On State-Space Reduction in Multi-Strain Pathogen Models, with an Application to Antigenic Drift in Influenza A
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{"title"=>"On state-space reduction in multi-strain pathogen models, with an application to antigenic drift in influenza A", "type"=>"journal", "authors"=>[{"first_name"=>"Sergey", "last_name"=>"Kryazhimskiy", "scopus_author_id"=>"20734712600"}, {"first_name"=>"Ulf", "last_name"=>"Dieckmann", "scopus_author_id"=>"7003359350"}, {"first_name"=>"Simon A.", "last_name"=>"Levin", "scopus_author_id"=>"26643470900"}, {"first_name"=>"Jonathan", "last_name"=>"Dushoff", "scopus_author_id"=>"55748313900"}], "year"=>2007, "source"=>"PLoS Computational Biology", "identifiers"=>{"pui"=>"47358060", "doi"=>"10.1371/journal.pcbi.0030159", "isbn"=>"1553-7358 (Electronic)\\r1553-734X (Linking)", "pmid"=>"17708677", "issn"=>"1553734X", "scopus"=>"2-s2.0-34548409169", "sgr"=>"34548409169"}, "id"=>"43f37354-8c77-3508-8b8b-9a7e88112285", "abstract"=>"Many pathogens exist in phenotypically distinct strains that interact with each other through competition for hosts. General models that describe such multi-strain systems are extremely difficult to analyze because their state spaces are enormously large. Reduced models have been proposed, but so far all of them necessarily allow for coinfections and require that immunity be mediated solely by reduced infectivity, a potentially problematic assumption. Here, we suggest a new state-space reduction approach that allows immunity to be mediated by either reduced infectivity or reduced susceptibility and that can naturally be used for models with or without coinfections. Our approach utilizes the general framework of status-based models. The cornerstone of our method is the introduction of immunity variables, which describe multi-strain systems more naturally than the traditional tracking of susceptible and infected hosts. Models expressed in this way can be approximated in a natural way by a truncation method that is akin to moment closure, allowing us to sharply reduce the size of the state space, and thus to consider models with many strains in a tractable manner. Applying our method to the phenomenon of antigenic drift in influenza A, we propose a potentially general mechanism that could constrain viral evolution to a one-dimensional manifold in a two-dimensional trait space. Our framework broadens the class of multi-strain systems that can be adequately described by reduced models. It permits computational, and even analytical, investigation and thus serves as a useful tool for understanding the evolution and ecology of multi-strain pathogens.", "link"=>"http://www.mendeley.com/research/statespace-reduction-multistrain-pathogen-models-application-antigenic-drift-influenza", "reader_count"=>96, "reader_count_by_academic_status"=>{"Unspecified"=>2, "Professor > Associate Professor"=>6, "Researcher"=>32, "Student > Doctoral Student"=>4, "Student > Ph. D. Student"=>26, "Student > Postgraduate"=>1, "Other"=>3, "Student > Master"=>4, "Student > Bachelor"=>3, "Lecturer"=>1, "Lecturer > Senior Lecturer"=>2, "Professor"=>12}, "reader_count_by_user_role"=>{"Unspecified"=>2, "Professor > Associate Professor"=>6, "Researcher"=>32, "Student > Doctoral Student"=>4, "Student > Ph. D. Student"=>26, "Student > Postgraduate"=>1, "Other"=>3, "Student > Master"=>4, "Student > Bachelor"=>3, "Lecturer"=>1, "Lecturer > Senior Lecturer"=>2, "Professor"=>12}, "reader_count_by_subject_area"=>{"Unspecified"=>11, "Agricultural and Biological Sciences"=>34, "Arts and Humanities"=>1, "Business, Management and Accounting"=>1, "Veterinary Science and Veterinary Medicine"=>1, "Chemistry"=>1, "Computer Science"=>3, "Earth and Planetary Sciences"=>1, "Engineering"=>1, "Environmental Science"=>3, "Biochemistry, Genetics and Molecular Biology"=>8, "Mathematics"=>15, "Medicine and Dentistry"=>8, "Physics and Astronomy"=>5, "Immunology and Microbiology"=>3}, "reader_count_by_subdiscipline"=>{"Medicine and Dentistry"=>{"Medicine and Dentistry"=>8}, "Physics and Astronomy"=>{"Physics and Astronomy"=>5}, "Mathematics"=>{"Mathematics"=>15}, "Unspecified"=>{"Unspecified"=>11}, "Environmental Science"=>{"Environmental Science"=>3}, "Arts and Humanities"=>{"Arts and Humanities"=>1}, "Engineering"=>{"Engineering"=>1}, "Chemistry"=>{"Chemistry"=>1}, "Earth and Planetary Sciences"=>{"Earth and Planetary Sciences"=>1}, "Immunology and Microbiology"=>{"Immunology and Microbiology"=>3}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>34}, "Computer Science"=>{"Computer Science"=>3}, "Business, Management and Accounting"=>{"Business, Management and Accounting"=>1}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>8}, "Veterinary Science and Veterinary Medicine"=>{"Veterinary Science and Veterinary Medicine"=>1}}, "reader_count_by_country"=>{"Vietnam"=>1, "Sweden"=>1, "United States"=>2, "United Kingdom"=>4, "Mexico"=>1, "Italy"=>1, "France"=>1, "Australia"=>1}, "group_count"=>3}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/946048"], "description"=>"<p>Solution of the full model, Gog and Grenfell's model, order-1 independence closure model, order-1 interpolation closure model, and order-2 independence closure model are denoted by F, GG, O1A, O1B, and O2, respectively. Parameter values: <i>m</i> = 10<sup>−6</sup>, <i>R</i><sub>0</sub> = 5, and <i>s</i> = 0.6.</p>", "links"=>[], "tags"=>["four-strain", "compared", "approximations"], "article_id"=>616409, "categories"=>["Ecology", "Infectious Diseases", "Evolutionary Biology"], "users"=>["Sergey Kryazhimskiy", "Ulf Dieckmann", "Simon A Levin", "Jonathan Dushoff"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.0030159.g003", "stats"=>{"downloads"=>2, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Dynamics_of_the_Full_Four_Strain_SIR_Model_Black_Curves_Compared_with_Its_Approximations_Thick_Gray_Curves_/616409", "title"=>"Dynamics of the Full Four-Strain SIR Model (Black Curves) Compared with Its Approximations (Thick Gray Curves)", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2007-08-17 01:46:49"}
  • {"files"=>["https://ndownloader.figshare.com/files/946120"], "description"=>"<div><p>Strains to which strain (0,0) confers full, 37%, 2%, or no cross-immunity are shown in white, light gray, dark gray, and black, respectively.</p><p>(A) Our model. (B) Gog and Grenfell's model [<a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0030159#pcbi-0030159-b015\" target=\"_blank\">15</a>].</p></div>", "links"=>[], "tags"=>["ecology", "Evolutionary biology", "Infectious diseases"], "article_id"=>616484, "categories"=>["Ecology", "Infectious Diseases", "Evolutionary Biology"], "users"=>["Sergey Kryazhimskiy", "Ulf Dieckmann", "Simon A Levin", "Jonathan Dushoff"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.0030159.g004", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Cross_Immunity_Structure_/616484", "title"=>"Cross-Immunity Structure", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2007-08-17 01:48:04"}
  • {"files"=>["https://ndownloader.figshare.com/files/945956"], "description"=>"<p>Details as in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0030159#pcbi-0030159-g001\" target=\"_blank\">Figure 1</a>.</p>", "links"=>[], "tags"=>["ecology", "Evolutionary biology", "Infectious diseases"], "article_id"=>616326, "categories"=>["Ecology", "Infectious Diseases", "Evolutionary Biology"], "users"=>["Sergey Kryazhimskiy", "Ulf Dieckmann", "Simon A Levin", "Jonathan Dushoff"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.0030159.g002", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Quantitative_Accuracy_Measure_916_for_the_Performance_of_Four_Models_/616326", "title"=>"Quantitative Accuracy Measure Δ for the Performance of Four Models", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2007-08-17 01:45:26"}
  • {"files"=>["https://ndownloader.figshare.com/files/946191"], "description"=>"<div><p>Parameter values: μ = 0, ν = 1, <i>m</i> = 10<sup>−4</sup>, and <i>a</i> = 1/\n\t\t\t\t\t\t\t; the heterogeneous transmission coefficients β<sub>(<i>i</i>,<i>j</i>)</sub> were drawn from a normal distribution with mean 3 and standard deviation 0.5. The numerical solution for the time interval <i>t</i> ∈ [0,100] was obtained for a strain space given by a 20 × 20 rectangular lattice. The initial condition was given by all state variables being zero except for <i>I</i><sub>(1,1)</sub>(0) = 0.01 and ξ<sub>(1,1)</sub>(0) = 0.01, corresponding to a healthy and fully susceptible host population with 1% of hosts infected with strain (1,1).\n\t\t\t\t\t\t</p><p>(A) Strains whose maximum epidemic size exceeded 0.01 are shown. The gray shade indicates the maximum epidemic size; the number above each shaded square indicates the time when the maximum of the epidemic for that particular strain was reached. Circles indicate strains whose transmission coefficients are less than 3; crosses indicate strains with transmission coefficients greater than 3.</p><p>(B) The sum of all proportions of infectious hosts as a function of time.</p></div>", "links"=>[], "tags"=>["antigenic", "influenza", "order-1", "interpolation"], "article_id"=>616551, "categories"=>["Ecology", "Infectious Diseases", "Evolutionary Biology"], "users"=>["Sergey Kryazhimskiy", "Ulf Dieckmann", "Simon A Levin", "Jonathan Dushoff"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.0030159.g005", "stats"=>{"downloads"=>0, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Approximate_Dynamics_of_Antigenic_Drift_in_Influenza_A_Based_on_the_Order_1_Interpolation_Closure_/616551", "title"=>"Approximate Dynamics of Antigenic Drift in Influenza A, Based on the Order-1 Interpolation Closure", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2007-08-17 01:49:11"}
  • {"files"=>["https://ndownloader.figshare.com/files/945883"], "description"=>"<p>Qualitative accuracy measure (vertical axis) is shown in dependence on the cross-immunity coefficient <i>s</i> (horizontal axis), the transmission coefficient <i>R</i><sub>0</sub> (rows), and the mutation probability <i>m</i> (columns). Each panel shows the performance of four models: Gog and Grenfell's model (circles), order-1 independence closure (pluses), order-1 interpolation closure (squares), and order-2 independence closure (triangles). The highlighted region indicates the parameter combination <i>m</i> = 10<sup>−6</sup>, <i>R</i><sub>0</sub> = 5, and <i>s</i> = 0.6, for which the system's dynamics are shown in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0030159#pcbi-0030159-g003\" target=\"_blank\">Figure 3</a>.</p>", "links"=>[], "tags"=>["ecology", "Evolutionary biology", "Infectious diseases"], "article_id"=>616249, "categories"=>["Ecology", "Infectious Diseases", "Evolutionary Biology"], "users"=>["Sergey Kryazhimskiy", "Ulf Dieckmann", "Simon A Levin", "Jonathan Dushoff"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.0030159.g001", "stats"=>{"downloads"=>0, "page_views"=>0, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Qualitative_Accuracy_Measure_961_for_the_Performance_of_Four_Models_/616249", "title"=>"Qualitative Accuracy Measure ρ for the Performance of Four Models", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2007-08-17 01:44:09"}
  • {"files"=>["https://ndownloader.figshare.com/files/466559"], "description"=>"<div><p>Many pathogens exist in phenotypically distinct strains that interact with each other through competition for hosts. General models that describe such multi-strain systems are extremely difficult to analyze because their state spaces are enormously large. Reduced models have been proposed, but so far all of them necessarily allow for coinfections and require that immunity be mediated solely by reduced infectivity, a potentially problematic assumption. Here, we suggest a new state-space reduction approach that allows immunity to be mediated by either reduced infectivity or reduced susceptibility and that can naturally be used for models with or without coinfections. Our approach utilizes the general framework of status-based models. The cornerstone of our method is the introduction of immunity variables, which describe multi-strain systems more naturally than the traditional tracking of susceptible and infected hosts. Models expressed in this way can be approximated in a natural way by a truncation method that is akin to moment closure, allowing us to sharply reduce the size of the state space, and thus to consider models with many strains in a tractable manner. Applying our method to the phenomenon of antigenic drift in influenza A, we propose a potentially general mechanism that could constrain viral evolution to a one-dimensional manifold in a two-dimensional trait space. Our framework broadens the class of multi-strain systems that can be adequately described by reduced models. It permits computational, and even analytical, investigation and thus serves as a useful tool for understanding the evolution and ecology of multi-strain pathogens.</p></div>", "links"=>[], "tags"=>["state-space", "multi-strain", "pathogen", "antigenic", "influenza"], "article_id"=>151959, "categories"=>["Cancer", "Ecology", "Evolutionary Biology"], "users"=>["Sergey Kryazhimskiy", "Ulf Dieckmann", "Simon A Levin", "Jonathan Dushoff"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.0030159", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/On_State_Space_Reduction_in_Multi_Strain_Pathogen_Models_with_an_Application_to_Antigenic_Drift_in_Influenza_A/151959", "title"=>"On State-Space Reduction in Multi-Strain Pathogen Models, with an Application to Antigenic Drift in Influenza A", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2007-08-17 00:32:39"}

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  • {"unique-ip"=>"2", "full-text"=>"2", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2012", "month"=>"10"}
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  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2013", "month"=>"7"}
  • {"unique-ip"=>"1", "full-text"=>"2", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2013", "month"=>"9"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2013", "month"=>"11"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2014", "month"=>"1"}
  • {"unique-ip"=>"2", "full-text"=>"1", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2014", "month"=>"3"}
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  • {"unique-ip"=>"7", "full-text"=>"6", "pdf"=>"3", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2014", "month"=>"8"}
  • {"unique-ip"=>"2", "full-text"=>"2", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"1", "cited-by"=>"0", "year"=>"2014", "month"=>"9"}
  • {"unique-ip"=>"6", "full-text"=>"4", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"1", "cited-by"=>"0", "year"=>"2014", "month"=>"10"}
  • {"unique-ip"=>"4", "full-text"=>"2", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"2", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2014", "month"=>"11"}
  • {"unique-ip"=>"4", "full-text"=>"4", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2014", "month"=>"12"}
  • {"unique-ip"=>"5", "full-text"=>"4", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"1"}
  • {"unique-ip"=>"5", "full-text"=>"4", "pdf"=>"4", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"2", "cited-by"=>"0", "year"=>"2015", "month"=>"2"}
  • {"unique-ip"=>"3", "full-text"=>"6", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"1", "cited-by"=>"0", "year"=>"2015", "month"=>"3"}
  • {"unique-ip"=>"4", "full-text"=>"5", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"4"}
  • {"unique-ip"=>"4", "full-text"=>"5", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"5"}
  • {"unique-ip"=>"3", "full-text"=>"3", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"6"}
  • {"unique-ip"=>"5", "full-text"=>"4", "pdf"=>"3", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"7"}
  • {"unique-ip"=>"2", "full-text"=>"3", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"8"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"9"}
  • {"unique-ip"=>"8", "full-text"=>"7", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"1", "year"=>"2015", "month"=>"10"}
  • {"unique-ip"=>"5", "full-text"=>"5", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"11"}
  • {"unique-ip"=>"5", "full-text"=>"4", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2015", "month"=>"12"}
  • {"unique-ip"=>"6", "full-text"=>"4", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"1", "cited-by"=>"0", "year"=>"2016", "month"=>"1"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"2"}
  • {"unique-ip"=>"5", "full-text"=>"4", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"3"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"4"}
  • {"unique-ip"=>"3", "full-text"=>"2", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"5"}
  • {"unique-ip"=>"3", "full-text"=>"3", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"6"}
  • {"unique-ip"=>"2", "full-text"=>"3", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"7"}
  • {"unique-ip"=>"4", "full-text"=>"3", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"9"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"10"}
  • {"unique-ip"=>"3", "full-text"=>"2", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"11"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2016", "month"=>"12"}
  • {"unique-ip"=>"1", "full-text"=>"0", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"1"}
  • {"unique-ip"=>"2", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"1", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"2"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"3"}
  • {"unique-ip"=>"3", "full-text"=>"2", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"2", "cited-by"=>"0", "year"=>"2017", "month"=>"4"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"5"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"7"}
  • {"unique-ip"=>"5", "full-text"=>"4", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"8"}
  • {"unique-ip"=>"4", "full-text"=>"2", "pdf"=>"0", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"9"}
  • {"unique-ip"=>"6", "full-text"=>"3", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"10"}
  • {"unique-ip"=>"3", "full-text"=>"2", "pdf"=>"1", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2017", "month"=>"11"}
  • {"unique-ip"=>"2", "full-text"=>"1", "pdf"=>"2", "abstract"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"1"}
  • {"unique-ip"=>"3", "full-text"=>"3", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"1"}
  • {"unique-ip"=>"2", "full-text"=>"2", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"6"}
  • {"unique-ip"=>"2", "full-text"=>"1", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"7"}
  • {"unique-ip"=>"2", "full-text"=>"3", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"8"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2018", "month"=>"9"}
  • {"unique-ip"=>"5", "full-text"=>"3", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"2", "cited-by"=>"0", "year"=>"2018", "month"=>"12"}
  • {"unique-ip"=>"3", "full-text"=>"2", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"1", "cited-by"=>"0", "year"=>"2018", "month"=>"11"}
  • {"unique-ip"=>"4", "full-text"=>"6", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"2"}
  • {"unique-ip"=>"1", "full-text"=>"1", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"3"}
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Relative Metric

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