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Delecroix C, van Nes EH, van de Leemput IA, Rotbarth R, Scheffer M, ten Bosch Q. The potential of resilience indicators to anticipate infectious disease outbreaks, a systematic review and guide. PLOS GLOBAL PUBLIC HEALTH 2023; 3:e0002253. [PMID: 37815958 PMCID: PMC10564242 DOI: 10.1371/journal.pgph.0002253] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2023] [Accepted: 07/12/2023] [Indexed: 10/12/2023]
Abstract
To reduce the consequences of infectious disease outbreaks, the timely implementation of public health measures is crucial. Currently used early-warning systems are highly context-dependent and require a long phase of model building. A proposed solution to anticipate the onset or termination of an outbreak is the use of so-called resilience indicators. These indicators are based on the generic theory of critical slowing down and require only incidence time series. Here we assess the potential for this approach to contribute to outbreak anticipation. We systematically reviewed studies that used resilience indicators to predict outbreaks or terminations of epidemics. We identified 37 studies meeting the inclusion criteria: 21 using simulated data and 16 real-world data. 36 out of 37 studies detected significant signs of critical slowing down before a critical transition (i.e., the onset or end of an outbreak), with a highly variable sensitivity (i.e., the proportion of true positive outbreak warnings) ranging from 0.03 to 1 and a lead time ranging from 10 days to 68 months. Challenges include low resolution and limited length of time series, a too rapid increase in cases, and strong seasonal patterns which may hamper the sensitivity of resilience indicators. Alternative types of data, such as Google searches or social media data, have the potential to improve predictions in some cases. Resilience indicators may be useful when the risk of disease outbreaks is changing gradually. This may happen, for instance, when pathogens become increasingly adapted to an environment or evolve gradually to escape immunity. High-resolution monitoring is needed to reach sufficient sensitivity. If those conditions are met, resilience indicators could help improve the current practice of prediction, facilitating timely outbreak response. We provide a step-by-step guide on the use of resilience indicators in infectious disease epidemiology, and guidance on the relevant situations to use this approach.
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Affiliation(s)
- Clara Delecroix
- Department of Environmental Sciences, Wageningen University, Wageningen, The Netherlands
- Quantitative Veterinary Epidemiology, Wageningen University, Wageningen, The Netherlands
| | - Egbert H. van Nes
- Department of Environmental Sciences, Wageningen University, Wageningen, The Netherlands
| | | | - Ronny Rotbarth
- Department of Environmental Sciences, Wageningen University, Wageningen, The Netherlands
| | - Marten Scheffer
- Department of Environmental Sciences, Wageningen University, Wageningen, The Netherlands
| | - Quirine ten Bosch
- Quantitative Veterinary Epidemiology, Wageningen University, Wageningen, The Netherlands
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2
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Tredennick AT, O'Dea EB, Ferrari MJ, Park AW, Rohani P, Drake JM. Anticipating infectious disease re-emergence and elimination: a test of early warning signals using empirically based models. JOURNAL OF THE ROYAL SOCIETY, INTERFACE 2022; 19:20220123. [PMID: 35919978 PMCID: PMC9346357 DOI: 10.1098/rsif.2022.0123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
Timely forecasts of the emergence, re-emergence and elimination of human infectious diseases allow for proactive, rather than reactive, decisions that save lives. Recent theory suggests that a generic feature of dynamical systems approaching a tipping point-early warning signals (EWS) due to critical slowing down (CSD)-can anticipate disease emergence and elimination. Empirical studies documenting CSD in observed disease dynamics are scarce, but such demonstration of concept is essential to the further development of model-independent outbreak detection systems. Here, we use fitted, mechanistic models of measles transmission in four cities in Niger to detect CSD through statistical EWS. We find that several EWS accurately anticipate measles re-emergence and elimination, suggesting that CSD should be detectable before disease transmission systems cross key tipping points. These findings support the idea that statistical signals based on CSD, coupled with decision-support algorithms and expert judgement, could provide the basis for early warning systems of disease outbreaks.
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Affiliation(s)
- Andrew T Tredennick
- Odum School of Ecology, University of Georgia, Athens, GA 30602, USA.,Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA 30602, USA.,Western EcoSystems Technology, Inc., 1610 East Reynolds Street, Laramie, WY 82070, USA
| | - Eamon B O'Dea
- Odum School of Ecology, University of Georgia, Athens, GA 30602, USA.,Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA 30602, USA
| | - Matthew J Ferrari
- The Center for Infectious Disease Dynamics and Department of Biology, The Pennsylvania State University, University Park, PA 16802, USA
| | - Andrew W Park
- Odum School of Ecology, University of Georgia, Athens, GA 30602, USA.,Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA 30602, USA.,Department of Infectious Diseases, University of Georgia, Athens, GA 30602, USA
| | - Pejman Rohani
- Odum School of Ecology, University of Georgia, Athens, GA 30602, USA.,Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA 30602, USA.,Department of Infectious Diseases, University of Georgia, Athens, GA 30602, USA
| | - John M Drake
- Odum School of Ecology, University of Georgia, Athens, GA 30602, USA.,Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA 30602, USA
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3
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O'Brien DA, Clements CF. Early warning signal reliability varies with COVID-19 waves. Biol Lett 2021; 17:20210487. [PMID: 34875183 PMCID: PMC8651412 DOI: 10.1098/rsbl.2021.0487] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2021] [Accepted: 11/15/2021] [Indexed: 01/07/2023] Open
Abstract
Early warning signals (EWSs) aim to predict changes in complex systems from phenomenological signals in time series data. These signals have recently been shown to precede the emergence of disease outbreaks, offering hope that policymakers can make predictive rather than reactive management decisions. Here, using a novel, sequential analysis in combination with daily COVID-19 case data across 24 countries, we suggest that composite EWSs consisting of variance, autocorrelation and skewness can predict nonlinear case increases, but that the predictive ability of these tools varies between waves based upon the degree of critical slowing down present. Our work suggests that in highly monitored disease time series such as COVID-19, EWSs offer the opportunity for policymakers to improve the accuracy of urgent intervention decisions but best characterize hypothesized critical transitions.
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Affiliation(s)
- Duncan A. O'Brien
- School of Biological Sciences, University of Bristol, Bristol BS8 1TQ, UK
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4
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Southall E, Brett TS, Tildesley MJ, Dyson L. Early warning signals of infectious disease transitions: a review. J R Soc Interface 2021; 18:20210555. [PMID: 34583561 PMCID: PMC8479360 DOI: 10.1098/rsif.2021.0555] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2021] [Accepted: 09/06/2021] [Indexed: 01/07/2023] Open
Abstract
Early warning signals (EWSs) are a group of statistical time-series signals which could be used to anticipate a critical transition before it is reached. EWSs are model-independent methods that have grown in popularity to support evidence of disease emergence and disease elimination. Theoretical work has demonstrated their capability of detecting disease transitions in simple epidemic models, where elimination is reached through vaccination, to more complex vector transmission, age-structured and metapopulation models. However, the exact time evolution of EWSs depends on the transition; here we review the literature to provide guidance on what trends to expect and when. Recent advances include methods which detect when an EWS becomes significant; the earlier an upcoming disease transition is detected, the more valuable an EWS will be in practice. We suggest that future work should firstly validate detection methods with synthetic and historical datasets, before addressing their performance with real-time data which is accruing. A major challenge to overcome for the use of EWSs with disease transitions is to maintain the accuracy of EWSs in data-poor settings. We demonstrate how EWSs behave on reported cases for pertussis in the USA, to highlight some limitations when detecting disease transitions with real-world data.
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Affiliation(s)
- Emma Southall
- The Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, School of Life Sciences and Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
- Mathematics for Real World Systems Centre for Doctoral Training, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
| | - Tobias S. Brett
- Odum School of Ecology, University of Georgia, Athens, GA, USA
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA, USA
| | - Michael J. Tildesley
- The Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, School of Life Sciences and Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
| | - Louise Dyson
- The Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, School of Life Sciences and Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
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5
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Southall E, Tildesley MJ, Dyson L. Prospects for detecting early warning signals in discrete event sequence data: Application to epidemiological incidence data. PLoS Comput Biol 2020; 16:e1007836. [PMID: 32960900 PMCID: PMC7531856 DOI: 10.1371/journal.pcbi.1007836] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2020] [Revised: 10/02/2020] [Accepted: 07/30/2020] [Indexed: 11/18/2022] Open
Abstract
Early warning signals (EWS) identify systems approaching a critical transition, where the system undergoes a sudden change in state. For example, monitoring changes in variance or autocorrelation offers a computationally inexpensive method which can be used in real-time to assess when an infectious disease transitions to elimination. EWS have a promising potential to not only be used to monitor infectious diseases, but also to inform control policies to aid disease elimination. Previously, potential EWS have been identified for prevalence data, however the prevalence of a disease is often not known directly. In this work we identify EWS for incidence data, the standard data type collected by the Centers for Disease Control and Prevention (CDC) or World Health Organization (WHO). We show, through several examples, that EWS calculated on simulated incidence time series data exhibit vastly different behaviours to those previously studied on prevalence data. In particular, the variance displays a decreasing trend on the approach to disease elimination, contrary to that expected from critical slowing down theory; this could lead to unreliable indicators of elimination when calculated on real-world data. We derive analytical predictions which can be generalised for many epidemiological systems, and we support our theory with simulated studies of disease incidence. Additionally, we explore EWS calculated on the rate of incidence over time, a property which can be extracted directly from incidence data. We find that although incidence might not exhibit typical critical slowing down properties before a critical transition, the rate of incidence does, presenting a promising new data type for the application of statistical indicators.
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Affiliation(s)
- Emma Southall
- EPSRC & MRC Centre for Doctoral Training in Mathematics for Real-World Systems, University of Warwick, Coventry, UK
| | - Michael J. Tildesley
- Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, Mathematics Institute and School of Life Sciences, University of Warwick, Coventry, UK
| | - Louise Dyson
- Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, Mathematics Institute and School of Life Sciences, University of Warwick, Coventry, UK
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Simoy MI, Aparicio JP. Ross-Macdonald models: Which one should we use? Acta Trop 2020; 207:105452. [PMID: 32302688 DOI: 10.1016/j.actatropica.2020.105452] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2019] [Revised: 02/07/2020] [Accepted: 03/16/2020] [Indexed: 11/19/2022]
Abstract
Ross-Macdonald models are the building blocks of most vector-borne disease models. Even for the same disease, different authors use different model formulations, but a study of the dynamical consequences of assuming different hypotheses is missing. In this work we present different formulations of the basic Ross-Macdonald model together with a careful discussion of the assumptions behind each model. The most general model presented is an agent based model for which arbitrary distributions for latency and infectious periods for both, host and vectors, is considered. At population level we also developed a deterministic Volterra integral equations model for which also arbitrary distributions in the waiting times are included. We compare the model solutions using different distributions for the infectious and latency periods using statistics, like the epidemic peak, or epidemic final size, to characterize the epidemic curves. The basic reproduction number (R0) for each formulation is computed and compared with empirical estimations obtained with the agent based models. The importance of considering realistic distributions for the latent and infectious periods is highlighted and discussed. We also show that seasonality is a key driver of vector-borne disease dynamics shaping the epidemic curve and its duration.
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Affiliation(s)
- Mario Ignacio Simoy
- Instituto de Investigaciones en Energía no Convencional (INENCO), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Universidad Nacional de Salta, Av. Bolivia 5100, Salta 4400, Argentina; Instituto Multidisciplinario sobre Ecosistemas y Desarrollo Sustentable, Universidad Nacional del Centro de la Provincia de Buenos Aires (UNICEN), Facultad de Ciencias Exactas, Paraje Arroyo Seco s/n, Tandil 7000, Argentina
| | - Juan Pablo Aparicio
- Instituto de Investigaciones en Energía no Convencional (INENCO), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Universidad Nacional de Salta, Av. Bolivia 5100, Salta 4400, Argentina; Simon A. Levin Mathematical, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904 Tempe, AZ 85287-1904, USA.
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7
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Phillips B, Anand M, Bauch CT. Spatial early warning signals of social and epidemiological tipping points in a coupled behaviour-disease network. Sci Rep 2020; 10:7611. [PMID: 32376908 PMCID: PMC7203335 DOI: 10.1038/s41598-020-63849-0] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2019] [Accepted: 04/06/2020] [Indexed: 01/12/2023] Open
Abstract
The resurgence of infectious diseases due to vaccine refusal has highlighted the role of interactions between disease dynamics and the spread of vaccine opinion on social networks. Shifts between disease elimination and outbreak regimes often occur through tipping points. It is known that tipping points can be predicted by early warning signals (EWS) based on characteristic dynamics near the critical transition, but the study of EWS in coupled behaviour-disease networks has received little attention. Here, we test several EWS indicators measuring spatial coherence and autocorrelation for their ability to predict a critical transition corresponding to disease outbreaks and vaccine refusal in a multiplex network model. The model couples paediatric infectious disease spread through a contact network to binary opinion dynamics of vaccine opinion on a social network. Through change point detection, we find that mutual information and join count indicators provided the best EWS. We also show the paediatric infectious disease natural history generates a discrepancy between population-level vaccine opinions and vaccine immunity status, such that transitions in the social network may occur before epidemiological transitions. These results suggest that monitoring social media for EWS of paediatric infectious disease outbreaks using these spatial indicators could be successful.
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Affiliation(s)
- Brendon Phillips
- University of Waterloo, Department of Mathematics, Waterloo, N2L 3G1, Canada.
| | - Madhur Anand
- University of Guelph, School of Environmental Sciences, Guelph, N1G 2W1, Canada
| | - Chris T Bauch
- University of Waterloo, Department of Mathematics, Waterloo, N2L 3G1, Canada
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8
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Harris MJ, Hay SI, Drake JM. Early warning signals of malaria resurgence in Kericho, Kenya. Biol Lett 2020; 16:20190713. [PMID: 32183637 PMCID: PMC7115183 DOI: 10.1098/rsbl.2019.0713] [Citation(s) in RCA: 14] [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: 10/11/2019] [Accepted: 02/19/2020] [Indexed: 11/12/2022] Open
Abstract
Campaigns to eliminate infectious diseases could be greatly aided by methods for providing early warning signals of resurgence. Theory predicts that as a disease transmission system undergoes a transition from stability at the disease-free equilibrium to sustained transmission, it will exhibit characteristic behaviours known as critical slowing down, referring to the speed at which fluctuations in the number of cases are dampened, for instance the extinction of a local transmission chain after infection from an imported case. These phenomena include increases in several summary statistics, including lag-1 autocorrelation, variance and the first difference of variance. Here, we report the first empirical test of this prediction during the resurgence of malaria in Kericho, Kenya. For 10 summary statistics, we measured the approach to criticality in a rolling window to quantify the size of effect and directions. Nine of the statistics increased as predicted and variance, the first difference of variance, autocovariance, lag-1 autocorrelation and decay time returned early warning signals of critical slowing down based on permutation tests. These results show that time series of disease incidence collected through ordinary surveillance activities may exhibit characteristic signatures prior to an outbreak, a phenomenon that may be quite general among infectious disease systems.
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Affiliation(s)
- Mallory J. Harris
- Odum School of Ecology, University of Georgia, Athens, GA 30602, USA
- Biology Department, Stanford University, 371 Serra Mall, Stanford, CA, USA
| | - Simon I. Hay
- Institute for Health Metrics and Evaluation, University of Washington, Seattle, WA 98121, USA
- Department of Health Metrics Sciences, School of Medicine, University of Washington, Seattle, WA 98121, USA
| | - John M. Drake
- Odum School of Ecology, University of Georgia, Athens, GA 30602, USA
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA 30602, USA
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9
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Brett T, Ajelli M, Liu QH, Krauland MG, Grefenstette JJ, van Panhuis WG, Vespignani A, Drake JM, Rohani P. Detecting critical slowing down in high-dimensional epidemiological systems. PLoS Comput Biol 2020; 16:e1007679. [PMID: 32150536 PMCID: PMC7082051 DOI: 10.1371/journal.pcbi.1007679] [Citation(s) in RCA: 26] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2019] [Revised: 03/19/2020] [Accepted: 01/23/2020] [Indexed: 01/05/2023] Open
Abstract
Despite medical advances, the emergence and re-emergence of infectious diseases continue to pose a public health threat. Low-dimensional epidemiological models predict that epidemic transitions are preceded by the phenomenon of critical slowing down (CSD). This has raised the possibility of anticipating disease (re-)emergence using CSD-based early-warning signals (EWS), which are statistical moments estimated from time series data. For EWS to be useful at detecting future (re-)emergence, CSD needs to be a generic (model-independent) feature of epidemiological dynamics irrespective of system complexity. Currently, it is unclear whether the predictions of CSD-derived from simple, low-dimensional systems-pertain to real systems, which are high-dimensional. To assess the generality of CSD, we carried out a simulation study of a hierarchy of models, with increasing structural complexity and dimensionality, for a measles-like infectious disease. Our five models included: i) a nonseasonal homogeneous Susceptible-Exposed-Infectious-Recovered (SEIR) model, ii) a homogeneous SEIR model with seasonality in transmission, iii) an age-structured SEIR model, iv) a multiplex network-based model (Mplex) and v) an agent-based simulator (FRED). All models were parameterised to have a herd-immunity immunization threshold of around 90% coverage, and underwent a linear decrease in vaccine uptake, from 92% to 70% over 15 years. We found evidence of CSD prior to disease re-emergence in all models. We also evaluated the performance of seven EWS: the autocorrelation, coefficient of variation, index of dispersion, kurtosis, mean, skewness, variance. Performance was scored using the Area Under the ROC Curve (AUC) statistic. The best performing EWS were the mean and variance, with AUC > 0.75 one year before the estimated transition time. These two, along with the autocorrelation and index of dispersion, are promising candidate EWS for detecting disease emergence.
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Affiliation(s)
- Tobias Brett
- Odum School of Ecology, University of Georgia, Athens, Georgia, United States of America
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America
| | - Marco Ajelli
- Laboratory for the Modeling of Biological and Socio-technical Systems, Northeastern University, Boston, Massachusetts, United States of America
- Bruno Kessler Foundation, Trento, Italy
| | - Quan-Hui Liu
- Laboratory for the Modeling of Biological and Socio-technical Systems, Northeastern University, Boston, Massachusetts, United States of America
- College of Computer Science, Sichuan University, Chengdu, China
| | - Mary G. Krauland
- University of Pittsburgh, Department of Health Policy and Management, Pittsburgh, Pennsylvania, United States of America
| | - John J. Grefenstette
- University of Pittsburgh, Department of Health Policy and Management, Pittsburgh, Pennsylvania, United States of America
| | - Willem G. van Panhuis
- University of Pittsburgh, Department of Epidemiology, Pittsburgh, Pennsylvania, United States of America
- University of Pittsburgh, Department of Biomedical Informatics, Pittsburgh, Pennsylvania, United States of America
| | - Alessandro Vespignani
- Laboratory for the Modeling of Biological and Socio-technical Systems, Northeastern University, Boston, Massachusetts, United States of America
- ISI Foundation, Turin, Italy
| | - John M. Drake
- Odum School of Ecology, University of Georgia, Athens, Georgia, United States of America
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America
| | - Pejman Rohani
- Odum School of Ecology, University of Georgia, Athens, Georgia, United States of America
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America
- Department of Infectious Diseases, College of Veterinary Medicine, University of Georgia, Athens, Georgia, United States of America
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Gama Dessavre A, Southall E, Tildesley MJ, Dyson L. The problem of detrending when analysing potential indicators of disease elimination. J Theor Biol 2019; 481:183-193. [PMID: 30980869 PMCID: PMC6859505 DOI: 10.1016/j.jtbi.2019.04.011] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/19/2018] [Revised: 04/08/2019] [Accepted: 04/10/2019] [Indexed: 01/04/2023]
Abstract
As we strive towards the elimination of many burdensome diseases, the question of when intervention efforts may cease is increasingly important. It can be very difficult to know when prevalences are low enough that the disease will die out without further intervention, particularly for diseases that lack accurate tests. The consequences of stopping an intervention prematurely can put back elimination efforts by decades. Critical slowing down theory predicts that as a dynamical system moves through a critical transition, deviations from the steady state return increasingly slowly. We study two potential indicators of disease elimination predicted by this theory, and investigate their response using a simple stochastic model. We compare our dynamical predictions to simulations of the fluctuation variance and coefficient of variation as the system moves through the transition to elimination. These comparisons demonstrate that the primary challenge facing the analysis of early warning signs in timeseries data is that of accurately 'detrending' the signal, in order to preserve the statistical properties of the fluctuations. We show here that detrending using the mean of even just four realisations of the process can give a significant improvement when compared to using a moving window average. Taking this idea further, we consider a 'metapopulation' model of an endemic disease, in which infection spreads in various separated areas with some movement between the subpopulations. We successfully predict the behaviour of both variance and the coefficient of variation in a metapopulation by using information from the other subpopulations to detrend the system.
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Affiliation(s)
| | - Emma Southall
- Mathematics Institute, University of Warwick, Coventry, UK
| | - Michael J Tildesley
- Mathematics Institute, University of Warwick, Coventry, UK; School of Life Sciences, University of Warwick, Coventry, UK
| | - Louise Dyson
- Mathematics Institute, University of Warwick, Coventry, UK; School of Life Sciences, University of Warwick, Coventry, UK.
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O'Dea EB, Park AW, Drake JM. Estimating the distance to an epidemic threshold. J R Soc Interface 2019; 15:rsif.2018.0034. [PMID: 29950512 PMCID: PMC6030631 DOI: 10.1098/rsif.2018.0034] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2018] [Accepted: 05/31/2018] [Indexed: 12/20/2022] Open
Abstract
The epidemic threshold of the susceptible-infected-recovered model is a boundary separating parameters that permit epidemics from those that do not. This threshold corresponds to parameters where the system's equilibrium becomes unstable. Consequently, we use the average rate at which deviations from the equilibrium shrink to define a distance to this threshold. However, the vital dynamics of the host population may occur slowly even when transmission is far from threshold levels. Here, we show analytically how such slow dynamics can prevent estimation of the distance to the threshold from fluctuations in the susceptible population. Although these results are exact only in the limit of long-term observation of a large system, simulations show that they still provide useful insight into systems with a range of population sizes, environmental noise and observation schemes. Having established some guidelines about when estimates are accurate, we then illustrate how multiple distance estimates can be used to estimate the rate of approach to the threshold. The estimation approach is general and may be applicable to zoonotic pathogens such as Middle East respiratory syndrome-related coronavirus (MERS-CoV) as well as vaccine-preventable diseases like measles.
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Affiliation(s)
- Eamon B O'Dea
- Department of Ecology, College of Veterinary Medicine, University of Georgia, Athens, GA, USA .,Center for the Ecology of Infectious Diseases, College of Veterinary Medicine, University of Georgia, Athens, GA, USA
| | - Andrew W Park
- Odum School of Ecology, College of Veterinary Medicine, University of Georgia, Athens, GA, USA.,Infectious Diseases, College of Veterinary Medicine, University of Georgia, Athens, GA, USA
| | - John M Drake
- Center for the Ecology of Infectious Diseases, College of Veterinary Medicine, University of Georgia, Athens, GA, USA.,Odum School of Ecology, College of Veterinary Medicine, University of Georgia, Athens, GA, USA
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12
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Abstract
Second-order statistics such as the variance and autocorrelation can be useful indicators of the stability of randomly perturbed systems, in some cases providing early warning of an impending, dramatic change in the system’s dynamics. One specific application area of interest is the surveillance of infectious diseases. In the context of disease (re-)emergence, a goal could be to have an indicator that is informative of whether the system is approaching the epidemic threshold, a point beyond which a major outbreak becomes possible. Prior work in this area has provided some proof of this principle but has not analytically treated the effect of imperfect observation on the behavior of indicators. This work provides expected values for several moments of the number of reported cases, where reported cases follow a binomial or negative binomial distribution with a mean based on the number of deaths in a birth-death-immigration process over some reporting interval. The normalized second factorial moment and the decay time of the number of reported cases are two indicators that are insensitive to the reporting probability. Simulation is used to show how this insensitivity could be used to distinguish a trend of increased reporting from a trend of increased transmission. The simulation study also illustrates both the high variance of estimates and the possibility of reducing the variance by averaging over an ensemble of estimates from multiple time series.
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Affiliation(s)
- Eamon B. O’Dea
- Odum School of Ecology and Center for the Ecology of Infectious Diseases, University of Georgia, 140 E. Green Street, Athens, GA, 30602, USA
| | - John M. Drake
- Odum School of Ecology and Center for the Ecology of Infectious Diseases, University of Georgia, 140 E. Green Street, Athens, GA, 30602, USA
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13
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Vector Preference Annihilates Backward Bifurcation and Reduces Endemicity. Bull Math Biol 2018; 81:4447-4469. [PMID: 30569327 DOI: 10.1007/s11538-018-00561-1] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2017] [Accepted: 12/12/2018] [Indexed: 10/27/2022]
Abstract
We propose and analyze a mathematical model of a vector-borne disease that includes vector feeding preference for carrier hosts and intrinsic incubation in hosts. Analysis of the model reveals the following novel results. We show theoretically and numerically that vector feeding preference for carrier hosts plays an important role for the existence of both the endemic equilibria and backward bifurcation when the basic reproduction number [Formula: see text] is less than one. Moreover, by increasing the vector feeding preference value, backward bifurcation is eliminated and endemic equilibria for hosts and vectors are diminished. Therefore, the vector protects itself and this benefits the host. As an example of these phenomena, we present a case of Andean cutaneous leishmaniasis in Peru. We use parameter values from previous studies, primarily from Peru to introduce bifurcation diagrams and compute global sensitivity of [Formula: see text] in order to quantify and understand the effects of the important parameters of our model. Global sensitivity analysis via partial rank correlation coefficient shows that [Formula: see text] is highly sensitive to both sandflies feeding preference and mortality rate of sandflies.
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14
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Arakala A, Hoover CM, Marshall JM, Sokolow SH, De Leo GA, Rohr JR, Remais JV, Gambhir M. Estimating the elimination feasibility in the 'end game' of control efforts for parasites subjected to regular mass drug administration: Methods and their application to schistosomiasis. PLoS Negl Trop Dis 2018; 12:e0006794. [PMID: 30418968 PMCID: PMC6258430 DOI: 10.1371/journal.pntd.0006794] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2017] [Revised: 11/26/2018] [Accepted: 08/27/2018] [Indexed: 11/18/2022] Open
Abstract
Progress towards controlling and eliminating parasitic worms, including schistosomiasis, onchocerciasis, and lymphatic filariasis, is advancing rapidly as national governments, multinational NGOs, and pharmaceutical companies launch collaborative chemotherapeutic control campaigns. Critical questions remain regarding the potential for achieving elimination of these infections, and analytical methods can help to quickly estimate progress towards-and the probability of achieving-elimination over specific timeframes. Here, we propose the effective reproduction number, Reff, as a proxy of elimination potential for sexually reproducing worms that are subject to poor mating success at very low abundance (positive density dependence, or Allee effects). Reff is the number of parasites produced by a single reproductive parasite at a given stage in the transmission cycle, over the parasite's lifetime-it is the generalized form of the more familiar basic reproduction number, R0, which only applies at the beginning of an epidemic-and it can be estimated in a 'model-free' manner by an estimator ('ε'). We introduce ε, demonstrate its estimation using simulated data, and discuss how it may be used in planning and evaluation of ongoing elimination efforts for a range of parasitic diseases.
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Affiliation(s)
- Arathi Arakala
- Department of Epidemiology and Preventive Medicine, Monash University, Melbourne, Australia
| | - Christopher M. Hoover
- Division of Environmental Health Sciences, School of Public Health, University of California, Berkeley, California, United States of America
| | - John M. Marshall
- Division of Epidemiology and Biostatistics, School of Public Health, University of California, Berkeley, California, United States of America
| | - Susanne H. Sokolow
- Department of Biology—Hopkins Marine Station, Stanford University, Pacific Grove, California, United States of America
| | - Giulio A. De Leo
- Department of Biology—Hopkins Marine Station, Stanford University, Pacific Grove, California, United States of America
| | - Jason R. Rohr
- Department of Integrative Biology, University of Southern Florida, Tampa, Florida, United States of America
| | - Justin V. Remais
- Division of Environmental Health Sciences, School of Public Health, University of California, Berkeley, California, United States of America
| | - Manoj Gambhir
- Department of Epidemiology and Preventive Medicine, Monash University, Melbourne, Australia
- Health Modelling and Analytics, IBM Research Australia, Melbourne, Australia
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15
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Brett TS, O'Dea EB, Marty É, Miller PB, Park AW, Drake JM, Rohani P. Anticipating epidemic transitions with imperfect data. PLoS Comput Biol 2018; 14:e1006204. [PMID: 29883444 PMCID: PMC6010299 DOI: 10.1371/journal.pcbi.1006204] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2017] [Revised: 06/20/2018] [Accepted: 05/14/2018] [Indexed: 11/18/2022] Open
Abstract
Epidemic transitions are an important feature of infectious disease systems. As the transmissibility of a pathogen increases, the dynamics of disease spread shifts from limited stuttering chains of transmission to potentially large scale outbreaks. One proposed method to anticipate this transition are early-warning signals (EWS), summary statistics which undergo characteristic changes as the transition is approached. Although theoretically predicted, their mathematical basis does not take into account the nature of epidemiological data, which are typically aggregated into periodic case reports and subject to reporting error. The viability of EWS for epidemic transitions therefore remains uncertain. Here we demonstrate that most EWS can predict emergence even when calculated from imperfect data. We quantify performance using the area under the curve (AUC) statistic, a measure of how well an EWS distinguishes between numerical simulations of an emerging disease and one which is stationary. Values of the AUC statistic are compared across a range of different reporting scenarios. We find that different EWS respond to imperfect data differently. The mean, variance and first differenced variance all perform well unless reporting error is highly overdispersed. The autocorrelation, autocovariance and decay time perform well provided that the aggregation period of the data is larger than the serial interval and reporting error is not highly overdispersed. The coefficient of variation, skewness and kurtosis are found to be unreliable indicators of emergence. Overall, we find that seven of ten EWS considered perform well for most realistic reporting scenarios. We conclude that imperfect epidemiological data is not a barrier to using EWS for many potentially emerging diseases.
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Affiliation(s)
- Tobias S Brett
- Odum School of Ecology, University of Georgia, Athens, Georgia, United States of America
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America
| | - Eamon B O'Dea
- Odum School of Ecology, University of Georgia, Athens, Georgia, United States of America
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America
| | - Éric Marty
- Odum School of Ecology, University of Georgia, Athens, Georgia, United States of America
| | - Paige B Miller
- Odum School of Ecology, University of Georgia, Athens, Georgia, United States of America
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America
| | - Andrew W Park
- Odum School of Ecology, University of Georgia, Athens, Georgia, United States of America
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America
- Department of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America
| | - John M Drake
- Odum School of Ecology, University of Georgia, Athens, Georgia, United States of America
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America
| | - Pejman Rohani
- Odum School of Ecology, University of Georgia, Athens, Georgia, United States of America
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America
- Department of Infectious Diseases, University of Georgia, Athens, Georgia, United States of America
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16
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O'Regan SM, Burton DL. How Stochasticity Influences Leading Indicators of Critical Transitions. Bull Math Biol 2018; 80:1630-1654. [PMID: 29713924 DOI: 10.1007/s11538-018-0429-z] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2017] [Accepted: 03/29/2018] [Indexed: 12/25/2022]
Abstract
Many complex systems exhibit critical transitions. Of considerable interest are bifurcations, small smooth changes in underlying drivers that produce abrupt shifts in system state. Before reaching the bifurcation point, the system gradually loses stability ('critical slowing down'). Signals of critical slowing down may be detected through measurement of summary statistics, but how extrinsic and intrinsic noises influence statistical patterns prior to a transition is unclear. Here, we consider a range of stochastic models that exhibit transcritical, saddle-node and pitchfork bifurcations. Noise was assumed to be either intrinsic or extrinsic. We derived expressions for the stationary variance, autocorrelation and power spectrum for all cases. Trends in summary statistics signaling the approach of each bifurcation depend on the form of noise. For example, models with intrinsic stochasticity may predict an increase in or a decline in variance as the bifurcation parameter changes, whereas models with extrinsic noise applied additively predict an increase in variance. The ability to classify trends of summary statistics for a broad class of models enhances our understanding of how critical slowing down manifests in complex systems approaching a transition.
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Affiliation(s)
- Suzanne M O'Regan
- Department of Mathematics, North Carolina A&T State University, Greensboro, NC, 27411, USA. .,National Institute for Mathematical and Biological Synthesis, University of Tennessee, Knoxville, TN, USA.
| | - Danielle L Burton
- Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA
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17
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Clements CF, Ozgul A. Indicators of transitions in biological systems. Ecol Lett 2018; 21:905-919. [PMID: 29601665 DOI: 10.1111/ele.12948] [Citation(s) in RCA: 46] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2017] [Revised: 11/22/2017] [Accepted: 02/22/2018] [Indexed: 12/13/2022]
Abstract
In the face of global biodiversity declines, predicting the fate of biological systems is a key goal in ecology. One popular approach is the search for early warning signals (EWSs) based on alternative stable states theory. In this review, we cover the theory behind nonlinearity in dynamic systems and techniques to detect the loss of resilience that can indicate state transitions. We describe the research done on generic abundance-based signals of instability that are derived from the phenomenon of critical slowing down, which represent the genesis of EWSs research. We highlight some of the issues facing the detection of such signals in biological systems - which are inherently complex and show low signal-to-noise ratios. We then document research on alternative signals of instability, including measuring shifts in spatial autocorrelation and trait dynamics, and discuss potential future directions for EWSs research based on detailed demographic and phenotypic data. We set EWSs research in the greater field of predictive ecology and weigh up the costs and benefits of simplicity vs. complexity in predictive models, and how the available data should steer the development of future methods. Finally, we identify some key unanswered questions that, if solved, could improve the applicability of these methods.
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Affiliation(s)
- Christopher F Clements
- School of Biosciences, The University of Melbourne, Parkville, Vic., 3010, Australia.,Department of Evolutionary Biology and Environmental Studies, University of Zurich, Zurich, 8057, Switzerland
| | - Arpat Ozgul
- Department of Evolutionary Biology and Environmental Studies, University of Zurich, Zurich, 8057, Switzerland
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18
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Dallas TA, Krkošek M, Drake JM. Experimental evidence of a pathogen invasion threshold. ROYAL SOCIETY OPEN SCIENCE 2018; 5:171975. [PMID: 29410876 PMCID: PMC5792953 DOI: 10.1098/rsos.171975] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2017] [Accepted: 12/11/2017] [Indexed: 05/15/2023]
Abstract
Host density thresholds to pathogen invasion separate regions of parameter space corresponding to endemic and disease-free states. The host density threshold is a central concept in theoretical epidemiology and a common target of human and wildlife disease control programmes, but there is mixed evidence supporting the existence of thresholds, especially in wildlife populations or for pathogens with complex transmission modes (e.g. environmental transmission). Here, we demonstrate the existence of a host density threshold for an environmentally transmitted pathogen by combining an epidemiological model with a microcosm experiment. Experimental epidemics consisted of replicate populations of naive crustacean zooplankton (Daphnia dentifera) hosts across a range of host densities (20-640 hosts l-1) that were exposed to an environmentally transmitted fungal pathogen (Metschnikowia bicuspidata). Epidemiological model simulations, parametrized independently of the experiment, qualitatively predicted experimental pathogen invasion thresholds. Variability in parameter estimates did not strongly influence outcomes, though systematic changes to key parameters have the potential to shift pathogen invasion thresholds. In summary, we provide one of the first clear experimental demonstrations of pathogen invasion thresholds in a replicated experimental system, and provide evidence that such thresholds may be predictable using independently constructed epidemiological models.
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Affiliation(s)
- Tad A. Dallas
- Department of Environmental Science and Policy, University of California, Davis, CA, USA
- Odum School of Ecology, University of Georgia, Athens, GA, USA
| | - Martin Krkošek
- Department of Ecology and Evolutionary Biology, University of Toronto, Toronto, Ontario, Canada
| | - John M. Drake
- Odum School of Ecology, University of Georgia, Athens, GA, USA
- Center for the Ecology of Infectious Disease, University of Georgia, Athens, GA, USA
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19
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Miller PB, O’Dea EB, Rohani P, Drake JM. Forecasting infectious disease emergence subject to seasonal forcing. Theor Biol Med Model 2017; 14:17. [PMID: 28874167 PMCID: PMC5586031 DOI: 10.1186/s12976-017-0063-8] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2017] [Accepted: 08/23/2017] [Indexed: 12/04/2022] Open
Abstract
BACKGROUND Despite high vaccination coverage, many childhood infections pose a growing threat to human populations. Accurate disease forecasting would be of tremendous value to public health. Forecasting disease emergence using early warning signals (EWS) is possible in non-seasonal models of infectious diseases. Here, we assessed whether EWS also anticipate disease emergence in seasonal models. METHODS We simulated the dynamics of an immunizing infectious pathogen approaching the tipping point to disease endemicity. To explore the effect of seasonality on the reliability of early warning statistics, we varied the amplitude of fluctuations around the average transmission. We proposed and analyzed two new early warning signals based on the wavelet spectrum. We measured the reliability of the early warning signals depending on the strength of their trend preceding the tipping point and then calculated the Area Under the Curve (AUC) statistic. RESULTS Early warning signals were reliable when disease transmission was subject to seasonal forcing. Wavelet-based early warning signals were as reliable as other conventional early warning signals. We found that removing seasonal trends, prior to analysis, did not improve early warning statistics uniformly. CONCLUSIONS Early warning signals anticipate the onset of critical transitions for infectious diseases which are subject to seasonal forcing. Wavelet-based early warning statistics can also be used to forecast infectious disease.
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Affiliation(s)
- Paige B. Miller
- University of Georgia, Odum School of Ecology, 140 E. Green Street, Athens, USA
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, USA
| | - Eamon B. O’Dea
- University of Georgia, Odum School of Ecology, 140 E. Green Street, Athens, USA
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, USA
| | - Pejman Rohani
- University of Georgia, Odum School of Ecology, 140 E. Green Street, Athens, USA
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, USA
- Department of Infectious Diseases, University of Georgia, Athens, USA
| | - John M. Drake
- University of Georgia, Odum School of Ecology, 140 E. Green Street, Athens, USA
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, USA
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20
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Brett TS, Drake JM, Rohani P. Anticipating the emergence of infectious diseases. J R Soc Interface 2017; 14:20170115. [PMID: 28679666 PMCID: PMC5550966 DOI: 10.1098/rsif.2017.0115] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2017] [Accepted: 06/09/2017] [Indexed: 12/02/2022] Open
Abstract
In spite of medical breakthroughs, the emergence of pathogens continues to pose threats to both human and animal populations. We present candidate approaches for anticipating disease emergence prior to large-scale outbreaks. Through use of ideas from the theories of dynamical systems and stochastic processes we develop approaches which are not specific to a particular disease system or model, but instead have general applicability. The indicators of disease emergence detailed in this paper can be classified into two parallel approaches: a set of early-warning signals based around the theory of critical slowing down and a likelihood-based approach. To test the reliability of these two approaches we contrast theoretical predictions with simulated data. We find good support for our methods across a range of different model structures and parameter values.
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Affiliation(s)
- Tobias S Brett
- Odum School of Ecology, University of Georgia, Athens, GA, USA
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA, USA
| | - John M Drake
- Odum School of Ecology, University of Georgia, Athens, GA, USA
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA, USA
| | - Pejman Rohani
- Odum School of Ecology, University of Georgia, Athens, GA, USA
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA, USA
- Department of Infectious Diseases, University of Georgia, Athens, GA, USA
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21
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Drake JM, Hay SI. Monitoring the Path to the Elimination of Infectious Diseases. Trop Med Infect Dis 2017; 2:E20. [PMID: 30270879 PMCID: PMC6082106 DOI: 10.3390/tropicalmed2030020] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2017] [Revised: 06/17/2017] [Accepted: 06/21/2017] [Indexed: 12/21/2022] Open
Abstract
During the endgame of elimination programs, parasite populations may exhibit dynamical phenomena not typical of endemic disease. Particularly, monitoring programs for tracking infection prevalence may be hampered by overall rarity, the sporadic and unpredictable timing and location of outbreaks, and under-reporting. A particularly important problem for monitoring is determining the distance that must be covered to achieve the elimination threshold at an effective reproduction number less than one. In this perspective, we suggest that this problem may be overcome by measuring critical slowing down. Critical slowing down is a phenomenon exhibited by nonlinear dynamical systems in the vicinity of a critical threshold. In infectious disease dynamics, critical slowing down is expressed as an increase in the coefficient of variation and other properties of the fluctuations in the number of cases. In simulations, we show the coefficient of variation to be insensitive to under-reporting error and therefore a robust measurement of the approach to elimination. Additionally, we show that there is an inevitable delay between the time at which the effective reproduction number is reduced to below one and complete elimination is achieved. We urge that monitoring programs include dynamical properties such as critical slowing down in their metrics for measuring achievement and avoid withdrawing control activities prematurely.
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Affiliation(s)
- John M Drake
- Odum School of Ecology, University of Georgia, Athens, GA 30602-2202, USA.
- Center for the Ecology of Infectious Diseases, University of Georgia, Athens, GA 30602-2202, USA.
- Department of Zoology, University of Oxford, Oxford OX2, UK.
| | - Simon I Hay
- Institute for Health Metrics and Evaluation, University of Washington, Seattle, WA 98121, USA.
- Big Data Institute, Li Ka Shing Centre for Health Information and Discovery, University of Oxford, Oxford OX3 7LF, UK.
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22
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Pepin D, Penn M. Legal Authority for Mosquito Control and Pesticide Use in the United States. Public Health Rep 2017; 132:389-391. [DOI: 10.1177/0033354917704628] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Affiliation(s)
- Dawn Pepin
- Chenega Professional and Technical Services, LLC, Atlanta, GA, USA
- Public Health Law Program, Office for State, Tribal, Local and Territorial Support, Centers for Disease Control and Prevention, Atlanta, GA, USA
| | - Matthew Penn
- Public Health Law Program, Office for State, Tribal, Local and Territorial Support, Centers for Disease Control and Prevention, Atlanta, GA, USA
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