Epidemiological Models for Influenza and COVID-19

Robert B. Nachbar
Original post: 11-Mar-2020
Revised: 15-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

    ◼
  • Use the subsection cells to navigate to the other notebooks

    • Part 1
    

    Part 2
    

    Part 3
    

    Part 4
    

    Part 5

    COVID-19

    ◼
  • Models
    • ◼
  • SEIsaRD Model with standard incidence
    • ◼
  • Hubei outbreak—SEIsaRD
    • ◼
  • Outbreak start 20 Dec 2019
    • ◼
  • Fitting data
    • ◼
  • Manual fit
    • ◼
  • Optimization—
    100000
    • ◼
  • Optimization—
    50000
    • ◼
  • Comparisons

    • References

    Initialization

    COVID-19

    Models

    SEIsaRD Model

    ◼
  • These are the ODEs
    • forceOfInfectionSEIsaRD=λ[t_]
    ℐ
    "s"
    [t]+
    ℐ
    "a"
    [t]
    -[t]
    ;​​{varsSEIsaRD,odesSEIsaRD}=Values@KineticCompartmentalModel
    βλ[t]
    →
    ℰ,ℰ
    ζρ
    →
    ℐ
    "s"
    ,ℰ
    ζ(1-ρ)
    →
    ℐ
    "a"
    ,
    ℐ
    "s"
    γ
    →
    ℛ
    "s"
    ,
    ℐ
    "a"
    γ
    →
    ℛ
    "a"
    ,
    ℐ
    "s"
    μ
    →
    ,t{1,-1};​​Column[odesSEIsaRD]
    Out[]=
    ′
    
    [t]μ
    ℐ
    s
    [t]
    ′
    ℰ
    [t]-ζ(1-ρ)ℰ[t]-ζρℰ[t]+β[t]λ[t]
    ′
    
    [t]-β[t]λ[t]
    ′
    ℐ
    a
    [t]ζ(1-ρ)ℰ[t]-γ
    ℐ
    a
    [t]
    ′
    ℐ
    s
    [t]ζρℰ[t]-γ
    ℐ
    s
    [t]-μ
    ℐ
    s
    [t]
    ′
    ℛ
    a
    [t]γ
    ℐ
    a
    [t]
    ′
    ℛ
    s
    [t]γ
    ℐ
    s
    [t]
    {susceptibleSEIsaRD,exposedSEIsaRD,infectedSymptomaticSEIsaRD,infectedAsymptomaticSEIsaRD,recoveredSymptomaticSEIsaRD,recoveredAsymptomaticSEIsaRD,deadSEIsaRD}={,ℰ,
    ℐ
    "s"
    ,
    ℐ
    "a"
    ,
    ℛ
    "s"
    ,
    ℛ
    "a"
    ,}/.ParametricNDSolve[Join[odesSEIsaRD/.forceOfInfectionSEIsaRD,{[0]-I0,ℰ[0]0,
    ℐ
    "s"
    [0]ρI0,
    ℐ
    "a"
    [0](1-ρ)I0,
    ℛ
    "s"
    [0]0,
    ℛ
    "a"
    [0]0,[0]0}],varsSEIsaRD,{t,0,500},{,I0,β,ζ,ρ,γ,μ}];

    Hubei outbreak—SEIsaRD

    Outbreak start 20 Dec 2019

    Fitting data

    In[]:=
    t0=DateObject["20 Dec 2019"]
    Out[]=
    Day: Fri 20 Dec 2019
    In[]:=
    fitData1=fitData=fitDataWDR
    Hubei, China
    ADMINISTRATIVE DIVISION
    ,t0,"makeCorrectionForHubei"True,"dateRange"{All,"26 Feb 2020"};​​Dimensions/@%
    Out[]=
    {{36,2},{36,2},{36,2}}
    In[]:=
    ListPlot[fitData,PlotLabel"Hubei",FrameLabel{"time (d)","# individuals"},PlotLegends{"confirmed"-("recovered"+"dead"),"recovered","dead"}]
    Out[]=
    confirmed-dead-recovered
    recovered
    dead

    Manual fit

    Out[]=
    ​
    effective population size
    50000
    100000
    500000
    1000000
    5000000
    β = 0.57 (infection)
    ζ = 0.15 (incubation)
    ρ = 0.92 (symp./asymp.)
    γ = 0.041 (recovery)
    μ = 0.028 (death)
    t
    max
    y
    max
    0.5
    ℰ
    ℐ
    s
    ℐ
    a
    ℛ
    s
    ℛ
    a
    

    Optimization—
    100000
    

    Optimization—
    50000
    

    Comparisons

    In[]:=
    {2.237574248580851`*^8,{logBeta-1.431643518034431`,logZeta0.17799734382500643`,logitRho0.037767138874603576`,logGamma-4.042140864793616`,logMu-5.461954934030664`}}//Append[MapThread[Compose,{{fromLog,fromLog,fromLogit,fromLog,fromLog},Values@Last@#}],First@#]&
    Out[]=
    {0.238916,1.19482,0.509441,0.0175598,0.00424525,2.23757×
    8
    10
    }
    Out[]//TableForm=
    startDate
    populationSize
    β
    ζ
    ρ
    γ
    μ
    SSE
    20 Dec 2019
    50000
    0.201151
    2.68809
    0.990566
    0.0128232
    0.00454096
    3.69765×
    8
    10
    20 Dec 2019
    100000
    0.238916
    1.19482
    0.509441
    0.0175598
    0.00424525
    2.23757×
    8
    10
    In[]:=
    MapThread[plotFitSEIsaRD[#,Insert[#2,10,2],ImageSize300]&,{{fitData1,fitData1(*,fitData2,fitData2*)},Most/@Rest/@bestFits}]//Partition[#,2]&//Grid​​closeCellGroup[]
    Out[]=
    1.
    ℰ
    ℐ
    s
    ℐ
    a
    ℛ
    s
    ℛ
    a
    
    0.5
    ℰ
    ℐ
    s
    ℐ
    a
    ℛ
    s
    ℛ
    a
    
    ◼
  • Increasing the population size increases the recovery rate
    • ◼
  • In all cases, the susceptible compartment goes to 0 and the epidemic size is 100% of the population
    • 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

    SEIsaRD