Epidemiological Models for Influenza and COVID-19—part 2

Robert B. Nachbar
Original post: 11-Mar-2020
Revised: 14-Mar-2020
Revised: 9-Apr-2020
◼
  • This set of notebooks uses a package which can be downloaded from https://www.wolframcloud.com/obj/rnachbar/Published/CompartmentalModeling--v1.wl
    • ◼
  • The package should be in the same directory as the notebooks, and is automatically loaded as part of the initialization.

    • Table of Contents

    Part 1
    

    Part 2
    

    Part 3
    

    Part 4
    

    Part 5
    

    COVID-19

    Background and notes
    

    Epidemic data from Wolfram Data Repository (WDR)

    ◼
  • The Johns Hopkins University data are available from the Wolfram Data Repository, and are described at https://www.wolframcloud.com/obj/resourcesystem/published/DataRepository/resources/Epidemic-Data-for-Novel-Coronavirus-COVID-19
    • In[]:=
    ResourceObject["Epidemic Data for Novel Coronavirus COVID-19"]
    Out[]=
    ResourceObject
    Name: Epidemic Data for Novel Coronavirus COVID-19
    »
    Type: DataResource
    Description:
    Estimated cases of novel coronavirus (COVID-19, formerly known as 2019-nCoV) inf
    
    In[]:=
    covid19Data=ResourceData["Epidemic Data for Novel Coronavirus COVID-19"]
    Out[]=
    DatasetContent1,Path{},Grid
    AdministrativeDivision
    Country
    GeoPosition
    ConfirmedCases
    RecoveredCases
    Deaths
    New York, United States
    United States
    42.96°N 75.53°W
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    -
    Spain
    40.0°N 4.0°W
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    -
    Italy
    42.83°N 12.83°E
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    -
    Germany
    51.0°N 9.0°E
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    -
    France
    46.23°N 2.214°E
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    Hubei, China
    China
    30.98°N 112.3°E
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    -
    Iran
    32.0°N 53.0°E
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    -
    United Kingdom
    55.38°N 3.436°W
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    TimeSeries
    Time: 22 Jan 2020 to 08 Apr 2020
    Data points: 78
    
    rows 1–8 of
    320
    ,StateInitialDataTypeSystem`CacheKey[10963875611834016894650152640171114932344370322697058
    240232],InitialShapeTypeSystem`PackageScope`HeaderShape[{All,AdministrativeDivision}1,{All,
    Country}1,{All,GeoPosition}1,{All,ConfirmedCases}1,{All,RecoveredCases}1,{All,Deaths}1
    ,
    TypeSystem`PackageScope`Limited[TypeSystem`PackageScope`ColumnShape[TypeSystem`PackageScope`Limited[TypeSystem`PackageScope`RowShape[
    AdministrativeDivisionTypeSystem`PackageScope`AtomShape[150],Country
    TypeSystem`PackageScope`AtomShape[90],GeoPositionTypeSystem`PackageScope`AtomShape[106],
    ConfirmedCasesTypeSystem`PackageScope`AtomShape[160],RecoveredCases
    TypeSystem`PackageScope`AtomShape[160],DeathsTypeSystem`PackageScope`AtomShape[160]],∞,
    6,{All}]],8,∞,{}]],InitialTypeTypeSystem`Vector[TypeSystem`Struct[{AdministrativeDivision,
    Country,GeoPosition,ConfirmedCases,RecoveredCases,
    Deaths},{TypeSystem`Atom[Entity[AdministrativeDivision]],TypeSystem`Atom[Entity[Country]],
    TypeSystem`Atom[GeoPosition],TypeSystem`Atom[TemporalData],TypeSystem`Atom[TemporalData],
    TypeSystem`Atom[TemporalData]}],320],MetaOriginHoldComplete[Join,
    DatasetHandle[212571014222362],DatasetHandle[212575309189658]],ID212579604156954,RowTarget8,
    ColumnTarget20,ShapeTypeSystem`PackageScope`HeaderShape[{All,AdministrativeDivision}1,{All,
    Country}1,{All,GeoPosition}1,{All,ConfirmedCases}1,{All,RecoveredCases}1,{All,Deaths}1
    ,
    TypeSystem`PackageScope`Limited[TypeSystem`PackageScope`ColumnShape[TypeSystem`PackageScope`Limited[TypeSystem`PackageScope`RowShape[
    AdministrativeDivisionTypeSystem`PackageScope`AtomShape[150],Country
    TypeSystem`PackageScope`AtomShape[90],GeoPositionTypeSystem`PackageScope`AtomShape[106],
    ConfirmedCasesTypeSystem`PackageScope`AtomShape[160],RecoveredCases
    TypeSystem`PackageScope`AtomShape[160],DeathsTypeSystem`PackageScope`AtomShape[160]],∞,
    6,{All}]],8,∞,{}]],TypeTypeSystem`Vector[TypeSystem`Struct[{AdministrativeDivision,Country,
    GeoPosition,ConfirmedCases,RecoveredCases,Deaths},{TypeSystem`Atom[Entity[AdministrativeDivision]],
    TypeSystem`Atom[Entity[Country]],TypeSystem`Atom[GeoPosition],TypeSystem`Atom[TemporalData],
    TypeSystem`Atom[TemporalData],TypeSystem`Atom[TemporalData]}],320],Path{},BaseIndices{},
    DisplayedRowCount8,DisplayedColumnCount6,DataRowCount320,DataColumnCount6,SortPaths{},
    SortDirections{},HiddenItemsMapNull,UpdateType1,VPosHold[FE`TypeSystem`NestedGrid`PackagePriv
    ate`$vPos$$4811581314151405808538690828468880594958],HPosHold[FE`TypeSystem`NestedGrid`PackagePrivate`
    $hPos$$4811581314151405808538690828468880594958],TopBar
    ,BottomBar
    
    In[]:=
    covid19Data[Counts/*KeySort,"Country"]
    Out[]=
    $Aborted
    ◼
  • The function
    fitDataWDR
    (defined in the Initialization section at the end of the notebook) provides convenient access to the data for modeling
    • ◼
  • Note: a correction to the confirmed cases data for Hubei is applied [BK]
    • In[]:=
    ?fitDataWDR
    Out[]=
    Symbol
    fitDataWDR[locale, startDate] gives data for infected, recovered, and dead relative to the given date for the onset of infection for the given locale, either an administrative division or a country. For reference, cases were first reported on 31 Dec 2019.
    In[]:=
    Options@fitDataWDR
    Out[]=
    {makeCorrectionForHubeiFalse,dateRangeAll}
    In[]:=
    ListPlotfitDataWDR
    Anhui, China
    ADMINISTRATIVE DIVISION
    ,"1 Jan 2020",PlotLabel"Anhui",FrameLabel{"time (d)","# individuals"},PlotLegends{"confirmed"-("recovered"+"dead"),"recovered","dead"}
    Out[]=
    confirmed-dead-recovered
    recovered
    dead
    ◼
  • The correction also applies data for the country China
    • ◼
  • This table gives a rough idea of the epidemic size in each province of mainland China
    • ◼
  • Get the population size for the top three provinces
    • ◼
  • Get the population size for the city of Wuhan

    • Models

    ◼
  • We’ll explore a number of models, hoping to find one with a reasonable structure that recapitulates the reported data
    • ◼
  • SEIRD: SEIR model with dead compartment
    • ◼
  • SEIQRD: SEIQR model with dead compartment
    • ◼
  • SEIsaRD: SEIR model with symptomatic and asymptomatic infectious compartments
    • ◼
  • We’ll explore all the the models for one location, then apply the “best” model to the other locations
    • ◼
  • We’ll examine alternative models with full and effective population sizes
    • ◼
  • Full population sizes will come from Wolfram|Alpha
    • ◼
  • We’ll examine alternative models with mass action or standard incidence
    • ◼
  • General assumptions:
    • ◼
  • No birth or natural mortality due to short time span of coverage of model
    • ◼
  • Death due to disease will be included
    • ◼
  • There is no migration into or out of the population
    • ◼
  • We don’t know with certainty when the first infection occurred
    • ◼
  • Du et al. assumed 1 Dec 2019 [DWC]
    • ◼
  • We will examine datasets assuming the first infections occurred on 20 Dec 2019, and 30 Dec 2019 (when the first reported infections)
    • ◼
  • Even though the severity of the infection and risk of death have been reported to be age- and gender-dependent, the available data for modeling does not have sufficient detail to support stratification by age and gender
    • ◼
  • Under reporting of cases is a serious concern, primarily due to the infection being asymptomatic in many individuals, especially children [CXL]; we will assume that all cases are reported
    • ◼
  • This work was carried out over a number of days, with the available WDR data keeping pace. However to assure a consistent analysis and interpretation of results we’ll work with data for the period 21 Jan 2020 to 26 Feb 2020.
    • ◼
  • More recent data can be used to test the models.

    • SEIRD Model with standard incidence

    ◼
  • SEIRD: SEIR model with dead compartment
    • ◼
  • This assumes that individuals in the population cannot come in contact with all other individuals in the population; this is in contrast to models for influenza and SARS [MM]
    • ◼
  • These are the transitions:
    • ◼
  • These are the ODEs

    • SEIQRD Model with standard incidence

    ◼
  • SEIQRD: SEIQR model with dead compartment
    • ◼
  • The duration of circulation of infected individuals is constant even though we know it was initially infinitely long and later became very short as isolation and quarantine measures were instituted, however no detailed information about when changes occurred is available
    • ◼
  • The duration of infection and the mortality rate for circulating and quarantined infectious individuals is the same
    • ◼
  • These are the transitions:
    • ◼
  • These are the ODEs

    • SEIsaRD Model

    ◼
  • SEIsaRD: SEIR model with symptomatic and asymptomatic infectious compartments
    • ◼
  • Standard incidence
    • ◼
  • The duration of infection for symptomatic and asymptomatic infectious individuals is the same
    • ◼
  • We have two different recovered compartments for book keeping
    • ◼
  • death due to disease occurs only for symptomatic infectious individuals
    • ◼
  • These are the transitions:
    • ◼
  • These are the ODEs
    • References
    [JHU] “Mapping 2019-nCoV”, https://systems.jhu.edu/research/public-health/ncov/
    [TG] T. Götz, “First attempts to model the dynamics of the coronavirus outbreak 2020”, https://arxiv.org/pdf/2002.03821.pdf
    [PYZ] L. Peng, W. Yang, D. Zhang, C. Zhuge, L. Hong “Epidemic analysis of COVID-19 in China by dynamical modeling”, https://www.medrxiv.org/content/10.1101/2020.02.16.20023465v1
    [ZCW] Y. Zhou, Z. Chen, X. Wu, Z. Tian, L. Cheng, L. Ye “The Outbreak Evaluation of COVID-19 in Wuhan District of China”, https://arxiv.org/pdf/2002.09640.pdf
    [JDL] J. Jia, J. Ding, S. Liu, G. Liao, J. Li, B. Duan, G. Wang, R. Zhang “Modeling the Control of COVID-19: Impact of
    Policy Interventions and Meteorological Factors”, https://arxiv.org/pdf/2003.02985.pdf
    [EGE] E. G. M E. “An SEIR like model that fits the coronavirus infection data”, https://community.wolfram.com/groups/-/m/t/1888335
    [AA] A. Antonov “Basic experiments workflow for simple epidemiological models”, https://community.wolfram.com/groups/-/m/t/1895686
    [AV] J. Arino, P. van den Driessche “Time delays in Epidemic Models; Modeling and Numerical Considerations” in “Delay Differential Equations and Applications”, O. Arino (ed.) Springer, 2006.
    [FB] F. Brauer “Reproduction numbers and final size relations”, https://www.fields.utoronto.ca/programs/scientific/10-11/drugresistance/emergence/fred1.pdf
    [BCR] M. Biggerstaff, S. Cauchemez, C. Reed, M. Gambhir, L. Finelli “Estimates of the reproduction number for seasonal, pandemic, and zoonotic influenza: a systematic review of the literature” BMC Infectious Diseases, 14, 480 (2014), http://www.biomedcentral.com/1471-2334/14/480
    [MM] M. Martcheva “An introduction to mathematical epidemiology” Springer, 2015.
    [A] Anonymous, Anonymous, Brit. Med. J., 1978, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1603269/pdf/brmedj00115-0064.pdf
    [HJR] H. J. Rose “The use of amantadine and influenza vaccine in a type A influenza epidemic in a boarding school”, Journal of Royal College of General Practitioners, 30, 619-621 (1980). PubMedCentral
    [FT] Z. Feng, H. R. Thieme “Recurrent Outbreaks of Childhood Diseases Revisited: The Impact of Isolation”, Math. Biosciences, 128, 93-130 (1995). https://doi.org/10.1016/0025-5564(94)00069-C
    [BK] S. Boseley, L. Kuo “Huge rise in coronavirus cases casts doubt over scale of epidemic”, The Guardian, 13 Feb 2020, https://www.theguardian.com/world/2020/feb/13/huge-rise-coronavirus-cases-raises-doubts-scale-epidemic-china
    [DWC] Z. Du, L. Wang, S. Cauchemex, X. Xu, X. Wang, B. J. Cowling, L. A. Meyers “Risk for Transportation of 2019 Novel Coronavirus (COVID-19) from Wuhan to Cities in China”, https://doi.org/10.1101/2020.01.28.20019299
    [CXL] J. Cai, J. Xu, D. Lin, Z. Yang, L. Xu, Z, Qu, Y. Zhang, H. Zhang, R. Jia, P. Liu, X. Wang, Y. Ge, A. Xia, H. Tian, H. Chang, C. Wang, J. Li, J. Wang, M. Zheng “A Case Series of children with 2019 novel coronavirus infection: clinical and epidemiological features”, Clinical Infectious Diseases, https://doi.org/10.1093/cid/ciaa198
    [CWB] B. J. Coburn, B. G. Wagner, S. Blower “Modeling influenza epidemics and pandemics: insights into the future of swine flu (H1N1)”, BMC Medicine, 7, (2009), http://www.biomedcentral.com/1741-7015/7/30
    Initialization
    ◼
  • the package can be downloaded from https://www.wolframcloud.com/obj/rnachbar/Published/CompartmentalModeling.wl

    • General

    Fitting data

    Fitting error