1
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Leung KY, Metting E, Ebbers W, Veldhuijzen I, Andeweg SP, Luijben G, de Bruin M, Wallinga J, Klinkenberg D. Effectiveness of a COVID-19 contact tracing app in a simulation model with indirect and informal contact tracing. Epidemics 2024; 46:100735. [PMID: 38128242 DOI: 10.1016/j.epidem.2023.100735] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2023] [Revised: 11/17/2023] [Accepted: 12/08/2023] [Indexed: 12/23/2023] Open
Abstract
During the COVID-19 pandemic, contact tracing was used to identify individuals who had been in contact with a confirmed case so that these contacted individuals could be tested and quarantined to prevent further spread of the SARS-CoV-2 virus. Many countries developed mobile apps to find these contacted individuals faster. We evaluate the epidemiological effectiveness of the Dutch app CoronaMelder, where we measure effectiveness as the reduction of the reproduction number R. To this end, we use a simulation model of SARS-CoV-2 spread and contact tracing, informed by data collected during the study period (December 2020 - March 2021) in the Netherlands. We show that the tracing app caused a clear but small reduction of the reproduction number, and the magnitude of the effect was found to be robust in sensitivity analyses. The app could have been more effective if more people had used it, and if notification of contacts could have been done directly by the user and thus reducing the time intervals between symptom onset and reporting of contacts. The model has two innovative aspects: i) it accounts for the clustered nature of social networks and ii) cases can alert their contacts informally without involvement of health authorities or the tracing app.
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Affiliation(s)
- Ka Yin Leung
- National Institute for Public Health and the Environment, Centre for Infectious Disease Control, Bilthoven, the Netherlands.
| | - Esther Metting
- University of Groningen, University Medical Center Groningen, Data Science Center in Health, the Netherlands; University of Groningen, University Medical Center Groningen, Department of Primary Care, the Netherlands; University of Groningen, faculty of Economics and Business, Department of Operations, the Netherlands
| | - Wolfgang Ebbers
- Erasmus School of Social and Behavioural Sciences, Department of Public Administration and Sociology, Erasmus University Rotterdam, Rotterdam, the Netherlands
| | - Irene Veldhuijzen
- National Institute for Public Health and the Environment, Centre for Infectious Disease Control, Bilthoven, the Netherlands
| | - Stijn P Andeweg
- National Institute for Public Health and the Environment, Centre for Infectious Disease Control, Bilthoven, the Netherlands
| | - Guus Luijben
- National Institute for Public Health and the Environment, Centre for Health and Society, Bilthoven, the Netherlands
| | - Marijn de Bruin
- National Institute for Public Health and the Environment, Centre for Health and Society, Bilthoven, the Netherlands; Radboud University Medical Centre, Radboud Institute of Health Sciences, IQ Healthcare, Nijmegen, the Netherlands
| | - Jacco Wallinga
- National Institute for Public Health and the Environment, Centre for Infectious Disease Control, Bilthoven, the Netherlands; Leiden University Medical Centre, Department of Biomedical Datasciences, Leiden, the Netherlands
| | - Don Klinkenberg
- National Institute for Public Health and the Environment, Centre for Infectious Disease Control, Bilthoven, the Netherlands
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2
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Burden CJ, Griffiths RC. Coalescence and sampling distributions for Feller diffusions. Theor Popul Biol 2024; 155:67-76. [PMID: 38092137 DOI: 10.1016/j.tpb.2023.12.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2022] [Revised: 09/11/2023] [Accepted: 12/09/2023] [Indexed: 12/29/2023]
Abstract
Consider the diffusion process defined by the forward equation ut(t,x)=12{xu(t,x)}xx-α{xu(t,x)}x for t,x≥0 and -∞<α<∞, with an initial condition u(0,x)=δ(x-x0). This equation was introduced and solved by Feller to model the growth of a population of independently reproducing individuals. We explore important coalescent processes related to Feller's solution. For any α and x0>0 we calculate the distribution of the random variable An(s;t), defined as the finite number of ancestors at a time s in the past of a sample of size n taken from the infinite population of a Feller diffusion at a time t since its initiation. In a subcritical diffusion we find the distribution of population and sample coalescent trees from time t back, conditional on non-extinction as t→∞. In a supercritical diffusion we construct a coalescent tree which has a single founder and derive the distribution of coalescent times.
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Affiliation(s)
- Conrad J Burden
- Mathematical Sciences Institute, Australian National University, Canberra, Australia.
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3
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Wang Y, Zhou JX, Pedrini E, Rubin I, Khalil M, Taramelli R, Qian H, Huang S. Cell population growth kinetics in the presence of stochastic heterogeneity of cell phenotype. J Theor Biol 2023; 575:111645. [PMID: 37863423 DOI: 10.1016/j.jtbi.2023.111645] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2023] [Revised: 09/29/2023] [Accepted: 10/13/2023] [Indexed: 10/22/2023]
Abstract
Recent studies at individual cell resolution have revealed phenotypic heterogeneity in nominally clonal tumor cell populations. The heterogeneity affects cell growth behaviors, which can result in departure from the idealized uniform exponential growth of the cell population. Here we measured the stochastic time courses of growth of an ensemble of populations of HL60 leukemia cells in cultures, starting with distinct initial cell numbers to capture a departure from the uniform exponential growth model for the initial growth ("take-off"). Despite being derived from the same cell clone, we observed significant variations in the early growth patterns of individual cultures with statistically significant differences in growth dynamics, which could be explained by the presence of inter-converting subpopulations with different growth rates, and which could last for many generations. Based on the hypothesis of existence of multiple subpopulations, we developed a branching process model that was consistent with the experimental observations.
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Affiliation(s)
- Yue Wang
- Department of Computational Medicine, University of California, Los Angeles, CA, United States of America; Department of Applied Mathematics, University of Washington, Seattle, WA, United States of America
| | - Joseph X Zhou
- Institute for Systems Biology, Seattle, WA, United States of America
| | - Edoardo Pedrini
- Institute for Systems Biology, Seattle, WA, United States of America
| | - Irit Rubin
- Institute for Systems Biology, Seattle, WA, United States of America
| | - May Khalil
- Institute for Systems Biology, Seattle, WA, United States of America
| | - Roberto Taramelli
- Department of Theoretical and Applied Science, University of Insubria, Italy
| | - Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, WA, United States of America
| | - Sui Huang
- Institute for Systems Biology, Seattle, WA, United States of America.
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4
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Boudreau MC, Allen AJ, Roberts NJ, Allard A, Hébert-Dufresne L. Temporal and Probabilistic Comparisons of Epidemic Interventions. Bull Math Biol 2023; 85:118. [PMID: 37857996 DOI: 10.1007/s11538-023-01220-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2023] [Accepted: 09/26/2023] [Indexed: 10/21/2023]
Abstract
Forecasting disease spread is a critical tool to help public health officials design and plan public health interventions. However, the expected future state of an epidemic is not necessarily well defined as disease spread is inherently stochastic, contact patterns within a population are heterogeneous, and behaviors change. In this work, we use time-dependent probability generating functions (PGFs) to capture these characteristics by modeling a stochastic branching process of the spread of a disease over a network of contacts in which public health interventions are introduced over time. To achieve this, we define a general transmissibility equation to account for varying transmission rates (e.g. masking), recovery rates (e.g. treatment), contact patterns (e.g. social distancing) and percentage of the population immunized (e.g. vaccination). The resulting framework allows for a temporal and probabilistic analysis of an intervention's impact on disease spread, which match continuous-time stochastic simulations that are much more computationally expensive. To aid policy making, we then define several metrics over which temporal and probabilistic intervention forecasts can be compared: Looking at the expected number of cases and the worst-case scenario over time, as well as the probability of reaching a critical level of cases and of not seeing any improvement following an intervention. Given that epidemics do not always follow their average expected trajectories and that the underlying dynamics can change over time, our work paves the way for more detailed short-term forecasts of disease spread and more informed comparison of intervention strategies.
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Affiliation(s)
- Mariah C Boudreau
- Vermont Complex Systems Center, University of Vermont, Burlington, VT, USA.
- Department of Mathematics & Statistics, University of Vermont, Burlington, VT, USA.
| | - Andrea J Allen
- Vermont Complex Systems Center, University of Vermont, Burlington, VT, USA
- Children's Hospital of Philadelphia, Applied Clinical Research Center, Philadelphia, PA, USA
| | - Nicholas J Roberts
- Vermont Complex Systems Center, University of Vermont, Burlington, VT, USA
| | - Antoine Allard
- Vermont Complex Systems Center, University of Vermont, Burlington, VT, USA
- Départment de Physique, de génie physique et d'optique, Université Laval, Québec, Québec, G1V 0A6, Canada
- Centre interdisciplinaire en modélisation mathématique, Université Laval, Québec, Québec, G1V 0A6, Canada
| | - Laurent Hébert-Dufresne
- Vermont Complex Systems Center, University of Vermont, Burlington, VT, USA
- Department of Mathematics & Statistics, University of Vermont, Burlington, VT, USA
- Départment de Physique, de génie physique et d'optique, Université Laval, Québec, Québec, G1V 0A6, Canada
- Department of Computer Science, University of Vermont, Burlington, VT, USA
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5
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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] [What about the content of this article? (0)] [Affiliation(s)] [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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6
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Zhang R, Ukogu OA, Bozic I. Waiting times in a branching process model of colorectal cancer initiation. Theor Popul Biol 2023; 151:44-63. [PMID: 37100121 DOI: 10.1016/j.tpb.2023.04.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/13/2022] [Revised: 04/13/2023] [Accepted: 04/19/2023] [Indexed: 04/28/2023]
Abstract
We study a multi-stage model for the development of colorectal cancer from initially healthy tissue. The model incorporates a complex sequence of driver gene alterations, some of which result in immediate growth advantage, while others have initially neutral effects. We derive analytic estimates for the sizes of premalignant subpopulations, and use these results to compute the waiting times to premalignant and malignant genotypes. This work contributes to the quantitative understanding of colorectal tumor evolution and the lifetime risk of colorectal cancer.
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Affiliation(s)
- Ruibo Zhang
- Department of Applied Mathematics, University of Washington, United States of America
| | - Obinna A Ukogu
- Department of Applied Mathematics, University of Washington, United States of America
| | - Ivana Bozic
- Department of Applied Mathematics, University of Washington, United States of America.
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7
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Cheek D, Johnston SGG. Ancestral reproductive bias in branching processes. J Math Biol 2023; 86:70. [PMID: 37027075 DOI: 10.1007/s00285-023-01907-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2022] [Revised: 02/08/2023] [Accepted: 03/06/2023] [Indexed: 04/08/2023]
Abstract
Consider a branching process whose reproduction law is homogeneous. Sampling a single cell uniformly from the population at a time [Formula: see text] and looking along the sampled cell's ancestral lineage, we find that the reproduction law is heterogeneous-the expected reproductive output of ancestral cells on the lineage from time 0 to time T continuously increases with time. This 'inspection paradox' is due to sampling bias, that cells with a larger number of offspring are more likely to have one of their descendants sampled by virtue of their prolificity. The bias's strength changes with the random population size and/or the sampling time T. Our main result explicitly characterises the evolution of reproduction rates and sizes along the sampled ancestral lineage as a mixture of Poisson processes, which simplifies in special cases. The ancestral bias helps to explain recently observed variation in mutation rates along lineages of the developing human embryo.
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Affiliation(s)
- David Cheek
- Massachusetts General Hospital and Harvard Medical School, 149 13th St., Charlestown, MA, 02129, USA
| | - Samuel G G Johnston
- Department of Mathematics, King's College London, Strand Building, London, WC2R 2LS, UK.
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8
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Southall E, Ogi-Gittins Z, Kaye AR, Hart WS, Lovell-Read FA, Thompson RN. A practical guide to mathematical methods for estimating infectious disease outbreak risks. J Theor Biol 2023; 562:111417. [PMID: 36682408 DOI: 10.1016/j.jtbi.2023.111417] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2022] [Revised: 01/13/2023] [Accepted: 01/16/2023] [Indexed: 01/21/2023]
Abstract
Mathematical models are increasingly used throughout infectious disease outbreaks to guide control measures. In this review article, we focus on the initial stages of an outbreak, when a pathogen has just been observed in a new location (e.g., a town, region or country). We provide a beginner's guide to two methods for estimating the risk that introduced cases lead to sustained local transmission (i.e., the probability of a major outbreak), as opposed to the outbreak fading out with only a small number of cases. We discuss how these simple methods can be extended for epidemiological models with any level of complexity, facilitating their wider use, and describe how estimates of the probability of a major outbreak can be used to guide pathogen surveillance and control strategies. We also give an overview of previous applications of these approaches. This guide is intended to help quantitative researchers develop their own epidemiological models and use them to estimate the risks associated with pathogens arriving in new host populations. The development of these models is crucial for future outbreak preparedness. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".
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Affiliation(s)
- E Southall
- Mathematics Institute, University of Warwick, Coventry, UK; Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, University of Warwick, Coventry, UK
| | - Z Ogi-Gittins
- Mathematics Institute, University of Warwick, Coventry, UK; Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, University of Warwick, Coventry, UK
| | - A R Kaye
- Mathematics Institute, University of Warwick, Coventry, UK; Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, University of Warwick, Coventry, UK
| | - W S Hart
- Mathematical Institute, University of Oxford, Oxford, UK
| | | | - R N Thompson
- Mathematics Institute, University of Warwick, Coventry, UK; Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, University of Warwick, Coventry, UK.
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9
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Müller J, Hösel V. Contact tracing & super-spreaders in the branching-process model. J Math Biol 2023; 86:24. [PMID: 36625934 PMCID: PMC9830628 DOI: 10.1007/s00285-022-01857-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2020] [Revised: 07/18/2021] [Accepted: 07/21/2021] [Indexed: 01/11/2023]
Abstract
In recent years, it became clear that super-spreader events play an important role, particularly in the spread of airborne infections. We investigate a novel model for super-spreader events, not based on a heterogeneous contact graph but on a random contact rate: Many individuals become infected synchronously in single contact events. We use the branching-process approach for contact tracing to analyze the impact of super-spreader events on the effect of contact tracing. Here we neglect a tracing delay. Roughly speaking, we find that contact tracing is more efficient in the presence of super-spreaders if the fraction of symptomatics is small, the tracing probability is high, or the latency period is distinctively larger than the incubation period. In other cases, the effect of contact tracing can be decreased by super-spreaders. Numerical analysis with parameters suited for SARS-CoV-2 indicates that super-spreaders do not decrease the effect of contact tracing crucially in case of that infection.
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Affiliation(s)
- Johannes Müller
- Center for Mathematics, Technische Universität München, 85748, Garching, Germany. .,Institute for Computational Biology, Helmholtz Center Munich, 85764, Neuherberg, Germany.
| | - Volker Hösel
- grid.6936.a0000000123222966Center for Mathematics, Technische Universität München, 85748 Garching, Germany
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10
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Kwok KO, Wei WI, Tang A, Wong SYS, Tang JW. Estimation of local transmissibility in the early phase of monkeypox epidemic in 2022. Clin Microbiol Infect 2022; 28:1653.e1-1653.e3. [PMID: 35817231 PMCID: PMC9675054 DOI: 10.1016/j.cmi.2022.06.025] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2022] [Revised: 06/21/2022] [Accepted: 06/23/2022] [Indexed: 11/03/2022]
Abstract
OBJECTIVES To estimate the basic reproductive number (Ro) to help us understand and control the spread of monkeypox in immunologically naive populations. METHODS Using three highest incidence populations including England, Portugal, and Spain as examples as of 18 June 2022, we employed the branching process with a Poisson likelihood and gamma-distributed serial interval to fit daily reported case data of monkeypox to estimate Ro. Sensitivity analyses were performed by varying mean serial interval from 6.8 to 12.8 days. RESULTS The median posterior estimates of Ro for monkeypox in the three study populations were statistically >1 (England: Ro = 1.60 [95% (credible interval) CrI, 1.50-1.70]; Portugal: Ro = 1.40 [95% CrI, 1.20-1.60]; Spain: Ro = 1.80 [95% CrI, 1.70-2.00]). Ro estimates varied over 1.30 to 2.10, depending on the serial interval. DISCUSSION The updated Ro estimates across different populations will inform policy makers' plans for public health control measures. Currently, monkeypox has a sustainable outbreak potential and may challenge healthcare systems, mainly due to declines in the population level immunity to Orthopoxviruses since the cessation of routine smallpox vaccination. Smallpox vaccination has been shown to be effective in protecting (≤85% effectiveness) against monkeypox infection in earlier times. So early postexposure vaccination is currently being offered in an attempt to control its spread.
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Affiliation(s)
- Kin On Kwok
- JC School of Public Health and Primary Care, The Chinese University of Hong Kong, Hong Kong Special Administrative Region, China; Stanley Ho Centre for Emerging Infectious Diseases, The Chinese University of Hong Kong, Hong Kong Special Administrative Region, China; Hong Kong Institute of Asia-Pacific Studies, The Chinese University of Hong Kong, Hong Kong Special Administrative Region, China; Shenzhen Research Institute, The Chinese University of Hong Kong, Shenzhen, China.
| | - Wan In Wei
- JC School of Public Health and Primary Care, The Chinese University of Hong Kong, Hong Kong Special Administrative Region, China
| | - Arthur Tang
- College of Computing and Informatics, Sungkyunkwan University, Seoul, Republic of Korea
| | - Samuel Yeung Shan Wong
- JC School of Public Health and Primary Care, The Chinese University of Hong Kong, Hong Kong Special Administrative Region, China
| | - Julian W Tang
- Respiratory Sciences, University of Leicester, Leicester, UK; Clinical Microbiology, Leicester Royal Infirmary, Leicester, UK.
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11
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Abstract
Consider a Markovian SIR epidemic model in a homogeneous community. To this model we add a rate at which individuals are tested, and once an infectious individual tests positive it is isolated and each of their contacts are traced and tested independently with some fixed probability. If such a traced individual tests positive it is isolated, and the contact tracing is iterated. This model is analysed using large population approximations, both for the early stage of the epidemic when the "to-be-traced components" of the epidemic behaves like a branching process, and for the main stage of the epidemic where the process of to-be-traced components converges to a deterministic process defined by a system of differential equations. These approximations are used to quantify the effect of testing and of contact tracing on the effective reproduction numbers (for the components as well as for the individuals), the probability of a major outbreak, and the final fraction getting infected. Using numerical illustrations when rates of infection and natural recovery are fixed, it is shown that Test-and-Trace strategy is effective in reducing the reproduction number. Surprisingly, the reproduction number for the branching process of components is not monotonically decreasing in the tracing probability, but the individual reproduction number is conjectured to be monotonic as expected. Further, in the situation where individuals also self-report for testing, the tracing probability is more influential than the screening rate (measured by the fraction infected being screened).
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Affiliation(s)
- Dongni Zhang
- Department of Mathematics, Stockholm University, Stockholm, Sweden.
| | - Tom Britton
- Department of Mathematics, Stockholm University, Stockholm, Sweden
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12
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Plank MJ. Minimising the use of costly control measures in an epidemic elimination strategy: A simple mathematical model. Math Biosci 2022;:108885. [PMID: 35907510 DOI: 10.1016/j.mbs.2022.108885] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/04/2022] [Revised: 07/20/2022] [Accepted: 07/21/2022] [Indexed: 12/02/2022]
Abstract
Countries such as New Zealand, Australia and Taiwan responded to the Covid-19 pandemic with an elimination strategy. This involves a combination of strict border controls with a rapid and effective response to eliminate border-related re-introductions. An important question for decision makers is, when there is a new re-introduction, what is the right threshold at which to implement strict control measures designed to reduce the effective reproduction number below 1. Since it is likely that there will be multiple re-introductions, responding at too low a threshold may mean repeatedly implementing controls unnecessarily for outbreaks that would self-eliminate even without control measures. On the other hand, waiting for too high a threshold to be reached creates a risk that controls will be needed for a longer period of time, or may completely fail to contain the outbreak. Here, we use a highly idealised branching process model of small border-related outbreaks to address this question. We identify important factors that affect the choice of threshold in order to minimise the expect time period for which control measures are in force. We find that the optimal threshold for introducing controls decreases with the effective reproduction number, and increases with overdispersion of the offspring distribution and with the effectiveness of control measures. Our results are not intended as a quantitative decision-making algorithm. However, they may help decision makers understand when a wait-and-see approach is likely to be preferable over an immediate response.
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13
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Ball F, Neal P. An epidemic model with short-lived mixing groups. J Math Biol 2022; 85:63. [PMID: 36315292 PMCID: PMC9622612 DOI: 10.1007/s00285-022-01822-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2022] [Revised: 09/23/2022] [Accepted: 10/03/2022] [Indexed: 12/05/2022]
Abstract
Almost all epidemic models make the assumption that infection is driven by the interaction between pairs of individuals, one of whom is infectious and the other of whom is susceptible. However, in society individuals mix in groups of varying sizes, at varying times, allowing one or more infectives to be in close contact with one or more susceptible individuals at a given point in time. In this paper we study the effect of mixing groups beyond pairs on the transmission of an infectious disease in an SIR (susceptible [Formula: see text] infective [Formula: see text] recovered) model, both through a branching process approximation for the initial stages of an epidemic with few initial infectives and a functional central limit theorem for the trajectories of the numbers of infectives and susceptibles over time for epidemics with many initial infectives. We also derive central limit theorems for the final size of (i) an epidemic with many initial infectives and (ii) a major outbreak triggered by few initial infectives. We show that, for a given basic reproduction number [Formula: see text], the distribution of the size of mixing groups has a significant impact on the probability and final size of a major epidemic outbreak. Moreover, the standard pair-based homogeneously mixing epidemic model is shown to represent the worst case scenario, with both the highest probability and the largest final size of a major epidemic.
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Affiliation(s)
- Frank Ball
- grid.4563.40000 0004 1936 8868School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD UK
| | - Peter Neal
- grid.4563.40000 0004 1936 8868School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD UK
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14
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Cloez B, de Saporta B, Roget T. Long-time behavior and Darwinian optimality for an asymmetric size-structured branching process. J Math Biol 2021; 83:69. [PMID: 34870743 DOI: 10.1007/s00285-021-01695-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/24/2020] [Revised: 09/03/2021] [Accepted: 11/17/2021] [Indexed: 11/25/2022]
Abstract
We study the long time behavior of an asymmetric size-structured measure-valued growth-fragmentation branching process that models the dynamics of a population of cells taking into account physiological and morphological asymmetry at division. We show that the process exhibits a Malthusian behavior; that is that the global population size grows exponentially fast and that the trait distribution of individuals converges to some stable distribution. The proof is based on a generalization of Lyapunov function techniques for non-conservative semi-groups. We then investigate the fluctuations of the growth rate with respect to the parameters guiding asymmetry. In particular, we exhibit that, under some special assumptions, symmetric division is sub-optimal in a Darwinian sense.
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Affiliation(s)
- Bertrand Cloez
- MISTEA, Univ Montpellier, INRAE, Institut Agro, Montpellier, France
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15
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Mugabi F, Duffy KJ, Mugisha JYT, Collins OC. Determining the effects of wind-aided midge movement on the outbreak and coexistence of multiple bluetongue virus serotypes in patchy environments. Math Biosci 2021; 342:108718. [PMID: 34666068 DOI: 10.1016/j.mbs.2021.108718] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2020] [Revised: 10/05/2021] [Accepted: 10/05/2021] [Indexed: 10/20/2022]
Abstract
Bluetongue virus (BTV) has 27 serotypes with some of them coexisting in different environments which make its control difficult. Wind-aided midge movement is a known mechanism in the spread of BTV. However, its effects on the dynamics of multiple BTV serotypes are not clear. Ordinary differential equation (ODE) and continuous-time Markov chain (CTMC) models for two BTV serotypes in an environment divided into two patches depending on the risk of infection are formulated and analysed. By approximating the CTMC model with a multitype branching process, an estimate for the probability of a major outbreak of two BTV serotypes is obtained. It is shown that without movement a major outbreak occurs in the high-risk patch, but with cattle or midge movement it occurs in both patches. When a major outbreak occurs, numerical simulations of the ODE model illustrate possible coexistence in both patches if the patches are connected by midge or cattle movement. Sensitivity analysis, based on the Latin hypercube sampling method, identified midge mortality and biting rates as being the most important in determining the magnitude of the probability of a major outbreak. These results indicate the significance of wind-aided midge movement on the outbreak and coexistence of multiple BTV serotypes in patchy environments.
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Affiliation(s)
- Francis Mugabi
- Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban, 4000, South Africa.
| | - Kevin J Duffy
- Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban, 4000, South Africa.
| | - Joseph Y T Mugisha
- Department of Mathematics, Makerere University, P.O. Box 7062, Kampala, Uganda.
| | - Obiora C Collins
- Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban, 4000, South Africa.
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16
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Azevedo RBR, Olofsson P. A branching process model of evolutionary rescue. Math Biosci 2021; 341:108708. [PMID: 34560091 DOI: 10.1016/j.mbs.2021.108708] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2021] [Revised: 09/05/2021] [Accepted: 09/13/2021] [Indexed: 12/01/2022]
Abstract
Evolutionary rescue is the process whereby a declining population may start growing again, thus avoiding extinction, via an increase in the frequency of fitter genotypes. These genotypes may either already be present in the population in small numbers, or arise by mutation as the population declines. We present a simple two-type discrete-time branching process model and use it to obtain results such as the probability of rescue, the shape of the population growth curve of a rescued population, and the time until the first rescuing mutation occurs. Comparisons are made to existing results in the literature in cases where both the mutation rate and the selective advantage of the beneficial mutations are small.
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Affiliation(s)
- Ricardo B R Azevedo
- Department of Biology and Biochemistry, University of Houston, Houston, TX, USA
| | - Peter Olofsson
- Department of Mathematics, Physics and Chemical Engineering, Jönköping University, Sweden.
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17
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Blath J, Hermann F, Slowik M. A branching process model for dormancy and seed banks in randomly fluctuating environments. J Math Biol 2021; 83:17. [PMID: 34279717 PMCID: PMC8289800 DOI: 10.1007/s00285-021-01639-6] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2020] [Revised: 06/17/2021] [Accepted: 06/27/2021] [Indexed: 11/25/2022]
Abstract
The goal of this article is to contribute towards the conceptual and quantitative understanding of the evolutionary benefits for (microbial) populations to maintain a seed bank consisting of dormant individuals when facing fluctuating environmental conditions. To this end, we discuss a class of '2-type' branching processes describing populations of individuals that may switch between 'active' and 'dormant' states in a random environment oscillating between a 'healthy' and a 'harsh' state. We incorporate different switching strategies and suggest a method of 'fair comparison' to incorporate potentially varying reproductive costs. We then use this concept to compare the fitness of the different strategies in terms of maximal Lyapunov exponents. This gives rise to a 'fitness map' depicting the environmental regimes where certain switching strategies are uniquely supercritical.
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Affiliation(s)
- Jochen Blath
- Present Address: Institute of Mathematics, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
| | - Felix Hermann
- Present Address: Institute of Mathematics, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
| | - Martin Slowik
- Present Address: Mathematical Institute, University of Mannheim, B6, 26, 68159 Mannheim, Germany
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18
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Allen LJS, Wang X. Stochastic models of infectious diseases in a periodic environment with application to cholera epidemics. J Math Biol 2021; 82:48. [PMID: 33830353 DOI: 10.1007/s00285-021-01603-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2020] [Revised: 11/20/2020] [Accepted: 03/29/2021] [Indexed: 10/21/2022]
Abstract
Seasonal variation affects the dynamics of many infectious diseases including influenza, cholera and malaria. The time when infectious individuals are first introduced into a population is crucial in predicting whether a major disease outbreak occurs. In this investigation, we apply a time-nonhomogeneous stochastic process for a cholera epidemic with seasonal periodicity and a multitype branching process approximation to obtain an analytical estimate for the probability of an outbreak. In particular, an analytic estimate of the probability of disease extinction is shown to satisfy a system of ordinary differential equations which follows from the backward Kolmogorov differential equation. An explicit expression for the mean (resp. variance) of the first extinction time given an extinction occurs is derived based on the analytic estimate for the extinction probability. Our results indicate that the probability of a disease outbreak, and mean and standard derivation of the first time to disease extinction are periodic in time and depend on the time when the infectious individuals or free-living pathogens are introduced. Numerical simulations are then carried out to validate the analytical predictions using two examples of the general cholera model. At the end, the developed theoretical results are extended to more general models of infectious diseases.
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19
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Abstract
Zoonotic infectious diseases are spread from animals to humans. It is estimated that over 60% of human infectious diseases are zoonotic and 75% of them are emerging zoonoses. The majority of emerging zoonotic infectious diseases are caused by viruses including avian influenza, rabies, Ebola, coronaviruses and hantaviruses. Spillover of infection from animals to humans depends on a complex transmission pathway, which is influenced by epidemiological and environmental processes. In this investigation, the focus is on direct transmission between animals and humans and the effects of seasonal variations on the transmission and recovery rates. Fluctuations in transmission and recovery, besides being influenced by physiological processes and behaviors of pathogen and host, are driven by seasonal variations in temperature, humidity or rainfall. A new time-nonhomogeneous stochastic process is formulated for infectious disease spread from animals to humans when transmission and recovery rates are time-periodic. A branching process approximation is applied near the disease-free state to predict the probability of the first spillover event from animals to humans. This probability is a periodic function of the time when infection is introduced into the animal population. It is shown that the highest risk of a spillover depends on a combination of animal to human transmission, animal to animal transmission and animal recovery. The results are applied to a stochastic model for avian influenza with spillover from domestic poultry to humans.
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Affiliation(s)
- Aadrita Nandi
- Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409-1042, USA.,Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, MI, 48108-5260, USA
| | - Linda J S Allen
- Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409-1042, USA
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20
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Garetto M, Leonardi E, Torrisi GL. A time-modulated Hawkes process to model the spread of COVID-19 and the impact of countermeasures. Annu Rev Control 2021; 51:551-563. [PMID: 33746561 PMCID: PMC7953674 DOI: 10.1016/j.arcontrol.2021.02.002] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/31/2020] [Revised: 02/23/2021] [Accepted: 02/27/2021] [Indexed: 05/04/2023]
Abstract
Motivated by the recent outbreak of coronavirus (COVID-19), we propose a stochastic model of epidemic temporal growth and mitigation based on a time-modulated Hawkes process. The model is sufficiently rich to incorporate specific characteristics of the novel coronavirus, to capture the impact of undetected, asymptomatic and super-diffusive individuals, and especially to take into account time-varying counter-measures and detection efforts. Yet, it is simple enough to allow scalable and efficient computation of the temporal evolution of the epidemic, and exploration of what-if scenarios. Compared to traditional compartmental models, our approach allows a more faithful description of virus specific features, such as distributions for the time spent in stages, which is crucial when the time-scale of control (e.g., mobility restrictions) is comparable to the lifetime of a single infection. We apply the model to the first and second wave of COVID-19 in Italy, shedding light onto several effects related to mobility restrictions introduced by the government, and to the effectiveness of contact tracing and mass testing performed by the national health service.
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Affiliation(s)
- Michele Garetto
- Università degli Studi di Torino, C.so Svizzera 185, Torino, Italy
| | - Emilio Leonardi
- Politecnico di Torino, C.so Duca degli Abruzzi 24, Torino, Italy
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21
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Nipa KF, Allen LJS. Disease Emergence in Multi-Patch Stochastic Epidemic Models with Demographic and Seasonal Variability. Bull Math Biol 2020; 82:152. [PMID: 33231753 DOI: 10.1007/s11538-020-00831-x] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2020] [Accepted: 11/02/2020] [Indexed: 11/24/2022]
Abstract
Factors such as seasonality and spatial connectivity affect the spread of an infectious disease. Accounting for these factors in infectious disease models provides useful information on the times and locations of greatest risk for disease outbreaks. In this investigation, stochastic multi-patch epidemic models are formulated with seasonal and demographic variability. The stochastic models are used to investigate the probability of a disease outbreak when infected individuals are introduced into one or more of the patches. Seasonal variation is included through periodic transmission and dispersal rates. Multi-type branching process approximation and application of the backward Kolmogorov differential equation lead to an estimate for the probability of a disease outbreak. This estimate is also periodic and depends on the time, the location, and the number of initial infected individuals introduced into the patch system as well as the magnitude of the transmission and dispersal rates and the connectivity between patches. Examples are given for seasonal transmission and dispersal in two and three patches.
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22
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Okolie A, Müller J. Exact and approximate formulas for contact tracing on random trees. Math Biosci 2020; 321:108320. [PMID: 32014418 DOI: 10.1016/j.mbs.2020.108320] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/16/2019] [Revised: 01/23/2020] [Accepted: 01/23/2020] [Indexed: 11/25/2022]
Abstract
We consider a stochastic susceptible-infected-recovered (SIR) model with contact tracing on random trees and on the configuration model. On a rooted tree, where initially all individuals are susceptible apart from the root which is infected, we are able to find exact formulas for the distribution of the infectious period. Thereto, we show how to extend the existing theory for contact tracing in homogeneously mixing populations to trees. Based on these formulas, we discuss the influence of randomness in the tree and the basic reproduction number. We find the well known results for the homogeneously mixing case as a limit of the present model (tree-shaped contact graph). Furthermore, we develop approximate mean field equations for the dynamics on trees, and - using the message passing method - also for the configuration model. The interpretation and implications of the results are discussed.
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Affiliation(s)
- Augustine Okolie
- Center for Mathematical Sciences, Technische Universität München, Garching 85748, Germany.
| | - Johannes Müller
- Center for Mathematical Sciences, Technische Universität München, Garching 85748, Germany; Institute for Computational Biology, Helmholtz Center Munich, Neuherberg 85764, Germany
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23
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Abstract
BACKGROUND A basic tool for studying the polyploidization history of a genome, especially in plants, is the distribution of duplicate gene similarities in syntenically aligned regions of a genome. This distribution can usually be decomposed into two or more components identifiable by peaks, or local maxima, each representing a different polyploidization event. The distributions may be generated by means of a discrete time branching process, followed by a sequence divergence model. The branching process, as well as the inference of fractionation rates based on it, requires knowledge of the ploidy level of each event, which cannot be directly inferred from the pair similarity distribution. RESULTS For a sequence of two events of unknown ploidy, either tetraploid, giving rise to whole genome doubling (WGD), or hexaploid, giving rise to whole genome tripling (WGT), we base our analysis on triples of similar genes. We calculate the probability of the four triplet types with origins in one or the other event, or both, and impose a mutational model so that the distribution resembles the original data. Using a ML transition point in the similarities between the two events as a discriminator for the hypothesized origin of each similarity, we calculate the predicted number of triplets of each type for each model combining WGT and/or WGD. This yields a predicted profile of triplet types for each model. We compare the observed and predicted triplet profiles for each model to confirm the polyploidization history of durian, poplar and cabbage. CONCLUSIONS We have developed a way of inferring the ploidy of up to three successive WGD and/or WGT events by estimating the time of origin of each of the similarities in triples of genes. This may be generalized to a larger number of events and to higher ploidies.
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Affiliation(s)
- Yue Zhang
- Department of Mathematics and Statistics, University of Ottawa, 150 Louis Pasteur pvt, Ottawa, K1N 6N5, Canada
| | - Chunfang Zheng
- Department of Mathematics and Statistics, University of Ottawa, 150 Louis Pasteur pvt, Ottawa, K1N 6N5, Canada
| | - David Sankoff
- Department of Mathematics and Statistics, University of Ottawa, 150 Louis Pasteur pvt, Ottawa, K1N 6N5, Canada.
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24
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Meehan MT, Cope RC, McBryde ES. On the probability of strain invasion in endemic settings: Accounting for individual heterogeneity and control in multi-strain dynamics. J Theor Biol 2019; 487:110109. [PMID: 31816294 PMCID: PMC7094110 DOI: 10.1016/j.jtbi.2019.110109] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2019] [Revised: 11/28/2019] [Accepted: 12/05/2019] [Indexed: 01/21/2023]
Abstract
Endemic infection can insulate host populations from invasion by mutant variants. The timing of control implementation strongly influences its efficacy. Controls that exacerbate host heterogeneity outperform those that curtail it. Differential control can facilitate strain invasion and eventual replacement.
Pathogen evolution is an imminent threat to global health that has warranted, and duly received, considerable attention within the medical, microbiological and modelling communities. Outbreaks of new pathogens are often ignited by the emergence and transmission of mutant variants descended from wild-type strains circulating in the community. In this work we investigate the stochastic dynamics of the emergence of a novel disease strain, introduced into a population in which it must compete with an existing endemic strain. In analogy with past work on single-strain epidemic outbreaks, we apply a branching process approximation to calculate the probability that the new strain becomes established. As expected, a critical determinant of the survival prospects of any invading strain is the magnitude of its reproduction number relative to that of the background endemic strain. Whilst in most circumstances this ratio must exceed unity in order for invasion to be viable, we show that differential control scenarios can lead to less-fit novel strains invading populations hosting a fitter endemic one. This analysis and the accompanying findings will inform our understanding of the mechanisms that have led to past instances of successful strain invasion, and provide valuable lessons for thwarting future drug-resistant strain incursions.
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Affiliation(s)
- Michael T Meehan
- James Cook University, Australian Institute of Tropical Health and Medicine, Townsville, Australia.
| | - Robert C Cope
- The University of Adelaide, School of Mathematical Sciences, Adelaide, Australia
| | - Emma S McBryde
- James Cook University, Australian Institute of Tropical Health and Medicine, Townsville, Australia
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25
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Burden CJ, Soewongsono AC. Coalescence in the diffusion limit of a Bienaymé-Galton-Watson branching process. Theor Popul Biol 2019; 130:50-59. [PMID: 31585138 DOI: 10.1016/j.tpb.2019.09.009] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/24/2019] [Revised: 08/27/2019] [Accepted: 09/25/2019] [Indexed: 10/25/2022]
Abstract
We consider the problem of estimating the elapsed time since the most recent common ancestor of a finite random sample drawn from a population which has evolved through a Bienaymé-Galton-Watson branching process. More specifically, we are interested in the diffusion limit appropriate to a supercritical process in the near-critical limit evolving over a large number of time steps. Our approach differs from earlier analyses in that we assume the only known information is the mean and variance of the number of offspring per parent, the observed total population size at the time of sampling, and the size of the sample. We obtain a formula for the probability that a finite random sample of the population is descended from a single ancestor in the initial population, and derive a confidence interval for the initial population size in terms of the final population size and the time since initiating the process. We also determine a joint likelihood surface from which confidence regions can be determined for simultaneously estimating two parameters, (1) the population size at the time of the most recent common ancestor, and (2) the time elapsed since the existence of the most recent common ancestor.
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Affiliation(s)
- Conrad J Burden
- Mathematical Sciences Institute, Australian National University, Canberra, Australia.
| | - Albert C Soewongsono
- Mathematical Sciences Institute, Australian National University, Canberra, Australia; School of Physical Sciences, University of Tasmania, Hobart, Australia.
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26
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Zhang Y, Zheng C, Sankoff D. A branching process for homology distribution-based inference of polyploidy, speciation and loss. Algorithms Mol Biol 2019; 14:18. [PMID: 31388348 PMCID: PMC6676528 DOI: 10.1186/s13015-019-0153-8] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2019] [Accepted: 07/19/2019] [Indexed: 11/10/2022] Open
Abstract
BACKGROUND The statistical distribution of the similarity or difference between pairs of paralogous genes, created by whole genome doubling, or between pairs of orthologous genes in two related species is an important source of information about genomic evolution, especially in plants. METHODS We derive the mixture of distributions of sequence similarity for duplicate gene pairs generated by repeated episodes of whole gene doubling. This involves integrating sequence divergence and gene pair loss through fractionation, using a branching process and a mutational model. We account not only for the timing of these events in terms of local modes, but also the amplitude and variance of the component distributions. This model is then extended to orthologous gene pairs. RESULTS We apply the model and inference procedures to the evolution of the Solanaceae, focusing on the genomes of economically important crops. We assess how consistent or variable fractionation rates are from species to species and over time.
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Affiliation(s)
- Yue Zhang
- Department of Mathematics and Statistics, University of Ottawa, 150 Louis Pasteur Pvt, Ottawa, K1N 6N5 Canada
| | - Chunfang Zheng
- Department of Mathematics and Statistics, University of Ottawa, 150 Louis Pasteur Pvt, Ottawa, K1N 6N5 Canada
| | - David Sankoff
- Department of Mathematics and Statistics, University of Ottawa, 150 Louis Pasteur Pvt, Ottawa, K1N 6N5 Canada
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27
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Abstract
We provide an analysis of a randomly grown 2-d network which models the morphological growth of dendritic and axonal arbors. From the stochastic geometry of this model we derive a dynamic graph of potential synaptic connections. We estimate standard network parameters such as degree distribution, average shortest path length and clustering coefficient, considering all these parameters as functions of time. Our results show that even a simple model with just a few parameters is capable of representing a wide spectra of architecture, capturing properties of well-known models, such as random graphs or small world networks, depending on the time of the network development. The introduced model allows not only rather straightforward simulations but it is also amenable to a rigorous analysis. This provides a base for further study of formation of synaptic connections on such networks and their dynamics due to plasticity.
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Affiliation(s)
- Fioralba Ajazi
- Department of Mathematical Statistics, Faculty of Science, Lund University, Sölvegatan 18, 22100, Lund, Sweden
- Faculty of Business and Economics, University of Lausanne, 1015, Lausanne, Switzerland
| | | | - Tatyana Turova
- Department of Mathematical Statistics, Faculty of Science, Lund University, Sölvegatan 18, 22100, Lund, Sweden.
- IMPB - The Branch of Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow, Russia.
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28
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Hautphenne S, Massaro M, Turner K. Fitting Markovian binary trees using global and individual demographic data. Theor Popul Biol 2019; 128:39-50. [PMID: 31059720 DOI: 10.1016/j.tpb.2019.04.007] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2018] [Revised: 04/19/2019] [Accepted: 04/22/2019] [Indexed: 10/26/2022]
Abstract
We consider a class of continuous-time branching processes called Markovian binary trees (MBTs), in which the individuals lifetime and reproduction epochs are modelled using a transient Markovian arrival process (TMAP). We develop methods for estimating the parameters of the TMAP by using either age-specific averages of reproduction and mortality rates, or age-specific individual demographic data. Depending on the degree of detail of the available information, we follow a weighted non-linear regression or a maximum likelihood approach. We discuss several criteria to determine the optimal number of states in the underlying TMAP. Our results improve the fit of an existing MBT model for human demography, and provide insights for the future conservation management of the threatened Chatham Island black robin population.
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Affiliation(s)
- Sophie Hautphenne
- The University of Melbourne, School of Mathematics and Statistics, 3010 Melbourne, Victoria, Australia; Ecole polythechnique fédérale de Lausanne, Switzerland.
| | - Melanie Massaro
- Charles Sturt University, Institute for Land, Water and Society and School of Environmental Sciences, Australia.
| | - Katharine Turner
- Ecole polythechnique fédérale de Lausanne, Switzerland; Australian National University, Australia.
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29
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Waxman D, Nouvellet P. Sub- or supercritical transmissibilities in a finite disease outbreak: Symmetry in outbreak properties of a disease conditioned on extinction. J Theor Biol 2019; 467:80-86. [PMID: 30711456 PMCID: PMC6408326 DOI: 10.1016/j.jtbi.2019.01.033] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2018] [Revised: 01/24/2019] [Accepted: 01/30/2019] [Indexed: 11/28/2022]
Abstract
This work is concerned with the transmissibility of a disease, on observation of an outbreak of limited size. When such an outbreak occurs, an accurate estimate of the transmissibility of the responsible pathogen is essential for an appropriate response to future outbreaks. Transmissibility is usually characterised in terms of the reproduction number, R, which is the mean number of new cases of infection produced by a single infectious individual. A subcritical reproduction number (R < 1) guarantees that an outbreak will eventually die out of its own accord. By contrast, a supercritical reproduction number (R > 1) does not guarantee spread of the disease, since even with appreciable transmissibility, an outbreak may become extinct due to stochastic effects associated with a small number of infected individuals. Once the number of infectious individuals is conditioned on extinction, we show that an exact symmetry of the underlying theory ensures two distinct values of R, one larger than unity, the other smaller than unity, for which all outbreak properties are identical, with no signature of difference. Therefore a disease with a subcritical R, or its supercritical counterpart, when conditioned on extinction, have, for a given outbreak, identical individual likelihoods. In the full likelihood, this symmetry is lost, since the individual likelihood for a subcritical R is weighted by an extinction probability of unity, but the individual likelihood of a supercritical R is weighted by a sub-unity extinction probability. However, the inference can still benefit from the underlying symmetry, since it yields a mapping of all supercritical reproduction numbers onto the subcritical domain (R < 1), thereby speeding up evaluation of the likelihood profile. The symmetry holds in the standard situation, where the distribution of secondary cases is Poisson, as well as where this distribution has a negative binomial form and super-spreading can occur.
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Affiliation(s)
- David Waxman
- Centre for Computational Systems Biology ISTBI, Fudan University, 220 Handan Road, Shanghai 200433, People's Republic of China
| | - Pierre Nouvellet
- School of Life Sciences, University of Sussex, Brighton BN1 9QG, UK.
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30
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Alsmeyer G, Raschel K. The extinction problem for a distylous plant population with sporophytic self-incompatibility. J Math Biol 2019; 78:1841-1874. [PMID: 30683998 DOI: 10.1007/s00285-019-01328-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2018] [Revised: 01/11/2019] [Indexed: 11/28/2022]
Abstract
In this paper, the extinction problem for a class of distylous plant populations is considered within the framework of certain nonhomogeneous nearest-neighbor random walks in the positive quadrant. For the latter, extinction means absorption at one of the axes. Despite connections with some classical probabilistic models (standard two-type Galton-Watson process, two-urn model), exact formulae for the probabilities of absorption seem to be difficult to come by and one must therefore resort to good approximations. In order to meet this task, we develop potential-theoretic tools and provide various sub- and super-harmonic functions which, for large initial populations, provide bounds which in particular improve those that have appeared earlier in the literature.
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Affiliation(s)
- Gerold Alsmeyer
- Institut für Mathematische Stochastik, Fachbereich Mathematik und Informatik, Universität Münster, Orléans-Ring 10, 48149, Münster, Germany
| | - Kilian Raschel
- CNRS, Institut Denis Poisson, Université de Tours, Parc de Grandmont, 37200, Tours, France.
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31
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Fabreti LG, Castro D, Gorzoni B, Janini LMR, Antoneli F. Stochastic Modeling and Simulation of Viral Evolution. Bull Math Biol 2018; 81:1031-1069. [PMID: 30552628 DOI: 10.1007/s11538-018-00550-4] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2017] [Accepted: 12/03/2018] [Indexed: 12/11/2022]
Abstract
RNA viruses comprise vast populations of closely related, but highly genetically diverse, entities known as quasispecies. Understanding the mechanisms by which this extreme diversity is generated and maintained is fundamental when approaching viral persistence and pathobiology in infected hosts. In this paper, we access quasispecies theory through a mathematical model based on the theory of multitype branching processes, to better understand the roles of mechanisms resulting in viral diversity, persistence and extinction. We accomplish this understanding by a combination of computational simulations and the theoretical analysis of the model. In order to perform the simulations, we have implemented the mathematical model into a computational platform capable of running simulations and presenting the results in a graphical format in real time. Among other things, we show that the establishment of virus populations may display four distinct regimes from its introduction into new hosts until achieving equilibrium or undergoing extinction. Also, we were able to simulate different fitness distributions representing distinct environments within a host which could either be favorable or hostile to the viral success. We addressed the most used mechanisms for explaining the extinction of RNA virus populations called lethal mutagenesis and mutational meltdown. We were able to demonstrate a correspondence between these two mechanisms implying the existence of a unifying principle leading to the extinction of RNA viruses.
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Affiliation(s)
- Luiza Guimarães Fabreti
- Programa de Pós-Graduação em Infectologia, Universidade Federal de São Paulo, São Paulo, SP, Brazil
| | - Diogo Castro
- Programa de Pós-Graduação em Infectologia, Universidade Federal de São Paulo, São Paulo, SP, Brazil
| | - Bruno Gorzoni
- Programa de Pós-Graduação em Infectologia, Universidade Federal de São Paulo, São Paulo, SP, Brazil
| | - Luiz Mario Ramos Janini
- Departamentos de Microbiologia, Imunologia, Parasitologia and Medicina, Laboratório de Retrovirologia, Universidade Federal de São Paulo, São Paulo, SP, Brazil
| | - Fernando Antoneli
- Departamento de Informática em Saúde, Laboratório de Biocomplexidade e Genômica Evolutiva, Universidade Federal de São Paulo, São Paulo, SP, Brazil.
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32
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Nandi A, Allen LJS. Stochastic two-group models with transmission dependent on host infectivity or susceptibility. J Biol Dyn 2018; 13:201-224. [PMID: 30381000 DOI: 10.1080/17513758.2018.1538462] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2018] [Accepted: 10/12/2018] [Indexed: 06/08/2023]
Abstract
Stochastic epidemic models with two groups are formulated and applied to emerging and re-emerging infectious diseases. In recent emerging diseases, disease spread has been attributed to superspreaders, highly infectious individuals that infect a large number of susceptible individuals. In some re-emerging infectious diseases, disease spread is attributed to waning immunity in susceptible hosts. We apply a continuous-time Markov chain (CTMC) model to study disease emergence or re-emergence from different groups, where the transmission rates depend on either the infectious host or the susceptible host. Multitype branching processes approximate the dynamics of the CTMC model near the disease-free equilibrium and are used to estimate the probability of a minor or a major epidemic. It is shown that the probability of a major epidemic is greater if initiated by an individual from the superspreader group or by an individual from the highly susceptible group. The models are applied to Severe Acute Respiratory Syndrome and measles.
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Affiliation(s)
- Aadrita Nandi
- a Department of Mathematics and Statistics, Texas Tech University , Lubbock , TX , USA
| | - Linda J S Allen
- a Department of Mathematics and Statistics, Texas Tech University , Lubbock , TX , USA
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33
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Luo Q. A minimally parametrized branching process explaining plateau phase of qPCR amplification. J Math Biol 2019; 78:767-76. [PMID: 30151697 DOI: 10.1007/s00285-018-1291-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2018] [Revised: 08/22/2018] [Indexed: 10/28/2022]
Abstract
Quantitative polymerase chain reaction (qPCR) is a commonly used molecular biology technique for measuring the concentration of a target nucleic acid sequence in a sample. The whole qPCR amplification process usually consists of an exponential, a linear and a plateau phase. In qPCR experiments, amplification curves of samples with different template concentrations often, even though not always, have the same plateau height. The biological theory for this phenomenon is that the plateau height is determined by reaction kinetics. Does it mean that the target concentration has no effect on the final plateau height? We proposed a branching process based on Michaelis-Menten kinetics. Our model can describe all phases of qPCR amplification despite its simplicity (it depends on only one parameter). We theoretically showed, through almost sure convergence, that amplification curves will eventually plateau at finite values in any experiment, under any condition. We conclude that the plateau height is largely determined by reaction kinetics but could also be affected by the template concentration. This is in accordance with the current biological theory.
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Yan AWC, Black AJ, McCaw JM, Rebuli N, Ross JV, Swan AJ, Hickson RI. The distribution of the time taken for an epidemic to spread between two communities. Math Biosci 2018; 303:139-47. [PMID: 30089576 DOI: 10.1016/j.mbs.2018.07.004] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2018] [Revised: 07/05/2018] [Accepted: 07/05/2018] [Indexed: 11/21/2022]
Abstract
Assessing the risk of disease spread between communities is important in our highly connected modern world. However, the impact of disease- and population-specific factors on the time taken for an epidemic to spread between communities, as well as the impact of stochastic disease dynamics on this spreading time, are not well understood. In this study, we model the spread of an acute infection between two communities ('patches') using a susceptible-infectious-removed (SIR) metapopulation model. We develop approximations to efficiently evaluate the probability of a major outbreak in a second patch given disease introduction in a source patch, and the distribution of the time taken for this to occur. We use these approximations to assess how interventions, which either control disease spread within a patch or decrease the travel rate between patches, change the spreading probability and median spreading time. We find that decreasing the basic reproduction number in the source patch is the most effective way of both decreasing the spreading probability, and delaying epidemic spread to the second patch should this occur. Moreover, we show that the qualitative effects of interventions are the same regardless of the approximations used to evaluate the spreading time distribution, but for some regions in parameter space, quantitative findings depend upon the approximations used. Importantly, if we neglect the possibility that an intervention prevents a large outbreak in the source patch altogether, then intervention effectiveness is not estimated accurately.
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35
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Mazoyer A. Fluctuation analysis on mutation models with birth-date dependence. Math Biosci 2018; 303:83-100. [PMID: 29932952 DOI: 10.1016/j.mbs.2018.06.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2017] [Revised: 06/13/2018] [Accepted: 06/18/2018] [Indexed: 10/28/2022]
Abstract
The classic Luria-Delbrück model can be interpreted as a Poisson compound (number of mutations) of exponential mixtures (developing time of mutant clones) of geometric distributions (size of a clone in a given time). This "three-ingredients" approach is generalized in this paper to the case where the split instant distributions of cells are not i.i.d. : the lifetime of each cell is assumed to depend on its birth date. This model takes also into account cell deaths and non-exponentially distributed lifetimes. Previous results on the convergence of the distribution of mutant counts are recovered. The particular case where the instantaneous division rates of normal and mutant cells are proportional is studied. The classic Luria-Delbrück and Haldane models are recovered. Probability computations and simulation algorithms are provided. Robust estimation methods developed for the classic mutation models are adapted to the new model: their properties of consistency and asymptotic normality remain true; their asymptotic variances are computed. Finally, the estimation biases induced by considering classic mutation models instead of an inhomogeneous model are studied with simulation experiments.
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Affiliation(s)
- Adrien Mazoyer
- Laboratoire Jean Kuntzmann, Bâtiment IMAG, 700 avenue centrale, Saint Martin d'Hères 38401, France.
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36
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Cui F, Yuan B. Fixation probability of a beneficial mutation conferring decreased generation time in changing environments. BMC Syst Biol 2018; 12:48. [PMID: 29745841 PMCID: PMC5998907 DOI: 10.1186/s12918-018-0575-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
BACKGROUND One central building block of population genetics is the fixation probability. It is a probabilistic understanding of the eventual fate of new mutations. Moreover, the fixation probability of new beneficial mutations plays an important effect on the adaptation of populations to environmental challenges. Great progress has been made in the study of the beneficial mutations that increases offspring number. However, the fixation probability of beneficial mutations with a shorter generation time under various genetic and ecological conditions has not been explored. RESULTS Here we extend the classical result of the fixation probability of beneficial mutations obtained by Haldane, and estimate the fixation probability of a beneficial mutation with a reduced generation time in a changing environment. Assuming that the selective advantage is very small, we concentrate all the changing factors of environment on a single quantity: effective selective advantage. Using a time-dependent branching process, we get the analytic approximation for the fixation probability of beneficial mutations that decrease the generation time. Then, we apply this approximation to four interesting biological cases. CONCLUSIONS In these instances, we show the comparison of the approximation with the accurate values. We find that they are consistent, demonstrating the effectiveness of our result for the fixation probability of beneficial mutations conferring a reduced replication time.
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Affiliation(s)
- Fangshu Cui
- Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240 China
| | - Bo Yuan
- Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240 China
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Abstract
Disease outbreaks in stochastic SIR epidemic models are characterized as either minor or major. When ℛ0<1, all epidemics are minor, whereas if ℛ0>1, they can be minor or major. In 1955, Whittle derived formulas for the probability of a minor or a major epidemic. A minor epidemic is distinguished from a major one in that a minor epidemic is generally of shorter duration and has substantially fewer cases than a major epidemic. In this investigation, analytical formulas are derived that approximate the probability density, the mean, and the higher-order moments for the duration of a minor epidemic. These analytical results are applicable to minor epidemics in stochastic SIR, SIS, and SIRS models with a single infected class. The probability density for minor epidemics in more complex epidemic models can be computed numerically applying multitype branching processes and the backward Kolmogorov differential equations. When ℛ0 is close to one, minor epidemics are more common than major epidemics and their duration is significantly longer than when ℛ0≪1 or ℛ0≫1.
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Affiliation(s)
- William Tritch
- Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA
| | - Linda J S Allen
- Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA
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Abstract
This study focuses on the extinction rate of a population that follows a continuous-time multi-type branching process in a random environment. Numerical computations in a particular example inspired by an epidemic model suggest an explicit formula for this extinction rate, but only for certain parameter values.
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McAvoy A, Fraiman N, Hauert C, Wakeley J, Nowak MA. Public goods games in populations with fluctuating size. Theor Popul Biol 2018; 121:72-84. [PMID: 29408219 DOI: 10.1016/j.tpb.2018.01.004] [Citation(s) in RCA: 25] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2017] [Revised: 01/16/2018] [Accepted: 01/24/2018] [Indexed: 11/18/2022]
Abstract
Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here, we consider evolutionary game dynamics in an extended Wright-Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Although defectors are favored for any mixed population, random drift could lead to their elimination and the resulting pure-cooperator population could survive. On the other hand, if the defectors remain, then the population will quickly go extinct because the frequency of cooperators steadily declines and defectors alone cannot survive. In a mutation-selection model, we find that (i) a steady supply of cooperators can enable long-term population survival, provided selection is sufficiently strong, and (ii) selection can increase the abundance of cooperators but reduce their relative frequency. Thus, evolutionary game dynamics in populations with variable size generate a multifaceted notion of what constitutes a trait's long-term success.
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Affiliation(s)
- Alex McAvoy
- Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, United States.
| | - Nicolas Fraiman
- Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States
| | - Christoph Hauert
- Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2
| | - John Wakeley
- Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, United States
| | - Martin A Nowak
- Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, United States; Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, United States; Department of Mathematics, Harvard University, Cambridge, MA 02138, United States
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40
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Bastogne T, Marchand JL, Pinel S, Vallois P. A branching process model of heterogeneous DNA damages caused by radiotherapy in in vitro cell cultures. Math Biosci 2017; 294:100-9. [PMID: 29054768 DOI: 10.1016/j.mbs.2017.09.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2017] [Revised: 07/21/2017] [Accepted: 09/23/2017] [Indexed: 11/22/2022]
Abstract
This paper deals with the dynamic modeling and simulation of cell damage heterogeneity and associated mutant cell phenotypes in the therapeutic responses of cancer cell populations submitted to a radiotherapy session during in vitro assays. Each cell is described by a finite number of phenotypic states with possible transitions between them. The population dynamics is then given by an age-dependent multi-type branching process. From this representation, we obtain formulas for the average size of the global survival population as well as the one of subpopulations associated with 10 mutation phenotypes. The proposed model has been implemented into Matlab© and the numerical results corroborate the ability of the model to reproduce four major types of cell responses: delayed growth, anti-proliferative, cytostatic and cytotoxic.
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Abstract
The classic Luria-Delbrück model for fluctuation analysis is extended to the case where the split instant distributions of cells are not i.i.d.: the lifetime of each cell is assumed to depend on its birth date. This model takes also into account cell deaths and non-exponentially distributed lifetimes. In particular, it is possible to consider subprobability distributions and to model non-exponential growth. The extended model leads to a family of probability distributions which depend on the expected number of mutations, the death probability of mutant cells, and the split instant distributions of normal and mutant cells. This is deduced from the Bellman-Harris integral equation, written for the birth date inhomogeneous case. A new theorem of convergence for the final mutant counts is proved, using an analytic method. Particular examples like the Haldane model or the case where hazard functions of the split-instant distributions are proportional are studied. The Luria-Delbrück distribution with cell deaths is recovered. A computation algorithm for the probabilities is provided.
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Affiliation(s)
- Adrien Mazoyer
- Laboratoire Jean Kuntzmann, Bâtiment IMAG, 700 Avenue Centrale, 38401, Saint Martin d'Hères, France.
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42
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Jain K, James A. Fixation probability of a nonmutator in a large population of asexual mutators. J Theor Biol 2017; 433:85-93. [PMID: 28870620 DOI: 10.1016/j.jtbi.2017.08.027] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2017] [Revised: 08/29/2017] [Accepted: 08/31/2017] [Indexed: 10/18/2022]
Abstract
In an adapted population of mutators in which most mutations are deleterious, a nonmutator that lowers the mutation rate is under indirect selection and can sweep to fixation. Using a multitype branching process, we calculate the fixation probability of a rare nonmutator in a large population of asexual mutators. We show that when beneficial mutations are absent, the fixation probability is a nonmonotonic function of the mutation rate of the mutator: it first increases sublinearly and then decreases exponentially. We also find that beneficial mutations can enhance the fixation probability of a nonmutator. Our analysis is relevant to an understanding of recent experiments in which a reduction in the mutation rates has been observed.
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Affiliation(s)
- Kavita Jain
- Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India.
| | - Ananthu James
- Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India
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Abstract
BACKGROUND The introduction of new malaria control interventions has often led to the evolution of resistance, both of the parasite to new drugs and of the mosquito vector to new insecticides, compromising the efficacy of the interventions. Recent progress in molecular and population biology raises the possibility of new genetic-based interventions, and the potential for resistance to evolve against these should be considered. Here, population modelling is used to determine the main factors affecting the likelihood that resistance will evolve against a synthetic, nuclease-based driving Y chromosome that produces a male-biased sex ratio. METHODS A combination of deterministic differential equation models and stochastic analyses involving branching processes and Gillespie simulations is utilized to assess the probability that resistance evolves against a driving Y that otherwise is strong enough to eliminate the target population. The model considers resistance due to changes at the target site such that they are no longer cleaved by the nuclease, and due to trans-acting autosomal suppressor alleles. RESULTS The probability that resistance evolves increases with the mutation rate and the intrinsic rate of increase of the population, and decreases with the strength of drive and any pleiotropic fitness costs of the resistant allele. In seasonally varying environments, the time of release can also affect the probability of resistance evolving. Trans-acting suppressor alleles are more likely to suffer stochastic loss at low frequencies than target site resistant alleles. CONCLUSIONS As with any other intervention, there is a risk that resistance will evolve to new genetic approaches to vector control, and steps should be taken to minimize this probability. Two design features that should help in this regard are to reduce the rate at which resistant mutations arise, and to target sequences such that if they do arise, they impose a significant fitness cost on the mosquito.
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Affiliation(s)
- Andrea Beaghton
- Life Sciences, Imperial College, Silwood Park, Ascot, Berkshire, SL5 7PY UK
| | | | - Austin Burt
- Life Sciences, Imperial College, Silwood Park, Ascot, Berkshire, SL5 7PY UK
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Abstract
This paper is concerned with the analysis of vaccination strategies in a stochastic susceptible \documentclass[12pt]{minimal}
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\begin{document}$$\rightarrow $$\end{document}→ infected \documentclass[12pt]{minimal}
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\begin{document}$$\rightarrow $$\end{document}→ removed model for the spread of an epidemic amongst a population of individuals with a random network of social contacts that is also partitioned into households. Under various vaccine action models, we consider both household-based vaccination schemes, in which the way in which individuals are chosen for vaccination depends on the size of the households in which they reside, and acquaintance vaccination, which targets individuals of high degree in the social network. For both types of vaccination scheme, assuming a large population with few initial infectives, we derive a threshold parameter which determines whether or not a large outbreak can occur and also the probability of a large outbreak and the fraction of the population infected by a large outbreak. The performance of these schemes is studied numerically, focusing on the influence of the household size distribution and the degree distribution of the social network. We find that acquaintance vaccination can significantly outperform the best household-based scheme if the degree distribution of the social network is heavy-tailed. For household-based schemes, when the vaccine coverage is insufficient to prevent a major outbreak and the vaccine is imperfect, we find situations in which both the probability and size of a major outbreak under the scheme which minimises the threshold parameter are larger than in the scheme which maximises the threshold parameter.
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Affiliation(s)
- Frank Ball
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
| | - David Sirl
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
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45
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Abstract
We study the spread of sexually transmitted infections (STIs) and other infectious diseases on a dynamic network by using a branching process approach. The nodes in the network represent the sexually active individuals, while connections represent sexual partnerships. This network is dynamic as partnerships are formed and broken over time and individuals enter and leave the sexually active population due to demography. We assume that individuals enter the sexually active network with a random number of partners, chosen according to a suitable distribution and that the maximal number of partners that an individual can have at a time is finite. We discuss two different branching process approximations for the initial stages of an outbreak of the STI. In the first approximation we ignore some dependencies between infected individuals. We compute the offspring mean of this approximating branching process and discuss its relation to the basic reproduction number \documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}R0. The second branching process approximation is asymptotically exact, but only defined if individuals can have at most one partner at a time. For this model we compute the probability of a minor outbreak of the epidemic starting with one or few initial cases. We illustrate complications caused by dependencies in the epidemic model by showing that if individuals have at most one partner at a time, the probabilities of extinction of the two approximating branching processes are different. This implies that ignoring dependencies in the epidemic model leads to a wrong prediction of the probability of a large outbreak. Finally, we analyse the first branching process approximation if the number of partners an individual can have at a given time is unbounded. In this model we show that the branching process approximation is asymptomatically exact as the population size goes to infinity.
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Affiliation(s)
- Abid Ali Lashari
- Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
| | - Pieter Trapman
- Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
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46
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Abstract
The factors that regulate the size of organs to ensure that they fit within an organism are not well understood. A simple organ, the ocular lens serves as a useful model with which to tackle this problem. In many systems, considerable variance in the organ growth process is tolerable. This is almost certainly not the case in the lens, which in addition to fitting comfortably within the eyeball, must also be of the correct size and shape to focus light sharply onto the retina. Furthermore, the lens does not perform its optical function in isolation. Its growth, which continues throughout life, must therefore be coordinated with that of other tissues in the optical train. Here, we review the lens growth process in detail, from pioneering clinical investigations in the late nineteenth century to insights gleaned more recently in the course of cell and molecular studies. During embryonic development, the lens forms from an invagination of surface ectoderm. Consequently, the progenitor cell population is located at its surface and differentiated cells are confined to the interior. The interactions that regulate cell fate thus occur within the obligate ellipsoidal geometry of the lens. In this context, mathematical models are particularly appropriate tools with which to examine the growth process. In addition to identifying key growth determinants, such models constitute a framework for integrating cell biological and optical data, helping clarify the relationship between gene expression in the lens and image quality at the retinal plane.
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Affiliation(s)
- Steven Bassnett
- Department of Ophthalmology & Visual Sciences, Washington University School of Medicine, USA.
| | - Hrvoje Šikić
- Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
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47
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Allen LJS. A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis. Infect Dis Model 2017; 2:128-42. [PMID: 29928733 DOI: 10.1016/j.idm.2017.03.001] [Citation(s) in RCA: 94] [Impact Index Per Article: 13.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/29/2016] [Revised: 03/05/2017] [Accepted: 03/07/2017] [Indexed: 11/23/2022] Open
Abstract
Some mathematical methods for formulation and numerical simulation of stochastic epidemic models are presented. Specifically, models are formulated for continuous-time Markov chains and stochastic differential equations. Some well-known examples are used for illustration such as an SIR epidemic model and a host-vector malaria model. Analytical methods for approximating the probability of a disease outbreak are also discussed.
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48
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Nouvellet P, Cori A, Garske T, Blake IM, Dorigatti I, Hinsley W, Jombart T, Mills HL, Nedjati-Gilani G, Van Kerkhove MD, Fraser C, Donnelly CA, Ferguson NM, Riley S. A simple approach to measure transmissibility and forecast incidence. Epidemics 2017; 22:29-35. [PMID: 28351674 PMCID: PMC5871640 DOI: 10.1016/j.epidem.2017.02.012] [Citation(s) in RCA: 44] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2016] [Revised: 02/01/2017] [Accepted: 02/17/2017] [Indexed: 11/25/2022] Open
Abstract
Our simple approach relies on very few parameters and minimal assumptions Subjective choice of best training period improved forecasts Despites its simplicity, our model forecasted well under a range scenarios. This approach can be a natural 'null model' for comparison with methods.
Outbreaks of novel pathogens such as SARS, pandemic influenza and Ebola require substantial investments in reactive interventions, with consequent implementation plans sometimes revised on a weekly basis. Therefore, short-term forecasts of incidence are often of high priority. In light of the recent Ebola epidemic in West Africa, a forecasting exercise was convened by a network of infectious disease modellers. The challenge was to forecast unseen “future” simulated data for four different scenarios at five different time points. In a similar method to that used during the recent Ebola epidemic, we estimated current levels of transmissibility, over variable time-windows chosen in an ad hoc way. Current estimated transmissibility was then used to forecast near-future incidence. We performed well within the challenge and often produced accurate forecasts. A retrospective analysis showed that our subjective method for deciding on the window of time with which to estimate transmissibility often resulted in the optimal choice. However, when near-future trends deviated substantially from exponential patterns, the accuracy of our forecasts was reduced. This exercise highlights the urgent need for infectious disease modellers to develop more robust descriptions of processes – other than the widespread depletion of susceptible individuals – that produce non-exponential patterns of incidence.
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Affiliation(s)
- Pierre Nouvellet
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK; National Institute for Health Research Health Protection Research Unit in Modelling Methodology, Imperial College London, Faculty of Medicine, London, UK
| | - Anne Cori
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK
| | - Tini Garske
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK
| | - Isobel M Blake
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK
| | - Ilaria Dorigatti
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK
| | - Wes Hinsley
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK
| | - Thibaut Jombart
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK; National Institute for Health Research Health Protection Research Unit in Modelling Methodology, Imperial College London, Faculty of Medicine, London, UK
| | - Harriet L Mills
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK
| | - Gemma Nedjati-Gilani
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK
| | - Maria D Van Kerkhove
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK; Center for Global Health, Institute Pasteur, Paris, France
| | - Christophe Fraser
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK; National Institute for Health Research Health Protection Research Unit in Modelling Methodology, Imperial College London, Faculty of Medicine, London, UK
| | - Christl A Donnelly
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK; National Institute for Health Research Health Protection Research Unit in Modelling Methodology, Imperial College London, Faculty of Medicine, London, UK
| | - Neil M Ferguson
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK; National Institute for Health Research Health Protection Research Unit in Modelling Methodology, Imperial College London, Faculty of Medicine, London, UK
| | - Steven Riley
- MRC Centre for Outbreak Analysis and Modelling, Imperial College London, Faculty of Medicine, London, UK; National Institute for Health Research Health Protection Research Unit in Modelling Methodology, Imperial College London, Faculty of Medicine, London, UK.
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49
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Abstract
Recent years have seen a large amount of interest in epidemics on networks as a way of representing the complex structure of contacts capable of spreading infections through the modern human population. The configuration model is a popular choice in theoretical studies since it combines the ability to specify the distribution of the number of contacts (degree) with analytical tractability. Here we consider the early real-time behaviour of the Markovian SIR epidemic model on a configuration model network using a multitype branching process. We find closed-form analytic expressions for the mean and variance of the number of infectious individuals as a function of time and the degree of the initially infected individual(s), and write down a system of differential equations for the probability of extinction by time t that are numerically fast compared to Monte Carlo simulation. We show that these quantities are all sensitive to the degree distribution—in particular we confirm that the mean prevalence of infection depends on the first two moments of the degree distribution and the variance in prevalence depends on the first three moments of the degree distribution. In contrast to most existing analytic approaches, the accuracy of these results does not depend on having a large number of infectious individuals, meaning that in the large population limit they would be asymptotically exact even for one initial infectious individual.
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Affiliation(s)
- Frank Ball
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD UK
| | - Thomas House
- School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL UK
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Britton T, Juher D, Saldaña J. A Network Epidemic Model with Preventive Rewiring: Comparative Analysis of the Initial Phase. Bull Math Biol 2016; 78:2427-2454. [PMID: 27800576 DOI: 10.1007/s11538-016-0227-4] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2016] [Accepted: 10/17/2016] [Indexed: 11/30/2022]
Abstract
This paper is concerned with stochastic SIR and SEIR epidemic models on random networks in which individuals may rewire away from infected neighbors at some rate [Formula: see text] (and reconnect to non-infectious individuals with probability [Formula: see text] or else simply drop the edge if [Formula: see text]), so-called preventive rewiring. The models are denoted SIR-[Formula: see text] and SEIR-[Formula: see text], and we focus attention on the early stages of an outbreak, where we derive the expressions for the basic reproduction number [Formula: see text] and the expected degree of the infectious nodes [Formula: see text] using two different approximation approaches. The first approach approximates the early spread of an epidemic by a branching process, whereas the second one uses pair approximation. The expressions are compared with the corresponding empirical means obtained from stochastic simulations of SIR-[Formula: see text] and SEIR-[Formula: see text] epidemics on Poisson and scale-free networks. Without rewiring of exposed nodes, the two approaches predict the same epidemic threshold and the same [Formula: see text] for both types of epidemics, the latter being very close to the mean degree obtained from simulated epidemics over Poisson networks. Above the epidemic threshold, pairwise models overestimate the value of [Formula: see text] computed from simulations, which turns out to be very close to the one predicted by the branching process approximation. When exposed individuals also rewire with [Formula: see text] (perhaps unaware of being infected), the two approaches give different epidemic thresholds, with the branching process approximation being more in agreement with simulations.
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Affiliation(s)
- Tom Britton
- Department of Mathematics, Stockholm University, Stockholm, Sweden
| | - David Juher
- Departament d'Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, Catalonia, Spain
| | - Joan Saldaña
- Departament d'Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, Catalonia, Spain.
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