WXML 2020 covid-modeling learning
This page provides a survey of some influential
mathematical models being used to track and forecast COVID-19 in
the United States. It was produced by Jarod Alper, an
Associate Professor of Mathematics at the University of Washington
whose research lies in algebraic geometry.
He is not an expert in mathematical epidemiology and has no
prior experience with infectious disease modeling.
This page was put together as a learning guide for a Spring 2020
Washington Experimental Mathematics Lab (WXML) project.
Last updated: May 6, 2020
Influential modeling groups
(in no particular order)
Institute for Health Metrics and Evaluation (IHME)
- Imperial College London (ICL)
reports: from WHO Collaborating Centre
for Infectious Disease Modeling, MRC Centre for Global Infectious
Disease Analysis, Abdul Latif Jameel Institute for Disease and
Emergency Analytics, Imperial College London, Vaccine Centre,
London School of Hygiene and Tropical Medicine, UK
- ICL March 16 model
- ICL March 26 model
- ICL March 30 model
Institute for Disease Modeling (IDM)
- The IHME model has 4 components:
- identifying and processing covid data,
- a statistical model where for each location,
a curve fit is applied to population death
rates as a function of time since the death rate
exceeds a given threshold in the location
- predicting the time a given location reaches
this given threshold if it hasn't already, and
- an individual age-structured microsimulation
to model hospital usage.
- For the curve fit in (ii),
two different S-shaped (or sigmoid) curves were considered each depending on 3 parameters.
You can see the curves and change the parameter values in this
- The model is regularly-updated and provides predictions on
deaths and hospital usage in the US and Europe.
Global Epidemic and Mobility Project (GLEAM) with contributions from Northeastern
research reports modeling transmission dynamics in
various locations including
King & Snohomish counties, Oregon and sub-Saharan Africa.
- King & Snohomish counties March 10 report:
Working paper – model-based estimates of COVID-19 burden in
King and Snohomish counties through April 7, 2020 by Klein, et al.
- Provides projections on infections and deaths through April 7
using a stochastic SEIR model with models parameters taken
from the scientific literature. To monitor hospital usage,
each model uses a discrete compartmentalized event care usage model.
- Not age-structured or spatially located but simulations
use a negative binomial transmission model with
heterogeneous parameter k = 0.54.
- Considers affects various social distancing measures
given as a percentage in the reduction of transmission rates
(usual, 25%, 50%, 75%)
- King county March 29 reports: Understanding the Impact of
COVID-19 Policy Change in the Greater Seattle Area using Mobility
Data by Burstein, et al and Social distancing and mobility
reductions have reduced COVID-19 transmission in King County, WA
by Thakkar, et al.
- Uses mobility data from Facebook Data For Good to determine
effect of policy changes. Namely, mobility data is used
determine (1) population flux between day & night,
(2) perecent difference in daytime residential occupancy,
(3) perecent difference in commuting, and
(4) percent difference in daytime non-residential occupancy.
- Uses a stochastic SEIR model to estimate the effective
reproductive rate R_e through March 24
from case data. Unknown parameters (assumed to depend on time)
such as the transmission rate and reporting rate
(% of infections detected) are fit to case data.
- King county April 10 report: Physical distancing is working and still needed to prevent
COVID-19 resurgence in King, Snohomish, and Pierce
counties by Thakkar, et al.
- Uses similar method as the March 29 reports with updated data
to estimate the effective reproductive rate through March 30.
- King county April 21 report: Short report: Updated analysis of
COVID-19 transmission in
King County, WA by Thakkar, et al.
- Uses similar method as the March 29 reports with updated data
to estimate the effective reproductive rate through April 9.
- King county April 29 report: Sustained reductions in transmission
have led to declining COVID- 19 prevalence in King County, WA
by Thakkar, et al.
- Estimates the effective reproductive rate through April 15.
- Uses a stochastic SEIR model as in the March 29 report but
incorporates a case detection rate p_t (the probability an infected case is
reported) with independent values before and after March 10.
It also assumes that the transmission rate varies over time
with the structure inferred by the positive vs negative testing data.
Assumes that an unknown quantity z_t of infections were imported
to King County on January 15. The model determines the unknown
parameters p_t and z_t by fitting them to the observed data.
Centre for the Mathematical Modeling of
Infectious Diseases (CMMID) and
the London School of Hygiene & Tropical Medicine (LSHTM)
COVID-Projections by Youyang Gu (YYG)
The effect of travel restrictions on the spread of the 2019 novel coronavirus
(COVID-19) outbreak (preprint March 6, published April 24), Science,
by Chinazzi M., Davis J.T. Additional details can be found in the
Model description: individual-based, stochastic, and spatial-based model
based on a metapopulation network approach which subdivides
the world's roughly 200 countries into 3200 geographic subpopulations.
- Uses airline transportation data from Official Aviation Guide
(OAG) and International Air Transport Association (IATA), and
ground mobility from statistics offices for 30 countries on 5 continents.
Visualization dashboard for number of infections,
deaths, hospital and ICU beds needed in the US.
- GLEAM also offers:
"EpiRisk is a computational platform designed to allow a
quick estimate of the probability of exporting infected
individuals from sites affected by a disease outbreak to
other areas in the world through the airline transportation
network and the daily commuting patters. It also lets the user to
explore the effects of potential restrictions applied to airline
traffic and commuting flows."
- GLEAMviz: a desktop application
allowing you to configure and run and analyze your own simulations.
Univisty of Texas COVID-19 Modeling Consortium
- Uses a basic SEIR/SEIS model to simulate the covid epidemic in each
country/state/region. The model is neither age-structure
nor models hospital usage.
The parameters of the model are learned using
machine learning techniques attempting to minimize the error between
the actual data on the number of deaths (as reported by John Hopkins CSSE)
and their projections.
model summary with comparison to IHME
projections of their model
tracker of the basic reproduction number R_0 and number of infections
Los Alamos National Laboratory (LANL)
Report: UT COVID-19 Mortality Forecasting Model (April 16)
- Uses a statistical-curve fitting approach to a fit a sigmoid curve
(in fact, the same curve as the IHME model depending on 3 parameters)
and a probabalistic error model to observed death rates. In comparison
to the IHME model, they "reformulated the approach in a generalized
linear model framework to correct a statistical flaw that leads
to the underestimation of uncertainty in the IHME forecasts."
- Uses local mobile-phone GPS data from SafeGraph from each state to
quantify the effects of social-distancing measures
reports and publications
- model description and forecasets
- Uses a basic SIR model combined with a statistical process
where for each state, the model
attempts to determine/learn (1) the "growth parameter"
(i.e. transmissibility rate) as a function
of time (assumed to decrease over time) based on trends in the number of
new cases reported, (2) the case fatality rate determined
by the number of new deaths and reported
cases, and (3) the discrepancy between the actual and reported number
of cases/deaths as a function of time.
- In forecasting, the model assumes the case fatality rate to be constant
and that deaths are synchronous with a positive test.
University of Minnesota
Severe COVID-19 Risk Mapping :
This tool shows projections for hospital demand for each US county
and the expected date of peak capacity.
paper summarizing results: Flattening the curve before it flattens us:
hospital critical care capacity limits and mortality from novel
coronavirus (SARS-CoV2) cases in US counties
- 3 and 6 week projections from April 2, 2020
paper with modeling details
(contains model details)
Simulation of SARS-CoV2 Spread and Intervention Effects in the
Continental US with Variable Contact Rates, March 24, 2020
- Uses a metapopulation SEIR model (similar to this
Science paper) with S, E, I, R components for each county incorporating
commuting and random movements of individuals using data from the US Census
Bureau on county-to-county commuting patterns. Transmissions are separated
into daytime (8 hours) and nighttime (16 hours).
Model with links to video briefing,
slides, faq, infographic and technical documentation
- Estimates number of daily cases & deaths and ICU usage in Minnesota
and predicts peak usage. Examines impact of social distancing
(assumed to reduce transmission rates by 50%) and shelter in place
(assumed to reduce transmission rates by 80%).
- Uses an age and comorbidity structured compartamentalized SEIR
COVID-19 Epidemic Forecasting Dashboard
- "We calculate the growth rate of cumulative cases
(resp. deaths) between two days ago and today. If it's greater
than 5%, we use an exponential model to forecast the cumulative
number of cases (resp. deaths), and then derive the daily number of
cases (resp. deaths).
If it's less than 5%, we use an linear model instead."
- No other descriptive model documentation is provided but there
is a public GitHub repository.
(in no particular order)
- Kissler (Harvard), et al. Science paper
Projecting the transmission dynamics of SARS-CoV-2 through
the postpandemic period (preprint March 6, published in Science April 14)
- Explores the role of seasonality, length of immunity, cross-immunity
(with two other coronavirus HCoV-OC43 and HCoV-HKU1),
social distancing and adding critical care capacity to
model the transmission dynamics through 2025.
- Uses a compartmentalized deterministic SEIRS model with
gamma-distributed waiting times.
- To estimate the incidence of each coronavirus,
used the percentage of positive tests multiplied by the population-weighted
proportion of influenza-like illnesses.
- homogeneous model: no age-structure, not spacially-located,
did not include school closings.
- Dandekar (MIT)--Barbastathis (MIT) paper in PNAS
Quantifying the effect of quarantine control in Covid-19 infectious
spread using machine learning article
(preprint April 3, 2020, published in PNAS)
- Uses a classical SIR approach with
"a neural network added as a non linear function approximator
(Rackauckas et al. 2020) informs the infected variable I in the SIR model.
This neural network encodes information about the quarantine strength
function in the locale where the model is implemented.
The neural network is trained from publicly available infection and
population data for Covid-19 for a specific region under study".
- The neural network is a two-layer fully-connected neural network
which trains Q(t), the percentage of the infected population I(t) that are in
quarantine at time t, so that Q(t)I(t) is the number of people in the
quarantine compartment. The funnction Q(t) is calculated from the
(S, I, R, T) vector by the formula Q = f o A_2 o f o A_1
where A_2:R^10 -> R and A_1:R^4 -> R^10 are affine-linear transformations
fitted to data,
and f is the nonlinear activation function ReLu. This is a rather small
neural network as it is training only 61 parameters.
- Compares their SIR + neural network controlling quarantine strength Q to a
classical SEIR approach (which does not incorporate
a compartment for quarantine)
- Giodano (Trento), et al paper in Nature Medicine
Modeling the COVID-19 epidemic and
implementation of population-wide interventions in Italy
(published Aprill 22, 2020)
- Uses a compartamentalized SIDARTHE model; this is a variation of the
classical SIR/SEIR epidemiological model but here
S=susceptible, I=infected, D=diagnosed, A=ailing, R=recognized,
T=threatened, H=healed, E=extinct. This distinguishes between infected
individuals on whether they have been diagnosed and the severity.
- They run simulations using data from Italy modeling affects of
social-distancing, population-wide testing ad contact tracing.
The CDC has has a very informative summary of
COVID-19 forecasts and models
Comparison of models
COVID Projections Tracker by David Yu
- Compares the various predictions of
IHME and LANL based on the update date of their projections.
- UMass Amherst Reich Lab
COVID Forecast Hub ensemble :Unifies multiple models to give more accurate predictions.
- Updated every Monday, this
provides an interactive visualization comparing the most recent projections
from various teams including ICL, IHME, UT, GLEAM, LANL
Where The Latest COVID-19 Models Think We're Headed — And Why They Disagree
- With help from UMass Amherst's Reich Lab,
compares predictions from IHME, Columbia, GLEAM, UT, MIT
- Keeling and Rohani,
Modeling infectious diseases in humans and animals
- This is one of the standard texts in
- Kiss, Miller, and Simon, Mathematics of Epidemics on Networks:
From Exact to Approximate Models
Online COVID-related lectures and courses
Imperial College London online course
- Recommend watching Weeks 1-3
and taking a look at the suggested readings
- UW seminar series:
Exploring and understanding the COVID-19 pandemic
Network Epidemiology Online Workshop Series
offered by the University of Maryland
Mathematics of the Corona outbreak (March 13, 35 min), Tom Britton (Stockholm)
COVID-19: The Exponential Power of Now (March 18, 1 hr), Jewell (Berkeley) hosted by MSRI
COVID Math: trailing indicators, leading indicators,
math models, and stats models (April 17, 1 hr),
Flaxman (IHME) hosted by UW Math
- UW panel:
Emerging from COVID-19: A Ground-Zero Perspective (April 23, 1 hr)
with Keith Jerome (Head of UW Virology), Elizabeth Halloran (UW Biostats) and
Jeffrey Duchin (UW Medicine).
with Adam Kucharski (LSHTM)
NYU seminar on "Introduction to Infectious Disease Modeling"
- Fun youtube videos:
Basic data and visualizations
Some educational and informative visualizations
Modeling COVID-19 Spread vs Healthcare Capacity
by Alison Hill (Harvard),
- has links to other recommended covid19 apps in
from NY Times regarding impacts of social distancing
- article by Nicholas Kristof and Stuart A. Thompson
- model created with Gabriel Goh, Steven De Keninck,
Ashleigh Tuite and David N. Fisman
An interactive visualization
by John Burn-Murdoch
- tracks the exponential spread of COVID-19
- by Richard Neher, et al
at the Biozentrum, University of Basel
- basic homogeneous SEIR model with compartments
including various hospital stages.
- individual model parameters can be varied.
- Impacts of social distancing measures are considered.
Washington post covid simulator
- article by Harry Stevens
- Most viewed Washington post article ever.
- Covid Trends
- By Aatish Bhatia in collaboration with Minute Physics.
- Visualization of spread by country and US state.
Breaking the wave
- by Jon McClure, a Reuters Graphics editor
- explains how the rate of increase in covid deaths is decreasing
Forecasting s-curves is hard by Constance Crozier
- Several interactive notebooks
by Yong-Yeol Ahn