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Earn DJD, Park SW, Bolker BM. Fitting Epidemic Models to Data: A Tutorial in Memory of Fred Brauer. Bull Math Biol 2024; 86:109. [PMID: 39052140 DOI: 10.1007/s11538-024-01326-9] [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: 03/19/2024] [Accepted: 06/04/2024] [Indexed: 07/27/2024]
Abstract
Fred Brauer was an eminent mathematician who studied dynamical systems, especially differential equations. He made many contributions to mathematical epidemiology, a field that is strongly connected to data, but he always chose to avoid data analysis. Nevertheless, he recognized that fitting models to data is usually necessary when attempting to apply infectious disease transmission models to real public health problems. He was curious to know how one goes about fitting dynamical models to data, and why it can be hard. Initially in response to Fred's questions, we developed a user-friendly R package, fitode, that facilitates fitting ordinary differential equations to observed time series. Here, we use this package to provide a brief tutorial introduction to fitting compartmental epidemic models to a single observed time series. We assume that, like Fred, the reader is familiar with dynamical systems from a mathematical perspective, but has limited experience with statistical methodology or optimization techniques.
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Affiliation(s)
- David J D Earn
- Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada.
| | - Sang Woo Park
- Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, 08544, USA
| | - Benjamin M Bolker
- Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada
- Department of Biology, McMaster University, Hamilton, ON, L8S 4K1, Canada
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Gunasekaran N, Vadivel R, Zhai G, Vinoth S. Finite-time stability analysis and control of stochastic SIR epidemic model: A study of COVID-19. Biomed Signal Process Control 2023; 86:105123. [PMID: 37337551 PMCID: PMC10261717 DOI: 10.1016/j.bspc.2023.105123] [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: 02/13/2023] [Revised: 05/20/2023] [Accepted: 06/08/2023] [Indexed: 06/21/2023]
Abstract
Finite-time stability analysis is a powerful tool for understanding the long-term behavior of epidemiological models and has been widely used to study the spread of infectious diseases such as COVID-19. In this paper, we present a finite-time stability analysis of a stochastic susceptible-infected-recovered (SIR) epidemic compartmental model with switching signals. The model includes a linear parameter variation (LPV) and switching system that represents the impact of external factors, such as changes in public health policies or seasonal variations, on the transmission rate of the disease. We use the Lyapunov stability theory to examine the long-term behavior of the model and determine conditions under which the disease is likely to die out or persist in the population. By taking advantage of the average dwell time method and Lyapunov functional (LF) method, and using novel inequality techniques the finite-time stability (FTS) criterion in linear matrix inequalities (LMIs) is developed. The finite-time stability of the resultant closed-loop system, with interval and linear parameter variation (LPV), is then guaranteed by state feedback controllers. By analyzing the modified SIR model with these interventions, we are able to examine the efficiency of different control measures and determine the most appropriate response to the COVID-19 pandemic and demonstrate the efficacy of the suggested strategy through simulation results.
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Affiliation(s)
- Nallappan Gunasekaran
- Eastern Michigan Joint College of Engineering, Beibu Gulf University, Qinzhou 535011, China
| | - R Vadivel
- Department of Mathematics, Faculty of Science and Technology, Phuket Rajabhat University, Phuket - 83000, Thailand
| | - Guisheng Zhai
- Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama 337-8570, Japan
| | - S Vinoth
- Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai - 600 069, Tamilnadu, India
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Servadio JL, Thai PQ, Choisy M, Boni MF. Repeatability and timing of tropical influenza epidemics. PLoS Comput Biol 2023; 19:e1011317. [PMID: 37467254 PMCID: PMC10389745 DOI: 10.1371/journal.pcbi.1011317] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2022] [Accepted: 06/29/2023] [Indexed: 07/21/2023] Open
Abstract
Much of the world experiences influenza in yearly recurring seasons, particularly in temperate areas. These patterns can be considered repeatable if they occur predictably and consistently at the same time of year. In tropical areas, including southeast Asia, timing of influenza epidemics is less consistent, leading to a lack of consensus regarding whether influenza is repeatable. This study aimed to assess repeatability of influenza in Vietnam, with repeatability defined as seasonality that occurs at a consistent time of year with low variation. We developed a mathematical model incorporating parameters to represent periods of increased transmission and then fitted the model to data collected from sentinel hospitals throughout Vietnam as well as four temperate locations. We fitted the model for individual (sub)types of influenza as well as all combined influenza throughout northern, central, and southern Vietnam. Repeatability was evaluated through the variance of the timings of peak transmission. Model fits from Vietnam show high variance (sd = 64-179 days) in peak transmission timing, with peaks occurring at irregular intervals and throughout different times of year. Fits from temperate locations showed regular, annual epidemics in winter months, with low variance in peak timings (sd = 32-57 days). This suggests that influenza patterns are not repeatable or seasonal in Vietnam. Influenza prevention in Vietnam therefore cannot rely on anticipation of regularly occurring outbreaks.
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Affiliation(s)
- Joseph L Servadio
- Center for Infectious Disease Dynamics and Department of Biology, Pennsylvania State University, University Park, Pennsylvania, United States of America
| | - Pham Quang Thai
- National Institute of Hygiene and Epidemiology, Hanoi, Vietnam
- School of Preventative Medicine and Public Health, Hanoi Medical University, Hanoi, Vietnam
| | - Marc Choisy
- Oxford University Clinical Research Unit, Ho Chi Minh City, Vietnam
- Centre for Tropical Medicine and Global Health, Nuffield Department of Medicine, University of Oxford, Oxford, United Kingdom
| | - Maciej F Boni
- Center for Infectious Disease Dynamics and Department of Biology, Pennsylvania State University, University Park, Pennsylvania, United States of America
- Centre for Tropical Medicine and Global Health, Nuffield Department of Medicine, University of Oxford, Oxford, United Kingdom
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Sharp JA, Browning AP, Burrage K, Simpson MJ. Parameter estimation and uncertainty quantification using information geometry. J R Soc Interface 2022; 19:20210940. [PMID: 35472269 PMCID: PMC9042578 DOI: 10.1098/rsif.2021.0940] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022] Open
Abstract
In this work, we: (i) review likelihood-based inference for parameter estimation and the construction of confidence regions; and (ii) explore the use of techniques from information geometry, including geodesic curves and Riemann scalar curvature, to supplement typical techniques for uncertainty quantification, such as Bayesian methods, profile likelihood, asymptotic analysis and bootstrapping. These techniques from information geometry provide data-independent insights into uncertainty and identifiability, and can be used to inform data collection decisions. All code used in this work to implement the inference and information geometry techniques is available on GitHub.
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Affiliation(s)
- Jesse A Sharp
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia.,ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Queensland, Australia
| | - Alexander P Browning
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia.,ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Queensland, Australia
| | - Kevin Burrage
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia.,ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Queensland, Australia.,Department of Computer Science, University of Oxford, Oxford, UK
| | - Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia.,Centre for Data Science, Queensland University of Technology, Brisbane, Queensland, Australia
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Bin M, Cheung PYK, Crisostomi E, Ferraro P, Lhachemi H, Murray-Smith R, Myant C, Parisini T, Shorten R, Stein S, Stone L. Post-lockdown abatement of COVID-19 by fast periodic switching. PLoS Comput Biol 2021; 17:e1008604. [PMID: 33476332 PMCID: PMC7861565 DOI: 10.1371/journal.pcbi.1008604] [Citation(s) in RCA: 26] [Impact Index Per Article: 8.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2020] [Revised: 02/04/2021] [Accepted: 12/03/2020] [Indexed: 11/18/2022] Open
Abstract
COVID-19 abatement strategies have risks and uncertainties which could lead to repeating waves of infection. We show-as proof of concept grounded on rigorous mathematical evidence-that periodic, high-frequency alternation of into, and out-of, lockdown effectively mitigates second-wave effects, while allowing continued, albeit reduced, economic activity. Periodicity confers (i) predictability, which is essential for economic sustainability, and (ii) robustness, since lockdown periods are not activated by uncertain measurements over short time scales. In turn-while not eliminating the virus-this fast switching policy is sustainable over time, and it mitigates the infection until a vaccine or treatment becomes available, while alleviating the social costs associated with long lockdowns. Typically, the policy might be in the form of 1-day of work followed by 6-days of lockdown every week (or perhaps 2 days working, 5 days off) and it can be modified at a slow-rate based on measurements filtered over longer time scales. Our results highlight the potential efficacy of high frequency switching interventions in post lockdown mitigation. All code is available on Github at https://github.com/V4p1d/FPSP_Covid19. A software tool has also been developed so that interested parties can explore the proof-of-concept system.
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Affiliation(s)
- Michelangelo Bin
- Department of Electrical and Electronic Engineering, Imperial College London, London, United Kingdom
| | - Peter Y. K. Cheung
- Dyson School of Design Engineering, Imperial College London, London, United Kingdom
| | - Emanuele Crisostomi
- Department of Energy, Systems, Territory and Constructions Engineering, University of Pisa, Pisa, Italy
| | - Pietro Ferraro
- Dyson School of Design Engineering, Imperial College London, London, United Kingdom
| | | | | | - Connor Myant
- Dyson School of Design Engineering, Imperial College London, London, United Kingdom
| | - Thomas Parisini
- Department of Electrical and Electronic Engineering, Imperial College London, London, United Kingdom
- Department of Engineering and Architecture, University of Trieste, Trieste, Italy
- KIOS Research and Innovation Center of Excellence, University of Cyprus, Nicosia, Cyprus
| | - Robert Shorten
- Dyson School of Design Engineering, Imperial College London, London, United Kingdom
- Department of Electrical and Electronic Engineering, University College Dublin, Dublin, Ireland
| | - Sebastian Stein
- School of Computing Science, University of Glasgow, Glasgow, Scotland
| | - Lewi Stone
- Mathematics, School of Science, RMIT University, Melbourne, Australia
- Faculty of Life Sciences, Tel Aviv University, Tel Aviv, Israel
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Kovács T. How can contemporary climate research help understand epidemic dynamics? Ensemble approach and snapshot attractors. J R Soc Interface 2020; 17:20200648. [PMID: 33292097 DOI: 10.1098/rsif.2020.0648] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023] Open
Abstract
Standard epidemic models based on compartmental differential equations are investigated under continuous parameter change as external forcing. We show that seasonal modulation of the contact parameter superimposed upon a monotonic decay needs a different description from that of the standard chaotic dynamics. The concept of snapshot attractors and their natural distribution has been adopted from the field of the latest climate change research. This shows the importance of the finite-time chaotic effect and ensemble interpretation while investigating the spread of a disease. By defining statistical measures over the ensemble, we can interpret the internal variability of the epidemic as the onset of complex dynamics-even for those values of contact parameters where originally regular behaviour is expected. We argue that anomalous outbreaks of the infectious class cannot die out until transient chaos is presented in the system. Nevertheless, this fact becomes apparent by using an ensemble approach rather than a single trajectory representation. These findings are applicable generally in explicitly time-dependent epidemic systems regardless of parameter values and time scales.
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Affiliation(s)
- T Kovács
- Institute for Theoretical Physics, Eötvös University, Pázmány P. s. 1A, H-1117 Budapest, Hungary
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Krylova O, Earn DJD. Patterns of smallpox mortality in London, England, over three centuries. PLoS Biol 2020; 18:e3000506. [PMID: 33347440 PMCID: PMC7751884 DOI: 10.1371/journal.pbio.3000506] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2019] [Accepted: 11/17/2020] [Indexed: 11/19/2022] Open
Abstract
Smallpox is unique among infectious diseases in the degree to which it devastated human populations, its long history of control interventions, and the fact that it has been successfully eradicated. Mortality from smallpox in London, England was carefully documented, weekly, for nearly 300 years, providing a rare and valuable source for the study of ecology and evolution of infectious disease. We describe and analyze smallpox mortality in London from 1664 to 1930. We digitized the weekly records published in the London Bills of Mortality (LBoM) and the Registrar General's Weekly Returns (RGWRs). We annotated the resulting time series with a sequence of historical events that might have influenced smallpox dynamics in London. We present a spectral analysis that reveals how periodicities in reported smallpox mortality changed over decades and centuries; many of these changes in epidemic patterns are correlated with changes in control interventions and public health policies. We also examine how the seasonality of reported smallpox mortality changed from the 17th to 20th centuries in London.
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Affiliation(s)
- Olga Krylova
- Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada
| | - David J. D. Earn
- Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada
- M.G. DeGroote Institute for Infectious Disease Research, McMaster University, Hamilton, Ontario, Canada
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