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Das P, Igoe M, Lacy A, Farthing T, Timsina A, Lanzas C, Lenhart S, Odoi A, Lloyd AL. Modeling county level COVID-19 transmission in the greater St. Louis area: Challenges of uncertainty and identifiability when fitting mechanistic models to time-varying processes. Math Biosci 2024; 371:109181. [PMID: 38537734 DOI: 10.1016/j.mbs.2024.109181] [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: 11/29/2023] [Accepted: 03/13/2024] [Indexed: 04/12/2024]
Abstract
We use a compartmental model with a time-varying transmission parameter to describe county level COVID-19 transmission in the greater St. Louis area of Missouri and investigate the challenges in fitting such a model to time-varying processes. We fit this model to synthetic and real confirmed case and hospital discharge data from May to December 2020 and calculate uncertainties in the resulting parameter estimates. We also explore non-identifiability within the estimated parameter set. We find that the death rate of infectious non-hospitalized individuals, the testing parameter and the initial number of exposed individuals are not identifiable based on an investigation of correlation coefficients between pairs of parameter estimates. We also explore how this non-identifiability ties back into uncertainties in the estimated parameters and find that it inflates uncertainty in the estimates of our time-varying transmission parameter. However, we do find that R0 is not highly affected by non-identifiability of its constituent components and the uncertainties associated with the quantity are smaller than those of the estimated parameters. Parameter values estimated from data will always be associated with some uncertainty and our work highlights the importance of conducting these analyses when fitting such models to real data. Exploring identifiability and uncertainty is crucial in revealing how much we can trust the parameter estimates.
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Affiliation(s)
- Praachi Das
- Biomathematics Graduate Program, Department of Mathematics, North Carolina State University, Raleigh, NC, USA
| | - Morganne Igoe
- Department of Mathematics, University of Tennessee, Knoxville, TN, USA
| | - Alexanderia Lacy
- Department of Mathematics, University of Tennessee, Knoxville, TN, USA
| | - Trevor Farthing
- Department of Population Health and Pathobiology and Comparative Medicine Institute, North Carolina State University, Raleigh, NC, USA
| | - Archana Timsina
- Department of Population Health and Pathobiology and Comparative Medicine Institute, North Carolina State University, Raleigh, NC, USA
| | - Cristina Lanzas
- Department of Population Health and Pathobiology and Comparative Medicine Institute, North Carolina State University, Raleigh, NC, USA
| | - Suzanne Lenhart
- Department of Mathematics, University of Tennessee, Knoxville, TN, USA
| | - Agricola Odoi
- Department of Biomedical and Diagnostics Sciences, University of Tennessee, Knoxville, TN, USA
| | - Alun L Lloyd
- Biomathematics Graduate Program, Department of Mathematics, North Carolina State University, Raleigh, NC, USA.
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Rakhshan SA, Zaj M, Ghane FH, Nejad MS. Exploring the potential of learning methods and recurrent dynamic model with vaccination: A comparative case study of COVID-19 in Austria, Brazil, and China. Phys Rev E 2024; 109:014212. [PMID: 38366403 DOI: 10.1103/physreve.109.014212] [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: 06/14/2023] [Accepted: 12/11/2023] [Indexed: 02/18/2024]
Abstract
In order to effectively manage infectious diseases, it is crucial to understand the interplay between disease dynamics and human conduct. Various factors can impact the control of an epidemic, including social interventions, adherence to health protocols, mask-wearing, and vaccination. This article presents the development of an innovative hybrid model, known as the Combined Dynamic-Learning Model, that integrates classical recurrent dynamic models with four different learning methods. The model is composed of two approaches: The first approach introduces a traditional dynamic model that focuses on analyzing the impact of vaccination on the occurrence of an epidemic, and the second approach employs various learning methods to forecast the potential outcomes of an epidemic. Furthermore, our numerical results offer an interesting comparison between the traditional approach and modern learning techniques. Our classic dynamic model is a compartmental model that aims to analyze and forecast the diffusion of epidemics. The model we propose has a recurrent structure with piecewise constant parameters and includes compartments for susceptible, exposed, vaccinated, infected, and recovered individuals. This model can accurately mirror the dynamics of infectious diseases, which enables us to evaluate the impact of restrictive measures on the spread of diseases. We conduct a comprehensive dynamic analysis of our model. Additionally, we suggest an optimal numerical design to determine the parameters of the system. Also, we use regression tree learning, bidirectional long short-term memory, gated recurrent unit, and a combined deep learning method for training and evaluation of an epidemic. In the final section of our paper, we apply these methods to recently published data on COVID-19 in Austria, Brazil, and China from 26 February 2021 to 4 August 2021, which is when vaccination efforts began. To evaluate the numerical results, we utilized various metrics such as RMSE and R-squared. Our findings suggest that the dynamic model is ideal for long-term analysis, data fitting, and identifying parameters that impact epidemics. However, it is not as effective as the supervised learning method for making long-term forecasts. On the other hand, supervised learning techniques, compared to dynamic models, are more effective for predicting the spread of diseases, but not for analyzing the behavior of epidemics.
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Affiliation(s)
- Seyed Ali Rakhshan
- Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
| | - Marzie Zaj
- Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
| | | | - Mahdi Soltani Nejad
- Department of Railway Engineering, Iran University of Science and Technology, Tehran, Iran
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3
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Zavrakli E, Parnell A, Malone D, Duffy K, Dey S. Optimal age-specific vaccination control for COVID-19: An Irish case study. PLoS One 2023; 18:e0290974. [PMID: 37669287 PMCID: PMC10479919 DOI: 10.1371/journal.pone.0290974] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Accepted: 08/21/2023] [Indexed: 09/07/2023] Open
Abstract
The outbreak of a novel coronavirus causing severe acute respiratory syndrome in December 2019 has escalated into a worldwide pandemic. In this work, we propose a compartmental model to describe the dynamics of transmission of infection and use it to obtain the optimal vaccination control. The model accounts for the various stages of the vaccination, and the optimisation is focused on minimising the infections to protect the population and relieve the healthcare system. As a case study, we selected the Republic of Ireland. We use data provided by Ireland's COVID-19 Data-Hub and simulate the evolution of the pandemic with and without the vaccination in place for two different scenarios, one representative of a national lockdown situation and the other indicating looser restrictions in place. One of the main findings of our work is that the optimal approach would involve a vaccination programme where the older population is vaccinated in larger numbers earlier while simultaneously part of the younger population also gets vaccinated to lower the risk of transmission between groups. We compare our simulated results with those of the vaccination policy taken by the Irish government to explore the advantages of our optimisation method. Our comparison suggests that a similar reduction in cases may have been possible even with a reduced set of vaccinations available for use.
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Affiliation(s)
- Eleni Zavrakli
- Hamilton Institute, Maynooth University, Maynooth, Co. Kildare, Ireland
- Department of Mathematics and Statistics, Maynooth University, Maynooth, Co. Kildare, Ireland
- I-Form, Advanced Manufacturing Research Centre, Maynooth, Ireland
| | - Andrew Parnell
- Hamilton Institute, Maynooth University, Maynooth, Co. Kildare, Ireland
- Department of Mathematics and Statistics, Maynooth University, Maynooth, Co. Kildare, Ireland
- I-Form, Advanced Manufacturing Research Centre, Maynooth, Ireland
| | - David Malone
- Hamilton Institute, Maynooth University, Maynooth, Co. Kildare, Ireland
- Department of Mathematics and Statistics, Maynooth University, Maynooth, Co. Kildare, Ireland
| | - Ken Duffy
- Hamilton Institute, Maynooth University, Maynooth, Co. Kildare, Ireland
| | - Subhrakanti Dey
- Department of Electrical Engineering, Uppsala University, Uppsala, Sweden
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Johnston MD, Pell B, Rubel DA. A two-strain model of infectious disease spread with asymmetric temporary immunity periods and partial cross-immunity. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:16083-16113. [PMID: 37920004 DOI: 10.3934/mbe.2023718] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/04/2023]
Abstract
We introduce a two-strain model with asymmetric temporary immunity periods and partial cross-immunity. We derive explicit conditions for competitive exclusion and coexistence of the strains depending on the strain-specific basic reproduction numbers, temporary immunity periods, and degree of cross-immunity. The results of our bifurcation analysis suggest that, even when two strains share similar basic reproduction numbers and other epidemiological parameters, a disparity in temporary immunity periods and partial or complete cross-immunity can provide a significant competitive advantage. To analyze the dynamics, we introduce a quasi-steady state reduced model which assumes the original strain remains at its endemic steady state. We completely analyze the resulting reduced planar hybrid switching system using linear stability analysis, planar phase-plane analysis, and the Bendixson-Dulac criterion. We validate both the full and reduced models with COVID-19 incidence data, focusing on the Delta (B.1.617.2), Omicron (B.1.1.529), and Kraken (XBB.1.5) variants. These numerical studies suggest that, while early novel strains of COVID-19 had a tendency toward dramatic takeovers and extinction of ancestral strains, more recent strains have the capacity for co-existence.
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Affiliation(s)
- Matthew D Johnston
- Department of Mathematics + Computer Science, Lawrence Technological University, 21000 W 10 Mile Rd, Southfield, MI 48075, USA
| | - Bruce Pell
- Department of Mathematics + Computer Science, Lawrence Technological University, 21000 W 10 Mile Rd, Southfield, MI 48075, USA
| | - David A Rubel
- Department of Mathematics + Computer Science, Lawrence Technological University, 21000 W 10 Mile Rd, Southfield, MI 48075, USA
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5
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Davarci OO, Yang EY, Viguerie A, Yankeelov TE, Lorenzo G. Dynamic parameterization of a modified SEIRD model to analyze and forecast the dynamics of COVID-19 outbreaks in the United States. ENGINEERING WITH COMPUTERS 2023:1-25. [PMID: 37362241 PMCID: PMC10129322 DOI: 10.1007/s00366-023-01816-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/20/2022] [Accepted: 03/24/2023] [Indexed: 06/28/2023]
Abstract
The rapid spread of the numerous outbreaks of the coronavirus disease 2019 (COVID-19) pandemic has fueled interest in mathematical models designed to understand and predict infectious disease spread, with the ultimate goal of contributing to the decision making of public health authorities. Here, we propose a computational pipeline that dynamically parameterizes a modified SEIRD (susceptible-exposed-infected-recovered-deceased) model using standard daily series of COVID-19 cases and deaths, along with isolated estimates of population-level seroprevalence. We test our pipeline in five heavily impacted states of the US (New York, California, Florida, Illinois, and Texas) between March and August 2020, considering two scenarios with different calibration time horizons to assess the update in model performance as new epidemiologic data become available. Our results show a median normalized root mean squared error (NRMSE) of 2.38% and 4.28% in calibrating cumulative cases and deaths in the first scenario, and 2.41% and 2.30% when new data are assimilated in the second scenario, respectively. Then, 2-week (4-week) forecasts of the calibrated model resulted in median NRMSE of cumulative cases and deaths of 5.85% and 4.68% (8.60% and 17.94%) in the first scenario, and 1.86% and 1.93% (2.21% and 1.45%) in the second. Additionally, we show that our method provides significantly more accurate predictions of cases and deaths than a constant parameterization in the second scenario (p < 0.05). Thus, we posit that our methodology is a promising approach to analyze the dynamics of infectious disease outbreaks, and that our forecasts could contribute to designing effective pandemic-arresting public health policies. Supplementary Information The online version contains supplementary material available at 10.1007/s00366-023-01816-9.
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Affiliation(s)
- Orhun O. Davarci
- Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E 24th St, Austin, TX 78712-1229 USA
- Department of Biomedical Engineering, The University of Texas at Austin, Austin, TX USA
| | - Emily Y. Yang
- Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E 24th St, Austin, TX 78712-1229 USA
| | | | - Thomas E. Yankeelov
- Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E 24th St, Austin, TX 78712-1229 USA
- Department of Biomedical Engineering, The University of Texas at Austin, Austin, TX USA
- Livestrong Cancer Institutes, Dell Medical School, The University of Texas at Austin, Austin, TX USA
- Department of Diagnostic Medicine, The University of Texas at Austin, Austin, TX USA
- Department of Oncology, The University of Texas at Austin, Austin, TX USA
- Department of Imaging Physics, MD Anderson Cancer Center, Houston, TX USA
| | - Guillermo Lorenzo
- Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E 24th St, Austin, TX 78712-1229 USA
- Department of Civil Engineering and Architecture, University of Pavia, Pavia, Italy
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6
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Díaz-Seoane S, Sellán E, Villaverde AF. Structural Identifiability and Observability of Microbial Community Models. Bioengineering (Basel) 2023; 10:bioengineering10040483. [PMID: 37106670 PMCID: PMC10135947 DOI: 10.3390/bioengineering10040483] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2023] [Revised: 04/10/2023] [Accepted: 04/14/2023] [Indexed: 04/29/2023] Open
Abstract
Biological communities are populations of various species interacting in a common location. Microbial communities, which are formed by microorganisms, are ubiquitous in nature and are increasingly used in biotechnological and biomedical applications. They are nonlinear systems whose dynamics can be accurately described by models of ordinary differential equations (ODEs). A number of ODE models have been proposed to describe microbial communities. However, the structural identifiability and observability of most of them-that is, the theoretical possibility of inferring their parameters and internal states by observing their output-have not been determined yet. It is important to establish whether a model possesses these properties, because, in their absence, the ability of a model to make reliable predictions may be compromised. Hence, in this paper, we analyse these properties for the main families of microbial community models. We consider several dimensions and measurements; overall, we analyse more than a hundred different configurations. We find that some of them are fully identifiable and observable, but a number of cases are structurally unidentifiable and/or unobservable under typical experimental conditions. Our results help in deciding which modelling frameworks may be used for a given purpose in this emerging area, and which ones should be avoided.
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Affiliation(s)
- Sandra Díaz-Seoane
- Department of Systems Engineering & Control, Universidade de Vigo, 36310 Vigo, Spain
| | - Elena Sellán
- Department of Systems Engineering & Control, Universidade de Vigo, 36310 Vigo, Spain
| | - Alejandro F Villaverde
- Department of Systems Engineering & Control, Universidade de Vigo, 36310 Vigo, Spain
- CITMAga, 15782 Santiago de Compostela, Spain
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Kiss IZ, Simon PL. On Parameter Identifiability in Network-Based Epidemic Models. Bull Math Biol 2023; 85:18. [PMID: 36705777 PMCID: PMC9880946 DOI: 10.1007/s11538-023-01121-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/15/2022] [Accepted: 01/03/2023] [Indexed: 01/28/2023]
Abstract
Modelling epidemics on networks represents an important departure from classical compartmental models which assume random mixing. However, the resulting models are high-dimensional and their analysis is often out of reach. It turns out that mean-field models, low-dimensional systems of differential equations, whose variables are carefully chosen expected quantities from the exact model provide a good approximation and incorporate explicitly some network properties. Despite the emergence of such mean-field models, there has been limited work on investigating whether these can be used for inference purposes. In this paper, we consider network-based mean-field models and explore the problem of parameter identifiability when observations about an epidemic are available. Making use of the analytical tractability of most network-based mean-field models, e.g. explicit analytical expressions for leading eigenvalue and final epidemic size, we set up the parameter identifiability problem as finding the solution or solutions of a system of coupled equations. More precisely, subject to observing/measuring growth rate and final epidemic size, we seek to identify parameter values leading to these measurements. We are particularly concerned with disentangling transmission rate from the network density. To do this, we give a condition for practical identifiability and we find that except for the simplest model, parameters cannot be uniquely determined, that is, they are practically unidentifiable. This means that there exist multiple solutions (a manifold of infinite measure) which give rise to model output that is close to the data. Identifying, formalising and analytically describing this problem should lead to a better appreciation of the complexity involved in fitting models with many parameters to data.
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Affiliation(s)
- István Z. Kiss
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH UK
| | - Péter L. Simon
- Institute of Mathematics, Eötvös Loránd University, Budapest, Hungary ,Numerical Analysis and Large Networks Research Group, ELKH-ELTE, Budapest, Hungary
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Multi-objective T-S fuzzy control of Covid-19 spread model: An LMI approach. Biomed Signal Process Control 2023; 79:104107. [PMID: 35996470 PMCID: PMC9385779 DOI: 10.1016/j.bspc.2022.104107] [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: 05/01/2022] [Revised: 08/12/2022] [Accepted: 08/15/2022] [Indexed: 11/26/2022]
Abstract
Due to the importance of control actions in spreading coronavirus disease, this paper is devoted to first modeling and then proposing an appropriate controller for this model. In the modeling procedure, we used a nonlinear mathematical model for the covid-19 outbreak to form a T-S fuzzy model. Then, for proposing the suitable controller, multiple optimization techniques including Linear Quadratic Regulator (LQR) and mixed H2-H∞ are taken into account. The mentioned controller is chosen because the model of corona-virus spread is not only full of disturbances like a sudden increase in infected people, but also noises such as unavailability of the exact number of each compartment. The controller is simulated accordingly to validate the results of mathematical calculations, and a comparative analysis is presented to investigate the different situations of the problem. Comparing the results of controlled and uncontrolled situations, it can be observed that we can tackle the devastating hazards of the covid-19 outbreak effectively if the suggested approaches and policies of controlling interventions are executed, appropriately.
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Naz R, Torrisi M. The Transmission Dynamics of a Compartmental Epidemic Model for COVID-19 with the Asymptomatic Population via Closed-Form Solutions. Vaccines (Basel) 2022; 10:vaccines10122162. [PMID: 36560572 PMCID: PMC9788203 DOI: 10.3390/vaccines10122162] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2022] [Revised: 12/08/2022] [Accepted: 12/13/2022] [Indexed: 12/23/2022] Open
Abstract
Unlike previous viral diseases, COVID-19 has an "asymptomatic" group that has no symptoms but can still spread the disease to others at the same rate as symptomatic patients who are infected. In the literature, the mass action or standard incidence rates are considered for compartmental models with asymptomatic compartment for studying the transmission dynamics of COVID-19, but the quarantined adjusted incidence rate is not. To bridge this gap, we developed a Susceptible Asymptomatic Infectious Quarantined (SAIQ) model with a Quarantine-Adjusted (QA) incidence to investigate the emergence and containment of COVID-19. COVID-19 models are investigated using various methods, but only a few studies take into account closed-form solutions. The knowledge of closed-form solutions simplifies the construction of the various epidemic indicators that describe the epidemic phenomenon and makes the sensitivity analysis to variations in the data under consideration possible. The closed-form solutions of the systems of four nonlinear first-order ordinary differential equations (ODEs) are established. The Epidemic Peak (EP), Force of Infection (FOI) and Rate of Infection (ROI) are the important indicators for the control and prevention of disease. We examined these indicators using closed-form solutions and particular parameter values. Different disease control scenarios are thoroughly examined. The four scenarios to analyze COVID-19 propagation and containment are (i) lockdown, (ii) quarantine and other preventative measures, (iii) stabilizing the basic reproduction rate to a level where the pandemic can be contained and (iv) containing the epidemic through an appropriate combination of lockdown, quarantine and other preventative measures.
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Affiliation(s)
- Rehana Naz
- Department of Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan
- Correspondence:
| | - Mariano Torrisi
- Dipartimento di Matematica ed Informatica, Università di Catania Viale A. Doria, 6, I-95125 Catania, Italy
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The Structural Identifiability of a Humidity-Driven Epidemiological Model of Influenza Transmission. Viruses 2022; 14:v14122795. [PMID: 36560799 PMCID: PMC9780807 DOI: 10.3390/v14122795] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2022] [Revised: 12/08/2022] [Accepted: 12/10/2022] [Indexed: 12/23/2022] Open
Abstract
Influenza epidemics cause considerable morbidity and mortality every year worldwide. Climate-driven epidemiological models are mainstream tools to understand seasonal transmission dynamics and predict future trends of influenza activity, especially in temperate regions. Testing the structural identifiability of these models is a fundamental prerequisite for the model to be applied in practice, by assessing whether the unknown model parameters can be uniquely determined from epidemic data. In this study, we applied a scaling method to analyse the structural identifiability of four types of commonly used humidity-driven epidemiological models. Specifically, we investigated whether the key epidemiological parameters (i.e., infectious period, the average duration of immunity, the average latency period, and the maximum and minimum daily basic reproductive number) can be uniquely determined simultaneously when prevalence data is observable. We found that each model is identifiable when the prevalence of infection is observable. The structural identifiability of these models will lay the foundation for testing practical identifiability in the future using synthetic prevalence data when considering observation noise. In practice, epidemiological models should be examined with caution before using them to estimate model parameters from epidemic data.
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Tuncer N, Timsina A, Nuno M, Chowell G, Martcheva M. Parameter identifiability and optimal control of an SARS-CoV-2 model early in the pandemic. JOURNAL OF BIOLOGICAL DYNAMICS 2022; 16:412-438. [PMID: 35635313 DOI: 10.1080/17513758.2022.2078899] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/11/2021] [Accepted: 04/28/2022] [Indexed: 06/15/2023]
Abstract
We fit an SARS-CoV-2 model to US data of COVID-19 cases and deaths. We conclude that the model is not structurally identifiable. We make the model identifiable by prefixing some of the parameters from external information. Practical identifiability of the model through Monte Carlo simulations reveals that two of the parameters may not be practically identifiable. With thus identified parameters, we set up an optimal control problem with social distancing and isolation as control variables. We investigate two scenarios: the controls are applied for the entire duration and the controls are applied only for the period of time. Our results show that if the controls are applied early in the epidemic, the reduction in the infected classes is at least an order of magnitude higher compared to when controls are applied with 2-week delay. Further, removing the controls before the pandemic ends leads to rebound of the infected classes.
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Affiliation(s)
- Necibe Tuncer
- Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL, USA
| | - Archana Timsina
- Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL, USA
| | - Miriam Nuno
- Department of Biostatistics, University of California, Davis, CA, USA
| | - Gerardo Chowell
- Department of Population Health Sciences, Georgia State University, Atlanta, GA, USA
| | - Maia Martcheva
- Department of Mathematics, University of Florida, Gainesville, FL, USA
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12
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Dankwa EA, Brouwer AF, Donnelly CA. Structural identifiability of compartmental models for infectious disease transmission is influenced by data type. Epidemics 2022; 41:100643. [PMID: 36308994 PMCID: PMC9772104 DOI: 10.1016/j.epidem.2022.100643] [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: 01/31/2022] [Revised: 09/05/2022] [Accepted: 10/14/2022] [Indexed: 12/29/2022] Open
Abstract
If model identifiability is not confirmed, inferences from infectious disease transmission models may not be reliable, so they might result in misleading recommendations. Structural identifiability analysis characterises whether it is possible to obtain unique solutions for all unknown model parameters, given the model structure. In this work, we studied the structural identifiability of some typical deterministic compartmental models for infectious disease transmission, focusing on the influence of the data type considered as model output on the identifiability of unknown model parameters, including initial conditions. We defined 26 model versions, each having a unique combination of underlying compartmental structure and data type(s) considered as model output(s). Four compartmental model structures and three common data types in disease surveillance (incidence, prevalence and detected vector counts) were studied. The structural identifiability of some parameters varied depending on the type of model output. In general, models with multiple data types as outputs had more structurally identifiable parameters, than did models with a single data type as output. This study highlights the importance of a careful consideration of data types as an integral part of the inference process with compartmental infectious disease transmission models.
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Affiliation(s)
- Emmanuelle A Dankwa
- Department of Statistics, University of Oxford, 24-29 St Giles', Oxford, United Kingdom
| | - Andrew F Brouwer
- Department of Epidemiology, University of Michigan, 1415 Washington Heights, Ann Arbor, MI 48109-2029, USA
| | - Christl A Donnelly
- Department of Statistics, University of Oxford, 24-29 St Giles', Oxford, United Kingdom; Department of Infectious Disease Epidemiology, Faculty of Medicine, School of Public Health, Imperial College London, United Kingdom.
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Valeriano JP, Cintra PH, Libotte G, Reis I, Fontinele F, Silva R, Malta S. Sequential time-window learning with approximate Bayesian computation: an application to epidemic forecasting. NONLINEAR DYNAMICS 2022; 111:549-558. [PMID: 36188164 PMCID: PMC9510304 DOI: 10.1007/s11071-022-07865-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/21/2022] [Accepted: 09/02/2022] [Indexed: 06/16/2023]
Abstract
UNLABELLED The long duration of the COVID-19 pandemic allowed for multiple bursts in the infection and death rates, the so-called epidemic waves. This complex behavior is no longer tractable by simple compartmental model and requires more sophisticated mathematical techniques for analyzing epidemic data and generating reliable forecasts. In this work, we propose a framework for analyzing complex dynamical systems by dividing the data in consecutive time-windows to be separately analyzed. We fit parameters for each time-window through an approximate Bayesian computation (ABC) algorithm, and the posterior distribution of parameters obtained for one window is used as the prior distribution for the next window. This Bayesian learning approach is tested with data on COVID-19 cases in multiple countries and is shown to improve ABC performance and to produce good short-term forecasting. SUPPLEMENTARY INFORMATION The online version contains supplementary material available at 10.1007/s11071-022-07865-x.
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Affiliation(s)
- João Pedro Valeriano
- Instituto de Física Teórica, Universidade Estadual Paulista, R. Dr. Bento Teobaldo Ferraz, 271, Bloco 2, Barra Funda, São Paulo, SP 01140-070 Brazil
| | - Pedro Henrique Cintra
- Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 777, Campinas, SP 13083-859 Brazil
| | - Gustavo Libotte
- Laboratório Nacional de Computção Científica, Av. Getulio Vargas, 333, Petrópolis, RJ 25651-076 Brazil
- Present Address: Department of Computational Modeling, Polytechnic Institute, Rio de Janeiro State University, Nova Friburgo, Brazil
| | - Igor Reis
- Instituto de Física de São Carlos, Universidade de São Paulo, Av. Trab. São Carlense, 400 - Parque Arnold Schimidt, São Carlos, SP 13566-590 Brazil
| | - Felipe Fontinele
- Department of Physics, University of Alberta, 116 St & 85 Ave, Edmonton, AB T6G 2E1 Canada
| | - Renato Silva
- Laboratório Nacional de Computção Científica, Av. Getulio Vargas, 333, Petrópolis, RJ 25651-076 Brazil
| | - Sandra Malta
- Laboratório Nacional de Computção Científica, Av. Getulio Vargas, 333, Petrópolis, RJ 25651-076 Brazil
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Brouwer AF. Why the Spectral Radius? An intuition-building introduction to the basic reproduction number. Bull Math Biol 2022; 84:96. [PMID: 35930076 PMCID: PMC9355935 DOI: 10.1007/s11538-022-01057-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2022] [Accepted: 07/21/2022] [Indexed: 11/26/2022]
Abstract
The basic reproduction number [Formula: see text] is a fundamental concept in mathematical epidemiology and infectious disease modeling. Loosely speaking, it describes the number of people that an infectious person is expected to infect. The basic reproduction number has profound implications for epidemic trajectories and disease control strategies. It is well known that the basic reproduction number can be calculated as the spectral radius of the next generation matrix, but why this is the case may not be intuitively obvious. Here, we walk through how the discrete, next generation process connects to the ordinary differential equation disease system of interest, linearized at the disease-free equilibrium. Then, we use linear algebra to develop a geometric explanation of why the spectral radius of the next generation matrix is an epidemic threshold. Finally, we work through a series of examples that help to build familiarity with the kinds of patterns that arise in parameter combinations produced by the next generation method. This article is intended to help new infectious disease modelers develop intuition for the form and interpretation of the basic reproduction number in their disease systems of interest.
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Affiliation(s)
- Andrew F Brouwer
- Department of Epidemiology, University of Michigan, 1415 Washington Heights, Ann Arbor, MI, 48109, USA.
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15
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Mullin S, Wyk BV, Asher JL, Compton SR, Allore HG, Zeiss CJ. Modeling pandemic to endemic patterns of SARS-CoV-2 transmission using parameters estimated from animal model data. PNAS NEXUS 2022; 1:pgac096. [PMID: 35799833 PMCID: PMC9254158 DOI: 10.1093/pnasnexus/pgac096] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/24/2022] [Accepted: 06/29/2022] [Indexed: 02/06/2023]
Abstract
The contours of endemic coronaviral disease in humans and other animals are shaped by the tendency of coronaviruses to generate new variants superimposed upon nonsterilizing immunity. Consequently, patterns of coronaviral reinfection in animals can inform the emerging endemic state of the SARS-CoV-2 pandemic. We generated controlled reinfection data after high and low risk natural exposure or heterologous vaccination to sialodacryoadenitis virus (SDAV) in rats. Using deterministic compartmental models, we utilized in vivo estimates from these experiments to model the combined effects of variable transmission rates, variable duration of immunity, successive waves of variants, and vaccination on patterns of viral transmission. Using rat experiment-derived estimates, an endemic state achieved by natural infection alone occurred after a median of 724 days with approximately 41.3% of the population susceptible to reinfection. After accounting for translationally altered parameters between rat-derived data and human SARS-CoV-2 transmission, and after introducing vaccination, we arrived at a median time to endemic stability of 1437 (IQR = 749.25) days with a median 15.4% of the population remaining susceptible. We extended the models to introduce successive variants with increasing transmissibility and included the effect of varying duration of immunity. As seen with endemic coronaviral infections in other animals, transmission states are altered by introduction of new variants, even with vaccination. However, vaccination combined with natural immunity maintains a lower prevalence of infection than natural infection alone and provides greater resilience against the effects of transmissible variants.
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Affiliation(s)
- Sarah Mullin
- Yale Center for Medical Informatics, Yale School of Medicine, New Haven, CT 06520, USA
| | - Brent Vander Wyk
- Department of Internal Medicine, Yale School of Medicine, New Haven, CT 06520, USA
| | - Jennifer L Asher
- Department of Comparative Medicine, Yale School of Medicine, New Haven, CT 06520, USA
| | - Susan R Compton
- Department of Comparative Medicine, Yale School of Medicine, New Haven, CT 06520, USA
| | - Heather G Allore
- Department of Internal Medicine, Yale School of Medicine, New Haven, CT 06520, USA
- Department of Biostatistics, Yale School of Public Health, New Haven, CT 06520, USA
| | - Caroline J Zeiss
- Department of Comparative Medicine, Yale School of Medicine, New Haven, CT 06520, USA
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16
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Mullin S, Wyk BV, Asher JL, Compton SR, Allore HG, Zeiss CJ. Modeling pandemic to endemic patterns of SARS-CoV-2 transmission using parameters estimated from animal model data. PNAS NEXUS 2022; 1:pgac096. [PMID: 35799833 PMCID: PMC9254158 DOI: 10.1093/pnasnexus/pgac096;] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Figures] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
Abstract
The contours of endemic coronaviral disease in humans and other animals are shaped by the tendency of coronaviruses to generate new variants superimposed upon nonsterilizing immunity. Consequently, patterns of coronaviral reinfection in animals can inform the emerging endemic state of the SARS-CoV-2 pandemic. We generated controlled reinfection data after high and low risk natural exposure or heterologous vaccination to sialodacryoadenitis virus (SDAV) in rats. Using deterministic compartmental models, we utilized in vivo estimates from these experiments to model the combined effects of variable transmission rates, variable duration of immunity, successive waves of variants, and vaccination on patterns of viral transmission. Using rat experiment-derived estimates, an endemic state achieved by natural infection alone occurred after a median of 724 days with approximately 41.3% of the population susceptible to reinfection. After accounting for translationally altered parameters between rat-derived data and human SARS-CoV-2 transmission, and after introducing vaccination, we arrived at a median time to endemic stability of 1437 (IQR = 749.25) days with a median 15.4% of the population remaining susceptible. We extended the models to introduce successive variants with increasing transmissibility and included the effect of varying duration of immunity. As seen with endemic coronaviral infections in other animals, transmission states are altered by introduction of new variants, even with vaccination. However, vaccination combined with natural immunity maintains a lower prevalence of infection than natural infection alone and provides greater resilience against the effects of transmissible variants.
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Affiliation(s)
- Sarah Mullin
- Yale Center for Medical Informatics, Yale School of Medicine, New Haven, CT 06520, USA
| | - Brent Vander Wyk
- Department of Internal Medicine, Yale School of Medicine, New Haven, CT 06520, USA
| | - Jennifer L Asher
- Department of Comparative Medicine, Yale School of Medicine, New Haven, CT 06520, USA
| | - Susan R Compton
- Department of Comparative Medicine, Yale School of Medicine, New Haven, CT 06520, USA
| | - Heather G Allore
- Department of Internal Medicine, Yale School of Medicine, New Haven, CT 06520, USA,Department of Biostatistics, Yale School of Public Health, New Haven, CT 06520, USA
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17
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Non-Linear Observer Design with Laguerre Polynomials. ENTROPY 2022; 24:e24070913. [PMID: 35885136 PMCID: PMC9316067 DOI: 10.3390/e24070913] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/22/2022] [Revised: 06/28/2022] [Accepted: 06/28/2022] [Indexed: 01/27/2023]
Abstract
In this paper, a methodology for a non-linear system state estimation is demonstrated, exploiting the input and parameter observability. For this purpose, the initial system is transformed into the canonical observability form, and the function that aggregates the non-linear dynamics of the system, which may be unknown or difficult to be computed, is approximated by a linear combination of Laguerre polynomials. Hence, the system identification translates into the estimation of the parameters involved in the linear combination in order for the system to be observable. For the validation of the elaborated observer, we consider a biological model from the literature, investigating whether it is practically possible to infer its states, taking into account the new coordinates to design the appropriate observer of the system states. Through simulations, we investigate the parameter settings under which the new observer can identify the state of the system. More specifically, as the parameter θ increases, the system converges more quickly to the steady-state, decreasing the respective distance from the system’s initial state. As for the first state, the estimation error is in the order of 10−2 for θ=15, and assuming c0={0,1},c1=1. Under the same conditions, the estimation error of the system’s second state is in the order of 10−1, setting a performance difference of 10−1 in relation to the first state. The outcomes show that the proposed observer’s performance can be further improved by selecting even higher values of θ. Hence, the system is observable through the measurement output.
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18
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Barbosa Libotte G, dos Anjos L, Célia Cerqueira de Almeida R, Mara Cardoso Malta S, de Andrade Medronho R. Impacts of a delayed and slow-paced vaccination on cases and deaths during the COVID-19 pandemic: a modelling study. J R Soc Interface 2022; 19:20220275. [PMID: 35611617 PMCID: PMC9131126 DOI: 10.1098/rsif.2022.0275] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2022] [Accepted: 05/03/2022] [Indexed: 11/21/2022] Open
Abstract
In Brazil, vaccination has always cut across party political and ideological lines, which has delayed its start and brought the whole process into disrepute. Such divergences put the immunization of the population in the background and create additional hurdles beyond the pandemic, mistrust and scepticism over vaccines. We conduct a mathematical modelling study to analyse the impacts of late vaccination along with slowly increasing coverage, as well as how harmful it would be if part of the population refused to get vaccinated or missed the second dose. We analyse data from confirmed cases, deaths and vaccination in the state of Rio de Janeiro in the period between 10 March 2020 and 27 October 2021. We estimate that if the start of vaccination had been 30 days earlier, combined with efforts to drive vaccination rates up, about 31 657 deaths could have been avoided. In addition, the slow pace of vaccination and the low demand for the second dose could cause a resurgence of cases as early as 2022. Even when reaching the expected vaccination coverage for the first dose, it is still challenging to increase adherence to the second dose and maintain a high vaccination rate to avoid new outbreaks.
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Affiliation(s)
| | - Lucas dos Anjos
- National Laboratory for Scientific Computing, Petrópolis, Rio de Janeiro, 25651-075 Brazil
| | | | | | - Roberto de Andrade Medronho
- Institute of Collective Health, School of Medicine, Federal University of Rio de Janeiro, Rio de Janeiro, 21941-598 Brazil
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19
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Libotte GB, dos Anjos L, Almeida RCC, Malta SMC, Silva RS. Framework for enhancing the estimation of model parameters for data with a high level of uncertainty. NONLINEAR DYNAMICS 2022; 107:1919-1936. [PMID: 35017792 PMCID: PMC8736321 DOI: 10.1007/s11071-021-07069-9] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2021] [Accepted: 11/15/2021] [Indexed: 05/07/2023]
Abstract
Reliable data are essential to obtain adequate simulations for forecasting the dynamics of epidemics. In this context, several political, economic, and social factors may cause inconsistencies in the reported data, which reflect the capacity for realistic simulations and predictions. In the case of COVID-19, for example, such uncertainties are mainly motivated by large-scale underreporting of cases due to reduced testing capacity in some locations. In order to mitigate the effects of noise in the data used to estimate parameters of models, we propose strategies capable of improving the ability to predict the spread of the diseases. Using a compartmental model in a COVID-19 study case, we show that the regularization of data by means of Gaussian process regression can reduce the variability of successive forecasts, improving predictive ability. We also present the advantages of adopting parameters of compartmental models that vary over time, in detriment to the usual approach with constant values.
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Affiliation(s)
- Gustavo B. Libotte
- National Laboratory for Scientific Computing, Getúlio Vargas Av., 333, Quitandinha, Petrópolis, Rio de Janeiro, Brazil
| | - Lucas dos Anjos
- National Laboratory for Scientific Computing, Getúlio Vargas Av., 333, Quitandinha, Petrópolis, Rio de Janeiro, Brazil
| | - Regina C. C. Almeida
- National Laboratory for Scientific Computing, Getúlio Vargas Av., 333, Quitandinha, Petrópolis, Rio de Janeiro, Brazil
| | - Sandra M. C. Malta
- National Laboratory for Scientific Computing, Getúlio Vargas Av., 333, Quitandinha, Petrópolis, Rio de Janeiro, Brazil
| | - Renato S. Silva
- National Laboratory for Scientific Computing, Getúlio Vargas Av., 333, Quitandinha, Petrópolis, Rio de Janeiro, Brazil
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20
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Péni T, Szederkényi G. Convex output feedback model predictive control for mitigation of COVID-19 pandemic. ANNUAL REVIEWS IN CONTROL 2021; 52:543-553. [PMID: 34720662 PMCID: PMC8549322 DOI: 10.1016/j.arcontrol.2021.10.003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/29/2021] [Revised: 07/07/2021] [Accepted: 10/05/2021] [Indexed: 06/13/2023]
Abstract
In this paper, a model predictive control approach is proposed for epidemic mitigation. The disease spreading dynamics is described by an 8-compartment smooth nonlinear model of the COVID-19 pandemic in Hungary known from the literature, where the manipulable control input is the stringency of the introduced non-pharmaceutical measures. It is assumed that only the number of hospitalized people is measured on-line, and the other state variables are computed using a state observer which is based on the dynamic inversion of a linear sub-system of the model. The objective function contains a measure of the direct harmful consequences of the restrictions, and the constraints refer to input bounds and to the capacity of the healthcare system. By exploiting the special properties of the model, the nonlinear optimization problem required by the control design is reformulated to convex tasks, allowing a computationally efficient solution. Two approaches are proposed: the first finds a suboptimal solution by geometric programming, while the second one further simplifies the problem and transforms it to a linear programming task. Simulations show that both suboptimal solutions fulfill the design specifications even in the presence of parameter uncertainties.
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Affiliation(s)
- T Péni
- Institute for Computer Science and Control (SZTAKI), Eötvös Lóránd Research Network (ELKH), H-1111, Kende u. 13-17., Budapest, Hungary
| | - G Szederkényi
- Pázmány Péter Catholic University, H-1083 Práter u. 50/a, Budapest, Hungary
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21
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Zhang S, Ponce J, Zhang Z, Lin G, Karniadakis G. An integrated framework for building trustworthy data-driven epidemiological models: Application to the COVID-19 outbreak in New York City. PLoS Comput Biol 2021; 17:e1009334. [PMID: 34495965 PMCID: PMC8452065 DOI: 10.1371/journal.pcbi.1009334] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/10/2021] [Revised: 09/20/2021] [Accepted: 08/10/2021] [Indexed: 12/11/2022] Open
Abstract
Epidemiological models can provide the dynamic evolution of a pandemic but they are based on many assumptions and parameters that have to be adjusted over the time the pandemic lasts. However, often the available data are not sufficient to identify the model parameters and hence infer the unobserved dynamics. Here, we develop a general framework for building a trustworthy data-driven epidemiological model, consisting of a workflow that integrates data acquisition and event timeline, model development, identifiability analysis, sensitivity analysis, model calibration, model robustness analysis, and projection with uncertainties in different scenarios. In particular, we apply this framework to propose a modified susceptible-exposed-infectious-recovered (SEIR) model, including new compartments and model vaccination in order to project the transmission dynamics of COVID-19 in New York City (NYC). We find that we can uniquely estimate the model parameters and accurately project the daily new infection cases, hospitalizations, and deaths, in agreement with the available data from NYC's government's website. In addition, we employ the calibrated data-driven model to study the effects of vaccination and timing of reopening indoor dining in NYC.
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Affiliation(s)
- Sheng Zhang
- Department of Mathematics, Purdue University, West Lafayette, Indiana, United States
| | - Joan Ponce
- Department of Mathematics, Purdue University, West Lafayette, Indiana, United States
| | - Zhen Zhang
- Division of Applied Mathematics and School of Engineering, Brown University, Providence, Rhode Island, United States
| | - Guang Lin
- Department of Mathematics, Purdue University, West Lafayette, Indiana, United States
- School of Mechanical Engineering, Purdue University, West Lafayette, Indiana, United States
| | - George Karniadakis
- Division of Applied Mathematics and School of Engineering, Brown University, Providence, Rhode Island, United States
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22
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Lu M, Ishwaran H. Cure and death play a role in understanding dynamics for COVID-19: Data-driven competing risk compartmental models, with and without vaccination. PLoS One 2021; 16:e0254397. [PMID: 34264960 PMCID: PMC8282006 DOI: 10.1371/journal.pone.0254397] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2021] [Accepted: 06/25/2021] [Indexed: 01/24/2023] Open
Abstract
Several factors have played a strong role in influencing the dynamics of COVID-19 in the U.S. One being the economy, where a tug of war has existed between lockdown measures to control disease versus loosening of restrictions to address economic hardship. A more recent effect has been availability of vaccines and the mass vaccination efforts of 2021. In order to address the challenges in analyzing this complex process, we developed a competing risk compartmental model framework with and without vaccination compartment. This framework separates instantaneous risk of removal for an infectious case into competing risks of cure and death, and when vaccinations are present, the vaccinated individual can also achieve immunity before infection. Computations are performed using a simple discrete time algorithm that utilizes a data driven contact rate. Using population level pre-vaccination data, we are able to identify and characterize three wave patterns in the U.S. Estimated mortality rates for second and third waves are 1.7%, which is a notable decrease from 8.5% of a first wave observed at onset of disease. This analysis reveals the importance cure time has on infectious duration and disease transmission. Using vaccination data from 2021, we find a fourth wave, however the effect of this wave is suppressed due to vaccine effectiveness. Parameters playing a crucial role in this modeling were a lower cure time and a signficantly lower mortality rate for the vaccinated.
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Affiliation(s)
- Min Lu
- Department of Public Health Sciences, Miller School of Medicine, University of Miami, Miami, FL, United States of America
| | - Hemant Ishwaran
- Department of Public Health Sciences, Miller School of Medicine, University of Miami, Miami, FL, United States of America
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23
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Bos WJT, Bertoglio JP, Gostiaux L. Modeling the role of clusters and diffusion in the evolution of COVID-19 infections during lock-down. COMPUTATIONAL MECHANICS 2021; 67:1485-1496. [PMID: 33746320 PMCID: PMC7957454 DOI: 10.1007/s00466-021-01999-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/06/2021] [Accepted: 02/24/2021] [Indexed: 06/12/2023]
Abstract
The dynamics of the spread of epidemics, such as the recent outbreak of the SARS-CoV-2 virus, is highly nonlinear and therefore difficult to predict. As time evolves in the present pandemic, it appears more and more clearly that a clustered dynamics is a key element of the description. This means that the disease rapidly evolves within spatially localized networks, that diffuse and eventually create new clusters. We improve upon the simplest possible compartmental model, the SIR model, by adding an additional compartment associated with the clustered individuals. This sophistication is compatible with more advanced compartmental models and allows, at the lowest level of complexity, to leverage the well-mixedness assumption. The so-obtained SBIR model takes into account the effect of inhomogeneity on epidemic spreading, and compares satisfactorily with results on the pandemic propagation in a number of European countries, during and immediately after lock-down. Especially, the decay exponent of the number of new cases after the first peak of the epidemic is captured without the need to vary the coefficients of the model with time. We show that this decay exponent is directly determined by the diffusion of the ensemble of clustered individuals and can be related to a global reproduction number, that overrides the classical, local reproduction number.
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Affiliation(s)
- Wouter J. T. Bos
- Univ Lyon, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, Laboratoire de Mécanique des Fluides et d’Acoustique, UMR 5509, 36 Avenue Guy de Collongue, F-69134 Ecully, France
| | - Jean-Pierre Bertoglio
- Univ Lyon, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, Laboratoire de Mécanique des Fluides et d’Acoustique, UMR 5509, 36 Avenue Guy de Collongue, F-69134 Ecully, France
| | - Louis Gostiaux
- Univ Lyon, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, Laboratoire de Mécanique des Fluides et d’Acoustique, UMR 5509, 36 Avenue Guy de Collongue, F-69134 Ecully, France
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24
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Niazi MUB, Kibangou A, Canudas-de-Wit C, Nikitin D, Tumash L, Bliman PA. Modeling and control of epidemics through testing policies. ANNUAL REVIEWS IN CONTROL 2021; 52:554-572. [PMID: 34664008 PMCID: PMC8514419 DOI: 10.1016/j.arcontrol.2021.09.004] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2020] [Revised: 07/21/2021] [Accepted: 09/16/2021] [Indexed: 05/02/2023]
Abstract
Testing is a crucial control mechanism in the beginning phase of an epidemic when the vaccines are not yet available. It enables the public health authority to detect and isolate the infected cases from the population, thereby limiting the disease transmission to susceptible people. However, despite the significance of testing in epidemic control, the recent literature on the subject lacks a control-theoretic perspective. In this paper, an epidemic model is proposed that incorporates the testing rate as a control input and differentiates the undetected infected from the detected infected cases, who are assumed to be removed from the disease spreading process in the population. After estimating the model on the data corresponding to the beginning phase of COVID-19 in France, two testing policies are proposed: the so-called best-effort strategy for testing (BEST) and constant optimal strategy for testing (COST). The BEST policy is a suppression strategy that provides a minimum testing rate that stops the growth of the epidemic when implemented. The COST policy, on the other hand, is a mitigation strategy that provides an optimal value of testing rate minimizing the peak value of the infected population when the total stockpile of tests is limited. Both testing policies are evaluated by their impact on the number of active intensive care unit (ICU) cases and the cumulative number of deaths for the COVID-19 case of France.
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Affiliation(s)
| | - Alain Kibangou
- Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-Lab, 38000 Grenoble, France
| | | | - Denis Nikitin
- Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-Lab, 38000 Grenoble, France
| | - Liudmila Tumash
- Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-Lab, 38000 Grenoble, France
| | - Pierre-Alexandre Bliman
- Sorbonne Université, Université Paris-Diderot SPC, Inria, CNRS, Laboratoire Jacques-Louis Lions, équipe Mamba, 75005 Paris, France
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25
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Tutsoy O, Çolak Ş, Polat A, Balikci K. A Novel Parametric Model for the Prediction and Analysis of the COVID-19 Casualties. IEEE ACCESS : PRACTICAL INNOVATIONS, OPEN SOLUTIONS 2020; 8:193898-193906. [PMID: 34976560 PMCID: PMC8675544 DOI: 10.1109/access.2020.3033146] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2020] [Accepted: 10/19/2020] [Indexed: 05/03/2023]
Abstract
Coronavirus disease (COVID-19) outbreak has affected billions of people, where millions of them have been infected and thousands of them have lost their lives. In addition, to constraint the spread of the virus, economies have been shut down, curfews and restrictions have interrupted the social lives. Currently, the key question in minds is the future impacts of the virus on the people. It is a fact that the parametric modelling and analyses of the pandemic viruses are able to provide crucial information about the character and also future behaviour of the viruses. This paper initially reviews and analyses the Susceptible-Infected-Recovered (SIR) model, which is extensively considered for the estimation of the COVID-19 casualties. Then, this paper introduces a novel comprehensive higher-order, multi-dimensional, strongly coupled, and parametric Suspicious-Infected-Death (SpID) model. The mathematical analysis results performed by using the casualties in Turkey show that the COVID-19 dynamics are inside the slightly oscillatory, stable (bounded) region, although some of the dynamics are close to the instability region (unbounded). However, analysis with the data just after lifting the restrictions reveals that the dynamics of the COVID-19 are moderately unstable, which would blow up if no actions are taken. The developed model estimates that the number of the infected and death individuals will converge zero around 300 days whereas the number of the suspicious individuals will require about a thousand days to be minimized under the current conditions. Even though the developed model is used to estimate the casualties in Turkey, it can be easily trained with the data from the other countries and used for the estimation of the corresponding COVID-19 casualties.
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Affiliation(s)
- Onder Tutsoy
- Department of Electrical–Electronics EngineeringAdana Alparslan Türkeş Science and Technology University01250AdanaTurkey
| | - Şule Çolak
- Department of Electrical–Electronics EngineeringAdana Alparslan Türkeş Science and Technology University01250AdanaTurkey
| | - Adem Polat
- Department of Electrical–Electronics EngineeringAdana Alparslan Türkeş Science and Technology University01250AdanaTurkey
| | - Kemal Balikci
- Department of Electrical and Electronics EngineeringOsmaniye Korkut Ata University80000OsmaniyeTurkey
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26
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Castro M, Ares S, Cuesta JA, Manrubia S. The turning point and end of an expanding epidemic cannot be precisely forecast. Proc Natl Acad Sci U S A 2020; 117:26190-26196. [PMID: 33004629 PMCID: PMC7585017 DOI: 10.1073/pnas.2007868117] [Citation(s) in RCA: 72] [Impact Index Per Article: 18.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023] Open
Abstract
Epidemic spread is characterized by exponentially growing dynamics, which are intrinsically unpredictable. The time at which the growth in the number of infected individuals halts and starts decreasing cannot be calculated with certainty before the turning point is actually attained; neither can the end of the epidemic after the turning point. A susceptible-infected-removed (SIR) model with confinement (SCIR) illustrates how lockdown measures inhibit infection spread only above a threshold that we calculate. The existence of that threshold has major effects in predictability: A Bayesian fit to the COVID-19 pandemic in Spain shows that a slowdown in the number of newly infected individuals during the expansion phase allows one to infer neither the precise position of the maximum nor whether the measures taken will bring the propagation to the inhibition regime. There is a short horizon for reliable prediction, followed by a dispersion of the possible trajectories that grows extremely fast. The impossibility to predict in the midterm is not due to wrong or incomplete data, since it persists in error-free, synthetically produced datasets and does not necessarily improve by using larger datasets. Our study warns against precise forecasts of the evolution of epidemics based on mean-field, effective, or phenomenological models and supports that only probabilities of different outcomes can be confidently given.
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Affiliation(s)
- Mario Castro
- Grupo Interdisciplinar de Sistemas Complejos, 28911 Madrid, Spain
- Instituto de Investigación Tecnológica, Universidad Pontificia Comillas, 28015 Madrid, Spain
| | - Saúl Ares
- Grupo Interdisciplinar de Sistemas Complejos, 28911 Madrid, Spain
- Departamento de Biología de Sistemas, Centro Nacional de Biotecnología, 28049 Madrid, Spain
| | - José A Cuesta
- Grupo Interdisciplinar de Sistemas Complejos, 28911 Madrid, Spain
- Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganes, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos, Campus Río Ebro, Universidad de Zaragoza, 50018 Zaragoza, Spain
- Universidad Carlos III de Madrid-Santander Big Data Institute, 28903 Getafe, Spain
| | - Susanna Manrubia
- Grupo Interdisciplinar de Sistemas Complejos, 28911 Madrid, Spain;
- Departamento de Biología de Sistemas, Centro Nacional de Biotecnología, 28049 Madrid, Spain
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