Epidemiological Models for Influenza and COVID-19—part 3

Robert B. Nachbar
Original post: 11-Mar-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

    COVID-19

    ◼
  • Models
    • ◼
  • SEIRD Model with standard incidence
    • ◼
  • Hubei outbreak—SEIRD
    • ◼
  • Outbreak start 20 Dec 2019
    • ◼
  • Fitting data
    • ◼
  • Manual fit
    • ◼
  • Optimization—
    100000
    • ◼
  • Optimization—
    50000
    • ◼
  • Outbreak start 30 Dec 2019
    • ◼
  • Fitting data
    • ◼
  • Manual fit
    • ◼
  • Optimization—
    100000
    • ◼
  • Optimization—
    50000
    • ◼
  • Comparisons

    • References

    Initialization

    Part 4
    

    Part 5
    

    COVID-19

    Models

    SEIRD Model with standard incidence

    ◼
  • These are the ODEs
    • forceOfInfectionSEIRD=λ[t_]
    ℐ[t]
    -[t]
    ;​​{varsSEIRD,odesSEIRD}=Values@KineticCompartmentalModel
    βλ[t]
    →
    ℰ,ℰ
    ζ
    →
    ℐ,ℐ
    γ
    →
    ℛ,ℐ
    μ
    →
    ,t{1,-1};​​Column[odesSEIRD]
    Out[]=
    ′
    
    [t]μℐ[t]
    ′
    ℰ
    [t]-ζℰ[t]+β[t]λ[t]
    ′
    ℐ
    [t]ζℰ[t]-γℐ[t]-μℐ[t]
    ′
    ℛ
    [t]γℐ[t]
    ′
    
    [t]-β[t]λ[t]
    In[]:=
    {susceptibleSEIRD,exposedSEIRD,infectedSEIRD,recoveredSEIRD,deadSEIRD}={,ℰ,ℐ,ℛ,}/.ParametricNDSolve[Join[odesSEIRD/.forceOfInfectionSEIRD,{[0]-I0,ℰ[0]0,ℐ[0]I0,ℛ[0]0,[0]0}],Head/@varsSEIRD,{t,0,500},{,I0,β,ζ,γ,μ}];

    Hubei outbreak—SEIRD

    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 = 0.32 (infection)
    ζ = 0.6 (incubation)
    γ = 0.072 (recovery)
    μ = 0.0028 (death)
    t
    max
    y
    max
    0.5
    ℰ
    ℐ
    ℛ
    

    Optimization—
    100000

    In[]:=
    fitErrorSEIRD[fitData,100000,10,logBeta,logZeta,logGamma,logMu]/.Thread[{logBeta,logZeta,logGamma,logMu}Log@{0.32,0.60,0.072,0.0028}]
    Out[]=
    1.48125×
    10
    10
    In[]:=
    randomSampleWithError=With[{delta=Log[10.]{-1,+1},init=Log@{0.32,0.60,0.072,0.0028}},{Thread[{β,ζ,γ,μ}Exp@{##}],fitErrorSEIRD[fitData,100000,10,##]}&@@@Prepend[Transpose[RandomReal[#+delta,50000]&/@init],init]​​]//SortBy[#,Last]&;//Timing
    Out[]=
    {535.716,Null}
    In[]:=
    Take[SortBy[Last]@randomSampleWithError,15]//tfn
    Out[]//TableForm=
    1
    2
    1
    {β0.783142,ζ0.0617432,γ0.0230569,μ0.00744279}
    1.60102×
    9
    10
    2
    {β0.80396,ζ0.0608227,γ0.0325909,μ0.00106566}
    1.69269×
    9
    10
    3
    {β0.68954,ζ0.0715426,γ0.0328715,μ0.00365816}
    1.74281×
    9
    10
    4
    {β0.781349,ζ0.0605596,γ0.0261109,μ0.0173146}
    2.01264×
    9
    10
    5
    {β0.662634,ζ0.0703356,γ0.0341459,μ0.00139895}
    2.0374×
    9
    10
    6
    {β0.754485,ζ0.0674952,γ0.0360232,μ0.000922667}
    2.03869×
    9
    10
    7
    {β0.65945,ζ0.0699785,γ0.0337624,μ0.00174329}
    2.08808×
    9
    10
    8
    {β0.824407,ζ0.0690598,γ0.0351951,μ0.0103829}
    2.14372×
    9
    10
    9
    {β0.734833,ζ0.0683832,γ0.0203583,μ0.00999414}
    2.14916×
    9
    10
    10
    {β0.778671,ζ0.0628293,γ0.0281187,μ0.000346511}
    2.15915×
    9
    10
    11
    {β0.791376,ζ0.0618965,γ0.0257106,μ0.00152217}
    2.18647×
    9
    10
    12
    {β0.733267,ζ0.0722545,γ0.0462848,μ0.0089711}
    2.2431×
    9
    10
    13
    {β0.690317,ζ0.0657344,γ0.0366964,μ0.00152161}
    2.24614×
    9
    10
    14
    {β0.524454,ζ0.108293,γ0.0281095,μ0.0143855}
    2.29754×
    9
    10
    15
    {β0.794401,ζ0.0703482,γ0.044285,μ0.0175903}
    2.3318×
    9
    10
    ◼
  • The table below shows the estimates for the epidemiological parameters
    • ◼
  • the average duration of infection takes into account both recovery and death
    • ◼
  • The table below shows the estimates for the epidemiological parameters
    • ◼
  • the average duration of infection takes into account both recovery and death
    • ◼
  • The first three and last two fits are essentially the same
    • ◼
  • The fourth fit under-predicts the deaths

    • Outbreak start 30 Dec 2019

    Fitting data

    Manual fit

    ◼
  • The table below shows the estimates for the epidemiological parameters
    • ◼
  • the average duration of infection takes into account both recovery and death
    • ◼
  • The first two fits are essentially the same
    • ◼
  • The table below shows the estimates for the epidemiological parameters
    • ◼
  • the average duration of infection takes into account both recovery and death
    • ◼
  • The are essentially the same

    • Comparisons

    ◼
  • increasing the population size makes ℐ peak broader
    • ◼
  • shifting t0 to a later date (closer to the data) does not have a noticeable effect
    • ◼
  • 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

    SEIRD