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S P Rajasekar, R Ramesh, Sabbar Y. Based on epidemiological parameter data, probe into a stochastically perturbed dominant variant of the COVID-19 pandemic model. Gene 2024; 926:148608. [PMID: 38823655 DOI: 10.1016/j.gene.2024.148608] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/29/2024] [Revised: 05/14/2024] [Accepted: 05/21/2024] [Indexed: 06/03/2024]
Abstract
During the COVID-19 pandemic, the SARS-CoV-2 gene mutation has been rapidly emerging and spreading all over the world. Experts worldwide regularly monitor genetic mutations and variants through genome-sequence-based surveillance, laboratory testing, outbreak investigation, and epidemiological probing. Clinical pathologists and medical laboratory scientists prefer developing or endorsing COVID-19 vaccines with a broader immune response involving various antibodies and cells to protect against mutations or new variants. Randomness plays an enormous role in pathology and epidemiology. Hence, based on epidemiological parameter data, we construct and probe a stochastically perturbed dominant variant of the coronavirus epidemic model with three nonlinear saturated incidence rates. We reveal the existence of a unique global positive solution to the constructed stochastic COVID-19 model. The Lyapunov function method is used to determine the presence of a stationary distribution of positive solutions. We derive sufficient conditions for the coronavirus to be eradicated. Eventually, numerical simulations validate the effectiveness of our theoretical outcomes.
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Affiliation(s)
- S P Rajasekar
- Department of Mathematics, Government Arts College for Women, Nilakottai 624 202, Tamil Nadu, India.
| | - R Ramesh
- Department of Mathematics, Arignar Anna Government Arts College, Musiri, Tamil Nadu, India
| | - Yassine Sabbar
- MAIS Laboratory, MAMCS Group, FST Errachidia, Moulay Ismail University of Meknes, Errachidia, Morocco
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2
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Wu P, Wang X, Wang H. Spatial heterogeneity analysis for the transmission of syphilis disease in China via a data-validated reaction-diffusion model. Math Biosci 2024; 375:109243. [PMID: 38964670 DOI: 10.1016/j.mbs.2024.109243] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2024] [Revised: 05/05/2024] [Accepted: 06/22/2024] [Indexed: 07/06/2024]
Abstract
Based on the distinctive spatial diffusion characteristics observed in syphilis transmission patterns, this paper introduces a novel reaction-diffusion model for syphilis disease dynamics, incorporating general incidence functions within a heterogeneous environment. We derive the basic reproduction number essential for threshold dynamics and investigate the uniform persistence of the model. We validate the model and estimate its parameters by employing the multi-objective Markov Chain Monte Carlo (MCMC) method, using real syphilis data from the years 2004 to 2018 in China. Furthermore, we explore the impact of spatial heterogeneity and intervention measures on syphilis transmission. Our findings reveal several key insights: (1) In addition to the original high-incidence areas of syphilis, Xinjiang, Guizhou, Hunan and Northeast China have also emerged as high-incidence regions for syphilis in China. (2) The latent syphilis cases represent the highest proportion of newly reported cases, highlighting the critical importance of considering their role in transmission dynamics to avoid underestimation of syphilis outbreaks. (3) Neglecting spatial heterogeneity results in an underestimation of disease prevalence and the number of syphilis-infected individuals, undermining effective disease prevention and control strategies. (4) The initial conditions have minimal impact on the long-term spatial distribution of syphilis-infected individuals in scenarios of varying diffusion rates. This study underscores the significance of spatial dynamics and intervention measures in assessing and managing syphilis transmission, which offers insights for public health policymakers.
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Affiliation(s)
- Peng Wu
- School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China.
| | - Xiunan Wang
- Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA.
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada.
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3
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Kwak M, Sun X, Wi Y, Nah K, Kim Y, Jin H. A novel indicator in epidemic monitoring through a case study of Ebola in West Africa (2014-2016). Sci Rep 2024; 14:12147. [PMID: 38802461 PMCID: PMC11130319 DOI: 10.1038/s41598-024-62719-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2024] [Accepted: 05/21/2024] [Indexed: 05/29/2024] Open
Abstract
The E/S (exposed/susceptible) ratio is analyzed in the SEIR model. The ratio plays a key role in understanding epidemic dynamics during the 2014-2016 Ebola outbreak in Sierra Leone and Guinea. The maximum value of the ratio occurs immediately before or after the time-dependent reproduction number (Rt) equals 1, depending on the initial susceptible population (S(0)). It is demonstrated that transmission rate curves corresponding to various incubation periods intersect at a single point referred to as the Cross Point (CP). At this point, the E/S ratio reaches an extremum, signifying a critical shift in transmission dynamics and aligning with the time when Rt approaches 1. By plotting transmission rate curves, β(t), for any two arbitrary incubation periods and tracking their intersections, we can trace CP over time. CP serves as an indicator of epidemic status, especially when Rt is close to 1. It provides a practical means of monitoring epidemics without prior knowledge of the incubation period. Through a case study, we estimate the transmission rate and reproduction number, identifying CP and Rt = 1 while examining the E/S ratio across various values of S(0).
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Affiliation(s)
- Minkyu Kwak
- Department of Mathematics and Statistics, Chonnam National University, Gwangju, South Korea
| | - Xiuxiu Sun
- Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Henan, China
| | - Yunju Wi
- Department of Mathematics and Statistics, Chonnam National University, Gwangju, South Korea
| | - Kyeongah Nah
- Busan Center for Medical Mathematics, National Institute of Mathematical Sciences, Busan, South Korea
| | - Yongkuk Kim
- Department of Mathematics, Kyungpook National University, Daegu, South Korea
| | - Hongsung Jin
- Department of Mathematics and Statistics, Chonnam National University, Gwangju, South Korea.
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4
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Yakovyna V, Shakhovska N, Szpakowska A. A novel hybrid supervised and unsupervised hierarchical ensemble for COVID-19 cases and mortality prediction. Sci Rep 2024; 14:9782. [PMID: 38684770 PMCID: PMC11059164 DOI: 10.1038/s41598-024-60637-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2023] [Accepted: 04/25/2024] [Indexed: 05/02/2024] Open
Abstract
Though COVID-19 is no longer a pandemic but rather an endemic, the epidemiological situation related to the SARS-CoV-2 virus is developing at an alarming rate, impacting every corner of the world. The rapid escalation of the coronavirus has led to the scientific community engagement, continually seeking solutions to ensure the comfort and safety of society. Understanding the joint impact of medical and non-medical interventions on COVID-19 spread is essential for making public health decisions that control the pandemic. This paper introduces two novel hybrid machine-learning ensembles that combine supervised and unsupervised learning for COVID-19 data classification and regression. The study utilizes publicly available COVID-19 outbreak and potential predictive features in the USA dataset, which provides information related to the outbreak of COVID-19 disease in the US, including data from each of 3142 US counties from the beginning of the epidemic (January 2020) until June 2021. The developed hybrid hierarchical classifiers outperform single classification algorithms. The best-achieved performance metrics for the classification task were Accuracy = 0.912, ROC-AUC = 0.916, and F1-score = 0.916. The proposed hybrid hierarchical ensemble combining both supervised and unsupervised learning allows us to increase the accuracy of the regression task by 11% in terms of MSE, 29% in terms of the area under the ROC, and 43% in terms of the MPP metric. Thus, using the proposed approach, it is possible to predict the number of COVID-19 cases and deaths based on demographic, geographic, climatic, traffic, public health, social-distancing-policy adherence, and political characteristics with sufficiently high accuracy. The study reveals that virus pressure is the most important feature in COVID-19 spread for classification and regression analysis. Five other significant features were identified to have the most influence on COVID-19 spread. The combined ensembling approach introduced in this study can help policymakers design prevention and control measures to avoid or minimize public health threats in the future.
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Affiliation(s)
- Vitaliy Yakovyna
- Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Ul. Oczapowskiego 2, 10-719, Olsztyn, Poland
- Artificial Intelligence Department, Lviv Polytechnic National University, 12 S. Bandery St, Lviv, 79013, Ukraine
| | - Nataliya Shakhovska
- Artificial Intelligence Department, Lviv Polytechnic National University, 12 S. Bandery St, Lviv, 79013, Ukraine.
- Universytet Rolniczy, 31120, Kraków, Poland.
| | - Aleksandra Szpakowska
- Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Ul. Oczapowskiego 2, 10-719, Olsztyn, Poland
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Espinosa O, Mora L, Sanabria C, Ramos A, Rincón D, Bejarano V, Rodríguez J, Barrera N, Álvarez-Moreno C, Cortés J, Saavedra C, Robayo A, Franco OH. Predictive models for health outcomes due to SARS-CoV-2, including the effect of vaccination: a systematic review. Syst Rev 2024; 13:30. [PMID: 38229123 PMCID: PMC10790449 DOI: 10.1186/s13643-023-02411-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 12/12/2022] [Accepted: 12/04/2023] [Indexed: 01/18/2024] Open
Abstract
BACKGROUND The interaction between modelers and policymakers is becoming more common due to the increase in computing speed seen in recent decades. The recent pandemic caused by the SARS-CoV-2 virus was no exception. Thus, this study aims to identify and assess epidemiological mathematical models of SARS-CoV-2 applied to real-world data, including immunization for coronavirus 2019 (COVID-19). METHODOLOGY PubMed, JSTOR, medRxiv, LILACS, EconLit, and other databases were searched for studies employing epidemiological mathematical models of SARS-CoV-2 applied to real-world data. We summarized the information qualitatively, and each article included was assessed for bias risk using the Joanna Briggs Institute (JBI) and PROBAST checklist tool. The PROSPERO registration number is CRD42022344542. FINDINGS In total, 5646 articles were retrieved, of which 411 were included. Most of the information was published in 2021. The countries with the highest number of studies were the United States, Canada, China, and the United Kingdom; no studies were found in low-income countries. The SEIR model (susceptible, exposed, infectious, and recovered) was the most frequently used approach, followed by agent-based modeling. Moreover, the most commonly used software were R, Matlab, and Python, with the most recurring health outcomes being death and recovery. According to the JBI assessment, 61.4% of articles were considered to have a low risk of bias. INTERPRETATION The utilization of mathematical models increased following the onset of the SARS-CoV-2 pandemic. Stakeholders have begun to incorporate these analytical tools more extensively into public policy, enabling the construction of various scenarios for public health. This contribution adds value to informed decision-making. Therefore, understanding their advancements, strengths, and limitations is essential.
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Affiliation(s)
- Oscar Espinosa
- Directorate of Analytical, Economic and Actuarial Studies in Health, Instituto de Evaluación Tecnológica en Salud (IETS) & Economic Models and Quantitative Methods Research Group, Centro de Investigaciones para el Desarrollo, Universidad Nacional de Colombia, Bogotá, D.C., Colombia.
| | - Laura Mora
- Directorate of Analytical, Economic and Actuarial Studies in Health, Instituto de Evaluación Tecnológica en Salud (IETS), Bogotá, Colombia
| | - Cristian Sanabria
- Directorate of Analytical, Economic and Actuarial Studies in Health, Instituto de Evaluación Tecnológica en Salud (IETS), Bogotá, Colombia
| | - Antonio Ramos
- Directorate of Analytical, Economic and Actuarial Studies in Health, Instituto de Evaluación Tecnológica en Salud (IETS) & Economic Models and Quantitative Methods Research Group, Centro de Investigaciones para el Desarrollo, Universidad Nacional de Colombia, Bogotá, D.C., Colombia
| | - Duván Rincón
- Directorate of Analytical, Economic and Actuarial Studies in Health, Instituto de Evaluación Tecnológica en Salud (IETS), Bogotá, Colombia
| | - Valeria Bejarano
- Directorate of Analytical, Economic and Actuarial Studies in Health, Instituto de Evaluación Tecnológica en Salud (IETS) & Economic Models and Quantitative Methods Research Group, Centro de Investigaciones para el Desarrollo, Universidad Nacional de Colombia, Bogotá, D.C., Colombia
| | - Jhonathan Rodríguez
- Directorate of Analytical, Economic and Actuarial Studies in Health, Instituto de Evaluación Tecnológica en Salud (IETS) & Economic Models and Quantitative Methods Research Group, Centro de Investigaciones para el Desarrollo, Universidad Nacional de Colombia, Bogotá, D.C., Colombia
| | - Nicolás Barrera
- Directorate of Analytical, Economic and Actuarial Studies in Health, Instituto de Evaluación Tecnológica en Salud (IETS), Bogotá, Colombia
| | | | - Jorge Cortés
- Faculty of Medicine, Universidad Nacional de Colombia, Bogotá, D.C., Colombia
| | - Carlos Saavedra
- Faculty of Medicine, Universidad Nacional de Colombia, Bogotá, D.C., Colombia
| | - Adriana Robayo
- Directorate of Analytical, Economic and Actuarial Studies in Health, Instituto de Evaluación Tecnológica en Salud (IETS), Bogotá, Colombia
| | - Oscar H Franco
- University Medical Center Utrecht, Utrecht University & Harvard T.H. Chan School of Public Health, Harvard University, Cambridge, USA
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Ramezani M, Takian A, Bakhtiari A, Rabiee HR, Ghazanfari S, Mostafavi H. The application of artificial intelligence in health policy: a scoping review. BMC Health Serv Res 2023; 23:1416. [PMID: 38102620 PMCID: PMC10722786 DOI: 10.1186/s12913-023-10462-2] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2023] [Accepted: 12/08/2023] [Indexed: 12/17/2023] Open
Abstract
BACKGROUND Policymakers require precise and in-time information to make informed decisions in complex environments such as health systems. Artificial intelligence (AI) is a novel approach that makes collecting and analyzing data in complex systems more accessible. This study highlights recent research on AI's application and capabilities in health policymaking. METHODS We searched PubMed, Scopus, and the Web of Science databases to find relevant studies from 2000 to 2023, using the keywords "artificial intelligence" and "policymaking." We used Walt and Gilson's policy triangle framework for charting the data. RESULTS The results revealed that using AI in health policy paved the way for novel analyses and innovative solutions for intelligent decision-making and data collection, potentially enhancing policymaking capacities, particularly in the evaluation phase. It can also be employed to create innovative agendas with fewer political constraints and greater rationality, resulting in evidence-based policies. By creating new platforms and toolkits, AI also offers the chance to make judgments based on solid facts. The majority of the proposed AI solutions for health policy aim to improve decision-making rather than replace experts. CONCLUSION Numerous approaches exist for AI to influence the health policymaking process. Health systems can benefit from AI's potential to foster the meaningful use of evidence-based policymaking.
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Affiliation(s)
- Maryam Ramezani
- Department of Health Management, Policy and Economics, School of Public Health, Tehran University of Medical Sciences, Tehran, Iran
- Health Equity Research Center (HERC), Tehran University of Medical Sciences, Tehran, Iran
| | - Amirhossein Takian
- Department of Health Management, Policy and Economics, School of Public Health, Tehran University of Medical Sciences, Tehran, Iran.
- Department of Global Health and Public Policy, School of Public Health, Tehran University of Medical Sciences, Tehran, Iran.
- Health Equity Research Center (HERC), Tehran University of Medical Sciences, Tehran, Iran.
| | - Ahad Bakhtiari
- Department of Health Management, Policy and Economics, School of Public Health, Tehran University of Medical Sciences, Tehran, Iran
- Health Equity Research Center (HERC), Tehran University of Medical Sciences, Tehran, Iran
| | - Hamid R Rabiee
- Department of Computer Engineering, Sharif University of Technology, Tehran, Iran
| | - Sadegh Ghazanfari
- Department of Health Management, Policy and Economics, School of Public Health, Tehran University of Medical Sciences, Tehran, Iran
| | - Hakimeh Mostafavi
- Health Equity Research Center (HERC), Tehran University of Medical Sciences, Tehran, Iran
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Ji J, Wang H, Wang L, Ramazi P, Kong JD, Watmough J. Climate-dependent effectiveness of nonpharmaceutical interventions on COVID-19 mitigation. Math Biosci 2023; 366:109087. [PMID: 37858753 DOI: 10.1016/j.mbs.2023.109087] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2023] [Revised: 10/09/2023] [Accepted: 10/12/2023] [Indexed: 10/21/2023]
Abstract
Environmental factors have a significant impact on the transmission of infectious diseases. Existing results show that the novel coronavirus can persist outside the host. We propose a susceptible-exposed-presymptomatic-infectious-asymptomatic-recovered-susceptible (SEPIARS) model with a vaccination compartment and indirect incidence to explore the effect of environmental conditions, temperature and humidity, on the transmission of the SARS-CoV-2 virus. Using climate data and daily confirmed cases data in two Canadian cities with different atmospheric conditions, we evaluate the mortality rates of the SARS-CoV-2 virus and further estimate the transmission rates by the inverse method, respectively. The numerical results show that high temperature or humidity can be helpful in mitigating the spread of COVID-19 during the warm summer months. Our findings verify that nonpharmaceutical interventions are less effective if the virus can persist for a long time on surfaces. Based on climate data, we can forecast the transmission rate and the infection cases up to four weeks in the future by a generalized boosting machine learning model.
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Affiliation(s)
- Juping Ji
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2R3, Canada; Interdisciplinary Lab for Mathematical Ecology and Epidemiology, University of Alberta, Edmonton, AB T6G 2R3, Canada
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2R3, Canada; Interdisciplinary Lab for Mathematical Ecology and Epidemiology, University of Alberta, Edmonton, AB T6G 2R3, Canada.
| | - Lin Wang
- Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB E3B 5A3, Canada
| | - Pouria Ramazi
- Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S 3A1, Canada
| | - Jude Dzevela Kong
- Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada
| | - James Watmough
- Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB E3B 5A3, Canada
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Bugalia S, Tripathi JP, Wang H. Estimating the time-dependent effective reproduction number and vaccination rate for COVID-19 in the USA and India. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:4673-4689. [PMID: 36896517 DOI: 10.3934/mbe.2023216] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
The effective reproduction number, $ R_t $, is a vital epidemic parameter utilized to judge whether an epidemic is shrinking, growing, or holding steady. The main goal of this paper is to estimate the combined $ R_t $ and time-dependent vaccination rate for COVID-19 in the USA and India after the vaccination campaign started. Accounting for the impact of vaccination into a discrete-time stochastic augmented SVEIR (Susceptible-Vaccinated-Exposed-Infectious-Recovered) model, we estimate the time-dependent effective reproduction number $ (R_t) $ and vaccination rate $ (\xi_t) $ for COVID-19 by using a low pass filter and the Extended Kalman Filter (EKF) approach for the period February 15, 2021 to August 22, 2022 in India and December 13, 2020 to August 16, 2022 in the USA. The estimated $ R_t $ and $ \xi_t $ show spikes and serrations with the data. Our forecasting scenario represents the situation by December 31, 2022 that the new daily cases and deaths are decreasing for the USA and India. We also noticed that for the current vaccination rate, $ R_t $ would remain greater than one by December 31, 2022. Our results are beneficial for the policymakers to track the status of the effective reproduction number, whether it is greater or less than one. As restrictions in these countries ease, it is still important to maintain safety and preventive measures.
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Affiliation(s)
- Sarita Bugalia
- Department of Mathematics, Central University of Rajasthan, Bandar Sindri, Kishangarh-305817, Ajmer, Rajasthan, India
| | - Jai Prakash Tripathi
- Department of Mathematics, Central University of Rajasthan, Bandar Sindri, Kishangarh-305817, Ajmer, Rajasthan, India
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada
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Deep learning for Covid-19 forecasting: State-of-the-art review. Neurocomputing 2022; 511:142-154. [PMID: 36097509 PMCID: PMC9454152 DOI: 10.1016/j.neucom.2022.09.005] [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: 04/11/2022] [Revised: 07/03/2022] [Accepted: 09/04/2022] [Indexed: 11/21/2022]
Abstract
The Covid-19 pandemic has galvanized scientists to apply machine learning methods to help combat the crisis. Despite the significant amount of research there exists no comprehensive survey devoted specifically to examining deep learning methods for Covid-19 forecasting. In this paper, we fill the gap in the literature by reviewing and analyzing the current studies that use deep learning for Covid-19 forecasting. In our review, all published papers and preprints, discoverable through Google Scholar, for the period from Apr 1, 2020 to Feb 20, 2022 which describe deep learning approaches to forecasting Covid-19 were considered. Our search identified 152 studies, of which 53 passed the initial quality screening and were included in our survey. We propose a model-based taxonomy to categorize the literature. We describe each model and highlight its performance. Finally, the deficiencies of the existing approaches are identified and the necessary improvements for future research are elucidated. The study provides a gateway for researchers who are interested in forecasting Covid-19 using deep learning.
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Li Z, Zhang T. Analysis of a COVID-19 Epidemic Model with Seasonality. Bull Math Biol 2022; 84:146. [PMID: 36367626 PMCID: PMC9651129 DOI: 10.1007/s11538-022-01105-4] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2022] [Accepted: 11/02/2022] [Indexed: 11/13/2022]
Abstract
The statistics of COVID-19 cases exhibits seasonal fluctuations in many countries. In this paper, we propose a COVID-19 epidemic model with seasonality and define the basic reproduction number [Formula: see text] for the disease transmission. It is proved that the disease-free equilibrium is globally asymptotically stable when [Formula: see text], while the disease is uniformly persistent and there exists at least one positive periodic solution when [Formula: see text]. Numerically, we observe that there is a globally asymptotically stable positive periodic solution in the case of [Formula: see text]. Further, we conduct a case study of the COVID-19 transmission in the USA by using statistical data.
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Affiliation(s)
- Zhimin Li
- grid.25055.370000 0000 9130 6822Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7 Canada
| | - Tailei Zhang
- grid.440661.10000 0000 9225 5078School of Science, Chang’an University, Xi’an, 710064 China
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