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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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2
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Bürger R, Chowell G, Kröker I, Lara-Díaz LY. A computational approach to identifiability analysis for a model of the propagation and control of COVID-19 in Chile. JOURNAL OF BIOLOGICAL DYNAMICS 2023; 17:2256774. [PMID: 37708159 PMCID: PMC10620014 DOI: 10.1080/17513758.2023.2256774] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/29/2022] [Accepted: 08/30/2023] [Indexed: 09/16/2023]
Abstract
A computational approach is adapted to analyze the parameter identifiability of a compartmental model. The model is intended to describe the progression of the COVID-19 pandemic in Chile during the initial phase in early 2020 when government declared quarantine measures. The computational approach to analyze the structural and practical identifiability is applied in two parts, one for synthetic data and another for some Chilean regional data. The first part defines the identifiable parameter sets when these recover the true parameters used to create the synthetic data. The second part compares the results derived from synthetic data, estimating the identifiable parameter sets from regional Chilean epidemic data. Experiments provide evidence of the loss of identifiability if some initial conditions are estimated, the period of time used to fit is before the peak, and if a significant proportion of the population is involved in quarantine periods.
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Affiliation(s)
- Raimund Bürger
- CI[Formula: see text]MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Concepción, Chile
| | - Gerardo Chowell
- School of Public Health, Georgia State University, Atlanta, GA, USA
| | - Ilja Kröker
- Stochastic Simulation & Safety Research for Hydrosystems (LS3), Institute for Modelling Hydraulic and Environmental Systems (IWS), Universität Stuttgart, Stuttgart, Germany
| | - Leidy Yissedt Lara-Díaz
- Departamento de Matemática, Física y Estadística, Facultad de Ciencias Básicas, Universidad Católica del Maule, Talca, Chile
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3
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Akuno AO, Ramírez-Ramírez LL, Espinoza JF. Inference on a Multi-Patch Epidemic Model with Partial Mobility, Residency, and Demography: Case of the 2020 COVID-19 Outbreak in Hermosillo, Mexico. ENTROPY (BASEL, SWITZERLAND) 2023; 25:968. [PMID: 37509915 PMCID: PMC10378648 DOI: 10.3390/e25070968] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2023] [Revised: 06/02/2023] [Accepted: 06/14/2023] [Indexed: 07/30/2023]
Abstract
Most studies modeling population mobility and the spread of infectious diseases, particularly those using meta-population multi-patch models, tend to focus on the theoretical properties and numerical simulation of such models. As such, there is relatively scant literature focused on numerical fit, inference, and uncertainty quantification of epidemic models with population mobility. In this research, we use three estimation techniques to solve an inverse problem and quantify its uncertainty for a human-mobility-based multi-patch epidemic model using mobile phone sensing data and confirmed COVID-19-positive cases in Hermosillo, Mexico. First, we utilize a Brownian bridge model using mobile phone GPS data to estimate the residence and mobility parameters of the epidemic model. In the second step, we estimate the optimal model epidemiological parameters by deterministically inverting the model using a Darwinian-inspired evolutionary algorithm (EA)-that is, a genetic algorithm (GA). The third part of the analysis involves performing inference and uncertainty quantification in the epidemic model using two Bayesian Monte Carlo sampling methods: t-walk and Hamiltonian Monte Carlo (HMC). The results demonstrate that the estimated model parameters and incidence adequately fit the observed daily COVID-19 incidence in Hermosillo. Moreover, the estimated parameters from the HMC method yield large credible intervals, improving their coverage for the observed and predicted daily incidences. Furthermore, we observe that the use of a multi-patch model with mobility yields improved predictions when compared to a single-patch model.
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Affiliation(s)
- Albert Orwa Akuno
- Departamento de Probabilidad y Estadística, Centro de Investigación en Matemáticas A.C., Jalisco s/n, Colonia Valenciana, Guanajuato C.P. 36023, Gto, Mexico
| | - L Leticia Ramírez-Ramírez
- Departamento de Probabilidad y Estadística, Centro de Investigación en Matemáticas A.C., Jalisco s/n, Colonia Valenciana, Guanajuato C.P. 36023, Gto, Mexico
| | - Jesús F Espinoza
- Departamento de Matemáticas, Universidad de Sonora, Rosales y Boulevard Luis Encinas, Hermosillo C.P. 83000, Sonora, Mexico
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4
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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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5
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Renardy M, Kirschner D, Eisenberg M. Structural identifiability analysis of age-structured PDE epidemic models. J Math Biol 2022; 84:9. [PMID: 34982260 PMCID: PMC8724244 DOI: 10.1007/s00285-021-01711-1] [Citation(s) in RCA: 9] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2021] [Revised: 10/21/2021] [Accepted: 12/22/2021] [Indexed: 11/24/2022]
Abstract
Computational and mathematical models rely heavily on estimated parameter values for model development. Identifiability analysis determines how well the parameters of a model can be estimated from experimental data. Identifiability analysis is crucial for interpreting and determining confidence in model parameter values and to provide biologically relevant predictions. Structural identifiability analysis, in which one assumes data to be noiseless and arbitrarily fine-grained, has been extensively studied in the context of ordinary differential equation (ODE) models, but has not yet been widely explored for age-structured partial differential equation (PDE) models. These models present additional difficulties due to increased number of variables and partial derivatives as well as the presence of boundary conditions. In this work, we establish a pipeline for structural identifiability analysis of age-structured PDE models using a differential algebra framework and derive identifiability results for specific age-structured models. We use epidemic models to demonstrate this framework because of their wide-spread use in many different diseases and for the corresponding parallel work previously done for ODEs. In our application of the identifiability analysis pipeline, we focus on a Susceptible-Exposed-Infected model for which we compare identifiability results for a PDE and corresponding ODE system and explore effects of age-dependent parameters on identifiability. We also show how practical identifiability analysis can be applied in this example.
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Affiliation(s)
- Marissa Renardy
- Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, USA
| | - Denise Kirschner
- Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, USA
| | - Marisa Eisenberg
- Department of Epidemiology, University of Michigan, Ann Arbor, USA
- Department of Mathematics, University of Michigan, Ann Arbor, USA
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6
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Lega J. Parameter estimation from ICC curves. JOURNAL OF BIOLOGICAL DYNAMICS 2021; 15:195-212. [PMID: 33827379 DOI: 10.1080/17513758.2021.1912419] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2020] [Accepted: 03/13/2021] [Indexed: 06/12/2023]
Abstract
Incidence vs. Cumulative Cases (ICC) curves are introduced and shown to provide a simple framework for parameter identification in the case of the most elementary epidemiological model, consisting of susceptible, infected, and removed compartments. This novel methodology is used to estimate the basic reproduction ratio of recent outbreaks, including those associated with the ongoing COVID-19 pandemic.
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Affiliation(s)
- Joceline Lega
- Department of Mathematics, University of Arizona, Tucson, AZ, USA
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7
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Massonis G, Banga JR, Villaverde AF. Structural identifiability and observability of compartmental models of the COVID-19 pandemic. ANNUAL REVIEWS IN CONTROL 2021; 51:441-459. [PMID: 33362427 PMCID: PMC7752088 DOI: 10.1016/j.arcontrol.2020.12.001] [Citation(s) in RCA: 27] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/26/2020] [Revised: 09/24/2020] [Accepted: 12/01/2020] [Indexed: 05/18/2023]
Abstract
The recent coronavirus disease (COVID-19) outbreak has dramatically increased the public awareness and appreciation of the utility of dynamic models. At the same time, the dissemination of contradictory model predictions has highlighted their limitations. If some parameters and/or state variables of a model cannot be determined from output measurements, its ability to yield correct insights - as well as the possibility of controlling the system - may be compromised. Epidemic dynamics are commonly analysed using compartmental models, and many variations of such models have been used for analysing and predicting the evolution of the COVID-19 pandemic. In this paper we survey the different models proposed in the literature, assembling a list of 36 model structures and assessing their ability to provide reliable information. We address the problem using the control theoretic concepts of structural identifiability and observability. Since some parameters can vary during the course of an epidemic, we consider both the constant and time-varying parameter assumptions. We analyse the structural identifiability and observability of all of the models, considering all plausible choices of outputs and time-varying parameters, which leads us to analyse 255 different model versions. We classify the models according to their structural identifiability and observability under the different assumptions and discuss the implications of the results. We also illustrate with an example several alternative ways of remedying the lack of observability of a model. Our analyses provide guidelines for choosing the most informative model for each purpose, taking into account the available knowledge and measurements.
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Affiliation(s)
- Gemma Massonis
- BioProcess Engineering Group, IIM-CSIC, Vigo 36208, Galicia, Spain
| | - Julio R Banga
- BioProcess Engineering Group, IIM-CSIC, Vigo 36208, Galicia, Spain
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Browning AP, Warne DJ, Burrage K, Baker RE, Simpson MJ. Identifiability analysis for stochastic differential equation models in systems biology. J R Soc Interface 2020; 17:20200652. [PMID: 33323054 PMCID: PMC7811582 DOI: 10.1098/rsif.2020.0652] [Citation(s) in RCA: 27] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/11/2020] [Accepted: 11/24/2020] [Indexed: 12/26/2022] Open
Abstract
Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.
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Affiliation(s)
- Alexander P. Browning
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Australia
| | - David J. Warne
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Australia
| | - Kevin Burrage
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Australia
- ARC Centre of Excellence for Plant Success in Nature and Agriculture, Queensland University of Technology, Brisbane, Australia
- Department of Computer Science, University of Oxford, Oxford, UK
| | - Ruth E. Baker
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Matthew J. Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Australia
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9
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Piazzola C, Tamellini L, Tempone R. A note on tools for prediction under uncertainty and identifiability of SIR-like dynamical systems for epidemiology. Math Biosci 2020; 332:108514. [PMID: 33217409 DOI: 10.1016/j.mbs.2020.108514] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2020] [Revised: 11/09/2020] [Accepted: 11/09/2020] [Indexed: 12/18/2022]
Abstract
We provide an overview of the methods that can be used for prediction under uncertainty and data fitting of dynamical systems, and of the fundamental challenges that arise in this context. The focus is on SIR-like models, that are being commonly used when attempting to predict the trend of the COVID-19 pandemic. In particular, we raise a warning flag about identifiability of the parameters of SIR-like models; often, it might be hard to infer the correct values of the parameters from data, even for very simple models, making it non-trivial to use these models for meaningful predictions. Most of the points that we touch upon are actually generally valid for inverse problems in more general setups.
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Affiliation(s)
- Chiara Piazzola
- Consiglio Nazionale delle Ricerche - Istituto di Matematica Applicata e Tecnologie Informatiche "E. Magenes" (CNR-IMATI), Via Ferrata 5/A, 27100 Pavia, Italy.
| | - Lorenzo Tamellini
- Consiglio Nazionale delle Ricerche - Istituto di Matematica Applicata e Tecnologie Informatiche "E. Magenes" (CNR-IMATI), Via Ferrata 5/A, 27100 Pavia, Italy.
| | - Raúl Tempone
- Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen University, Pontdriesch 14-16, 52062, Aachen, Germany; King Abdullah University of Science and Technology (KAUST) - Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE), Thuwal, 23955-6900, Saudi Arabia.
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10
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Bertozzi AL, Franco E, Mohler G, Short MB, Sledge D. The challenges of modeling and forecasting the spread of COVID-19. Proc Natl Acad Sci U S A 2020; 117:16732-16738. [PMID: 32616574 PMCID: PMC7382213 DOI: 10.1073/pnas.2006520117] [Citation(s) in RCA: 242] [Impact Index Per Article: 60.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
The coronavirus disease 2019 (COVID-19) pandemic has placed epidemic modeling at the forefront of worldwide public policy making. Nonetheless, modeling and forecasting the spread of COVID-19 remains a challenge. Here, we detail three regional-scale models for forecasting and assessing the course of the pandemic. This work demonstrates the utility of parsimonious models for early-time data and provides an accessible framework for generating policy-relevant insights into its course. We show how these models can be connected to each other and to time series data for a particular region. Capable of measuring and forecasting the impacts of social distancing, these models highlight the dangers of relaxing nonpharmaceutical public health interventions in the absence of a vaccine or antiviral therapies.
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Affiliation(s)
- Andrea L Bertozzi
- Department of Mathematics, University of California, Los Angeles, CA 90095;
- Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095
| | - Elisa Franco
- Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095
- Department of Bioengineering, University of California, Los Angeles, CA 90095
| | - George Mohler
- Department of Computer Science, Indiana University-Purdue University Indianapolis, Indianapolis, IN 46202
| | - Martin B Short
- Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332
| | - Daniel Sledge
- Department of Political Science, University of Texas at Arlington, Arlington, TX 76019
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11
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Havumaki J, Meza R, Phares CR, Date K, Eisenberg MC. Comparing alternative cholera vaccination strategies in Maela refugee camp: using a transmission model in public health practice. BMC Infect Dis 2019; 19:1075. [PMID: 31864298 PMCID: PMC6925891 DOI: 10.1186/s12879-019-4688-6] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/13/2019] [Accepted: 12/08/2019] [Indexed: 12/11/2022] Open
Abstract
Background Cholera is a major public health concern in displaced-person camps, which often contend with overcrowding and scarcity of resources. Maela, the largest and longest-standing refugee camp in Thailand, located along the Thai-Burmese border, experienced four cholera outbreaks between 2005 and 2010. In 2013, a cholera vaccine campaign was implemented in the camp. To assist in the evaluation of the campaign and planning for subsequent campaigns, we developed a mathematical model of cholera in Maela. Methods We formulated a Susceptible-Infectious-Water-Recovered-based transmission model and estimated parameters using incidence data from 2010. We next evaluated the reduction in cases conferred by several immunization strategies, varying timing, effectiveness, and resources (i.e., vaccine availability). After the vaccine campaign, we generated case forecasts for the next year, to inform on-the-ground decision-making regarding whether a booster campaign was needed. Results We found that preexposure vaccination can substantially reduce the risk of cholera even when <50% of the population is given the full two-dose series. Additionally, the preferred number of doses per person should be considered in the context of one vs. two dose effectiveness and vaccine availability. For reactive vaccination, a trade-off between timing and effectiveness was revealed, indicating that it may be beneficial to give one dose to more people rather than two doses to fewer people, given that a two-dose schedule would incur a delay in administration of the second dose. Forecasting using realistic coverage levels predicted that there was no need for a booster campaign in 2014 (consistent with our predictions, there was not a cholera epidemic in 2014). Conclusions Our analyses suggest that vaccination in conjunction with ongoing water sanitation and hygiene efforts provides an effective strategy for controlling cholera outbreaks in refugee camps. Effective preexposure vaccination depends on timing and effectiveness. If a camp is facing an outbreak, delayed distribution of vaccines can substantially alter the effectiveness of reactive vaccination, suggesting that quick distribution of vaccines may be more important than ensuring every individual receives both vaccine doses. Overall, this analysis illustrates how mathematical models can be applied in public health practice, to assist in evaluating alternative intervention strategies and inform decision-making.
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Affiliation(s)
- Joshua Havumaki
- Department of Epidemiology, School of Public Health, University of Michigan, 1415 Washington Heights, Ann Arbor, 48109, MI, USA
| | - Rafael Meza
- Department of Epidemiology, School of Public Health, University of Michigan, 1415 Washington Heights, Ann Arbor, 48109, MI, USA
| | - Christina R Phares
- US Centers for Disease Control and Prevention; National Center for Emerging and Zoonotic Infectious Diseases; Division of Global Migration and Quarantine and Prevention, 1600 Clifton Road, Atlanta, 30329, GA, USA
| | - Kashmira Date
- US Centers for Disease Control and Prevention; Global Immunization Division - Center for Global Health, 1600 Clifton Road, Atlanta, 30329, GA, USA
| | - Marisa C Eisenberg
- Department of Epidemiology, School of Public Health, University of Michigan, 1415 Washington Heights, Ann Arbor, 48109, MI, USA.
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12
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Ducrot A, Magal P, Nguyen T, Webb GF. Identifying the number of unreported cases in SIR epidemic models. MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA 2019; 37:243-261. [DOI: 10.1093/imammb/dqz013] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2018] [Revised: 05/13/2019] [Accepted: 05/21/2019] [Indexed: 11/13/2022]
Abstract
Abstract
An SIR epidemic model is analysed with respect to the identification of its parameters and initial values, based upon reported case data from public health sources. The objective of the analysis is to understand the relationship of unreported cases to reported cases. In many epidemic diseases the reported cases are a small fraction of the unreported cases. This fraction can be estimated by the identification of parameters for the model from reported case data. The analysis is applied to the Hong Kong seasonal influenza epidemic in New York City in 1968–1969.
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Affiliation(s)
- A Ducrot
- Normandie University, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, Le Havre, France
| | - P Magal
- University of Bordeaux, CNRS, Bordeaux INP, IMB, UMR, Talence, France
| | - T Nguyen
- University of Bordeaux, CNRS, Bordeaux INP, IMB, UMR, Talence, France
| | - G F Webb
- Mathematics Department, Vanderbilt University, Nashville, TN, USA
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13
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Kao YH, Eisenberg MC. Practical unidentifiability of a simple vector-borne disease model: Implications for parameter estimation and intervention assessment. Epidemics 2018; 25:89-100. [PMID: 29903539 PMCID: PMC6264791 DOI: 10.1016/j.epidem.2018.05.010] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/17/2017] [Revised: 05/18/2018] [Accepted: 05/24/2018] [Indexed: 12/25/2022] Open
Abstract
Mathematical modeling has an extensive history in vector-borne disease epidemiology, and is increasingly used for prediction, intervention design, and understanding mechanisms. Many studies rely on parameter estimation to link models and data, and to tailor predictions and counterfactuals to specific settings. However, few studies have formally evaluated whether vector-borne disease models can properly estimate the parameters of interest given the constraints of a particular dataset. Identifiability analysis allows us to examine whether model parameters can be estimated uniquely-a lack of consideration of such issues can result in misleading or incorrect parameter estimates and model predictions. Here, we evaluate both structural (theoretical) and practical identifiability of a commonly used compartmental model of mosquito-borne disease, using the 2010 dengue epidemic in Taiwan as a case study. We show that while the model is structurally identifiable, it is practically unidentifiable under a range of human and mosquito time series measurement scenarios. In particular, the transmission parameters form a practically identifiable combination and thus cannot be estimated separately, potentially leading to incorrect predictions of the effects of interventions. However, in spite of the unidentifiability of the individual parameters, the basic reproduction number was successfully estimated across the unidentifiable parameter ranges. These identifiability issues can be resolved by directly measuring several additional human and mosquito life-cycle parameters both experimentally and in the field. While we only consider the simplest case for the model, we show that a commonly used model of vector-borne disease is unidentifiable from human and mosquito incidence data, making it difficult or impossible to estimate parameters or assess intervention strategies. This work illustrates the importance of examining identifiability when linking models with data to make predictions and inferences, and particularly highlights the importance of combining laboratory, field, and case data if we are to successfully estimate epidemiological and ecological parameters using models.
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Affiliation(s)
- Yu-Han Kao
- Department of Epidemiology, University of Michigan, Ann Arbor, MI, United States
| | - Marisa C Eisenberg
- Department of Epidemiology, University of Michigan, Ann Arbor, MI, United States; Department of Mathematics, University of Michigan, Ann Arbor, MI, United States.
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14
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Cazelles B, Champagne C, Dureau J. Accounting for non-stationarity in epidemiology by embedding time-varying parameters in stochastic models. PLoS Comput Biol 2018; 14:e1006211. [PMID: 30110322 PMCID: PMC6110518 DOI: 10.1371/journal.pcbi.1006211] [Citation(s) in RCA: 31] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2017] [Revised: 08/27/2018] [Accepted: 05/18/2018] [Indexed: 11/19/2022] Open
Abstract
The spread of disease through human populations is complex. The characteristics of disease propagation evolve with time, as a result of a multitude of environmental and anthropic factors, this non-stationarity is a key factor in this huge complexity. In the absence of appropriate external data sources, to correctly describe the disease propagation, we explore a flexible approach, based on stochastic models for the disease dynamics, and on diffusion processes for the parameter dynamics. Using such a diffusion process has the advantage of not requiring a specific mathematical function for the parameter dynamics. Coupled with particle MCMC, this approach allows us to reconstruct the time evolution of some key parameters (average transmission rate for instance). Thus, by capturing the time-varying nature of the different mechanisms involved in disease propagation, the epidemic can be described. Firstly we demonstrate the efficiency of this methodology on a toy model, where the parameters and the observation process are known. Applied then to real datasets, our methodology is able, based solely on simple stochastic models, to reconstruct complex epidemics, such as flu or dengue, over long time periods. Hence we demonstrate that time-varying parameters can improve the accuracy of model performances, and we suggest that our methodology can be used as a first step towards a better understanding of a complex epidemic, in situation where data is limited and/or uncertain.
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Affiliation(s)
- Bernard Cazelles
- Institut de Biologie de l’Ecole Normale Supérieure (IBENS), Ecole Normale Supérieure, CNRS UMR 8197, Paris, France
- International Center for Mathematical and Computational Modeling of Complex Systems (UMMISCO), UMI 209, UPMC/IRD, France
- Hosts, Vectors and Infectious Agents, CNRS URA 3012, Institut Pasteur, Paris, France
| | - Clara Champagne
- Institut de Biologie de l’Ecole Normale Supérieure (IBENS), Ecole Normale Supérieure, CNRS UMR 8197, Paris, France
- CREST, ENSAE, Université Paris Saclay, Palaiseau, France
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15
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Tuncer N, Le TT. Structural and practical identifiability analysis of outbreak models. Math Biosci 2018; 299:1-18. [PMID: 29477671 DOI: 10.1016/j.mbs.2018.02.004] [Citation(s) in RCA: 34] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2017] [Revised: 02/06/2018] [Accepted: 02/17/2018] [Indexed: 02/02/2023]
Abstract
Estimating the reproduction number of an emerging infectious disease from an epidemiological data is becoming more essential in evaluating the current status of an outbreak. However, these studies are lacking the fundamental prerequisite in parameter estimation problem, namely the structural identifiability of the epidemic model, which determines the possibility of uniquely determining the model parameters from the epidemic data. In this paper, we perform both structural and practical identifiability analysis to classical epidemic models such as SIR (Susceptible-Infected-Recovered), SEIR (Susceptible-Exposed-Infected-Recovered) and an epidemic model with the treatment class (SITR). We performed structural identifiability analysis on these epidemic models using a differential algebra approach to investigate the well-posedness of the parameter estimation problem. Parameters of these models are estimated from different data types, namely prevalence, cumulative incidences and treated individuals. Furthermore, we carried out practical identifiability analysis on these models using Monte Carlo simulations and Fisher's Information Matrix. Our study shows that the SIR model is both structurally and practically identifiable from the prevalence data. It is also structurally identifiable to cumulative incidence observations, but due to high correlations of the parameters, it is practically unidentifiable from the cumulative incidence data. Furthermore, we found that none of these simple epidemic models are practically identifiable from the cumulative incidence data which is the standard type of epidemiological data provided by CDC or WHO. Our analysis with simple SIR model suggest that the health agencies, if possible, should report prevalence rather than incidence data.
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Affiliation(s)
- Necibe Tuncer
- Department of Mathematical Sciences, Florida Atlantic University, Science Building, Room 234, 777 Glades Road, Boca Raton, FL 33431, USA.
| | - Trang T Le
- Department of Mathematics, University of Tulsa, 800 S Tucker Drive, Tulsa, OK 74104, USA.
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16
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Angulo MT, Velasco-Hernandez JX. Robust qualitative estimation of time-varying contact rates in uncertain epidemics. Epidemics 2018; 24:98-104. [PMID: 29567063 DOI: 10.1016/j.epidem.2018.03.001] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2017] [Revised: 09/15/2017] [Accepted: 03/07/2018] [Indexed: 11/20/2022] Open
Abstract
We will inevitably face new epidemics where the lack of long time-series data and the uncertainty about the outbreak dynamics make difficult to obtain quantitative predictions. Here we present an algorithm to qualitatively infer time-varying contact rates from short time-series data, letting us predict the start, relative magnitude and decline of epidemic outbreaks. Using real time-series data of measles, dengue, and the current zika outbreak, we demonstrate our algorithm can outperform existing algorithms based on estimating reproductive numbers.
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Affiliation(s)
- Marco Tulio Angulo
- CONACyT - Institute of Mathematics, Universidad Nacional Autónoma de México (UNAM), Juriquilla 76230, Mexico.
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17
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The parameter identification problem for SIR epidemic models: identifying unreported cases. J Math Biol 2018; 77:1629-1648. [PMID: 29330615 DOI: 10.1007/s00285-017-1203-9] [Citation(s) in RCA: 29] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2017] [Revised: 12/25/2017] [Indexed: 10/18/2022]
Abstract
A SIR epidemic model is analyzed with respect to identification of its parameters, based upon reported case data from public health sources. The objective of the analysis is to understand the relation of unreported cases to reported cases. In many epidemic diseases the ratio of unreported to reported cases is very high, and of major importance in implementing measures for controlling the epidemic. This ratio can be estimated by the identification of parameters for the model from reported case data. The analysis is applied to three examples: (1) the Hong Kong seasonal influenza epidemic in New York City in 1968-1969, (2) the bubonic plague epidemic in Bombay, India in 1906, and (3) the seasonal influenza epidemic in Puerto Rico in 2016-2017.
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18
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Shutt DP, Manore CA, Pankavich S, Porter AT, Del Valle SY. Estimating the reproductive number, total outbreak size, and reporting rates for Zika epidemics in South and Central America. Epidemics 2017; 21:63-79. [DOI: 10.1016/j.epidem.2017.06.005] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2017] [Revised: 06/14/2017] [Accepted: 06/27/2017] [Indexed: 11/16/2022] Open
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19
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Grandjean TRB, Chappell MJ, Yates JWT, Evans ND. Structural identifiability analyses of candidate models for in vitro Pitavastatin hepatic uptake. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2014; 114:e60-e69. [PMID: 23870173 DOI: 10.1016/j.cmpb.2013.06.013] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2012] [Revised: 06/17/2013] [Accepted: 06/18/2013] [Indexed: 06/02/2023]
Abstract
In this paper a review of the application of four different techniques (a version of the similarity transformation approach for autonomous uncontrolled systems, a non-differential input/output observable normal form approach, the characteristic set differential algebra and a recent algebraic input/output relationship approach) to determine the structural identifiability of certain in vitro nonlinear pharmacokinetic models is provided. The Organic Anion Transporting Polypeptide (OATP) substrate, Pitavastatin, is used as a probe on freshly isolated animal and human hepatocytes. Candidate pharmacokinetic non-linear compartmental models have been derived to characterise the uptake process of Pitavastatin. As a prerequisite to parameter estimation, structural identifiability analyses are performed to establish that all unknown parameters can be identified from the experimental observations available.
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Affiliation(s)
| | | | | | - Neil D Evans
- School of Engineering, University of Warwick, Coventry CV4 7AL, UK.
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20
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ZHANG YI, ZHANG QINGLING, ZHANG FUZHEN, BAI FENGLAN. CHAOS ANALYSIS AND CONTROL FOR A CLASS OF SIR EPIDEMIC MODEL WITH SEASONAL FLUCTUATION. INT J BIOMATH 2013. [DOI: 10.1142/s1793524512500635] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, the problems of chaos and chaos control for a class of susceptible-infected-removed (SIR) epidemic model with seasonal fluctuation are investigated. The seasonality in outbreak is natural among infectious diseases, as the common influenza, A type H1N1 influenza and so on. It is shown that there exist chaotic phenomena in the epidemic model. Furthermore, the tracking control method is used to control chaotic motions in the epidemic model. A feedback controller is designed to achieve tracking of an ideal output. Thus, the density of infected individuals can converge to zero, in other words, the disease can be disappeared. Finally, numerical simulations illustrate that the controller is effective.
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Affiliation(s)
- YI ZHANG
- Institute of Systems Science, Northeastern University, Shenyang, Liaoning 110819, P. R. China
- School of Science, Shenyang University of Technology, Shenyang 110870, P. R. China
| | - QINGLING ZHANG
- Institute of Systems Science, Northeastern University, Shenyang, Liaoning 110819, P. R. China
| | - FUZHEN ZHANG
- Division of Math, Science and Technology, Nova Southeastern University, Fort Lauderdale, Florida 33314, USA
| | - FENGLAN BAI
- School of Science, Dalian Jiaotong University, Dalian 116028, P. R. China
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21
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Eisenberg MC, Robertson SL, Tien JH. Identifiability and estimation of multiple transmission pathways in cholera and waterborne disease. J Theor Biol 2013; 324:84-102. [PMID: 23333764 DOI: 10.1016/j.jtbi.2012.12.021] [Citation(s) in RCA: 91] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2011] [Revised: 12/10/2012] [Accepted: 12/29/2012] [Indexed: 11/20/2022]
Abstract
Cholera and many waterborne diseases exhibit multiple characteristic timescales or pathways of infection, which can be modeled as direct and indirect transmission. A major public health issue for waterborne diseases involves understanding the modes of transmission in order to improve control and prevention strategies. An important epidemiological question is: given data for an outbreak, can we determine the role and relative importance of direct vs. environmental/waterborne routes of transmission? We examine whether parameters for a differential equation model of waterborne disease transmission dynamics can be identified, both in the ideal setting of noise-free data (structural identifiability) and in the more realistic setting in the presence of noise (practical identifiability). We used a differential algebra approach together with several numerical approaches, with a particular emphasis on identifiability of the transmission rates. To examine these issues in a practical public health context, we apply the model to a recent cholera outbreak in Angola (2006). Our results show that the model parameters-including both water and person-to-person transmission routes-are globally structurally identifiable, although they become unidentifiable when the environmental transmission timescale is fast. Even for water dynamics within the identifiable range, when noisy data are considered, only a combination of the water transmission parameters can practically be estimated. This makes the waterborne transmission parameters difficult to estimate, leading to inaccurate estimates of important epidemiological parameters such as the basic reproduction number (R0). However, measurements of pathogen persistence time in environmental water sources or measurements of pathogen concentration in the water can improve model identifiability and allow for more accurate estimation of waterborne transmission pathway parameters as well as R0. Parameter estimates for the Angola outbreak suggest that both transmission pathways are needed to explain the observed cholera dynamics. These results highlight the importance of incorporating environmental data when examining waterborne disease.
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Affiliation(s)
- Marisa C Eisenberg
- Mathematical Biosciences Institute, The Ohio State University, United States.
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22
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Capaldi A, Behrend S, Berman B, Smith J, Wright J, Lloyd AL. Parameter estimation and uncertainty quantification for an epidemic model. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2012; 9:553-576. [PMID: 22881026 DOI: 10.3934/mbe.2012.9.553] [Citation(s) in RCA: 30] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
We examine estimation of the parameters of Susceptible-Infective-Recovered (SIR) models in the context of least squares. We review the use of asymptotic statistical theory and sensitivity analysis to obtain measures of uncertainty for estimates of the model parameters and the basic reproductive number (R0)---an epidemiologically significant parameter grouping. We find that estimates of different parameters, such as the transmission parameter and recovery rate, are correlated, with the magnitude and sign of this correlation depending on the value of R0. Situations are highlighted in which this correlation allows R0 to be estimated with greater ease than its constituent parameters. Implications of correlation for parameter identifiability are discussed. Uncertainty estimates and sensitivity analysis are used to investigate how the frequency at which data is sampled affects the estimation process and how the accuracy and uncertainty of estimates improves as data is collected over the course of an outbreak. We assess the informativeness of individual data points in a given time series to determine when more frequent sampling (if possible) would prove to be most beneficial to the estimation process. This technique can be used to design data sampling schemes in more general contexts.
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Affiliation(s)
- Alex Capaldi
- Center for Quantitative Sciences in Biomedicine and Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States.
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23
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Chapman JD, Chappell MJ, Evans ND. The use of a formal sensitivity analysis on epidemic models with immune protection from maternally acquired antibodies. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2011; 104:37-49. [PMID: 21067842 DOI: 10.1016/j.cmpb.2010.08.019] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2009] [Revised: 08/13/2010] [Accepted: 08/31/2010] [Indexed: 05/30/2023]
Abstract
This paper considers the outcome of a formal sensitivity analysis on a series of epidemic model structures developed to study the population level effects of maternal antibodies. The analysis is used to compare the potential influence of maternally acquired immunity on various age and time domain observations of infection and serology, with and without seasonality. The results of the analysis indicate that time series observations are largely insensitive to variations in the average duration of this protection, and that age related empirical data are likely to be most appropriate for estimating these characteristics.
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Affiliation(s)
- J D Chapman
- School of Engineering, University of Warwick, Coventry, CV4 7AL, United Kingdom.
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24
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Grandjean TRB, Chappell MJ, Yates JTW, Jones K, Wood G, Coleman T. Compartmental modelling of the pharmacokinetics of a breast cancer resistance protein. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2011; 104:81-92. [PMID: 20971524 DOI: 10.1016/j.cmpb.2010.08.018] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2009] [Revised: 07/21/2010] [Accepted: 08/31/2010] [Indexed: 05/30/2023]
Abstract
A mathematical model for the pharmacokinetics of Hoechst 33342 following administration into a culture medium containing a population of transfected cells (HEK293 hBCRP) with a potent breast cancer resistance protein inhibitor, Fumitremorgin C (FTC), present is described. FTC is reported to almost completely annul resistance mediated by BCRP in vitro. This non-linear compartmental model has seven macroscopic sub-units, with 14 rate parameters. It describes the relationship between the concentration of Hoechst 33342 and FTC, initially spiked in the medium, and the observed change in fluorescence due to Hoechst 33342 binding to DNA. Structural identifiability analysis has been performed using two methods, one based on the similarity transformation/exhaustive modelling approach and the other based on the differential algebra approach. The analyses demonstrated that all models derived are uniquely identifiable for the experiments/observations available. A kinetic modelling software package, namely FACSIMILE (MPCA Software, UK), was used for parameter fitting and to obtain numerical solutions for the system equations. Model fits gave very good agreement with in vitro data provided by AstraZeneca across a variety of experimental scenarios.
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25
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Ogungbenro K, Aarons L. Structural identifiability analysis of pharmacokinetic models using DAISY: semi-mechanistic gastric emptying models for 13C-octanoic acid. J Pharmacokinet Pharmacodyn 2011; 38:279-92. [PMID: 21347679 DOI: 10.1007/s10928-011-9193-5] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/23/2010] [Accepted: 02/11/2011] [Indexed: 11/27/2022]
Abstract
Structural identifiability analysis is necessary for efficient parameter estimation and it is concerned with determination of whether the parameters in a model can be identified from specified experiments with perfect input-output data. Structural identifiability analysis is very important in mathematical modelling of biological and biomedical experiments and should be considered at the design stage of these experiments. There are three possible outcomes from a structural identifiability analysis; globally/uniquely identifiable, locally/non-uniquely identifiable or non-identifiable/unidentifiable. An ideal outcome is a globally/uniquely identifiable model, however a locally/non-uniquely identifiable outcome can help to identify areas of the model or experiment that need improvement. Despite the importance of structural identifiability analysis, it is still not widely used due to the heavy computational burden involved and the lack of software. A new software package, DAISY, that implemented differential algebra for identifiability analysis was recently released. DAISY is freely available, easy to use and does not require any high-level programming skill. The (13)C-octanoic acid breath test is now widely used for assessing the rate of gastric emptying in patients. Unlike scintigraphy, which is the gold standard and is a direct measure of the rate of gastric emptying, the (13)C-octanoic acid breath test is an indirect method for assessing the rate of gastric emptying. However the (13)C-octanoic acid breath test is cheaper, safer and easy to perform. Because the rate of excretion of (13)CO(2) in breath does not only reflect the rate of gastric emptying but other processes involved between the ingestion of (13)C-octanoic acid and elimination of (13)CO(2) in breath, the parameters commonly derived from the excretion data are not direct measures of gastric emptying. The aim of this paper was to propose a new semi-mechanistic model for the analysis of (13)C-octanoic acid breath excretion data and demonstrate the use of DAISY to assess the identifiability of the model. One- and two-compartment disposition models were linked to a model which has separate compartments for the stomach, intestine and breath. To obtain a globally identifiable model, a repeated (13)C-octanoic breath test in the same individual experimental design was also investigated and this adds a separate stomach compartment to the model. Finally the gastric emptying rate constant from the first (13)C-octanoic breath test was constrained to be the same as the absorption rate constant from the intestine. From the structural identifiability analysis carried out in DAISY, the model based on two experiments (baseline and treatment) and a constraint is globally identifiable. In summary, the present work describes a new semi-mechanistic model that will allow efficient and reliable assessment of the rate of gastric emptying from the (13)C-octanoic breath test.
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Affiliation(s)
- Kayode Ogungbenro
- Centre for Applied Pharmacokinetic Research, The University of Manchester, Oxford Road, Manchester M13 9PL, UK.
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26
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The structural identifiability of susceptible–infective–recovered type epidemic models with incomplete immunity and birth targeted vaccination. Biomed Signal Process Control 2009. [DOI: 10.1016/j.bspc.2009.02.003] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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27
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Jiménez-Hornero JE, Santos-Dueñas IM, Garci A-Garci A I. Structural identifiability of a model for the acetic acid fermentation process. Math Biosci 2008; 216:154-62. [PMID: 18848572 DOI: 10.1016/j.mbs.2008.09.004] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/26/2008] [Revised: 09/15/2008] [Accepted: 09/19/2008] [Indexed: 10/21/2022]
Abstract
Modelling has proved an essential tool for addressing research into biotechnological processes, particularly with a view to their optimization and control. Parameter estimation via optimization approaches is among the major steps in the development of biotechnology models. In fact, one of the first tasks in the development process is to determine whether the parameters concerned can be unambiguously determined and provide meaningful physical conclusions as a result. The analysis process is known as 'identifiability' and presents two different aspects: structural or theoretical identifiability and practical identifiability. While structural identifiability is concerned with model structure alone, practical identifiability takes into account both the quantity and quality of experimental data. In this work, we discuss the theoretical identifiability of a new model for the acetic acid fermentation process and review existing methods for this purpose.
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Affiliation(s)
- Jorge E Jiménez-Hornero
- Computing and Numerical Analysis Department, University of Córdoba, P.O. Box 14071, Córdoba, Spain.
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28
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White LJ, Mandl JN, Gomes MGM, Bodley-Tickell AT, Cane PA, Perez-Brena P, Aguilar JC, Siqueira MM, Portes SA, Straliotto SM, Waris M, Nokes DJ, Medley GF. Understanding the transmission dynamics of respiratory syncytial virus using multiple time series and nested models. Math Biosci 2007; 209:222-39. [PMID: 17335858 PMCID: PMC3724053 DOI: 10.1016/j.mbs.2006.08.018] [Citation(s) in RCA: 64] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2005] [Revised: 08/02/2006] [Accepted: 08/24/2006] [Indexed: 11/23/2022]
Abstract
The nature and role of re-infection and partial immunity are likely to be important determinants of the transmission dynamics of human respiratory syncytial virus (hRSV). We propose a single model structure that captures four possible host responses to infection and subsequent reinfection: partial susceptibility, altered infection duration, reduced infectiousness and temporary immunity (which might be partial). The magnitude of these responses is determined by four homotopy parameters, and by setting some of these parameters to extreme values we generate a set of eight nested, deterministic transmission models. In order to investigate hRSV transmission dynamics, we applied these models to incidence data from eight international locations. Seasonality is included as cyclic variation in transmission. Parameters associated with the natural history of the infection were assumed to be independent of geographic location, while others, such as those associated with seasonality, were assumed location specific. Models incorporating either of the two extreme assumptions for immunity (none or solid and lifelong) were unable to reproduce the observed dynamics. Model fits with either waning or partial immunity to disease or both were visually comparable. The best fitting structure was a lifelong partial immunity to both disease and infection. Observed patterns were reproduced by stochastic simulations using the parameter values estimated from the deterministic models.
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Affiliation(s)
- L J White
- Department of Biological Sciences, University of Warwick, Coventry CV4 7AL, UK.
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