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Asif Jeelani
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Misbah Ferooz Kawoosa
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Mehvish Afzal Khan
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Mosin Saleem Khan
- Department of Biochemistry, Government Medical College Srinagar, Srinagar, India
| | - Rafiya Kousar
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Ab Aziz Lone
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Shahroz Nabi
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Tanzeela Bashir Qazi
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Rouf Hussain Rather
- Directorate of Health Services Kashmir, Government of Jammu and Kashmir, Srinagar, India
| | - Iram Sabah
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
| | - Ishtiyaq Ahmad Sumji
- Department of Community Medicine, Government Medical College Srinagar, Srinagar, India
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Silva DM, Secchi AR. Recursive state and parameter estimation of COVID-19 circulating variants dynamics. Sci Rep 2022; 12:15879. [PMID: 36151226 PMCID: PMC9508243 DOI: 10.1038/s41598-022-18208-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2021] [Accepted: 08/08/2022] [Indexed: 11/29/2022] Open
Abstract
COVID-19 pandemic response with non-pharmaceutical interventions is an intrinsic control problem. Governments weigh social distancing policies to avoid overload in the health system without significant economic impact. The mutability of the SARS-CoV-2 virus, vaccination coverage, and mobility restriction measures change epidemic dynamics over time. A model-based control strategy requires reliable predictions to be efficient on a long-term basis. In this paper, a SEIR-based model is proposed considering dynamic feedback estimation. State and parameter estimations are performed on state estimators using augmented states. Three methods were implemented: constrained extended Kalman filter (CEKF), CEKF and smoother (CEKF & S), and moving horizon estimator (MHE). The parameters estimation was based on vaccine efficacy studies regarding transmissibility, severity of the disease, and lethality. Social distancing was assumed as a measured disturbance calculated using Google mobility data. Data from six federative units from Brazil were used to evaluate the proposed strategy. State and parameter estimations were performed from 1 October 2020 to 1 July 2021, during which Zeta and Gamma variants emerged. Simulation results showed that lethality increased between 11 and 30% for Zeta mutations and between 44 and 107% for Gamma mutations. In addition, transmissibility increased between 10 and 37% for the Zeta variant and between 43 and 119% for the Gamma variant. Furthermore, parameter estimation indicated temporal underreporting changes in hospitalized and deceased individuals. Overall, the estimation strategy showed to be suitable for dynamic feedback as simulation results presented an efficient detection and dynamic characterization of circulating variants.
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Affiliation(s)
- Daniel Martins Silva
- Chemical Engineering Program/COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 21941-942, Brazil.
| | - Argimiro Resende Secchi
- Chemical Engineering Program/COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 21941-942, Brazil
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SARS-CoV-2 Seroconversion in Response to Infection and Vaccination: a Time Series Local Study in Brazil. Microbiol Spectr 2022; 10:e0102622. [PMID: 35770982 PMCID: PMC9430992 DOI: 10.1128/spectrum.01026-22] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022] Open
Abstract
The investigation of antibodies raised against different severe acute respiratory syndrome coronavirus (SARS-CoV-2) antigens can help to determine the extent of previous SARS-CoV-2 infections in the population and track the humoral response to vaccination. Therefore, serological surveys can provide key information to better manage the pandemic and/or to implement the most effective vaccination program. Here we describe a time series anti-nucleocapsid, anti-spike IgG serological survey analysis in the city of Matinhos, PR, Brazil during the year of 2021. Seroconversion rates to the nucleocapsid antigen were not influenced by gender or age. The serological data support that the coronavirus disease 2019 (COVID-19) infection rate is ~50% higher than official numbers. Furthermore, by applying serological data, the corrected infection fatality rate was estimated to be lower than 2.4% in contrast with the official estimative of 3.6%. The rates of IgG reactive to spike antigen resembled the curve of the fraction the population that had taken the second vaccine dose. Up to 82% of spike seroconversion was detected in the end of 2021, confirming the effectiveness of the COVID-19 vaccination program in the city. This SARS-CoV-2 serological study unraveled the SARS-CoV-2 infection rates and the response to vaccination in the city of Matinhos. IMPORTANCE The investigation of antibodies raised against SARS-CoV-2 can help to determine the extent of previous SARS-CoV-2 infections and track the humoral response to vaccination. Here we describe a time series anti-nucleocapsid, anti-spike IgG serological survey in the city of Matinhos, PR, Brazil during the year of 2021. The data depicted the progression of SARS-CoV-2 infections in the city allowing the correction of the number of citizens who experienced COVID-19 and the disease fatality rate. The seroconversion rates to the spike antigen resembled the curve of the fraction of the population that had taken the second vaccine dose, thereby confirming the effectiveness of the COVID-19 vaccination program in the city.
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Lin L, Chen B, Zhao Y, Wang W, He D. Two waves of COIVD-19 in Brazilian cities and vaccination impact. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:4657-4671. [PMID: 35430833 DOI: 10.3934/mbe.2022216] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
BACKGROUNDS Brazil has suffered two waves of Coronavirus Disease 2019 (COVID-19). The second wave, coinciding with the spread of the Gamma variant, was more severe than the first wave. Studies have not yet reached a conclusion on some issues including the extent of reinfection, the infection fatality rate (IFR), the infection attack rate (IAR) and the effects of the vaccination campaign in Brazil, though it was reported that confirmed reinfection was at a low level. METHODS We modify the classical Susceptible-Exposed-Infectious-Recovered (SEIR) model with additional class for severe cases, vaccination and time-varying transmission rates. We fit the model to the severe acute respiratory infection (SARI) deaths, which is a proxy of the COVID-19 deaths, in 20 Brazilian cities with the large number of death tolls. We evaluate the vaccination effect by a contrast of "with" vaccination actual scenario and "without" vaccination in a counterfactual scenario. We evaluate the model performance when the reinfection is absent in the model. RESULTS In the 20 Brazilian cities, the model simulated death matched the reported deaths reasonably well. The effect of the vaccination varies across cities. The estimated median IFR is around 1.2%. CONCLUSION Overall, through this modeling exercise, we conclude that the effects of vaccination campaigns vary across cites and the reinfection is not crucial for the second wave. The relatively high IFR could be due to the breakdown of medical system in many cities.
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Affiliation(s)
- Lixin Lin
- Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong 999077, China
| | - Boqiang Chen
- Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong 999077, China
| | - Yanji Zhao
- Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong 999077, China
| | - Weiming Wang
- School of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, China
| | - Daihai He
- Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong 999077, China
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Pietzonka P, Brorson E, Bankes W, Cates ME, Jack RL, Adhikari R. Bayesian inference across multiple models suggests a strong increase in lethality of COVID-19 in late 2020 in the UK. PLoS One 2021; 16:e0258968. [PMID: 34818345 PMCID: PMC8612566 DOI: 10.1371/journal.pone.0258968] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2021] [Accepted: 10/10/2021] [Indexed: 12/04/2022] Open
Abstract
We apply Bayesian inference methods to a suite of distinct compartmental models of generalised SEIR type, in which diagnosis and quarantine are included via extra compartments. We investigate the evidence for a change in lethality of COVID-19 in late autumn 2020 in the UK, using age-structured, weekly national aggregate data for cases and mortalities. Models that allow a (step-like or graded) change in infection fatality rate (IFR) have consistently higher model evidence than those without. Moreover, they all infer a close to two-fold increase in IFR. This value lies well above most previously available estimates. However, the same models consistently infer that, most probably, the increase in IFR preceded the time window during which variant B.1.1.7 (alpha) became the dominant strain in the UK. Therefore, according to our models, the caseload and mortality data do not offer unequivocal evidence for higher lethality of a new variant. We compare these results for the UK with similar models for Germany and France, which also show increases in inferred IFR during the same period, despite the even later arrival of new variants in those countries. We argue that while the new variant(s) may be one contributing cause of a large increase in IFR in the UK in autumn 2020, other factors, such as seasonality, or pressure on health services, are likely to also have contributed.
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Affiliation(s)
- Patrick Pietzonka
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, United Kingdom
- * E-mail:
| | - Erik Brorson
- Quantitative Research, JPMorgan Chase & Co., London, United Kingdom
| | - William Bankes
- Applied Machine Learning and Artificial Intelligence, JPMorgan Chase & Co., London, United Kingdom
| | - Michael E. Cates
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, United Kingdom
| | - Robert L. Jack
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, United Kingdom
- Department of Chemistry, University of Cambridge, Cambridge, United Kingdom
| | - Ronojoy Adhikari
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, United Kingdom
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