Epidemiological Models for Influenza and COVID-19—part 1

Robert B. Nachbar
Original post: 11-Mar-2020
Revised: 17-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

    ◼
  • Introduction

    • Influenza

    ◼
  • Models
    • ◼
  • SEIR
    • ◼
  • SEIQR
    • ◼
  • Outbreak 1
    • ◼
  • Outbreak 2
    • ◼
  • Summary
    • ◼
  • Suggestions for follow-up and improvements

    • References

    Initialization

    Part 2
    

    Part 3
    

    Part 4
    

    Part 5
    

    Introduction

    The COVID-19 outbreak, initially in China and now throughout the world, has captured the interest of a large number of organizations and individuals alike. Some effort has been spent to model (or at least visualize) the geographic spread [JHU][JEP]. There have also been some reports of epidemiological models [TG][PYZ][ZCW][JDL][EGE][AA] that have been developed in an effort to estimate parameters that can be used to project the severity of the outbreak, its duration, and the mortality rate.
    This report has to main goals. 1) It aims to put some of these modeling efforts into perspective so that conclusions and predictions can be better understood. We will be using compartmental models that allow one to describe the flow of individuals from one health state to another. We will attempt to employ just the right kinds of compartments and connections that are supported by the available data. As the noted statistician G. E. P. Box admonished us, “All models are wrong, but some models are useful.” It is also important know the assumptions on which the models are predicated. 2) It demonstrates the breadth of the Wolfram Language which makes these analyses relatively straight forward, from the retrieval of data from the web, to modeling and data fitting, to exposition and presentation, all in a single interactive document.
    It is instructive to first examine two influenza outbreaks that occurred in 1978 in two different boarding schools. The two populations were well defined in terms of size and the assumption of rapid and uniform mixing. These two examples demonstrate that the reported retrospective data can be analyzed satisfactorily by mechanistically different models, and that limitations of the models sometimes preclude the explanation of all the observations.
    Next we explore several epidemiological models for the COVID-19 outbreak, and use data from various provinces to estimate model parameters. We primarily want to see how similar or dissimilar the the outbreaks are to each other, point out the short comings, and make some suggestions for improvements.

    Compartmental models

    We will be using compartmental models, which have had numerous applications in biology, ecology, chemistry, and medicine. For our purposes, each compartment represents a group of individuals in the same health state, for example susceptible or infectious. The connections between compartments indication the direction and rate of movement from one health state to another. One of the simplest compartmental models for epidemiology has three compartments: susceptible, infectious, and recovered, or SIR. Susceptible individuals come in contact with infectious individuals and become infected. After some period of time, infectious individuals recover, are not longer infectious, and have permanent immunity. The process can be described as a set of transitions
    Out[]=
    
    βλ[t]
    →
    ℐ
    infection
    ℐ
    γ
    →
    ℛ
    recovery
    Schematically, the model looks like this
    Out[]=
    For clarity, we will use a slightly different form of schematic diagram known as a Petri net, which looks like this
    Out[]=
    There is code in the Initialization section at the end of the notebook for generating these graphs.
    Each compartment becomes a time-dependent variable in the model. The coefficient
    β
    is the transmission rate constant, and
    λ[t]
    is the force of infection. It is a function of all the infectious compartments in the model (in this case, just
    ℐ[t]
    ), and thus is time-dependent. The parameter
    γ
    is the recovery rate constant, and is defined as
    γ
    1
    τ
    , where
    τ
    is the average duration of infection.
    This transmission model describes the rate of change of each of the compartments, which can be modeled mathematically as a system of ordinary differential equations (ODEs).
    Out[]=
    ′
    
    [t]-β[t]λ[t]
    ′
    ℐ
    [t]-γℐ[t]+β[t]λ[t]
    ′
    ℛ
    [t]γℐ[t]
    λ[t]ℐ[t]
    The number of infectious individuals in the population
    ℐ[t]
    is the prevalence of the disease. The number of individuals that become infected per unit time is called the incidence.
    Many epidemiologic models include population demographics, that is, birth and natural death. For comparison, this is the SIR model with demographics
    Out[]=
    Λ
    ↦
    
    birth
    
    βλ[t]
    →
    ℐ
    infection
    ℐ
    γ
    →
    ℛ
    recovery
    
    μ
    ⇥
    death
    ℐ
    μ
    ⇥
    death
    ℛ
    μ
    ⇥
    death
    Out[]=
    Out[]=
    ′
    
    [t]Λ-μ[t]-β[t]λ[t]
    ′
    ℐ
    [t]-γℐ[t]-μℐ[t]+β[t]λ[t]
    ′
    ℛ
    [t]γℐ[t]-μℛ[t]
    λ[t]ℐ[t]
    The shape for
    Λ
    in the schematic diagram is different because its units are
    persons
    time
    , whereas all the other rates are
    1
    time
    .
    Because the duration of the influenza and COVID-19 outbreaks that are discussed here are so short (weeks or months), we can make the assumption that births and natural deaths can be ignored.

    Basic reproduction number

    The rate constants i these models can be used to estimate various epidemiological parameters, such as average recover time
    1
    γ
    . Another parameter that is often used to characterize epidemics is the basic reproduction number, or R0. It represents the number of individuals a single infected individual will infect in a completely susceptible population during her/his lifetime. If the value is less than 1, the outbreak will die out. If it is greater than 1, then the epidemic will persist. A very good description of this parameter, its uses, and what various values greater than 1 mean was recently published [EY].

    Delay differential equations

    While delay differential equations (DDE) are often used in population models for ecology, and many other areas as well, they can lead to unphysical artefacts. For example, shown below is the SIR model solution (solid curves) and the solution for the corresponding DDE model (dashed curves). While the total population size (
    [t][t]+ℐ[t]+ℛ[t]
    ) is constant, the infected compartment in the DDE model (dashed red) has several excursions below 0 and the recovered compartment (dashed blue) has corresponding excursions above . Adding a scaling factor to those terms in the DDEs will only minimize and not eliminate the issue. The oscillations are also of concern.
    While DDE models have been used for epidemiological models [AV], they must be used with care.

    Parameter optimization

    All of the parameters in these model need to remain in the range
    (0,∞)
    or
    (0,1)
    , so constraints should be used. Wolfram Language has many efficient functions for optimization with constraints, and for many systems they work well. From many years of experience with these kinds of models (epidemiologic, pharmacokinetic, and viral dynamics), a simpler passive constraint method (transformation) with unconstrained optimization works better.
    Values for parameters that should fall between 0 and ∞ are log-transformed and the parameters in the ODEs are back-transformed with
    Exp
    . Similarly, for parameters whose values should be between 0 and 1 we use the logit transformation and its inverse. To facilitate coding, the functions
    toLog
    ,
    fromLog
    ,
    toLogit
    , and
    fromLogit
    are defined in the initialization section.

    toLog
    ,
    fromLog

    Out[]=
    Influenza
    There are two textbook examples of influenza outbreaks in boarding schools. They are retrospective reports because the studies were unplanned, however, from a modeling perspective they present several advantages:
    ◼
  • The populations were closed (no migration in or out)
    • ◼
  • There were no births or deaths, so models without demographics can be used
    • ◼
  • The populations were well mixed as the students were attending classes together
    • ◼
  • There is a daily record of the number of students with influenza

    • The data

    Outbreak 1

    Dataset

    ◼
  • From a textbook problem, Martcheva. pp. 126-139 [MM][A]
    • …, in January-February 1978, an epidemic of influenza occurred in a boarding school in the north of England. The boarding school housed a total of 763 boys, who were at risk during the epidemic. On January 22, three boys were sick.
    512 boys (67%) spent between three and seven days away from class, …

    Fitting data

    Outbreak 2

    Dataset

    ◼
  • From another textbook problem, Martcheva, pp. 144-145 [MM][HJR]
    • The West Country English Boarding School housed 578 boys. An epidemic of influenza began on 15 January 1978.
    One hundred and sixty-six boys (29 per cent) were treated in the sick bays by bed rest and aspirin. Certainly this is not the total number with influenza, as the older boys had their own rooms in which some remained, treating themselves and avoiding detection and supervision as a result of the dislocated curriculum.

    Fitting data

    SEIR Model

    ◼
  • SEIR: SIR model with exposed compartment
    • ◼
  • No birth or natural mortality due to short time span of coverage of model
    • ◼
  • Mass action incidence
    • ◼
  • These are the transitions:
    • ◼
  • These are the ODEs

    • First dataset

    Manual fit

    Statistical fit

    Sensitivity analysis

    Total incidence & epidemic size

    ◼
  • Nearly every boy is predicted to become infected, whereas only 512 boys (67%) were reported to have been confined to bed

    • Second dataset

    Manual fit

    Statistical fit

    Sensitivity analysis

    Total incidence & epidemic size

    ◼
  • Again, grossly overestimating the epidemic size

    • SEIQR Model

    Since the data report the “number of boys confined to bed”, they are actually quarantined or isolated and not freely circulating among the student body
    ◼
  • SEIQR: SEIR model with quarantined compartment
    • ◼
  • No birth or natural mortality due to short time span of coverage of model
    • ◼
  • Mass action incidence
    • ◼
  • These are the transitions:
    • ◼
  • These are the ODEs

    • First dataset

    Manual fit

    Statistical fit

    Sensitivity analysis

    Total incidence & epidemic size

    ◼
  • Again, grossly overestimating the epidemic size

    • Second dataset

    Manual fit

    Statistical fit

    Sensitivity analysis

    Total incidence & epidemic size

    ◼
  • Again, overestimating the epidemic size, but much improved over the model without quarantine

    • Summary

    Parameter values and statistics

    Goodness of fit

    ◼
  • For the first influenza dataset, the SEIR model is somewhat better than the SEIQR model
    • ◼
  • For the second influenza dataset, the SEIQR model is much better than the SEIR model

    • Epidemiological parameters

    ◼
  • SEIR model
    • ◼
  • SEIQR model
    • ◼
  • Biggerstaff et al. [BCR] report much smaller values for the basic reproduction number for influenza:

    • General observations & conclusions

    ◼
  • Retrospective data without full details of collection can be interpreted in different ways
    • ◼
  • Recapitulating heterogenous data (e.g., prevalence and epidemic size) can be difficult
    • ◼
  • Different models may do better for different datasets for the same disease

    • Suggestions for follow-up and improvements

    ◼
  • Recent reading [MM][FT] suggests that standard incidence should be used for SEIQR models
    • ◼
  • Investigate the effect on model fitting
    • ◼
  • investigate the effect on the basic reproduction number
    • ◼
  • Try Jacobian method [MM]
    • ◼
  • Try next generation method [MM]
    • 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
    [EY] E. Yong “The Deceptively Simple Number Sparking Coronavirus Fears”, The Atlantic, 28 Jan 2020, https://www.theatlantic.com/science/archive/2020/01/how-fast-and-far-will-new-coronavirus-spread/605632/
    [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

    Fit visualization

    Fitting error