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Pakkanen MS, Miscouridou X, Penn MJ, Whittaker C, Berah T, Mishra S, Mellan TA, Bhatt S. Unifying incidence and prevalence under a time-varying general branching process. J Math Biol 2023; 87:35. [PMID: 37526739 PMCID: PMC10393927 DOI: 10.1007/s00285-023-01958-w] [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: 03/02/2022] [Revised: 12/23/2022] [Accepted: 04/29/2023] [Indexed: 08/02/2023]
Abstract
Renewal equations are a popular approach used in modelling the number of new infections, i.e., incidence, in an outbreak. We develop a stochastic model of an outbreak based on a time-varying variant of the Crump-Mode-Jagers branching process. This model accommodates a time-varying reproduction number and a time-varying distribution for the generation interval. We then derive renewal-like integral equations for incidence, cumulative incidence and prevalence under this model. We show that the equations for incidence and prevalence are consistent with the so-called back-calculation relationship. We analyse two particular cases of these integral equations, one that arises from a Bellman-Harris process and one that arises from an inhomogeneous Poisson process model of transmission. We also show that the incidence integral equations that arise from both of these specific models agree with the renewal equation used ubiquitously in infectious disease modelling. We present a numerical discretisation scheme to solve these equations, and use this scheme to estimate rates of transmission from serological prevalence of SARS-CoV-2 in the UK and historical incidence data on Influenza, Measles, SARS and Smallpox.
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Affiliation(s)
- Mikko S Pakkanen
- Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada.
- Department of Mathematics, Imperial College London, London, UK.
| | | | - Matthew J Penn
- Department of Statistics, University of Oxford, Oxford, UK
| | - Charles Whittaker
- Department of Infectious Disease Epidemiology, Imperial College London, London, UK
| | - Tresnia Berah
- Department of Mathematics, Imperial College London, London, UK
| | - Swapnil Mishra
- Department of Infectious Disease Epidemiology, Imperial College London, London, UK
- Section of Epidemiology, Department of Public Health, University of Copenhagen, Copenhagen, Denmark
| | - Thomas A Mellan
- Department of Infectious Disease Epidemiology, Imperial College London, London, UK
| | - Samir Bhatt
- Department of Infectious Disease Epidemiology, Imperial College London, London, UK.
- Section of Epidemiology, Department of Public Health, University of Copenhagen, Copenhagen, Denmark.
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Messina E, Pezzella M, Vecchio A. Nonlocal finite difference discretization of a class of renewal equation models for epidemics. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:11656-11675. [PMID: 37501414 DOI: 10.3934/mbe.2023518] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 07/29/2023]
Abstract
In this paper we consider a non-standard discretization to a Volterra integro-differential system which includes a number of age-of-infection models in the literature. The aim is to provide a general framework to analyze the proposed scheme for the numerical solution of a class of problems whose continuous dynamic is well known in the literature and allow a deeper analysis in cases where the theory lacks.
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Affiliation(s)
- Eleonora Messina
- Department of Mathematics and Applications, University of Naples Federico II, Via Cintia, I-80126 Naples, Italy
- Member of the Italian INdAM Research group GNCS
| | - Mario Pezzella
- Department of Mathematics and Applications, University of Naples Federico II, Via Cintia, I-80126 Naples, Italy
- Member of the Italian INdAM Research group GNCS
| | - Antonia Vecchio
- Member of the Italian INdAM Research group GNCS
- C.N.R. National Research Council of Italy, Institute for Computational Application "Mauro Picone", Via P. Castellino, 111 - 80131 Naples, Italy
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Park SW, Bolker BM, Funk S, Metcalf CJE, Weitz JS, Grenfell BT, Dushoff J. The importance of the generation interval in investigating dynamics and control of new SARS-CoV-2 variants. J R Soc Interface 2022; 19:20220173. [PMID: 35702867 PMCID: PMC9198506 DOI: 10.1098/rsif.2022.0173] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022] Open
Abstract
Inferring the relative strength (i.e. the ratio of reproduction numbers) and relative speed (i.e. the difference between growth rates) of new SARS-CoV-2 variants is critical to predicting and controlling the course of the current pandemic. Analyses of new variants have primarily focused on characterizing changes in the proportion of new variants, implicitly or explicitly assuming that the relative speed remains fixed over the course of an invasion. We use a generation-interval-based framework to challenge this assumption and illustrate how relative strength and speed change over time under two idealized interventions: a constant-strength intervention like idealized vaccination or social distancing, which reduces transmission rates by a constant proportion, and a constant-speed intervention like idealized contact tracing, which isolates infected individuals at a constant rate. In general, constant-strength interventions change the relative speed of a new variant, while constant-speed interventions change its relative strength. Differences in the generation-interval distributions between variants can exaggerate these changes and modify the effectiveness of interventions. Finally, neglecting differences in generation-interval distributions can bias estimates of relative strength.
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Affiliation(s)
- Sang Woo Park
- Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, USA
| | - Benjamin M Bolker
- Department of Biology, McMaster University, Hamilton, Ontario, Canada.,Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada.,M. G. DeGroote Institute for Infectious Disease Research, McMaster University, Hamilton, Ontario, Canada
| | - Sebastian Funk
- Department for Infectious Disease Epidemiology, London School of Hygiene & Tropical Medicine, London, UK.,Centre for Mathematical Modelling of Infectious Diseases, London School of Hygiene & Tropical Medicine, London, UK
| | - C Jessica E Metcalf
- Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, USA.,Princeton School of Public and International Affairs, Princeton University, Princeton, NJ, USA
| | - Joshua S Weitz
- School of Biological Sciences, Georgia Institute of Technology, Atlanta, GA, USA.,School of Physics, Georgia Institute of Technology, Atlanta, GA, USA.,Institut de Biologie, École Normale Supérieure, Paris, France
| | - Bryan T Grenfell
- Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, USA.,Princeton School of Public and International Affairs, Princeton University, Princeton, NJ, USA
| | - Jonathan Dushoff
- Department of Biology, McMaster University, Hamilton, Ontario, Canada.,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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Abstract
In this paper, we study a dynamically consistent numerical method for the approximation of a nonlinear integro-differential equation modeling an epidemic with age of infection. The discrete scheme is based on direct quadrature methods with Gregory convolution weights and preserves, with no restrictive conditions on the step-length of integration h, some of the essential properties of the continuous system. In particular, the numerical solution is positive and bounded and, in cases of interest in applications, it is monotone. We prove an order of convergence theorem and show by numerical experiments that the discrete final size tends to its continuous equivalent as h tends to zero.
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Abstract
An epidemic can be characterized by its strength (i.e., the reproductive number R) and speed (i.e., the exponential growth rate r). Disease modellers have historically placed much more emphasis on strength, in part because the effectiveness of an intervention strategy is typically evaluated on this scale. Here, we develop a mathematical framework for the classic, strength-based paradigm and show that there is a dual speed-based paradigm which can provide complementary insights. In particular, we note that r = 0 is a threshold for disease spread, just like R=1 [
1], and show that we can measure the strength and speed of an intervention on the same scale as the strength and speed of an epidemic, respectively. We argue that, while the strength-based paradigm provides the clearest insight into certain questions, the speed-based paradigm provides the clearest view in other cases. As an example, we show that evaluating the prospects of ‘test-and-treat’ interventions against the human immunodeficiency virus (HIV) can be done more clearly on the speed than strength scale, given uncertainty in the proportion of HIV spread that happens early in the course of infection. We also discuss evaluating the effects of the importance of pre-symptomatic transmission of the SARS-CoV-2 virus. We suggest that disease modellers should avoid over-emphasizing the reproductive number at the expense of the exponential growth rate, but instead look at these as complementary measures.
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Affiliation(s)
- Jonathan Dushoff
- Department of Biology, McMaster University, Hamilton, Ontario, Canada.,Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada.,M. G. DeGroote Institute for Infectious Disease Research, McMaster University, Hamilton, Ontario, Canada
| | - Sang Woo Park
- Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, USA
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Forward-looking serial intervals correctly link epidemic growth to reproduction numbers. Proc Natl Acad Sci U S A 2021; 118:2011548118. [PMID: 33361331 PMCID: PMC7812760 DOI: 10.1073/pnas.2011548118] [Citation(s) in RCA: 36] [Impact Index Per Article: 12.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022] Open
Abstract
The generation and serial interval distributions are key, but different, quantities in outbreak analyses. Recent studies suggest that the two distributions give different estimates of the reproduction number R as inferred from the observed growth rate r. Here, we show that estimating R based on r and the serial interval distribution, when defined from the correct reference cohort, gives the same estimate as using r and the generation interval distribution. We apply our framework to COVID-19 serial interval data from China, outside Hubei province (January 21 to February 8, 2020), revealing systematic biases in prior inference methods. Our study provides the theoretical basis for practical changes to the principled use of serial interval distributions in estimating R during epidemics. The reproduction number R and the growth rate r are critical epidemiological quantities. They are linked by generation intervals, the time between infection and onward transmission. Because generation intervals are difficult to observe, epidemiologists often substitute serial intervals, the time between symptom onset in successive links in a transmission chain. Recent studies suggest that such substitution biases estimates of R based on r. Here we explore how these intervals vary over the course of an epidemic, and the implications for R estimation. Forward-looking serial intervals, measuring time forward from symptom onset of an infector, correctly describe the renewal process of symptomatic cases and therefore reliably link R with r. In contrast, backward-looking intervals, which measure time backward, and intrinsic intervals, which neglect population-level dynamics, give incorrect R estimates. Forward-looking intervals are affected both by epidemic dynamics and by censoring, changing in complex ways over the course of an epidemic. We present a heuristic method for addressing biases that arise from neglecting changes in serial intervals. We apply the method to early (21 January to February 8, 2020) serial interval-based estimates of R for the COVID-19 outbreak in China outside Hubei province; using improperly defined serial intervals in this context biases estimates of initial R by up to a factor of 2.6. This study demonstrates the importance of early contact tracing efforts and provides a framework for reassessing generation intervals, serial intervals, and R estimates for COVID-19.
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Park SW, Champredon D, Dushoff J. Inferring generation-interval distributions from contact-tracing data. J R Soc Interface 2020; 17:20190719. [PMID: 32574542 PMCID: PMC7328397 DOI: 10.1098/rsif.2019.0719] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Generation intervals, defined as the time between when an individual is infected and when that individual infects another person, link two key quantities that describe an epidemic: the initial reproductive number, Rinitial, and the initial rate of exponential growth, r. Generation intervals can be measured through contact tracing by identifying who infected whom. We study how realized intervals differ from ‘intrinsic’ intervals that describe individual-level infectiousness and identify both spatial and temporal effects, including truncating (due to observation time), and the effects of susceptible depletion at various spatial scales. Early in an epidemic, we expect the variation in the realized generation intervals to be mainly driven by truncation and by the population structure near the source of disease spread; we predict that correcting realized intervals for the effect of temporal truncation but not for spatial effects will provide the initial forward generation-interval distribution, which is spatially informed and correctly links r and Rinitial. We develop and test statistical methods for temporal corrections of generation intervals, and confirm our prediction using individual-based simulations on an empirical network.
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Affiliation(s)
- Sang Woo Park
- Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, USA.,Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada
| | - David Champredon
- Department of Biology, McMaster University, Hamilton, ON, Canada.,Department of Pathology and Laboratory Medicine, University of Western Ontario, London, ON, Canada
| | - Jonathan Dushoff
- Department of Biology, McMaster University, Hamilton, ON, Canada.,Michael G. DeGroote Institute for Infectious Disease Research, McMaster University, Hamilton, ON, Canada
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Costantino V, Kunasekaran MP, Chughtai AA, MacIntyre CR. How Valid Are Assumptions About Re-emerging Smallpox? A Systematic Review of Parameters Used in Smallpox Mathematical Models. Mil Med 2019; 183:e200-e207. [PMID: 29425329 DOI: 10.1093/milmed/usx092] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2017] [Accepted: 11/09/2017] [Indexed: 01/16/2023] Open
Abstract
Background Globally eradicated in 1980, smallpox is listed as a category A bioterrorism agent. If smallpox were to re-emerge, it may be due to an act of bioterrorism or a laboratory accident, and the impact is likely to be severe. Preparedness against smallpox is subject to more uncertainty than other infectious diseases because it is eradicated, there is uncertainty about population immunity, and the current global health workforce has no practical experience or living memory of smallpox. In the event of re-emergence of smallpox, mathematical modeling plays a crucial role in improving the evidence base to inform preparedness, mitigation, and response activities. However, the predictions of mathematical models about outbreak magnitude and impact depend critically on the assumptions and disease parameters used. We aimed to identify modeling studies that would be applicable to re-emerging smallpox and to evaluate consistency and the certainty of the evidence published about the key parameters used. Methods We conducted a systematic review using PRISMA criteria, of assumptions used in modeling studies on duration of latent, prodromal, and infectious period, as well as the choice of the basic reproduction number (R0) for re-emerging smallpox. We performed a literature search using PubMED, Scopus, Web of Science, and EMBASE and included peer-reviewed articles that focused on smallpox models, stated at least three of the aforementioned parameters and published in English. Findings A total of 42 studies were selected for inclusion. There was general agreement on the duration of latent and prodromal periods, being 11-12 d (88%) and 3 d (59%), respectively. The duration of the infectious period varied from 4 to 20 d. Most models assumed 16 d (19%), 12 d (16.7%), and 8.6 d (12%) of infectiousness. In 25/34 studies, R0 ranged between 3 and 5, generally lower than the R0 calculated from past outbreaks. Discussion Models of smallpox re-emergence also tend to use the same limited available historical data sources but assume a wide range of different estimates for key parameters. Models use very optimistic assumptions of decreased population immunity, despite high uncertainty about duration and magnitude of post-vaccination immunity. This review reveals a paradox. A substantial proportion of the modern population is unvaccinated, never exposed to boosting from wild-type smallpox, or immunocompromised; furthermore, vaccine-induced immunity wanes over time. Failure to consider these factors in a model will lead to underestimating the true impact of a re-emergent smallpox epidemic in the contemporary population.
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Affiliation(s)
- Valentina Costantino
- School of Public Health and Community Medicine, Faculty of Medicine, University of New South Wales, Sydney, NSW, Australia
| | - Mohana P Kunasekaran
- School of Public Health and Community Medicine, Faculty of Medicine, University of New South Wales, Sydney, NSW, Australia
| | - Abrar A Chughtai
- School of Public Health and Community Medicine, Faculty of Medicine, University of New South Wales, Sydney, NSW, Australia
| | - Chandini R MacIntyre
- School of Public Health and Community Medicine, Faculty of Medicine, University of New South Wales, Sydney, NSW, Australia.,College of Public Service and Community Solutions, Arizona State University, 411 Central Avenue #750, Phoenix, AZ
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9
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Park SW, Champredon D, Weitz JS, Dushoff J. A practical generation-interval-based approach to inferring the strength of epidemics from their speed. Epidemics 2019; 27:12-18. [PMID: 30799184 DOI: 10.1016/j.epidem.2018.12.002] [Citation(s) in RCA: 39] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2018] [Revised: 12/18/2018] [Accepted: 12/28/2018] [Indexed: 11/16/2022] Open
Abstract
Infectious disease outbreaks are often characterized by the reproduction number R and exponential rate of growth r. R provides information about outbreak control and predicted final size, but estimating R is difficult, while r can often be estimated directly from incidence data. These quantities are linked by the generation interval - the time between when an individual is infected by an infector, and when that infector was infected. It is often infeasible to obtain the exact shape of a generation-interval distribution, and to understand how this shape affects estimates of R. We show that estimating generation interval mean and variance provides insight into the relationship between R and r. We use examples based on Ebola, rabies and measles to explore approximations based on gamma-distributed generation intervals, and find that use of these simple approximations are often sufficient to capture the r-R relationship and provide robust estimates of R.
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Affiliation(s)
- Sang Woo Park
- Department of Mathematics & Statistics, McMaster University, Hamilton, Ontario, Canada
| | - David Champredon
- Department of Biology, McMaster University, Hamilton, Ontario, Canada; Department of Mathematics & Statistics, Agent-Based Modelling Laboratory, York University, Toronto, Ontario, Canada
| | - Joshua S Weitz
- School of Biological Sciences, Georgia Institute of Technology, Atlanta, Georgia, United States; School of Physics, Georgia Institute of Technology, Atlanta, Georgia, United States
| | - Jonathan Dushoff
- Department of Biology, McMaster University, Hamilton, Ontario, Canada.
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10
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Graeden E, Fielding R, Steinhouse KE, Rubin IN. Modeling the Effect of Herd Immunity and Contagiousness in Mitigating a Smallpox Outbreak. Med Decis Making 2014; 35:648-59. [DOI: 10.1177/0272989x14561681] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2013] [Accepted: 10/18/2014] [Indexed: 01/13/2023]
Abstract
The smallpox antiviral tecovirimat has recently been purchased by the U.S. Strategic National Stockpile. Given significant uncertainty regarding both the contagiousness of smallpox in a contemporary outbreak and the efficiency of a mass vaccination campaign, vaccine prophylaxis alone may be unable to control a smallpox outbreak following a bioterror attack. Here, we present the results of a compartmental epidemiological model that identifies conditions under which tecovirimat is required to curtail the epidemic by exploring how the interaction between contagiousness and prophylaxis coverage of the affected population affects the ability of the public health response to control a large-scale smallpox outbreak. Each parameter value in the model is based on published empirical data. We describe contagiousness parametrically using a novel method of distributing an assumed R-value over the disease course based on the relative rates of daily viral shedding from human and animal studies of cognate orthopoxvirus infections. Our results suggest that vaccination prophylaxis is sufficient to control the outbreak when caused either by a minimally contagious virus or when a very high percentage of the population receives prophylaxis. As vaccination coverage of the affected population decreases below 70%, vaccine prophylaxis alone is progressively less capable of controlling outbreaks, even those caused by a less contagious virus (R0 less than 4). In these scenarios, tecovirimat treatment is required to control the outbreak (total number of cases under an order of magnitude more than the number of initial infections). The first study to determine the relative importance of smallpox prophylaxis and treatment under a range of highly uncertain epidemiological parameters, this work provides public health decision-makers with an evidence-based guide for responding to a large-scale smallpox outbreak.
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Pellis L, Ferguson NM, Fraser C. Threshold parameters for a model of epidemic spread among households and workplaces. J R Soc Interface 2009; 6:979-87. [PMID: 19324683 PMCID: PMC2827443 DOI: 10.1098/rsif.2008.0493] [Citation(s) in RCA: 41] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2008] [Accepted: 01/07/2009] [Indexed: 11/12/2022] Open
Abstract
The basic reproduction number R0 is one of the most important concepts in modern infectious disease epidemiology. However, for more realistic and more complex models than those assuming homogeneous mixing in the population, other threshold quantities can be defined that are sometimes more useful and easily derived in terms of model parameters. In this paper, we present a model for the spread of a permanently immunizing infection in a population socially structured into households and workplaces/schools, and we propose and discuss a new household-to-household reproduction number RH for it. We show how RH overcomes some of the limitations of a previously proposed threshold parameter, and we highlight its relationship with the effort required to control an epidemic when interventions are targeted at randomly selected households.
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Affiliation(s)
- L Pellis
- Department of Infectious Disease Epidemiology, Imperial College London, London SW7 2AZ, UK.
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12
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Nishiura H, Brockmann SO, Eichner M. Extracting key information from historical data to quantify the transmission dynamics of smallpox. Theor Biol Med Model 2008; 5:20. [PMID: 18715509 PMCID: PMC2538509 DOI: 10.1186/1742-4682-5-20] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/11/2008] [Accepted: 08/20/2008] [Indexed: 11/19/2022] Open
Abstract
Background Quantification of the transmission dynamics of smallpox is crucial for optimizing intervention strategies in the event of a bioterrorist attack. This article reviews basic methods and findings in mathematical and statistical studies of smallpox which estimate key transmission parameters from historical data. Main findings First, critically important aspects in extracting key information from historical data are briefly summarized. We mention different sources of heterogeneity and potential pitfalls in utilizing historical records. Second, we discuss how smallpox spreads in the absence of interventions and how the optimal timing of quarantine and isolation measures can be determined. Case studies demonstrate the following. (1) The upper confidence limit of the 99th percentile of the incubation period is 22.2 days, suggesting that quarantine should last 23 days. (2) The highest frequency (61.8%) of secondary transmissions occurs 3–5 days after onset of fever so that infected individuals should be isolated before the appearance of rash. (3) The U-shaped age-specific case fatality implies a vulnerability of infants and elderly among non-immune individuals. Estimates of the transmission potential are subsequently reviewed, followed by an assessment of vaccination effects and of the expected effectiveness of interventions. Conclusion Current debates on bio-terrorism preparedness indicate that public health decision making must account for the complex interplay and balance between vaccination strategies and other public health measures (e.g. case isolation and contact tracing) taking into account the frequency of adverse events to vaccination. In this review, we summarize what has already been clarified and point out needs to analyze previous smallpox outbreaks systematically.
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Affiliation(s)
- Hiroshi Nishiura
- Theoretical Epidemiology, University of Utrecht, Yalelaan 7, 3584CL, Utrecht, The Netherlands.
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13
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Roberts MG, Heesterbeek JAP. Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection. J Math Biol 2007; 55:803-16. [PMID: 17684743 PMCID: PMC2782110 DOI: 10.1007/s00285-007-0112-8] [Citation(s) in RCA: 73] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2006] [Revised: 05/23/2007] [Indexed: 11/10/2022]
Abstract
We investigate the merit of deriving an estimate of the basic reproduction number \documentclass[12pt]{minimal}
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, and we quantify the discrepancies that arise.
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Affiliation(s)
- M. G. Roberts
- Centre for Mathematical Biology, Institute of Information and Mathematical Sciences, Massey University, Private Bag 102 904, North Shore Mail Centre, Auckland, New Zealand
| | - J. A. P. Heesterbeek
- Faculty of Veterinary Medicine, University of Utrecht, Yalelaan 7, 3584 CL Utrecht, The Netherlands
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14
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Roberts MG, Baker M, Jennings LC, Sertsou G, Wilson N. A model for the spread and control of pandemic influenza in an isolated geographical region. J R Soc Interface 2007; 4:325-30. [PMID: 17251145 PMCID: PMC2359860 DOI: 10.1098/rsif.2006.0176] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2006] [Accepted: 10/22/2006] [Indexed: 11/12/2022] Open
Abstract
In the event of an influenza pandemic, the most probable way in which the virus would be introduced to an isolated geographical area is by an infected traveller. We use a mathematical model, structured on the location at which infection occurs and based on published parameters for influenza, to describe an epidemic in a community of one million people. The model is then modified to reflect a variety of control strategies based on social distancing measures, targeted antiviral treatment and antiviral prophylaxis and home quarantine, and the effectiveness of the strategies is compared. The results suggest that the only single strategy that would be successful in preventing an epidemic (with R0=2.0) is targeted antiviral treatment and prophylaxis, and that closing schools combined with either closing work places or home quarantine would only prevent such an epidemic if these strategies were combined with a modest level of antiviral coverage.
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Affiliation(s)
- M G Roberts
- Centre for Mathematical Biology, Institute of Information & Mathematical Sciences, Massey University, Auckland, New Zealand.
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15
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Klinkenberg D, Fraser C, Heesterbeek H. The effectiveness of contact tracing in emerging epidemics. PLoS One 2006; 1:e12. [PMID: 17183638 PMCID: PMC1762362 DOI: 10.1371/journal.pone.0000012] [Citation(s) in RCA: 145] [Impact Index Per Article: 8.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2006] [Accepted: 09/15/2006] [Indexed: 11/30/2022] Open
Abstract
Background Contact tracing plays an important role in the control of emerging infectious diseases, but little is known yet about its effectiveness. Here we deduce from a generic mathematical model how effectiveness of tracing relates to various aspects of time, such as the course of individual infectivity, the (variability in) time between infection and symptom-based detection, and delays in the tracing process. In addition, the possibility of iteratively tracing of yet asymptomatic infecteds is considered. With these insights we explain why contact tracing was and will be effective for control of smallpox and SARS, only partially effective for foot-and-mouth disease, and likely not effective for influenza. Methods and Findings We investigate contact tracing in a model of an emerging epidemic that is flexible enough to use for most infections. We consider isolation of symptomatic infecteds as the basic scenario, and express effectiveness as the proportion of contacts that need to be traced for a reproduction ratio smaller than 1. We obtain general results for special cases, which are interpreted with respect to the likely success of tracing for influenza, smallpox, SARS, and foot-and-mouth disease epidemics. Conclusions We conclude that (1) there is no general predictive formula for the proportion to be traced as there is for the proportion to be vaccinated; (2) variability in time to detection is favourable for effective tracing; (3) tracing effectiveness need not be sensitive to the duration of the latent period and tracing delays; (4) iterative tracing primarily improves effectiveness when single-step tracing is on the brink of being effective.
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Affiliation(s)
- Don Klinkenberg
- Theoretical Epidemiology, Department of Farm Animal Health, Faculty of Veterinary Medicine, Utrecht University, Utrecht, Netherlands.
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