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KhudaBukhsh WR, Rempała GA. How to correctly fit an SIR model to data from an SEIR model? Math Biosci 2024; 375:109265. [PMID: 39089573 DOI: 10.1016/j.mbs.2024.109265] [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: 12/19/2023] [Revised: 06/24/2024] [Accepted: 07/26/2024] [Indexed: 08/04/2024]
Abstract
In epidemiology, realistic disease dynamics often require Susceptible-Exposed-Infected-Recovered (SEIR)-like models because they account for incubation periods before individuals become infectious. However, for the sake of analytical tractability, simpler Susceptible-Infected-Recovered (SIR) models are commonly used, despite their lack of biological realism. Bridging these models is crucial for accurately estimating parameters and fitting models to observed data, particularly in population-level studies of infectious diseases. This paper investigates stochastic versions of the SEIR and SIR frameworks and demonstrates that the SEIR model can be effectively approximated by a SIR model with time-dependent infection and recovery rates. The validity of this approximation is supported by the derivation of a large-population Functional Law of Large Numbers (FLLN) limit and a finite-population concentration inequality. To apply this approximation in practice, the paper introduces a parameter inference methodology based on the Dynamic Survival Analysis (DSA) survival analysis framework. This method enables the fitting of the SIR model to data simulated from the more complex SEIR dynamics, as illustrated through simulated experiments.
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Affiliation(s)
- Wasiur R KhudaBukhsh
- School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham, NG7 2RD, Nottinghamshire, United Kingdom.
| | - Grzegorz A Rempała
- Division of Biostatistics, College of Public Health, The Ohio State University, 1841 Neil Avenue, Cunz Hall, Columbus, 43210, OH, United States of America.
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Liu L, Wang X, Liu O, Li Y, Jin Z, Tang S, Wang X. Valuation and comparison of the actual and optimal control strategy in an emerging infectious disease: Implication from a COVID-19 transmission model. Infect Dis Model 2024; 9:354-372. [PMID: 38385019 PMCID: PMC10879675 DOI: 10.1016/j.idm.2024.02.003] [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: 11/17/2023] [Revised: 02/03/2024] [Accepted: 02/03/2024] [Indexed: 02/23/2024] Open
Abstract
To effectively combat emerging infectious diseases like COVID-19, it is crucial to adopt strict prevention and control measures promptly to effectively contain the spread of the epidemic. In this paper, we propose a transmission model to investigate the influence of two control strategies: reducing contact numbers and improving medical resources. We examine these strategies in terms of constant control and time-varying control. Through sensitivity analysis on two reproduction numbers of the model with constant control, we demonstrate that reducing contact numbers is more effective than improving medical resources. Furthermore, these two constant controls significantly influence the peak values and timing of infections. Specifically, intensifying control measures can reduce peak values, albeit at the expense of delaying the peak time. In the model with time-varying control, we initially explore the corresponding optimal control problem and derive the characteristic expression of optimal control. Subsequently, we utilize real data from January 10th to April 12th, 2020, in Wuhan city as a case study to perform parameter estimation by using our proposed improved algorithm. Our findings illustrate that implementing optimal control measures can effectively reduce infections and deaths, and shorten the duration of the epidemic. Then, we numerically explore that implementing control measures promptly and increasing intensity to reduce contact numbers can make actual control be more closer to optimized control. Finally, we utilize the real data from October 31st to November 18th, 2021, in Hebei province as a second case study to validate the feasibility of our proposed suggestions.
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Affiliation(s)
- Lili Liu
- Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Complex Systems Research Center, Shanxi University, Taiyuan, 030006, China
| | - Xi Wang
- Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Complex Systems Research Center, Shanxi University, Taiyuan, 030006, China
- School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, China
| | - Ou Liu
- School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, China
| | - Yazhi Li
- School of Mathematics and Statistics, Qiannan Normal University for Nationalities, Guizhou, Duyun, 558000, China
| | - Zhen Jin
- Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Complex Systems Research Center, Shanxi University, Taiyuan, 030006, China
| | - Sanyi Tang
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710119, China
| | - Xia Wang
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710119, China
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Holm RH, Rempala GA, Choi B, Brick JM, Amraotkar AR, Keith RJ, Rouchka EC, Chariker JH, Palmer KE, Smith T, Bhatnagar A. Dynamic SARS-CoV-2 surveillance model combining seroprevalence and wastewater concentrations for post-vaccine disease burden estimates. COMMUNICATIONS MEDICINE 2024; 4:70. [PMID: 38594350 PMCID: PMC11004132 DOI: 10.1038/s43856-024-00494-y] [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: 06/13/2023] [Accepted: 03/28/2024] [Indexed: 04/11/2024] Open
Abstract
BACKGROUND Despite wide scale assessments, it remains unclear how large-scale severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) vaccination affected the wastewater concentration of the virus or the overall disease burden as measured by hospitalization rates. METHODS We used weekly SARS-CoV-2 wastewater concentration with a stratified random sampling of seroprevalence, and linked vaccination and hospitalization data, from April 2021-August 2021 in Jefferson County, Kentucky (USA). Our susceptible ( S ), vaccinated ( V ), variant-specific infected (I 1 andI 2 ), recovered ( R ), and seropositive ( T ) model ( S V I 2 R T ) tracked prevalence longitudinally. This was related to wastewater concentration. RESULTS Here we show the 64% county vaccination rate translate into about a 61% decrease in SARS-CoV-2 incidence. The estimated effect of SARS-CoV-2 Delta variant emergence is a 24-fold increase of infection counts, which correspond to an over 9-fold increase in wastewater concentration. Hospitalization burden and wastewater concentration have the strongest correlation (r = 0.95) at 1 week lag. CONCLUSIONS Our study underscores the importance of continuing environmental surveillance post-vaccine and provides a proof-of-concept for environmental epidemiology monitoring of infectious disease for future pandemic preparedness.
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Grants
- P20 GM103436 NIGMS NIH HHS
- P30 ES030283 NIEHS NIH HHS
- This study was supported by Centers for Disease Control and Prevention (75D30121C10273), Louisville Metro Government, James Graham Brown Foundation, Owsley Brown II Family Foundation, Welch Family, Jewish Heritage Fund for Excellence, the National Institutes of Health, (P20GM103436), the Rockefeller Foundation, the National Sciences Foundation (DMS-2027001), and the Basic Science Research Program National Research Foundation of Korea (NRF) (RS-2023-00245056).
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Affiliation(s)
- Rochelle H Holm
- Christina Lee Brown Envirome Institute, School of Medicine, University of Louisville, Louisville, KY, 40202, USA
| | - Grzegorz A Rempala
- Division of Biostatistics, College of Public Health, The Ohio State University, Columbus, OH, 43210, USA
| | - Boseung Choi
- Division of Biostatistics, College of Public Health, The Ohio State University, Columbus, OH, 43210, USA
- Division of Big Data Science, Korea University, Sejong, South Korea
- Biomedical Mathematics Group, Institute for Basic Science, Daejeon, South Korea
| | | | - Alok R Amraotkar
- Christina Lee Brown Envirome Institute, School of Medicine, University of Louisville, Louisville, KY, 40202, USA
| | - Rachel J Keith
- Christina Lee Brown Envirome Institute, School of Medicine, University of Louisville, Louisville, KY, 40202, USA
| | - Eric C Rouchka
- Department of Biochemistry and Molecular Genetics, School of Medicine, University of Louisville, Louisville, KY, 40202, USA
- KY INBRE Bioinformatics Core, University of Louisville, Louisville, KY, 40202, USA
| | - Julia H Chariker
- Department of Biochemistry and Molecular Genetics, School of Medicine, University of Louisville, Louisville, KY, 40202, USA
- KY INBRE Bioinformatics Core, University of Louisville, Louisville, KY, 40202, USA
| | - Kenneth E Palmer
- Center for Predictive Medicine for Biodefense and Emerging Infectious Diseases, University of Louisville, Louisville, KY, 40202, USA
- Department of Pharmacology and Toxicology, School of Medicine, University of Louisville, Louisville, KY, 40202, USA
| | - Ted Smith
- Christina Lee Brown Envirome Institute, School of Medicine, University of Louisville, Louisville, KY, 40202, USA
| | - Aruni Bhatnagar
- Christina Lee Brown Envirome Institute, School of Medicine, University of Louisville, Louisville, KY, 40202, USA.
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Kiss IZ, Berthouze L, KhudaBukhsh WR. Towards Inferring Network Properties from Epidemic Data. Bull Math Biol 2023; 86:6. [PMID: 38063898 PMCID: PMC10709280 DOI: 10.1007/s11538-023-01235-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2023] [Accepted: 11/09/2023] [Indexed: 12/18/2023]
Abstract
Epidemic propagation on networks represents an important departure from traditional mass-action models. However, the high-dimensionality of the exact models poses a challenge to both mathematical analysis and parameter inference. By using mean-field models, such as the pairwise model (PWM), the high-dimensionality becomes tractable. While such models have been used extensively for model analysis, there is limited work in the context of statistical inference. In this paper, we explore the extent to which the PWM with the susceptible-infected-recovered (SIR) epidemic can be used to infer disease- and network-related parameters. Data from an epidemics can be loosely categorised as being population level, e.g., daily new cases, or individual level, e.g., recovery times. To understand if and how network inference is influenced by the type of data, we employed the widely-used MLE approach for population-level data and dynamical survival analysis (DSA) for individual-level data. For scenarios in which there is no model mismatch, such as when data are generated via simulations, both methods perform well despite strong dependence between parameters. In contrast, for real-world data, such as foot-and-mouth, H1N1 and COVID19, whereas the DSA method appears fairly robust to potential model mismatch and produces parameter estimates that are epidemiologically plausible, our results with the MLE method revealed several issues pertaining to parameter unidentifiability and a lack of robustness to exact knowledge about key quantities such as population size and/or proportion of under reporting. Taken together, however, our findings suggest that network-based mean-field models can be used to formulate approximate likelihoods which, coupled with an efficient inference scheme, make it possible to not only learn about the parameters of the disease dynamics but also that of the underlying network.
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Affiliation(s)
- Istvan Z Kiss
- Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, UK.
- Network Science Institute, Northeastern University London, London, E1W 1LP, UK.
| | - Luc Berthouze
- Department of Informatics, University of Sussex, Falmer, Brighton, BN1 9QH, UK
| | - Wasiur R KhudaBukhsh
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
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Kiss IZ, Kenah E, Rempała GA. Necessary and sufficient conditions for exact closures of epidemic equations on configuration model networks. J Math Biol 2023; 87:36. [PMID: 37532967 PMCID: PMC10397147 DOI: 10.1007/s00285-023-01967-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/11/2022] [Revised: 05/09/2023] [Accepted: 07/11/2023] [Indexed: 08/04/2023]
Abstract
We prove that it is possible to obtain the exact closure of SIR pairwise epidemic equations on a configuration model network if and only if the degree distribution follows a Poisson, binomial, or negative binomial distribution. The proof relies on establishing the equivalence, for these specific degree distributions, between the closed pairwise model and a dynamical survival analysis (DSA) model that was previously shown to be exact. Specifically, we demonstrate that the DSA model is equivalent to the well-known edge-based Volz model. Using this result, we also provide reductions of the closed pairwise and Volz models to a single equation that involves only susceptibles. This equation has a useful statistical interpretation in terms of times to infection. We provide some numerical examples to illustrate our results.
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Affiliation(s)
- István Z Kiss
- Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, UK.
- Network Science Institute, Northeastern University London, London, E1W 1LP, UK.
| | - Eben Kenah
- Division of Biostatistics, College of Public Health and Mathematical Biosciences Institute, The Ohio State University, Columbus, OH, USA
| | - Grzegorz A Rempała
- Division of Biostatistics, College of Public Health and Mathematical Biosciences Institute, The Ohio State University, Columbus, OH, USA
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KhudaBukhsh WR, Khalsa SK, Kenah E, Rempała GA, Tien JH. COVID-19 dynamics in an Ohio prison. Front Public Health 2023; 11:1087698. [PMID: 37064663 PMCID: PMC10098107 DOI: 10.3389/fpubh.2023.1087698] [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: 11/02/2022] [Accepted: 02/20/2023] [Indexed: 03/31/2023] Open
Abstract
Incarcerated individuals are a highly vulnerable population for infection with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Understanding the transmission of respiratory infections within prisons and between prisons and surrounding communities is a crucial component of pandemic preparedness and response. Here, we use mathematical and statistical models to analyze publicly available data on the spread of SARS-CoV-2 reported by the Ohio Department of Rehabilitation and Corrections (ODRC). Results from mass testing conducted on April 16, 2020 were analyzed together with time of first reported SARS-CoV-2 infection among Marion Correctional Institution (MCI) inmates. Extremely rapid, widespread infection of MCI inmates was reported, with nearly 80% of inmates infected within 3 weeks of the first reported inmate case. The dynamical survival analysis (DSA) framework that we use allows the derivation of explicit likelihoods based on mathematical models of transmission. We find that these data are consistent with three non-exclusive possibilities: (i) a basic reproduction number >14 with a single initially infected inmate, (ii) an initial superspreading event resulting in several hundred initially infected inmates with a reproduction number of approximately three, or (iii) earlier undetected circulation of virus among inmates prior to April. All three scenarios attest to the vulnerabilities of prisoners to COVID-19, and the inability to distinguish among these possibilities highlights the need for improved infection surveillance and reporting in prisons.
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Affiliation(s)
- Wasiur R. KhudaBukhsh
- School of Mathematical Sciences, The University of Nottingham, Nottingham, United Kingdom
| | - Sat Kartar Khalsa
- Wexner Medical Center, The Ohio State University, Columbus, OH, United States
| | - Eben Kenah
- Division of Biostatistics, The Ohio State University, Columbus, OH, United States
| | - Gregorz A. Rempała
- Division of Biostatistics, Department of Mathematics, The Ohio State University, Columbus, OH, United States
| | - Joseph H. Tien
- Division of Epidemiology, Department of Mathematics, The Ohio State University, Columbus, OH, United States
- *Correspondence: Joseph H. Tien
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